F-characteristic cycle of a rank one sheaf
on an arithmetic surface

Ryosuke Ooe

Abstract

We prove the rationality of the characteristic form for a degree one character of the Galois group of an abelian extension of henselian discrete valuation fields. We prove the integrality of the characteristic form for a rank one sheaf on a regular excellent scheme. These properties are shown by reducing to the corresponding properties of the refined Swan conductor proved by Kato.

We define the F-characteristic cycle of a rank one sheaf on an arithmetic surface as a cycle on the FW-cotangent bundle using the characteristic form on the basis of the computation of the characteristic cycle in the equal characteristic case by Yatagawa. The rationality and the integrality of the characteristic form are necessary for the definition of the F-characteristic cycle. We prove the intersection of the F-characteristic cycle with the 0-section computes the Swan conductor of cohomology of the generic fiber.

Introduction

Let $K$ be a henselian discrete valuation field with residue field $F$ of characteristic $p>0$ and let $L$ be a finite abelian extension of $K$ . Kato [6] defined the refined Swan conductor of a character of the Galois group $\operatorname{Gal}(L/K)$ as an injection to the $F$ -vector space $\Omega^{1}_{F}(\log)$ . Recently, Saito [17] defined the characteristic form of such a character as a non-logarithmic variant of the refined Swan conductor. The characteristic form takes value in the $\overline{F}$ -vector space $\mathrm{H}_{1}(L_{\overline{F}/\mathcal{O}_{K}})$ where $\mathcal{O}_{K}$ denotes the valuation ring of $K$ and $\mathrm{H}_{1}(L_{\overline{F}/\mathcal{O}_{K}})$ denotes the first homology group of the cotangent complex. In the equal characteristic case, the non-logarithmic theory played an important role in the computation of the characteristic cycle [14, Section 7.3].

In Section 4, we show two properties of the characteristic form for rank one sheaves. The first property is the rationality of the characteristic form.

Theorem 0.1 (rationality, Theorem 1.3).

Let $\chi\colon\operatorname{Gal}(L/K)\to\mathbf{Q}/\mathbf{Z}$ be a character. Let $m$ be the total dimension of $\chi$ . Then the image of the characteristic form $\operatorname{char}\chi\colon\mathfrak{m}^{m}_{K}/\mathfrak{m}^{m+1}_{K}\to\mathrm{H}_{1}(L_{\overline{F}/\mathcal{O}_{K}})=\mathrm{H}_{1}(L_{F/\mathcal{O}_{K}})\otimes_{F}\overline{F}$ of $\chi$ is contained in $\mathrm{H}_{1}(L_{F/\mathcal{O}_{K}})\otimes_{F}F^{1/p}$ .

The second property is the integrality of the characteristic form for rank one sheaves. Let $D$ be a divisor with simple normal crossings on a regular excellent scheme $X$ . Let $\{D_{i}\}_{i\in I}$ be the set of irreducible components of $D$ and let $K_{i}$ be the local field at the generic point $\mathfrak{p}_{i}$ of $D_{i}$ . Let $U$ be the complement of $D$ . Let $\chi$ be an element of $\mathrm{H}^{1}(U,\mathbf{Q}/\mathbf{Z})$ . Let $Z_{\chi}$ be the union of $D_{i}$ such that $\chi|_{K_{i}}$ is wildly ramified and $R_{\chi}=\sum_{i\in I}\operatorname{dt}(\chi|_{K_{i}})D_{i}$ be the total dimension divisor.

Theorem 0.2 (integrality, Theorem 1.5, cf. (5.1)).

There exists a unique global section $\operatorname{char}(\chi)\in\Gamma(Z_{\chi},F\Omega^{1}_{X}(pR_{\chi})|_{Z_{\chi}})$ such that the germ at $\mathfrak{p}_{i}$ is equal to the linear combination with coefficients that are the $p$ -th powers of the coefficients in the characteristic form $\operatorname{char}(\chi|_{K_{i}})$ .

In the case where the characteristic of $K$ is $p$ , these properties have been already proved by using the Artin-Schreier-Witt theory by Matsuda [10] and Yatagawa [20]. In the case where the characteristic of $K$ is zero, the Artin-Schreier-Witt theory does not work, so we need to take a different method. The strategy of the proof of Theorem 0.1 and Theorem 0.2 is to reduce to the corresponding properties of the refined Swan conductor proved by Kato [6]. To do this, we compare the refined Swan conductor with the characteristic form.

The relation between the refined Swan conductor and the characteristic form is explained as follows. Let $\chi\colon\operatorname{Gal}(L/K)\to\mathbf{Q}/\mathbf{Z}$ be a character. The characters are divided into two types. If $\chi$ is of type I (for example, the residue field extension is separable), the characteristic form of $\chi$ is the image of the refined Swan conductor of $\chi$ . On the other hand, if $\chi$ is of type II (for example, the ramification index of $L/K$ is 1 and the residue field extension is inseparable), the refined Swan conductor of $\chi$ is the image of the characteristic form of $\chi$ . A large part of the proof of these relations is due to Saito. The author thanks him for kindly suggesting the author to include the proof in this paper.

For a character of type I, Theorem 0.1 holds since the characteristic form is the image of the refined Swan conductor and the refined Swan conductor takes value in the $F$ -vector space $\Omega^{1}_{F}(\log)$ . For a character of type II, we would like to change the character to a character of type I. The typical case where a character is of type I is when the residue field $F$ is perfect. Hence we would like to take an extension $K^{\prime}$ of $K$ such that the residue field of $K^{\prime}$ is perfect. In fact, it suffices to consider the field $K^{\prime}$ with the $p$ -th power roots of a lifting of a $p$ -basis of $F$ , though the residue field of $K^{\prime}$ may not be perfect.

Similarly as in the proof of Theorem 0.1, we prove Theorem 0.2 using the integrality of the refined Swan conductor, but the proof is more complicated.

In Section 5, we consider the theory of the characteristic cycle. The characteristic cycle of an étale sheaf on a smooth scheme over a perfect field of positive characteristic is defined by Saito [14]. The characteristic cycle is defined as a cycle on the cotangent bundle. By the index formula, the intersection with the 0-section computes the Euler characteristic if the scheme is projective. The characteristic cycle was computed on the closed subset of codimension $<2$ by using the characteristic form. Yatagawa [21] gave an explicit computation of the characteristic cycle of a rank one sheaf on a scheme of dimension 2.

The existence of the cotangent bundle on a scheme of mixed characteristic has not been known. Instead, Saito [15] defined the FW-cotangent bundle $FT^{*}X|_{X_{F}}$ of a regular noetherian scheme $X$ over a discrete valuation ring $\mathcal{O}_{K}$ of mixed characteristic $(0,p)$ to be the vector bundle of rank $\dim X$ on the closed fiber $X_{F}$ to consider the characteristic cycle of an étale sheaf on a scheme of mixed characteristic. The characteristic cycle in the mixed characteristic case has not been defined in general.

Let $D$ be a divisor with simple normal crossings on $X$ and let $j\colon U=X-D\to X$ be the open immersion. Let $\Lambda$ be a finite field of characteristic different from $p$ and let $\mathcal{F}$ be a smooth sheaf of $\Lambda$ -modules of rank one. In the case $\dim X=2$ , we define the F-characteristic cycle $FCCj_{!}\mathcal{F}$ of $j_{!}\mathcal{F}$ as a cycle on the FW-cotangent bundle on the basis of the computation in the equal characteristic case by Yatagawa.

On the closed subset of codimension $<2$ , we define the F-characteristic cycle using the characteristic form. To determine the coefficients of the fibers at closed points, we use both the refined Swan conductor and the characteristic form. The main reason for using both non-log and log theories is that after successive blowups, the refined Swan conductor becomes a locally split injection but the characteristic form has no such properties. The rationality (Theorem 0.1) and the integrality (Theorem 0.2) of the characteristic form are crucial to determine the coefficients of the fibers.

In analogy with the index formula, we prove the intersection of the F-characteristic cycle with the 0-section computes the Swan conductor of cohomology of the generic fiber.

Theorem 0.3 (Theorem 5.15).

Assume $\dim X=2$ and $X$ is proper over $\mathcal{O}_{K}$ . Then we have

(FCCj_{!}\mathcal{F}-FCCj_{!}\Lambda,FT^{*}_{X}X|_{X_{F}})_{FT^{*}X|_{X_{F}}}=p\cdot(\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\mathcal{F})-\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\Lambda)).

Abbes [1] found the formula computing the Swan conductor of cohomology of the generic fiber of an arithmetic surface under the assumption that a coefficient sheaf has no fierce ramification. Our formula restricts to a coefficient sheaf of rank one but needs no assumption on ramification.

We prove Theorem 0.3 using Kato-Saito’s conductor formula [8]. We study the relation between the F-characteristic cycle and the pullback of the logarithmic characteristic cycle defined by Kato [7]. This step is similar to the computation by Yatagawa in the equal characteristic case.

We give an outline of the paper. In Section 1, we briefly recall the definition of the characteristic form and state the rationality and the integrality of the characteristic form explained above. In Section 2, we recall the definition and properties of the refined Swan conductor in parallel with the characteristic form. In Section 3, we give relations between the refined Swan conductor and the characteristic form. In Section 4, we prove the rationality and the integrality established in Section 1 using the results in Section 3. In Section 5, we define the F-characteristic cycle of a rank one sheaf on an arithmetic surface. We prove the main theorem, which gives a formula computing the Swan conductor of cohomology of the generic fiber. We give an example of the F-characteristic cycle.

Acknowledgment

The author would like to express his sincere gratitude to his advisor Professor Takeshi Saito for suggesting the problem, giving a lot of helpful advice, and showing his unpublished book on ramification theory, which contains the contents of Section 2 and the proof of Lemma 3.3 and Proposition 3.4. The author thanks an anonymous referee for careful reading and comments.

1 Characteristic form

In this section, we recall the notion of characteristic form and state the integrality of the characteristic form.

1.1 Cotangent complex and FW-differential

We briefly recall the properties on cotangent complexes from [17].

Let $K$ be a discrete valuation field with valuation ring $\mathcal{O}_{K}$ and with residue field $F$ of characteristic $p>0$ . Let $E$ be a field containing $F$ . For an element $u\in\mathcal{O}_{K}$ , we write $\overline{u}$ for the image of $u$ in $F$ . If there exists a $p$ -th root of $\overline{u}$ in $E$ , the element $\tilde{d}u$ in $\mathrm{H}_{1}(L_{E/\mathcal{O}_{K}})$ is defined in [17, (1.9)]. We write $wu$ for this element instead of $\tilde{d}u$ .

Proposition 1.1.

Let $\pi$ be a uniformizer of $K$ and $(v_{i})_{i\in I}$ be a $p$ -basis of $F$ . Assume that the field $E$ contains $F^{1/p}$ . Then, $\{w\pi,wv_{i}\}_{i\in I}$ forms a basis of the $E$ -vector space $\mathrm{H}_{1}(L_{E/\mathcal{O}_{K}})$ .

Proof.

By [17, Proposition 1.1.3.2], we have an exact sequence

0\to\mathfrak{m}_{K}/\mathfrak{m}_{K}^{2}\otimes_{F}E\xrightarrow{w}\mathrm{H}_{1}(L_{E/\mathcal{O}_{K}})\to\Omega^{1}_{F}\otimes_{F}E\to 0

of $E$ -vector spaces. Then $\pi$ defines a basis of $\mathfrak{m}_{K}/\mathfrak{m}_{K}^{2}\otimes_{F}E$ and $\{dv_{i}\}_{i\in I}$ forms a basis of $\Omega^{1}_{F}\otimes_{F}E$ . The assertion follows since the map $\mathrm{H}_{1}(L_{E/\mathcal{O}_{K}})\to\Omega^{1}_{F}\otimes_{F}E$ sends $wv_{i}$ to $dv_{i}$ by [17, Proposition 1.1.4.2]. ∎

Let $L$ be a finite separable extension of $K$ with residue field $E$ . The morphism $\operatorname{Spec}E\to\operatorname{Spec}\mathcal{O}_{L}\to\operatorname{Spec}\mathcal{O}_{K}$ of schemes defines the distinguished triangle $L_{\mathcal{O}_{L}/\mathcal{O}_{K}}\otimes_{\mathcal{O}_{L}}^{\mathrm{L}}E\to L_{E/\mathcal{O}_{K}}\to L_{E/\mathcal{O}_{L}}\to$ . Since we have quasi-isomorphisms $L_{E/\mathcal{O}_{L}}\cong N_{E/\mathcal{O}_{L}}[1]$ and $L_{\mathcal{O}_{L}/\mathcal{O}_{K}}\cong\Omega^{1}_{\mathcal{O}_{L}/\mathcal{O}_{K}}[0]$ by [17, Lemma 1.2.6.4], we have an injection

\operatorname{Tor}_{1}^{\mathcal{O}_{L}}(\Omega^{1}_{\mathcal{O}_{L}/\mathcal{O}_{K}},E)\to\mathrm{H}_{1}(L_{E/\mathcal{O}_{K}})

(1.1)

of $E$ -modules.

The Frobenius-Witt differential was introduced by Saito [16] to define the cotangent bundle of a scheme over $\mathbf{Z}_{(p)}$ . The following relation between the cotangent complex and the FW-differential is known.

Proposition 1.2 ([16, Corollary 4.12]).

Let $A$ be a local ring with residue field $k$ of characteristic $p>0$ . Let $L_{k/A}$ denote the cotangent complex for the composition $\operatorname{Spec}k\xrightarrow{\mathrm{F}}\operatorname{Spec}k\to\operatorname{Spec}A$ where $\mathrm{F}$ is the Frobenius. Then, the canonical morphism $F\Omega^{1}_{A}\otimes_{A}k\to\mathrm{H}_{1}(L_{k/A})$ is an isomorphism.

1.2 Characteristic form

We briefly recall the construction of the characteristic form in [17]. Let $K$ be a henselian discrete valuation field with residue field $F$ of characteristic $p>0$ . Let $G_{K}$ be the absolute Galois group of $K$ and let $(G^{r}_{K})_{r\in\mathbf{Q}_{>0}}$ be Abbes-Saito’s non-logarithmic upper ramification filtration [2, Definition 3.4]. For an element $\chi\in\mathrm{H}^{1}(G_{K},\mathbf{Q}/\mathbf{Z})$ , we define the total dimension $\operatorname{dt}\chi$ to be the smallest rational number $r$ satisfying $\chi(G^{s}_{K})=0$ for all $s>r$ . The total dimension is an integer by [19, Theorem 4.3.5], [17, Theorem 4.3.1].

We fix some notations. Let $L$ be a finite separable extension of $K$ and let $K^{\prime}$ be a separable extension of $K$ of ramification index $e$ . Let $E,F^{\prime}$ be the residue fields of $L,K^{\prime}$ respectively. Let $S,S^{\prime},T$ be the spectrums of the valuation rings $\mathcal{O}_{K},\mathcal{O}_{K^{\prime}},\mathcal{O}_{L}$ respectively. Take a closed immersion $T\to P$ to a smooth scheme over $S$ . For a rational number $r>0$ such that $er$ is an integer, we define the scheme $P^{[r]}_{S^{\prime}}$ to be the dilatation $P^{[D_{r}\cdot T_{S^{\prime}}]}$ of $P_{S^{\prime}}=P\times_{S}{S^{\prime}}$ with respect to the Cartier divisor $D_{r}$ defined by $\mathfrak{m}^{er}_{K^{\prime}}$ and the closed subscheme $T_{S^{\prime}}=T\times_{S}S^{\prime}$ . (See [17, Definition 3.1.1] for the definition of the dilatation). Let $P^{(r)}_{S^{\prime}}$ be the normalization of $P^{[r]}_{S^{\prime}}$ . Let $P^{[r]}_{F^{\prime}}$ and $P^{(r)}_{F^{\prime}}$ be the closed fibers of $P^{[r]}_{S^{\prime}}$ and $P^{(r)}_{S^{\prime}}$ respectively.

For an immersion $T\to P$ to a smooth scheme over $S$ , we have an exact sequence

0\to N_{T/P}\to\Omega^{1}_{T/S}\otimes_{\mathcal{O}_{P}}\mathcal{O}_{T}\to\Omega^{1}_{T/S}\to 0

(1.2)

of $\mathcal{O}_{L}$ -modules. We say that an immersion $T\to P$ to a smooth scheme over $S$ is minimal if the map

\operatorname{Tor}_{1}^{\mathcal{O}_{L}}(\Omega^{1}_{T/S},E)\to N_{T/P}\otimes_{\mathcal{O}_{L}}E

(1.3)

induced by (1.2) is an isomorphism. There exists a minimal immersion by [17, Lemma 1.2.3.1].

Let $L/K$ be a finite Galois extension and let $G=\operatorname{Gal}(L/K)$ be the Galois group. Let $r>1$ be a rational number such that $G^{r+}=\cup_{s>r}G^{s}=1$ . By the reduced fiber theorem [4], there exists a separable extension $K^{\prime}$ of $K$ of ramification index $e$ such that $er$ is an integer and the geometric closed fiber $P_{S^{\prime}}^{(r)}\times_{S^{\prime}}\overline{F}$ is reduced. We define the scheme $\Theta_{L/K,F^{\prime}}^{(r)}$ to be the vector bundle $\operatorname{Hom}_{F^{\prime}}(\mathfrak{m}^{er}_{K^{\prime}}/\mathfrak{m}^{er+}_{K^{\prime}},\operatorname{Tor}_{1}^{\mathcal{O}_{L}}(\Omega^{1}_{\mathcal{O}_{L}/\mathcal{O}_{K}},E))$ over $\operatorname{Spec}(E\otimes_{F}F^{\prime})_{\operatorname{red}}$ . If we take a minimal immersion $T\to P$ to a smooth scheme over $S$ , the isomorphism (1.3) induces an isomorphism $P_{F^{\prime},\operatorname{red}}^{[r]}\to\Theta_{L/K,F^{\prime}}^{(r)}$ by [17, Proposition 3.1.3.2]. We define the scheme $\Phi_{L/K,F^{\prime}}^{(r)}$ to be $P_{F^{\prime}}^{(r)}$ . The definition does not depend on the choice of a minimal immersion $T\to P$ by [17, Lemma 3.3.7].

We fix a morphism $i_{0}\colon L\to K_{s}$ to a separable closure of $K$ . Let $\overline{F}$ be the algebraic closure of $F$ and let $\overline{T}_{\overline{F}}$ be the normalization of $T_{\overline{F}}=T\times_{F}\overline{F}$ . Let $\Theta_{L/K,\overline{F}}^{(r)\circ},\Phi_{L/K,\overline{F}}^{(r)\circ}$ denote the connected component of $\Theta_{L/K,\overline{F}}^{(r)},\Phi_{L/K,\overline{F}}^{(r)}$ containing the image of the closed point of $\overline{T}_{\overline{F}}$ corresponding to $i_{0}$ . Then $\Phi_{L/K,\overline{F}}^{(r)\circ}$ is an additive $G^{r}$ -torsor over $\Theta_{L/K,\overline{F}}^{(r)\circ}$ by [17, Theorem 4.3.3.1] in the sense of [17, Definition 2.1.4.1]. By [17, Proposition 2.1.6], there exists a group scheme structure on $\Phi_{L/K,\overline{F}}^{(r)\circ}$ such that

0\to G^{r}\to\Phi_{L/K,\overline{F}}^{(r)\circ}\to\Theta_{L/K,\overline{F}}^{(r)\circ}\to 0

is an extension of smooth group schemes. We define the map $[\Phi]$ by

[\Phi]\colon\operatorname{Hom}(G^{r},\mathbf{F}_{p})\to\mathrm{H}^{1}(\Theta_{L/K,\overline{F}}^{(r)\circ},\mathbf{F}_{p})

sending a character $\chi$ to the image $\chi_{*}[\Phi_{L/K,\overline{F}}^{(r)\circ}]$ of $[\Phi_{L/K,\overline{F}}^{(r)\circ}]$ by $\chi_{*}\colon\mathrm{H}^{1}(\Theta_{L/K,\overline{F}}^{(r)\circ},G^{r})\to\mathrm{H}^{1}(\Theta_{L/K,\overline{F}}^{(r)\circ},\mathbf{F}_{p})$ . By [17, Proposition 2.1.6], the morphism $[\Phi]$ is an injection and the image of $[\Phi]$ is contained in $\operatorname{Ext}(\Theta_{L/K,\overline{F}}^{(r)\circ},\mathbf{F}_{p})$ .

Let $\mathfrak{m}_{K_{s}}^{r}$ be the ideal $\{x\in K_{s}\mid\operatorname{ord}_{K}x\geq r\}$ and let $\mathfrak{m}_{K_{s}}^{r+}$ be the ideal $\{x\in K_{s}\mid\operatorname{ord}_{K}x>r\}$ for the extension of the valuation of $K$ to $K_{s}$ . If we identify $\operatorname{Ext}(\Theta_{L/K,\overline{F}}^{(r)\circ},\mathbf{F}_{p})$ with $({\Theta_{L/K,\overline{F}}^{(r)\circ}})^{\vee}=\operatorname{Hom}_{\overline{F}}(\mathfrak{m}^{r}_{K_{s}}/\mathfrak{m}^{r+}_{K_{s}},\operatorname{Tor}^{1}(\Omega^{1}_{T/S},E))$ by the isomorphism [17, (2.1)], we have a commutative diagram

(1.4)

of extensions of smooth group schemes where the lower extension is the Artin-Schreier extension.

We define the characteristic form to be the composition of injections

\operatorname{char}\colon\operatorname{Hom}(G^{r},\mathbf{F}_{p})\xrightarrow{[\Phi]}\operatorname{Hom}_{\overline{F}}(\mathfrak{m}^{r}_{K_{s}}/\mathfrak{m}^{r+}_{K_{s}},\operatorname{Tor}_{1}^{\mathcal{O}_{L}}(\Omega^{1}_{T/S},\overline{F}))\to\operatorname{Hom}_{\overline{F}}(\mathfrak{m}^{r}_{K_{s}}/\mathfrak{m}^{r+}_{K_{s}},\mathrm{H}_{1}(L_{\overline{F}/\mathcal{O}_{K}}))

where the second morphism is induced by the injectoin (1.1). For a character $\chi\colon G^{r}\to\mathbf{F}_{p}$ , we call $\operatorname{char}\chi$ the characteristic form of $\chi$ .

We state the rationality of the characteristic form.

Theorem 1.3.

Let $\chi\in\mathrm{H}^{1}(K,\mathbf{Q}/\mathbf{Z})$ be a character of total dimension $m$ . Then the image of the characteristic form $\operatorname{char}\chi\colon\mathfrak{m}^{m}_{K}/\mathfrak{m}^{m+1}_{K}\to\mathrm{H}_{1}(L_{\overline{F}/\mathcal{O}_{K}})=\mathrm{H}_{1}(L_{F/\mathcal{O}_{K}})\otimes_{F}\overline{F}$ of $\chi$ is contained in $\mathrm{H}_{1}(L_{F/\mathcal{O}_{K}})\otimes_{F}F^{1/p}$ .

We give a proof of Theorem 1.3 in Section 4.

Remark 1.4.

We have an example such that the image of the characteristic form is not contained in $\mathrm{H}_{1}(L_{{F/\mathcal{O}_{K}}})$ when we assume that the characteristic of $F$ is 2. Consider the Kummer character $\chi$ defined by $t^{2}=1+\pi^{2(e-1)}u$ where $\pi$ is a uniformizer of $K$ , $e=\operatorname{ord}_{K}2$ and $u\in\mathcal{O}_{K}$ such that $\sqrt{\overline{u}}\notin F$ . Then, the computation in [15, Lemma 3.2.5.3] shows that we have

\operatorname{char}\chi=\frac{wu-\sqrt{\overline{u}}\cdot w(2/\pi^{e-1})}{\pi^{2}}\in\mathrm{H}_{1}(L_{F/\mathcal{O}_{K}})\otimes_{F}\mathfrak{m}_{K}^{-2}/\mathfrak{m}_{K}^{-1}.

When the characteristic of the residue field is not 2, we can expect from the results in the equal characteristic [10, Proposition 3.2.3], [20, Proposition 1.17] that the image of the characteristic form is contained in $\mathrm{H}_{1}(L_{F/\mathcal{O}_{K}})$ , but the author does not how to prove.

We state the integrality of the characteristic form.

Theorem 1.5.

Let $A$ be an excellent regular local ring of dimension $d$ with fraction field $K$ and with residue field $k$ of characteristic $p>0$ . We assume $c=[k:k^{p}]<\infty$ and fix a lifting $(x_{l})_{l=1,\dots,c}$ of a p-basis of $k$ to $A$ . Let $(\pi_{i})_{i=1,\dots,d}$ be a regular system of parameters of $A$ and let $K_{i}$ be the local field at the prime ideal generated by $\pi_{i}$ . We fix an integer $r$ satisfying $1\leq r\leq d$ . Let $D_{i}$ be a divisor on $X=\operatorname{Spec}A$ defined by $\pi_{i}$ and let $U$ be the complement of $D=\cup_{i=1}^{r}D_{i}$ . Let $\chi$ be an element of $\mathrm{H}^{1}(U,\mathbf{Q}/\mathbf{Z})$ and let $\chi|_{K_{i}}$ be the pullback of $\chi$ by $\operatorname{Spec}K_{i}\to U$ . We put $m_{i}=\operatorname{dt}(\chi|_{K_{i}})$ . By Proposition 1.1 and Theorem 1.3, we may write

\operatorname{char}(\chi|_{K_{j}})=(\sum_{1\leq i\leq d}\alpha_{i,j}w\pi_{i}+\sum_{1\leq l\leq c}\beta_{l,j}wx_{l})/\pi_{1}^{m_{1}}\cdots\pi_{r}^{m_{r}}

with $\alpha_{i,j},\beta_{l,j}\in{\operatorname{Frac}(A/\pi_{j})}^{1/p}$ where $1\leq i\leq d$ , $1\leq j\leq r$ satisfying $m_{j}\geq 2$ , and $1\leq l\leq c$ . Then, we have the following properties:

(1)

$\alpha_{i,j}^{p},\beta_{l,j}\in A/\pi_{j}$ .
(2)

For integers $j,j^{\prime}$ satisfying $1\leq j,j^{\prime}\leq r$ , the images of $\alpha_{i,j}^{p},\alpha_{i,j^{\prime}}^{p}$ in $A/(\pi_{j})+(\pi_{j^{\prime}})$ are equal for each $i$ and the images of $\beta_{l,j},\beta_{l,j^{\prime}}$ in $A/(\pi_{j})+(\pi_{j^{\prime}})$ are equal for each $l$ .

We give a proof of Theorem 1.5 in Section 4.

2 Refined Swan conductor

In this section, we recall the notion of refined Swan conductor. The refined Swan conductor was defined by Kato [6] as an injection from the dual of the graded quotients to twisted cotangent spaces with logarithmic poles. Using Abbes-Saito’s (logarithmic) ramification theory [2], Saito defined [13] another injection from the dual of the graded quotients to twisted cotangent spaces with logarithmic poles. The coincidence of these two notions of refined Swan conductor is verified by Kato and Saito [9, Theorem 1.5]. In this paper, we use the definition by Saito, but we slightly change the construction to compare with the characteristic form. The construction here is also given by Saito.

We use the terminologies and symbols on log geometry. We refer to [3, Section 3], [12] for the definition. In this article, we consider the log structure on the spectrum of a discrete valuation ring defined by the closed point.

Let $K$ be a henselian discrete valuation field with residue field $F$ of characteristic $p>0$ . Let $G_{K}$ be the absolute Galois group of $K$ and let $(G^{r}_{K,\log})_{r\in\mathbf{Q}_{>0}}$ be Abbes-Saito’s logarithmic upper ramification filtration [2, Definition 3.12]. For an element $\chi\in\mathrm{H}^{1}(G_{K},\mathbf{Q}/\mathbf{Z})$ , we define the Swan conductor $\operatorname{sw}\chi$ to be the smallest rational number $r$ satisfying $\chi(G^{s}_{K,\log})=0$ for all $s>r$ . The Swan conductor is an integer since Kato’s definition [6] of the Swan conductor coincides with the definition here by [9, Theorem 1.3] and the Swan conductor is defined as an integer in Kato’s definition.

We use the same notations as in Subsection 1.2. Take an exact closed immersion $T\to Q$ to a log smooth scheme over $S$ . For a rational number $r>0$ such that $er$ is an integer, we define the scheme $Q^{[r]}_{S^{\prime}}$ to be the dilatation $Q^{[D_{r}\cdot T_{S^{\prime}}]}$ of $Q_{S^{\prime}}$ with respect to the Cartier divisor $D_{r}$ defined by $\mathfrak{m}^{er}_{K^{\prime}}$ and the closed subscheme $T_{S^{\prime}}$ . (See [17, Definition 3.1.1] for the definition of the dilatation).

Let $Q^{(r)}_{S^{\prime}}$ be the normalization of $Q^{[r]}_{S^{\prime}}$ . Let $Q^{[r]}_{F^{\prime}}$ and $Q^{(r)}_{F^{\prime}}$ be the closed fibers of $Q^{[r]}_{S^{\prime}}$ and $Q^{(r)}_{S^{\prime}}$ respectively.

For an exact immersion $T\to Q$ to a log smooth scheme over $S$ , we have an exact sequence

0\to N_{T/Q}\to\Omega^{1}_{Q/S}(\log/\log)\otimes_{\mathcal{O}_{Q}}\mathcal{O}_{L}\to\Omega^{1}_{T/S}(\log/\log)\to 0

(2.1)

of $\mathcal{O}_{L}$ -modules by [12, Proposition 2.3.2].

We say that an exact immersion $T\to Q$ to a log smooth scheme over $S$ is minimal if the map

\operatorname{Tor}_{1}^{\mathcal{O}_{L}}(\Omega^{1}_{T/S}(\log/\log),E)\to N_{T/Q}\otimes_{\mathcal{O}_{L}}E

(2.2)

induced by (2.1) is an isomorphism.

Lemma 2.1 (cf. [17, Lemma 1.2.3.1]).

There exists a minimal exact immersion $T\to Q$ to a log smooth scheme over $S$ .

Proof.

Let $\pi$ be a uniformizer of $K$ and $m$ be the ramification index of $L/K$ . Take a system of generators $a_{1},\dots a_{n}\in\mathcal{O}_{L}$ over $\mathcal{O}_{K}$ and put $ua_{1}^{m}=\pi$ with $u\in\mathcal{O}_{K}^{\times}$ . We define an exact closed immersion

T=\operatorname{Spec}\mathcal{O}_{L}\to Q^{\prime}=\operatorname{Spec}\mathcal{O}_{K}[X_{1},\dots,X_{n},U^{\pm 1}]/(UX_{1}^{m}-\pi)

to a log smooth scheme sending $X_{1},\dots X_{n},U$ to $a_{1},\dots,a_{n},u$ . Let $I$ be the kernel of the map

\mathcal{O}_{K}[X_{1},\dots,X_{n},U^{\pm 1}]/(UX_{1}^{m}-\pi)\to\mathcal{O}_{L}.

Take a lifting $f_{1},\dots,f_{s}\in I$ of a basis of the image of $N_{T/Q^{\prime}}\otimes_{\mathcal{O}_{L}}E\to\Omega^{1}_{Q^{\prime}/S}(\log/\log)\otimes_{\mathcal{O}_{Q^{\prime}}}E$ . Then the closed subscheme $Q$ of $Q^{\prime}$ defined by the ideal $(f_{1},\dots,f_{s})$ is log smooth over $S$ on a neighborhood of $T$ . We show that the immersion $T\to Q$ is minimal.

The construction of $Q$ shows that $N_{T/Q}\otimes_{\mathcal{O}_{L}}E\to\Omega^{1}_{Q/S}(\log/\log)\otimes_{\mathcal{O}_{Q}}E$ is a zero map. Hence the exact sequence (2.1) induces the isomorphism $\operatorname{Tor}_{1}^{\mathcal{O}_{L}}(\Omega^{1}_{T/S}(\log/\log),E)\to N_{T/Q}\otimes_{\mathcal{O}_{L}}E$ . ∎

We define the morphism

\operatorname{Tor}_{1}^{\mathcal{O}_{L}}(\Omega^{1}_{T/S}(\log/\log),E)\to\Omega^{1}_{F}(\log)\otimes_{F}E

(2.3)

as follows. For an exact immersion $T\to Q$ to a log smooth scheme over $S$ , we have an injection $N_{T/Q}\otimes_{\mathcal{O}_{T}}E\to\Omega^{1}_{Q}(\log)\otimes_{\mathcal{O}_{Q}}E$ . The composition of this injection and (2.2) is independent of the choice of exaxt immersion $T\to Q$ . The image is contained in $\Omega^{1}_{F}(\log)\otimes_{F}E$ , so this defines the map (2.3).

Let $L/K$ be a finite Galois extension and let $G=\operatorname{Gal}(L/K)$ be Galois group. Let $r>0$ be a rational number such that $G^{r+}_{\log}=\cup_{s>r}G^{s}_{\log}=1$ . By the reduced fiber theorem [4], there exists a separable extension $K^{\prime}$ of $K$ of ramification index $e$ such that $er$ is an integer and the geometric closed fiber $Q_{S^{\prime}}^{(r)}\times_{S^{\prime}}\overline{F}$ is reduced. We define the scheme $\Theta_{L/K,\log,F^{\prime}}^{(r)}$ to be the vector bundle $\operatorname{Hom}_{F^{\prime}}(\mathfrak{m}^{er}_{K^{\prime}}/\mathfrak{m}^{er+}_{K^{\prime}},\operatorname{Tor}_{1}^{\mathcal{O}_{L}}(\Omega^{1}_{\mathcal{O}_{L}/\mathcal{O}_{K}}(\log/\log),E))$ over $\operatorname{Spec}(E\otimes_{F}F^{\prime})_{\operatorname{red}}$ . If we take a minimal exact immersion $T\to Q$ to a log smooth scheme over $S$ , the isomorphism (2.2) induces an isomorphism $Q_{F^{\prime},\operatorname{red}}^{[r]}\to\Theta_{L/K,\log,F^{\prime}}^{(r)}$ by [17, Proposition 3.1.3.2]. We define the scheme $\Phi_{L/K,\log,F^{\prime}}^{(r)}$ to be $Q_{F^{\prime}}^{(r)}$ . As a logarithmic variant of [17, Lemma 3.3.7], the definition of $\Phi_{L/K,\log,F^{\prime}}^{(r)}$ does not depend on the choice of a minimal exact immersion $T\to Q$ , and for every exact immersion $T=\operatorname{Spec}\mathcal{O}_{L}\to Q$ to a log smooth scheme $Q$ over $S$ , we have a cartesian diagram

(2.4)

We fix a morphism $i_{0}\colon L\to K_{s}$ to a separable closure of $K$ . Let $\Theta_{L/K,\log,\overline{F}}^{(r)\circ},\Phi_{L/K,\log,\overline{F}}^{(r)\circ}$ denote the connected components of $\Theta_{L/K,\log,\overline{F}}^{(r)},\Phi_{L/K,\log,\overline{F}}^{(r)}$ containing the image of the closed point of $\overline{T}_{\overline{F}}$ corresponding to $i_{0}$ respectively.

Proposition 2.2 (cf. [17, Theorem 4.3.3]).

The $G^{r}_{\log}$ -torsor $\Phi_{L/K,\log,\overline{F}}^{(r)\circ}$ over $\Theta_{L/K,\log,\overline{F}}^{(r)\circ}$ is additive. Hence, there exists a group scheme structure on $\Phi_{L/K,\log,\overline{F}}^{(r)\circ}$ such that the sequence

0\to G^{r}_{\log}\to\Phi_{L/K,\log,\overline{F}}^{(r)\circ}\to\Theta_{L/K,\log,\overline{F}}^{(r)\circ}\to 0

is an extension of smooth group schemes.

Proof (Saito).

We reduce the assertion to the case where the ramification index $e_{L/K}=1$ . By [9, Theorem 3.1], there exists an extension $K^{\prime}/K$ such that $e_{L^{\prime}/K^{\prime}}=1$ and $\Omega^{1}_{F}(\log)\to\Omega^{1}_{F^{\prime}}(\log)$ is injective where $L^{\prime}=LK^{\prime}$ denotes the composite field and $F^{\prime}$ denotes the residue field of $K^{\prime}$ . From the functoriality of the construction of $\Phi_{\log}\to\Theta_{\log}$ , we have the commutative diagram

Hence it suffices to show that the ${G^{\prime}}^{r}_{\log}$ -torsor $\Phi_{L^{\prime}/K^{\prime},\log,\overline{F}}^{(r)\circ}$ over $\Theta_{L^{\prime}/K^{\prime},\log,\overline{F}}^{(r)\circ}$ is additive by [17, Corollary 2.1.8.3] where $G^{\prime}=\operatorname{Gal}(L^{\prime}/K^{\prime})$ .

If $e_{L/K}$ is 1, then we have $G^{r}=G^{r}_{\log}$ and the morphism $\Phi_{L/K,\log,\overline{F}}^{(r)\circ}\to\Theta_{L/K,\log,\overline{F}}^{(r)\circ}$ is equal to $\Phi_{L/K,\overline{F}}^{(r)\circ}\to\Theta_{L/K,\overline{F}}^{(r)\circ}$ since every immersion $T\to P$ to a smooth scheme is exact under the assumption $e_{L/K}=1$ . Hence the assertion follows from the fact that $\Phi_{L/K,\overline{F}}^{(r)\circ}$ is an additive torsor over $\Theta_{L/K,\overline{F}}^{(r)\circ}$ [17, Theorem 4.3.3.1]. ∎

In the same way as in Subsection 1.2, we get a morphism

[\Phi_{\log}]\colon\operatorname{Hom}_{\mathbf{F}_{p}}(G^{r}_{\log},\mathbf{F}_{p})\to\operatorname{Hom}_{\overline{F}}(\mathfrak{m}^{r}_{K_{s}}/\mathfrak{m}^{r+}_{K_{s}},\operatorname{Tor}_{1}^{\mathcal{O}_{L}}(\Omega^{1}_{T/S}(\log/\log),\overline{F})).

We define the refined Swan conductor to be the composition of injections

\operatorname{rsw}\colon\operatorname{Hom}(G^{r},\mathbf{F}_{p})\xrightarrow{[\Phi_{\log}]}\operatorname{Hom}_{\overline{F}}(\mathfrak{m}^{r}_{K_{s}}/\mathfrak{m}^{r+}_{K_{s}},\operatorname{Tor}_{1}^{\mathcal{O}_{L}}(\Omega^{1}_{T/S}(\log/\log),\overline{F}))\\ \to\operatorname{Hom}_{\overline{F}}(\mathfrak{m}^{r}_{K_{s}}/\mathfrak{m}^{r+}_{K_{s}},\Omega^{1}_{F}(\log)\otimes_{F}\overline{F})

(2.5)

where the second morphism is induced by the map (2.3). We call $\operatorname{rsw}\chi$ the refined Swan conductor of $\chi$ for $\chi\colon G^{r}\to\mathbf{F}_{p}$ .

Remark 2.3.

The construction of the refined Swan conductor here coincides with the construction in [13]. Indeed, using the notation in [13], we have the following diagram

by (2.4) where the right vertical map is induced by the second morphism of (2.5).

Proposition 2.4.

Let $\chi\in\mathrm{H}^{1}(K,\mathbf{Q}/\mathbf{Z})$ be a character of Swan conductor $n$ . Then, the image of the refined Swan conductor $\operatorname{rsw}\chi\colon\mathfrak{m}^{n}_{K}/\mathfrak{m}^{n+1}_{K}\to\Omega^{1}_{F}(\log)\otimes_{F}\overline{F}$ is contained in $\Omega^{1}_{F}(\log)$ .

Proof.

The assertion follows from [9, Theorem 1.5] since the refined Swan conductor by Kato is defined as a map to $\Omega^{1}_{F}(\log)$ . ∎

We recall the integrality of the refined Swan conductor proved by Kato.

Theorem 2.5 ([6, Theorem 7.1, 7.3]).

Let $A$ be an excellent regular local ring of dimension $d$ with fraction field $K$ and with residue field $k$ of characteristic $p>0$ . We assume $c=[k:k^{p}]<\infty$ and fix a lifting $(x_{l})_{l=1,\dots,c}$ of a $p$ -basis of $k$ to $A$ . Let $(\pi_{i})_{i=1,\dots,d}$ be a regular system of parameters of $A$ and let $K_{i}$ be the local field at the prime ideal generated by $\pi_{i}$ . We fix an integer $r$ satisfying $1\leq r\leq d$ . Let $D_{i}$ be a divisor on $X=\operatorname{Spec}A$ defined by $\pi_{i}$ and let $U$ be the complement of $D=\cup_{i=1}^{r}D_{i}$ . Let $\chi$ be an element of $\mathrm{H}^{1}(U,\mathbf{Q}/\mathbf{Z})$ and put $n_{i}=\operatorname{sw}(\chi|_{K_{i}})$ . Write

\operatorname{rsw}(\chi|_{K_{j}})=(\sum_{1\leq i\leq r}\alpha_{i,j}d\log\pi_{i}+\sum_{r+1\leq i\leq d}\alpha_{i,j}d\pi_{i}+\sum_{1\leq l\leq c}\beta_{l,j}dx_{l})/\pi_{1}^{n_{1}}\cdots\pi_{r}^{n_{r}}.

with $\alpha_{i,j},\beta_{l,j}\in\operatorname{Frac}(A/\pi_{j})$ where $1\leq i\leq d$ , $1\leq j\leq r$ satisfying $n_{j}\geq 1$ , and $1\leq l\leq c$ . Then, we have the following properties:

(1)

$\alpha_{i,j},\beta_{l,j}\in A/\pi_{j}$ .
(2)

For integers $j,j^{\prime}$ satisfying $1\leq j,j^{\prime}\leq r$ , the images of $\alpha_{i,j},\alpha_{i,j^{\prime}}$ in $A/(\pi_{j})+(\pi_{j^{\prime}})$ are equal for each $i$ and the images of $\beta_{l,j},\beta_{l,j^{\prime}}$ in $A/(\pi_{j})+(\pi_{j^{\prime}})$ are equal for each $l$ .

3 Comparison

In this section, we compare the refined Swan conductor with the characteristic form.

Let $K$ be a henselian discrete valuation field with residue field $F$ of characteristic $p>0$ . Let $\chi$ be an element of $\mathrm{H}^{1}(K,\mathbf{Q}/\mathbf{Z})$ and let $L$ be a finite abelian Galois extension of $K$ such that $\chi$ factors through $G=\operatorname{Gal}(L/K)$ . Since we have $G^{r}\supset G^{r}_{\log}\supset G^{r+1}$ for a rational number $r>0$ by [3, Lemma 5.3] and the Swan conductor and the total dimension are integers, we have $\operatorname{dt}(\chi)=\operatorname{sw}(\chi)+1$ or $\operatorname{dt}(\chi)=\operatorname{sw}(\chi)$ . We say that $\chi$ is of type I if $\operatorname{dt}(\chi)=\operatorname{sw}(\chi)+1$ and $\chi$ is of type II if $\operatorname{dt}(\chi)=\operatorname{sw}(\chi)$ . If the residue field $F$ of $K$ is perfect, the character $\chi$ is of type I by [3, Proposition 6.3.1].

Proposition 3.1 ([17, Proposition 1.1.8.2, 1.1.10]).

Let $K$ be a discrete valuation field with residue field $F$ . There exists an extension $K^{\prime}$ of $K$ with perfect residue field $F^{\prime}$ such that

\mathrm{H}_{1}(L_{\overline{F}/\mathcal{O}_{K}})\to\mathrm{H}_{1}(L_{\overline{F^{\prime}}/\mathcal{O}_{K^{\prime}}})

(3.1)

is injective and $e_{K^{\prime}/K}$ is equal to 1.

Proposition 3.2.

Let $K$ be a henselian discrete valuation field with residue field $F$ . Let $L$ be a finite Galois extension of $K$ of Galois group $G$ . Let $r>0$ be a rational number and assume $G^{r+}_{\log}=1$ and $G^{r}_{\log}=G^{r+1}$ . Then, there exists a commutative diagram

where the right vertical map is induced by the composition of the maps

\Omega^{1}_{F}(\log)\otimes_{F}\overline{F}\xrightarrow{\operatorname{res}\otimes 1}\overline{F}\cong\overline{F}\otimes_{F}\mathfrak{m}_{K}/\mathfrak{m}^{2}_{K}\otimes_{\mathcal{O}_{K}}\mathfrak{m}_{K}^{-1}\ \xrightarrow{w}\mathrm{H}_{1}(L_{\overline{F}/\mathcal{O}_{K}})\otimes_{\mathcal{O}_{K}}\mathfrak{m}_{K}^{-1}.

We reduce the proof of Proposition 3.2 to the following case, which is proved by Saito.

Lemma 3.3.

Proposition 3.2 holds if the residue field $E$ of $L$ is a separable extension of $F$ . (In this case, the equality $G^{r}_{\log}=G^{r+1}$ holds by [3, Proposition 6.3.1]).

Proof (Saito).

If $L/K$ is tamely ramified, then we have $G^{r}_{\log}=G^{r+1}=1$ and the assertion is trivial. Hence we may assume that $L/K$ is wildly ramified. Let $m=pn$ be the ramification index of $L/K$ . Since $E$ is a separable extension of $F$ and thus $\mathcal{O}_{L}$ is generated by a single element over $\mathcal{O}_{K}$ , we may take a minimal immersion $T=\operatorname{Spec}\mathcal{O}_{L}\to P$ to a smooth scheme of relative dimension of 1 over $S=\operatorname{Spec}\mathcal{O}_{K}$ . We prove that the dilatation $P^{[1]}$ contains an open subscheme $Q$ such that $Q$ is log smooth over $S$ and the immersion $T\to Q$ is a minimal exact immersion. Let $x\in P$ the image of the closed point of $T$ . Since the assertion is local at $x$ , after replacing $P$ by an open neighborhood of $x$ , we may assume $P=\operatorname{Spec}A$ is affine and $\mathcal{O}_{L}=A/f$ with $f\in A$ . Let $\pi$ be a uniformizer of $K$ and let $s\in A$ be a lifting of a uniformizer of $L$ . Further replacing $P$ , we may assume that a morphism $P\to\mathbf{A}^{1}_{S}=\operatorname{Spec}\mathcal{O}_{K}[s]$ is étale. Let $\mathfrak{m}_{x}=(f,s)$ be the maximal ideal of $A$ at $x$ . Since $\pi$ is divisible by $s^{m}$ in $\mathcal{O}_{L}=A/f$ and $\pi\in\mathfrak{m}_{x}-\mathfrak{m}_{x}^{2}$ , we have $f\equiv\pi\mod\mathfrak{m}_{x}^{2}$ and $\mathfrak{m}_{x}=(\pi,s)$ . We may write $f=a\pi+bs^{m}$ with $a,b\in A$ . Since $f$ is not in $\mathfrak{m}_{x}^{2}$ , we see that $a$ is not in $\mathfrak{m}_{x}$ . Hence we may assume $a$ is a unit in $A$ by replacing $P$ . Since we have $a\pi+bs^{m}=0\in\mathcal{O}_{L}$ , $b$ is not in $\mathfrak{m}_{x}$ and we may also assume $b$ is a unit. Then we have an equality $(f,\pi)=(\pi,s^{m})$ of ideals of $A$ . We have $P^{[1]}=\operatorname{Spec}A[s^{m}/\pi]=\operatorname{Spec}A[v]/(s^{m}-v\pi)$ and $P^{[1]}$ contains an open subscheme $Q=\operatorname{Spec}A[u^{\pm 1}]/(us^{m}-\pi)=P\times_{\mathbf{A}^{1}_{S}}\mathcal{O}_{K}[s,u^{\pm 1}]/(us^{m}-\pi)$ , which is log smooth over $S$ since $P\to\mathbf{A}^{1}_{S}$ is étale. Since the closed subscheme $Q_{F,\operatorname{red}}$ of $Q$ is defined by $s$ , the inverse image $T\times_{Q}Q_{F,\operatorname{red}}$ is $E$ and $T\to Q$ is an exact immersion. We note that $Q\to P$ induces an isomorphism $N_{T/P}\otimes_{\mathcal{O}_{K}}\mathfrak{m}_{K}^{-1}\to N_{T/Q}$ .

Let $K^{\prime}$ be a finite separable extension of $K$ such that the closed fibers of $P_{S^{\prime}}^{(r)}$ and $Q_{S^{\prime}}^{(r)}$ are reduced. By the functoriality of dilatations and normalizations, the middle square of the diagram

(3.2)

is commutative and we have a commutative diagram

of extensions of smooth group schemes. Hence we have a commutative diagram

It suffices to show that the diagram

(3.3)

is commutative. Since $g=f/\pi=a+bv$ defines a basis of $N_{T/Q}$ , we consider the image of this basis. The left vertical map sends $g$ to $f\otimes\pi^{-1}$ . The lower horizontal map sends $f\otimes\pi^{-1}$ to $wf\otimes\pi^{-1}=a\cdot w\pi\otimes\pi^{-1}$ . The right horizontal map sends $g=f/\pi$ to $da+vdb+bvd\log v$ . This is equal to $da+vdb+ad\log\pi$ in $\Omega^{1}_{F}(\log)\otimes E$ since we have $md\log s-d\log v-d\log\pi=0$ in $\Omega^{1}_{Q}(\log)$ and $p$ divides $m$ and $g=a+bv=0$ in $\mathcal{O}_{L}=A[g]/g$ . The right vertical map sends $da+vdb+ad\log\pi$ to $a\cdot w\pi\otimes\pi^{-1}$ since $a,b$ are units in $A$ and $da+vdb\in\Omega^{1}_{E}$ . ∎

Proof of Proposition 3.2.

Let $K^{\prime}$ be an extension as in Proposition 3.1 and let $L^{\prime}=LK^{\prime}$ be the composition field and $G^{\prime}=\operatorname{Gal}(L^{\prime}/K^{\prime})$ be the Galois group. Then, we have $G^{r+1}={G^{\prime}}^{r+1}$ by [17, Proposition 4.2.4.1]. Since the residue field $F^{\prime}$ of $K^{\prime}$ is perfect, we have ${G^{\prime}}^{r}_{\log}={G^{\prime}}^{r+1}$ by [3, Proposition 6.3.1]. Since we assume $G^{r}_{\log}=G^{r+1}$ , we have $G^{r}_{\log}=G^{\prime r}_{\log}$ .

By the commutative diagram

it suffices to show that the diagram

is commutative since the map $\mathrm{H}_{1}(L_{\overline{F}/\mathcal{O}_{K}})\to\mathrm{H}_{1}(L_{\overline{F^{\prime}}/\mathcal{O}_{K^{\prime}}})$ is injective. The upper square and the lower square are commutative by the functoriality of the refined Swan conductor and the characteristic form respectively. The middle square is commutative by Lemma 3.3 since $F^{\prime}$ is perfect. ∎

The following proposition is proved by Saito.

Proposition 3.4.

Let $K$ be a henselian discrete valuation field with residue field $F$ . Let $L$ be a finite Galois extension of $K$ of Galois group $G$ . Let $r>1$ be a rational number and assume $G^{r+}=1$ . Then, there exists a commutative diagram

where the right vertical map is induced from the composition of the maps

\mathrm{H}_{1}(L_{\overline{F}/\mathcal{O}_{K}})\to\Omega^{1}_{F}\otimes_{F}\overline{F}\to\Omega^{1}_{F}(\log)\otimes_{F}\overline{F}.

Proof (Saito).

We show that there exists a commutative diagram

(3.4)

of extensions of smooth group schemes. We may take a minimal immersion $T\to P$ to a smooth scheme over $S=\operatorname{Spec}\mathcal{O}_{K}$ and a minimal exact immersion $T\to Q$ to a log smooth scheme over $S$ . By replacing $Q$ by an étale neighborhood, we may assume that there exists a morphism $Q\to P$ . Let $K^{\prime}$ be a finite separable extension of $K$ such that the closed fibers of $P_{S^{\prime}}^{(r)}$ and $Q_{S^{\prime}}^{(r)}$ are reduced. By the functoriality of dilatations and normalizations, we have a commutative diagram

(3.5)

Since $Q\to P$ induces a morphism

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of free resolutions, we obtain a commutative diagram

(3.6)

where the isomorphisms are (2.2) and (1.3). The diagram

(3.7)

is commutative by (3.5), (3.6) and the functoriality of normalizations and dilatations. Hence the diagram (3.4) is commutative and defines a commutative diagram

Hence it suffices to show that the diagram

is commutative where $E$ denotes the residue field of $L$ . By the injectivity of the map $\Omega^{1}_{F}(\log)\otimes_{F}E\to\Omega^{1}_{Q}(\log)\otimes_{\mathcal{O}_{Q}}E$ and the commutative diagrams

it suffices to show that the diagram

is commutative. The left square is commutative by (3.6). The middle square is commutative by the functoriality of conormal sheaves. The right square is commutative since the diagram

is commutative by the functoriality of cotangent complexes. ∎

4 Proof of the rationality and the integrality

In this section, we prove Theorem 1.3 and Theorem 1.5.

Proof of Theorem 1.3.

Let $L$ be a finite abelian extension such that $\chi$ factors through $G=\operatorname{Gal}(L/K)$ . Let $\pi$ be a uniformizer and let $(v_{i})_{i\in I}$ be a family of elements of $\mathcal{O}_{K}$ such that $(dv_{i})_{i\in I}$ forms a basis of $\Omega^{1}_{F}$ . We put $m=\operatorname{dt}(\chi)$ . First we consider the case where the character $\chi$ is of type I. If we put

\operatorname{rsw}(\chi)=(\alpha d\log\pi+\sum_{i\in I}\beta_{i}dv_{i})/\pi^{m-1},

then we have

\operatorname{char}(\chi)=(\alpha w\pi)/\pi^{m}

by Proposition 3.2 and the assertion follows from Proposition 2.4.

Second we consider the case where the character $\chi$ is of type II. If we put

\operatorname{char}(\chi)=(\alpha w\pi+\sum_{i\in I}\beta_{i}wv_{i})/\pi^{m},

then we have

\operatorname{rsw}(\chi)=(\sum_{i\in I}\beta_{i}dv_{i})/\pi^{m}

by Proposition 3.4. We see that $\beta_{i}$ are contained in $F$ by Proposition 2.4. We show that $\alpha$ is contained in $F^{1/p}$ . We define the discrete valuation ring $\mathcal{O}_{K^{\prime}}$ by

\mathcal{O}_{K^{\prime}}=\mathcal{O}_{K}[w_{i}]_{i\in I}/(w_{i}^{p}-v_{i})_{i\in I}

and let $K^{\prime}$ be the fraction field of $\mathcal{O}_{K^{\prime}}$ . Then the residue field $F^{\prime}$ of $K^{\prime}$ is $F^{1/p}$ . Let $L^{\prime}=LK^{\prime}$ be the composite field and $G^{\prime}=\operatorname{Gal}(L^{\prime}/K^{\prime})$ be the Galois group. The map $\mathrm{H}_{1}(L_{\overline{F}/\mathcal{O}_{K}})\to\mathrm{H}_{1}(L_{\overline{F^{\prime}}/\mathcal{O}_{K^{\prime}}})$ sends $w\pi$ to $w\pi$ and the other basis to 0. The diagram

\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 33.15959pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\&\crcr}}}\ignorespaces{\hbox{\kern-32.3896pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{Hom}_{\mathbf{F}_{p}}(G^{m},\mathbf{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 35.42697pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\operatorname{char}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 57.5116pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 0.0pt\raise-30.52054pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 57.5116pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{Hom}_{\overline{F}}(\mathfrak{m}^{m}_{K_{s}}/\mathfrak{m}^{m+}_{K_{s}},\mathrm{H}_{1}(L_{\overline{F}/\mathcal{O}_{K}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 115.62692pt\raise-31.40941pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 198.09425pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-33.15959pt\raise-41.90941pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{Hom}_{\mathbf{F}_{p}}(G^{\prime m},\mathbf{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 57.15959pt\raise-41.90941pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 57.15959pt\raise-41.90941pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{Hom}_{\overline{F^{\prime}}}(\mathfrak{m}^{m}_{K^{\prime}_{s}}/\mathfrak{m}^{m+}_{K^{\prime}_{s}},\mathrm{H}_{1}(L_{\overline{F^{\prime}}/\mathcal{O}_{K^{\prime}}}))}$}}}}}}}\ignorespaces}}}}\ignorespaces.

is commutative by the functoriality [17, (4.17)]. If the coefficient of $w\pi$ is not zero, then the image of $\operatorname{char}\chi$ by the right vertical arrow is not zero. Hence $G^{\prime m}$ is not trivial and thus we have $\operatorname{dt}(\chi^{\prime})=m$ . Let $\chi^{\prime}$ be the image of the character $\chi$ by the left vertical arrow. Then the characteristic form $\operatorname{char}(\chi^{\prime})$ is of the form

\operatorname{char}(\chi^{\prime})=\alpha\cdot w\pi/\pi^{m}.

If the character $\chi^{\prime}$ is of type II, then the refined Swan conductor of $\chi^{\prime}$ is zero and this is a contradiction. Hence the character $\chi^{\prime}$ is of type I. By the first case, we have $\alpha\in{F^{\prime}}=F^{1/p}$ . ∎

Next, we prove Theorem 1.5. We prepare the following lemma.

Lemma 4.1.

We use the notation as in Theorem 1.5 and assume the dimension of $A$ is $2$ . We define the regular local ring $A^{\prime}$ of dimension $2$ by

A^{\prime}=A[u_{2},y_{l}]_{1\leq l\leq c}/(u_{2}^{p}-\pi_{2},y_{l}^{p}-x_{l})_{1\leq l\leq c}.

The maximal ideal of $A^{\prime}$ is generated by $\pi_{1}$ and $u_{2}$ . Let $K^{\prime}_{1}$ be the local field of $A^{\prime}$ at the prime ideal generated by $\pi_{1}$ and let $K^{\prime}_{2}$ be the local field of $A^{\prime}$ at the prime ideal generated by $u_{2}$ . Let $L_{i}$ be a finite abelian extension of $K_{i}$ $(i=1,2)$ such that $\chi|_{K_{i}}$ factors through $G_{i}=\operatorname{Gal}(L_{i}/K_{i})$ . Let $L^{\prime}_{i}=L_{i}K^{\prime}_{i}$ be the composite field and put $G^{\prime}_{i}=\operatorname{Gal}(L^{\prime}_{i}/K^{\prime}_{i})$ . Let $F_{i}$ and $F^{\prime}_{i}$ be the residue fields of $K_{i}$ and $K^{\prime}_{i}$ respectively. Let $U^{\prime}$ be the pullback of $U$ by $\operatorname{Spec}A^{\prime}\to\operatorname{Spec}A$ and let $\chi^{\prime}\in\mathrm{H}^{1}(U^{\prime},\mathbf{Q}/\mathbf{Z})$ be the pullback of $\chi$ . We put $m^{\prime}_{i}=\operatorname{dt}(\chi^{\prime}|_{K^{\prime}_{i}})$ .

1. Assume that $\chi|_{K_{1}}$ is wildly ramified. If the coefficient of $w\pi_{1}$ in $\operatorname{char}(\chi|_{K_{1}})$ is not zero, then we have $m^{\prime}_{1}=m_{1}$ and the character $\chi^{\prime}|_{K^{\prime}_{1}}$ is of type I. If the coefficient of $w\pi_{1}$ in $\operatorname{char}(\chi|_{K_{1}})$ is zero, then we have $m^{\prime}_{1}<m_{1}$ .

2. Assume that $\chi|_{K_{2}}$ is of type II. If the coefficient of $w\pi_{1}$ in $\operatorname{char}(\chi|_{K_{2}})$ is not zero, then we have $m^{\prime}_{2}=pm_{2}$ and the character $\chi^{\prime}|_{K^{\prime}_{2}}$ is of type II. If the coefficient of $w\pi_{1}$ is zero, we have $\operatorname{sw}(\chi^{\prime}|_{K^{\prime}_{2}})<pm_{2}$ .

Proof.

1. The map $\mathrm{H}_{1}(L_{\overline{F_{1}}/\mathcal{O}_{K_{1}}})\to\mathrm{H}_{1}(L_{\overline{F^{\prime}_{1}}/\mathcal{O}_{K^{\prime}_{1}}})$ sends $w\pi_{1}$ to $w\pi_{1}$ and the other basis to 0. The diagram

is commutative by the functoriality [17, (4.17)]. The first assertion follows from the same argument as in the proof of Proposition 1.3. If the coefficient of $w\pi_{1}$ in $\operatorname{char}(\chi|_{K_{1}})$ is zero, then the image of $\operatorname{char}\chi|_{K_{1}}$ by the right vertical arrow is zero. Hence $G^{\prime m_{1}}_{1}$ is trivial and we have $m^{\prime}_{1}<m_{1}$ .

2. The map $\Omega^{1}_{F_{2}}(\log)\to\Omega^{1}_{F^{\prime}_{2}}(\log)$ sends $d\log\pi_{1}$ to $d\log\pi_{1}$ and the other basis to 0. Since $\chi|_{K_{2}}$ is of type II, we have $\operatorname{sw}(\chi|_{K_{2}})=m_{2}$ . The diagram

is commutative by the functoriality [9, (4.17)]. Since we assume the coefficient of $w\pi_{1}$ is not zero, the coefficient of $d\log\pi_{1}$ is not zero and the image of $\operatorname{rsw}\chi|_{K_{2}}$ by the right vertical arrow is not zero. Hence $G^{\prime pm_{2}}_{2,\log}$ is not trivial and thus we have $\operatorname{sw}(\chi^{\prime}|_{K^{\prime}_{2}})=pm_{2}$ . Since the inequality $m^{\prime}_{2}\leq pm_{2}$ holds, we obtain $m^{\prime}_{2}=pm_{2}$ and the character $\chi|_{K_{2}}$ is of type II. If the coefficient of $w\pi_{1}$ in $\operatorname{char}(\chi|_{K_{2}})$ is zero, then the image of $\operatorname{rsw}\chi|_{K_{2}}$ by the right vertical arrow is zero. Hence $G^{\prime pm_{2}}_{2,\log}$ is trivial and we have $\operatorname{sw}(\chi^{\prime}|_{K^{\prime}_{2}})<pm_{2}$ . ∎

Proof of Theorem 1.5.

Since $A/\pi_{j}$ is regular, it suffices to show that $\alpha_{i,j}^{p},\beta_{l,j}$ are elements of $(A/\pi_{j})_{\mathfrak{q}}$ for any prime ideal $\mathfrak{q}$ of height one of $A/\pi_{j}$ . By replacing $A$ by $A_{\mathfrak{q}}$ , we may assume $\dim A=2$ . We use the notation as in Lemma 4.1.

We divide the proof into six cases.

(a) The case where $r=1$ and $\chi|_{K_{1}}$ is of type I.

In this case, the characteristic form $\operatorname{char}(\chi|_{K_{1}})$ is the image of the refined Swan conductor $\operatorname{rsw}(\chi|_{K_{1}})$ by Proposition 3.2. If we put

\operatorname{rsw}(\chi|_{K_{1}})=(\alpha_{1}d\log\pi_{1}+\alpha_{2}d\pi_{2}+\sum_{1\leq l\leq c}\beta_{l}dx_{l})/\pi_{1}^{m_{1}-1},

then we have

\operatorname{char}(\chi|_{K_{1}})=\alpha_{1}w\pi_{1}/\pi_{1}^{m_{1}}.

Since $\alpha_{1}$ is in $A/\pi_{1}$ from Theorem 2.5, the assertion follows.

(b) The case where $r=1$ and $\chi|_{K_{1}}$ is of type II.

In this case, the refined Swan conductor $\operatorname{rsw}(\chi|_{K_{1}})$ is the image of the characteristic form $\operatorname{char}(\chi|_{K_{1}})$ by Proposition 3.4. Hence, if we put

\operatorname{char}(\chi|_{K_{1}})=(\alpha_{1}w\pi_{1}+\alpha_{2}w\pi_{2}+\sum_{1\leq l\leq c}\beta_{l}wx_{l})/\pi_{1}^{m_{1}},

then we have

\operatorname{rsw}(\chi|_{K_{1}})=(\alpha_{2}d\pi_{2}+\sum_{1\leq l\leq c}\beta_{l}dx_{l})/\pi_{1}^{m_{1}}.

This implies that $\alpha_{2},\beta_{l}\in A/\pi_{1}$ by Theorem 2.5. It remains to prove $\alpha_{1}^{p}\in A/\pi_{1}$ . If $\alpha_{1}=0$ , then the assertion holds, so we may assume $\alpha_{1}$ is not $0$ . Then we have $m^{\prime}_{1}=m_{1}$ and

\operatorname{char}(\chi^{\prime}|_{K^{\prime}_{1}})=\alpha_{1}\cdot w\pi_{1}/\pi_{1}^{m_{1}}

and $\chi^{\prime}|_{K^{\prime}_{1}}$ is of type I by Lemma 4.1.1. Hence we have $\alpha_{1}\in A^{\prime}/\pi_{1}$ by the case (a) applied to the pair $(A^{\prime},U^{\prime},\chi^{\prime})$ . Since $A/\pi_{1}$ is of characteristic $p$ , we obtain $\alpha_{1}^{p}\in A/\pi_{1}$ .

(c) The case where $r=2$ and $\chi|_{K_{1}}$ or $\chi|_{K_{2}}$ is tamely ramified.

In this case, we can prove the assertion by a similar argument as that in the case (a) and (b).

(d) The case where $r=2$ and $\chi|_{K_{1}}$ and $\chi|_{K_{2}}$ are both of type I.

If we put

\operatorname{rsw}(\chi|_{K_{1}})=(\alpha_{1,1}d\log\pi_{1}+\alpha_{2,1}d\pi_{2}+\sum_{1\leq l\leq c}\beta_{l,1}dx_{l})/\pi_{1}^{m_{1}-1}\pi_{2}^{m_{2}-1},

\operatorname{rsw}(\chi|_{K_{2}})=(\alpha_{1,2}d\pi_{1}+\alpha_{2,2}d\log\pi_{2}+\sum_{1\leq l\leq c}\beta_{l,2}dx_{l})/\pi_{1}^{m_{1}-1}\pi_{2}^{m_{2}-1},

then we have

\operatorname{char}(\chi|_{K_{1}})=\pi_{2}\alpha_{1,1}w\pi_{1}/\pi_{1}^{m_{1}}\pi_{2}^{m_{2}},

\operatorname{char}(\chi|_{K_{2}})=\pi_{1}\alpha_{2,2}w\pi_{2}/\pi_{1}^{m_{1}}\pi_{2}^{m_{2}}

by Proposition 3.2. Since $\alpha_{1,1}$ is in $A/\pi_{1}$ and $\alpha_{2,2}$ is in $A/\pi_{2}$ by Theorem 2.5, the assertion follows. We note that the coefficient of $w\pi_{1}$ in $\operatorname{char}(\chi|_{K_{1}})$ is contained in $\pi_{2}\cdot(A/\pi_{1})$ .

(e) The case where $r=2$ and $\chi|_{K_{1}}$ is of type II and $\chi|_{K_{2}}$ is of type I, or $\chi|_{K_{1}}$ is of type I and $\chi|_{K_{2}}$ is of type II.

We may only consider the case where $\chi|_{K_{1}}$ is of type II and $\chi|_{K_{2}}$ is of type I. If we put

\operatorname{char}(\chi|_{K_{1}})=(\alpha_{1,1}w\pi_{1}+\alpha_{2,1}w\pi_{2}+\sum_{1\leq l\leq c}\beta_{l,1}wx_{l})/\pi_{1}^{m_{1}}\pi_{2}^{m_{2}},

\operatorname{rsw}(\chi|_{K_{2}})=(\alpha_{1,2}d\log\pi_{1}+\alpha_{2,2}d\log\pi_{2}+\sum_{1\leq l\leq c}\beta_{l,2}dx_{l})/\pi_{1}^{m_{1}}\pi_{2}^{m_{2}-1},

then we have

\operatorname{rsw}(\chi|_{K_{1}})=(\alpha_{2,1}d\log\pi_{2}+\sum_{1\leq l\leq c}\pi_{2}^{-1}\beta_{l,1}dx_{l})/\pi_{1}^{m_{1}}\pi_{2}^{m_{2}-1},

\operatorname{char}(\chi|_{K_{2}})=\alpha_{2,2}w\pi_{2}/\pi_{1}^{m_{1}}\pi_{2}^{m_{2}}

by Proposition 3.2 and Proposition 3.4. By Theorem 2.5, we have $\alpha_{2,1},\beta_{l,1}\in A/\pi_{1}$ and $\alpha_{2,2}\in A/\pi_{2}$ and $\alpha_{2,1}=\alpha_{2,2}$ and $\beta_{l,1}=0$ in $A/(\pi_{1})+(\pi_{2})$ . Hence it suffices to show that $\alpha_{1,1}^{p}\in A/\pi_{1}$ and $\alpha_{1,1}^{p}=0\in A/(\pi_{1})+(\pi_{2})$ .

If $\alpha_{1,1}=0$ , then the assertion holds, so we may assume $\alpha_{1,1}$ is not $0$ . Then the characteristic form $\operatorname{char}(\chi^{\prime}|_{K^{\prime}_{1}})$ is of the form

\operatorname{char}(\chi^{\prime}|_{K^{\prime}_{1}})=\alpha_{1,1}\cdot w\pi_{1}/\pi_{1}^{m_{1}}u_{2}^{pm_{2}}=u_{2}^{m^{\prime}_{2}-pm_{2}}\alpha_{1,1}\cdot w\pi_{1}/\pi_{1}^{m_{1}}u_{2}^{m^{\prime}_{2}}

and $\chi^{\prime}|_{K^{\prime}_{1}}$ is of type I by Lemma 4.1.1. Since we assume $\chi|_{K_{2}}$ is of type I, we have $\operatorname{sw}(\chi|_{K_{2}})=m_{2}-1$ . Since the ramification index of the extension $K^{\prime}_{2}/K_{2}$ is $p$ , we have $\operatorname{sw}(\chi^{\prime}|_{K^{\prime}_{2}})\leq p(m_{2}-1)$ by [9, Proposition 5.1.1]. Thus we have $m^{\prime}_{2}-pm_{2}<0$ . Hence we have $\alpha_{1,1}\in u_{2}\cdot(A^{\prime}/\pi_{1})$ by the case (c) applied to the pair $(A^{\prime},U^{\prime},\chi^{\prime})$ if $\chi^{\prime}|_{K^{\prime}_{2}}$ is tamely ramified, by the case (d) if $\chi^{\prime}|_{K^{\prime}_{2}}$ is of type I, and by the first half of the argument in the case (e) if $\chi^{\prime}|_{K^{\prime}_{2}}$ is of type II. Hence we obtain $\alpha_{1,1}^{p}\in\pi_{2}\cdot(A/\pi_{1})$ .

(f) The case where $r=2$ and $\chi|_{K_{1}}$ and $\chi|_{K_{2}}$ are both of type II.

If we put

\operatorname{char}(\chi|_{K_{1}})=(\alpha_{1,1}w\pi_{1}+\alpha_{2,1}w\pi_{2}+\sum_{1\leq l\leq c}\beta_{l,1}wx_{l})/\pi_{1}^{m_{1}}\pi_{2}^{m_{2}},

\operatorname{char}(\chi|_{K_{2}})=(\alpha_{1,2}w\pi_{1}+\alpha_{2,2}w\pi_{2}+\sum_{1\leq l\leq c}\beta_{l,2}wx_{l})/\pi_{1}^{m_{1}}\pi_{2}^{m_{2}},

then we have

\operatorname{rsw}(\chi|_{K_{1}})=(\pi_{2}\alpha_{2,1}d\log\pi_{2}+\sum_{1\leq l\leq c}\beta_{l,1}dx_{l})/\pi_{1}^{m_{1}}\pi_{2}^{m_{2}},

\operatorname{rsw}(\chi|_{K_{2}})=(\pi_{1}\alpha_{1,2}d\log\pi_{1}+\sum_{1\leq l\leq c}\beta_{l,2}dx_{l})/\pi_{1}^{m_{1}}\pi_{2}^{m_{2}}

by Proposition 3.4. By Theorem 2.5, we have $\beta_{l,1}=\beta_{l,2}$ in $A/(\pi_{1})+(\pi_{2})$ . It suffices to show $\alpha_{1,1}^{p}\in A/\pi_{1}$ , $\alpha_{1,2}^{p}\in A/\pi_{2}$ and $\alpha_{1,1}^{p}=\alpha_{1,2}^{p}\in A/(\pi_{1})+(\pi_{2})$ since the assertion corresponding to $\alpha_{2,1},\alpha_{2,2}$ is proved by switching $\pi_{1}$ and $\pi_{2}$ .

If $\alpha_{1,1}\neq 0$ and $\alpha_{1,2}\neq 0$ , then we have $\operatorname{dt}(\chi^{\prime}|_{K^{\prime}_{1}})=m_{1}$ and $\operatorname{dt}(\chi^{\prime}|_{K^{\prime}_{2}})=pm_{2}$ by Lemma 4.1. Further, $\operatorname{char}(\chi^{\prime}|_{K^{\prime}_{1}})$ is of type I and $\operatorname{char}(\chi^{\prime}|_{K^{\prime}_{2}})$ is of type II and we have

\operatorname{char}(\chi^{\prime}|_{K^{\prime}_{1}})=\alpha_{1,1}w\pi_{1}/\pi_{1}^{m_{1}}u_{2}^{pm_{2}},

\operatorname{char}(\chi^{\prime}|_{K^{\prime}_{2}})=\alpha_{1,2}w\pi_{1}/\pi_{1}^{m_{1}}u_{2}^{pm_{2}}

by Lemma 4.1. By the case (e), we have $\alpha_{1,1}\in A^{\prime}/\pi_{1}$ , $\alpha_{1,2}\in A^{\prime}/u_{2}$ and $\alpha_{1,1}=\alpha_{1,2}\in A^{\prime}/(\pi_{1})+(u_{2})$ . Hence we obtain $\alpha_{1,1}^{p}\in A/\pi_{1},\alpha_{1,2}^{p}\in A/\pi_{2}$ and $\alpha_{1,1}^{p}=\alpha_{1,2}^{p}\in A/(\pi_{1})+(\pi_{2})$ .

If $\alpha_{1,1}\neq 0$ and $\alpha_{1,2}=0$ , then we have

\operatorname{char}(\chi^{\prime}|_{K^{\prime}_{1}})=\alpha_{1,1}w\pi_{1}/\pi_{1}^{m_{1}}u_{2}^{pm_{2}}=u_{2}^{m^{\prime}_{2}-pm_{2}}\alpha_{1,1}w\pi_{1}/\pi_{1}^{m_{1}}u_{2}^{m^{\prime}_{2}}

and $\chi^{\prime}|_{K^{\prime}_{1}}$ is of type I by Lemma 4.1.1. If $\chi^{\prime}|_{K^{\prime}_{2}}$ is tamely ramified, then we have $u_{2}^{m^{\prime}_{2}-pm_{2}}\alpha_{1,1}\in A^{\prime}/\pi_{1}$ by the case (c) and $m^{\prime}_{2}-pm_{2}=1-pm_{2}<0$ . If $\chi^{\prime}|_{K^{\prime}_{2}}$ is of type I, then we have $u_{2}^{m^{\prime}_{2}-pm_{2}}\alpha_{1,1}\in u_{2}\cdot(A^{\prime}/\pi_{1})$ by the last note in the case (d) and $m^{\prime}_{2}-pm_{2}=1+\operatorname{sw}(\chi^{\prime}|_{K^{\prime}_{2}})-pm_{2}\leq 0$ by Lemma 4.1.2. If $\chi^{\prime}|_{K^{\prime}_{2}}$ is of type II, then we have $u_{2}^{m^{\prime}_{2}-pm_{2}}\alpha_{1,1}\in u_{2}\cdot(A^{\prime}/\pi_{1})$ by the case (e) and $m^{\prime}_{2}-pm_{2}=\operatorname{sw}(\chi^{\prime}|_{K^{\prime}_{2}})-pm_{2}<0$ by Lemma 4.1.2. Hence we have $\alpha_{1,1}\in u_{2}\cdot(A^{\prime}/\pi_{1})$ in any case and we obtain $\alpha_{1,1}^{p}\in\pi_{2}\cdot(A/\pi_{1})$ .

If $\alpha_{1,1}=0$ and $\alpha_{1,2}\neq 0$ , then we prove $\alpha_{1,2}^{p}\in\pi_{1}\cdot(A/\pi_{2})$ by the induction on $m_{1}=\operatorname{dt}(\chi|_{K_{1}})>1$ . By Lemma 4.1.1, we have $m^{\prime}_{1}<m_{1}$ and by Lemma 4.1.2, we have

\operatorname{char}(\chi^{\prime}|_{K^{\prime}_{2}})=\alpha_{1,2}w\pi_{1}/\pi_{1}^{m_{1}}u_{2}^{pm_{2}}=\pi_{1}^{m^{\prime}_{1}-m_{1}}\alpha_{1,2}w\pi_{1}/\pi_{1}^{m^{\prime}_{1}}u_{2}^{pm_{2}}

and $\chi^{\prime}|_{K^{\prime}_{2}}$ is of type II. If $\chi^{\prime}|_{K^{\prime}_{1}}$ is tamely ramified or of type I, the assertion is true by the case (c) or (e) respectively. If $\chi^{\prime}|_{K^{\prime}_{1}}$ is of type II, we have $\pi_{1}^{p(m^{\prime}_{1}-m_{1})}\alpha_{1,2}^{p}\in\pi_{1}\cdot(A^{\prime}/u_{2})$ by the induction hypothesis. Hence we have $\alpha_{1,2}\in\pi_{1}\cdot(A^{\prime}/u_{2})$ and we obtain $\alpha_{1,2}^{p}\in\pi_{1}\cdot(A/\pi_{2})$ .

∎

5 F-Characteristic cycle

In this section, we define the F-characteristic cycle of a rank one sheaf on a regular surface as a cycle on the FW-cotangent bundle. We prove that the intersection with 0-section computes the Swan conductor of cohomology. We give an example of the F-characteristic cycle.

5.1 Refined Swan conductor and characteristic form of a rank one sheaf

Let $K$ be a discrete valuation field of characteristic 0 with residue field $F$ of characteristic $p>0$ . Let $X$ be a regular flat separated scheme of finite type over the valuation ring $\mathcal{O}_{K}$ of $K$ and let $D$ be a divisor with simple normal crossings. Let $\{D_{i}\}_{i\in I}$ be the irreducible components of $D$ and let $K_{i}$ be the local field at the generic point $\mathfrak{p}_{i}$ of $D_{i}$ . Let $U$ be the complement of $D$ . Let $\chi$ be an element of $\mathrm{H}^{1}(U,\mathbf{Q}/\mathbf{Z})$ . We define the Swan conductor divisor $R_{\chi}$ of $\chi$ by

R_{\chi}=\sum_{i\in I}\operatorname{sw}(\chi|_{K_{i}})D_{i}

and write the support of $R_{\chi}$ for $Z_{\chi}$ . We note that $Z_{\chi}$ is contained in the closed fiber of $X$ . Indeed, if $D_{i}$ intersects the generic fiber of $X$ , the character $\chi|_{K_{i}}$ is tamely ramified since the characteristic of $K$ is zero.

By Theorem 2.5, there exists a unique global section

\operatorname{rsw}(\chi)\in\Gamma(Z_{\chi},\Omega^{1}_{X}(\log D)(R_{\chi})|_{Z_{\chi}})

such that the germ $\operatorname{rsw}(\chi)_{\mathfrak{p}_{i}}$ of $\operatorname{rsw}(\chi)$ is $\operatorname{rsw}(\chi|_{K_{i}})$ for every $i\in I_{W,\chi}$ . We call $\operatorname{rsw}(\chi)$ the refined Swan conductor of $\chi$ .

Definition 5.1 ([7, Definition 4.2]).

Let $x$ be a closed point of $Z_{\chi}$ . For $i\in I$ satisfying $x\in D_{i}\subset Z_{\chi}$ , we define $\operatorname{ord}(\chi;x,D_{i})$ to be the maximal integer $n\geq 0$ such that

\operatorname{rsw}(\chi)|_{D_{i},x}\in\mathfrak{m}_{x}^{n}\Omega^{1}_{X}(\log D)(R_{\chi})|_{D_{i},x}

where $m_{x}$ is the maximal ideal of $\mathcal{O}_{X,x}$ . We say that $(X,U,\chi)$ is clean at $x$ if the integer $\operatorname{ord}(\chi;x,D_{i})$ is zero for every $i\in I$ satisfying $x\in D_{i}\subset Z_{\chi}$ . We say that $(X.U,\chi)$ is clean if $(X,U,\chi)$ is clean at every closed point in $Z_{\chi}$ .

We define the total dimension divisor $R^{\prime}_{\chi}$ by

R^{\prime}_{\chi}=\sum_{i\in I}\operatorname{dt}(\chi|_{K_{i}})D_{i}.

By proposition 1.2 and Theorem 1.5, there exists a unique global section

\operatorname{char}(\chi)\in\Gamma(Z_{\chi},F\Omega^{1}_{X}(pR^{\prime}_{\chi})|_{Z_{\chi}})

(5.1)

such that the germ $\operatorname{char}(\chi)_{\mathfrak{p}_{j}}$ of $\operatorname{char}(\chi)$ is

(\sum_{1\leq i\leq d}\alpha_{i,j}^{p}w\pi_{i}+\sum_{1\leq l\leq c}\beta_{l,j}^{p}wx_{l})/\pi_{1}^{pm_{1}}\cdots\pi_{r}^{pm_{r}}

for every $j\in I_{W,\chi}$ using the notation as in Theorem 1.5. We call $\operatorname{char}(\chi)$ the characteristic form of $\chi$ .

Definition 5.2.

Let $x$ be a closed point of $Z_{\chi}$ . For $i\in I$ satisfying $x\in D_{i}\subset Z_{\chi}$ , we define $n^{\prime}$ to be the maximal integer $n^{\prime}\geq 0$ such that

\operatorname{char}(\chi)|_{D_{i},x}\in\mathfrak{m}_{x}^{n^{\prime}}F\Omega^{1}_{X}(pR^{\prime}_{\chi})|_{D_{i},x}

where $m_{x}$ is the maximal ideal of $\mathcal{O}_{X,x}$ . We define $\operatorname{ord}^{\prime}(\chi;x,D_{i})$ by $\operatorname{ord}^{\prime}(\chi;x,D_{i})=n^{\prime}/p$ . We say that $(X,U,\chi)$ is non-degenerate at $x$ if $\operatorname{ord}^{\prime}(\chi;x,D_{i})$ is zero for every $i\in I$ satisfying $x\in D_{i}\subset Z_{\chi}$ . We say that $(X,U,\chi)$ is non-degenerate if $(X,U,\chi)$ is non-degenerate at every point at $x\in Z_{\chi}$ .

Remark 5.3.

By the definition, $p\cdot\operatorname{ord}^{\prime}(\chi;x,D_{i})$ is an integer but $\operatorname{ord}^{\prime}(\chi;x,D_{i})$ may not be an integer. Assume that the characteristic of the residue field of $K$ is 2 and put $e=\operatorname{ord}_{K}2$ . We consider the scheme $X=\operatorname{Spec}\mathcal{O}_{K}[T,(1+\pi^{2(e-1)}T^{3})^{-1}]$ . Let $U$ be the generic fiber $\operatorname{Spec}K[T,(1+\pi^{2(e-1)}T^{3})^{-1}]$ and let $\chi\in\mathrm{H}^{1}(U,\mathbf{F}_{2})$ be the Kummer character defined by $t^{2}=1+\pi^{2(e-1)}T^{3}$ . Then we have

\operatorname{char}(\chi)=\frac{T^{4}\cdot wT-T^{3}\cdot w(2/\pi^{e-1})}{\pi^{4}}

and we have $\operatorname{ord}^{\prime}(\chi,x,X_{F})=3/2$ where $x$ denotes the closed point defined by $(\pi,T)$ .

Similarly to Remark 1.4, we can expect that $\operatorname{ord}^{\prime}(\chi;x,D_{i})$ is an integer if the characteristic of the residue field of $K$ is not 2.

5.2 F-characteristic cycle

Let $K$ be a complete discrete valuation field of characteristic 0 with perfect residue field $F$ of characteristic $p>0$ . Let $X$ be a regular flat separated scheme of finite type over the valuation ring $\mathcal{O}_{K}$ of $K$ and let $D$ be a divisor with simple normal crossings. We assume that $X$ is purely of dimension 2. Let $D_{1},\dots,D_{n}$ be the irreducible components of $D$ and let $K_{i}$ be the local field at the generic point $\mathfrak{p}_{i}$ of $D_{i}$ . We put $U=X-D$ and let $j\colon U\to X$ be the open immersion. Let $X_{F}$ and $D_{F}$ be the closed fibers of $X$ and $D$ . We fix a finite field $\Lambda$ of characteristic $l\neq p$ . Let $\mathcal{F}$ be a locally constant constructible sheaf of $\Lambda$ -modules of rank 1 on $U$ and let $\chi\colon\pi_{1}^{\operatorname{ab}}(U)\to\Lambda^{\times}$ be the corresponding character. We fix an inclusion $\Lambda^{\times}\to\mathbf{Q}/\mathbf{Z}$ and regard $\chi$ as an element of $\mathrm{H}^{1}(U,\mathbf{Q}/\mathbf{Z})$ .

Let $I_{T,\chi},I_{W,\chi},I_{\textup{I},\chi},I_{\textup{II},\chi}$ be the subsets of $I$ consisting of $i\in I$ such that $\chi|_{K_{i}}$ is tamely ramified, wildly ramified, of type I and of type II respectively. For a closed point $x$ in $D$ , let $I_{x}$ be the subset of $I$ consisting of $i\in I$ such that $x\in D_{i}$ and $I_{*,\chi,x}$ be $I_{*,\chi}\cap I_{x}$ where $*=W,T,\textup{I},\textup{II}$ . Let $Z_{\textup{II},\chi}$ be the union $\cup_{i\in I_{\textup{II},\chi}}D_{i}$ .

We define the sub vector bundle $L_{i,\chi}$ of $T^{*}X(\log D)|_{D_{i}}$ for $i\in I_{W,\chi}$ by the image of the multiplication by the refined Swan conductor of $\chi$ :

\times\operatorname{rsw}(\chi)|_{D_{i}}\colon\mathcal{O}_{X}(-R_{\chi})\otimes_{\mathcal{O}_{X}}\mathcal{O}_{D_{i}}(\sum_{x\in D_{i}}\operatorname{ord}(\chi;x,D_{i})[x])\to\Omega^{1}_{X}(\log D)|_{D_{i}}.

Definition 5.4 ([6, (3.4.4)]).

Assume that $(X,U,\chi)$ is clean. We define the logarithmic characteristic cycle $CC^{\log}j_{!}\mathcal{F}$ as a cycle on the logarithmic cotangent bundle $T^{*}X(\log D)|_{D_{F}}$ by

CC^{\log}j_{!}\mathcal{F}=[T^{*}_{X}X(\log D)|_{D_{F}}]+\sum_{i\in I_{W,\chi}}\operatorname{sw}(\chi|_{K_{i}})[L_{i,\chi}]

where $T^{*}_{X}X(\log D)|_{D_{F}}$ denotes the 0-section of $T^{*}X(\log D)|_{D_{F}}$ .

In the case $\dim X=2$ , we define the logarithmic characteristic cycle without the assumption on the cleanness of $(X,U,\chi)$ . By [7, Theorem 4.1], there exist successive blowups $f\colon X^{\prime}\to X$ at closed points such that $(X^{\prime},f^{-1}(U),f^{*}\chi)$ is clean. Let $D^{\prime}$ be the inverse image of $D$ and let

T^{*}X(\log D)|_{D_{F}}\xleftarrow{\mathrm{pr}}T^{*}X(\log D)|_{D_{F}}\times_{D_{F}}{D^{\prime}_{F}}\xrightarrow{df^{D}}T^{*}X^{\prime}(\log D^{\prime})|_{D^{\prime}_{F}}

be the algebraic correspondence. We define $CC^{\log}j_{!}\mathcal{F}-[T^{*}_{X}X(\log D)|_{D_{F}}]$ to be the pushforward by pr of the pullback of $CC^{\log}j^{\prime}_{!}f^{*}\mathcal{F}-[T^{*}_{X^{\prime}}{X^{\prime}}(\log D^{\prime})|_{D^{\prime}_{F}}]$ by $df^{D}$ . This is independent of the choice of blowups by [7, Remark 5.7]. We put the logarithmic characteristic cycle as

CC^{\log}j_{!}\mathcal{F}=[T^{*}_{X}X(\log D)|_{D_{F}}]+\sum_{i\in I_{W,\chi}}\operatorname{sw}(\chi|_{K_{i}})[L_{i,\chi}]+\sum_{x\in D_{F}}s_{x}[T_{x}^{*}X(\log D)]

(5.2)

where $T_{x}^{*}X(\log D)$ denotes the fiber at $x$ .

Theorem 5.5 (Conductor formula, [8, Corollary 7.5.3, Theorem 8.3.7]).

Assume $\dim X=2$ and $X$ is proper over $\mathcal{O}_{K}$ . Then we have

(CC^{\log}j_{!}\mathcal{F}-[T^{*}_{X}X(\log D)|_{D_{F}}],T^{*}_{X}X(\log D)|_{D_{F}})_{T^{*}X(\log D)|_{D_{F}}}=-\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\mathcal{F})+\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\Lambda)

where $\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\mathcal{F})$ denotes the alternating sum $\sum_{m\geq 0}(-1)^{m}\operatorname{Sw}_{K}\mathrm{H}^{m}(X_{\overline{K}},j_{!}\mathcal{F})$ .

Proof.

There exists successive blowups $f\colon X^{\prime}\to X$ at closed points such that $(X^{\prime},f^{-1}(U),f^{*}\chi)$ is clean. Since the both sides do not change after replacing $X$ by $X^{\prime}$ , we may assume $(X,U,\chi)$ is clean. The intersection

(CC^{\log}j_{!}\mathcal{F}-[T^{*}_{X}X(\log D)|_{D_{F}}],T^{*}_{X}X(\log D)|_{D_{F}})_{T^{*}X(\log D)|_{D_{F}}}

equals to $-\deg c_{\chi}$ with the notation in [8, (8.3.0.1)]. Hence the assertion follows by [8, Corollary 7.5.3] and [8, Theorem 8.3.7]. ∎

We define the sub vector bundle $L^{\prime}_{i,\chi}$ of $FT^{*}X|_{D_{i}}$ for $i\in I_{W,\chi}$ by the image of the multiplication by the characteristic form of $\chi$ :

\times\operatorname{char}(\chi)|_{D_{i}}\colon\mathcal{O}_{X}(-pR^{\prime}_{\chi})\otimes_{\mathcal{O}_{X}}\mathcal{O}_{D_{i}}(\sum_{x\in D_{i}}p\operatorname{ord}^{\prime}(\chi;x,D_{i})[x])\to F\Omega^{1}_{X}|_{D_{i}},

For $i\in I_{T,\chi}$ , we define $L^{\prime}_{i,\chi}$ by $\mathrm{F}^{*}(T^{*}_{D_{i}}X|_{D_{i,F}})$ where $\mathrm{F}^{*}$ is the pullback by the Frobenius $\mathrm{F}\colon D_{i,F}\to D_{i,F}$ .

Definition 5.6.

Assume $\dim X=2$ . We define the F-characteristic cycle $FCCj_{!}\mathcal{F}$ as a cycle on the FW-cotangent bundle $FT^{*}X|_{X_{F}}$ by

FCCj_{!}\mathcal{F}=-\Big{(}\frac{1}{p}[FT_{X}^{*}X|_{X_{F}}]+\sum_{i\in I}\operatorname{dt}(\chi|_{K_{i}})[L^{\prime}_{i,\chi}]+\sum_{x\in D_{F}}pt_{x}[\mathrm{F}^{*}T_{x}^{*}X]\Big{)}

(5.3)

where $FT_{X}^{*}X|_{X_{F}}$ denotes the 0-section of $FT^{*}X|_{X_{F}}$ and

t_{x}=\#I_{x}-1+s_{x}+\sum_{i\in I_{W,x}}\operatorname{sw}(\chi|_{K_{i}})(\operatorname{ord}^{\prime}(\chi;x,D_{i})-\operatorname{ord}(\chi;x,D_{i}))\\ +\sum_{i\in I_{\textup{II},x}}(\operatorname{ord}(\chi;x,D_{i})+1-\#I_{x}).

(5.4)

Here, the integer $s_{x}$ is the coefficient of the fiber at $x$ in $CC^{\log}j_{!}\mathcal{F}$ (5.2).

The integrality of the characteristic form (Theorem 1.5) is necessary to define $t_{x}$ for all closed points $x\in D_{F}$ .

Lemma 5.7.

Let $h\colon W\to X$ be an étale morphism and let $j^{\prime}\colon W\times_{X}U\to W$ be the base change of $j$ . Then we have

FCCj^{\prime}_{!}h^{*}\mathcal{F}=h^{*}FCCj_{!}\mathcal{F}.

Proof.

Let $K^{\prime}_{i}$ be the local field at the generic point of $h^{*}D_{i}$ . Then we have $\operatorname{sw}(\chi|_{K_{i}})=\operatorname{sw}((h^{*}\chi)|_{K^{\prime}_{i}})$ , $\operatorname{dt}(\chi|_{K_{i}})=\operatorname{dt}((h^{*}\chi)|_{K^{\prime}_{i}})$ and $h^{*}\operatorname{rsw}(\chi)=\operatorname{rsw}(h^{*}\chi)$ , $h^{*}\operatorname{char}(\chi)=\operatorname{char}(h^{*}\chi)$ . Hence the assertion follows from Definition 5.6. ∎

Remark 5.8.

The F-characteristic cycle $FCCj_{!}\mathcal{F}$ is equal to

-\Big{(}\frac{1}{p}[FT_{X}^{*}X|_{X_{F}}]+\sum_{i\in J}[\mathrm{F}^{*}(T^{*}_{D_{i}}X|_{D_{i,F}})]+\sum_{i\in J^{\prime}}\operatorname{dt}(\chi|_{K_{i}})[L^{\prime}_{i,\chi}]+\sum_{x\in D_{F}}pt_{x}[\mathrm{F}^{*}T_{x}^{*}X]\Big{)}

where $J$ denotes the subset of $I$ consisting of $i\in I$ such that $D_{i}\cap X_{K}$ is not empty and $J^{\prime}$ denotes $I-J$ . Then $(1/p)[FT_{X}^{*}X|_{X_{F}}]+\sum_{i\in J}[\mathrm{F}^{*}(T^{*}_{D_{i}}X|_{D_{i,F}})]$ is a 1-cycle and $\sum_{i\in J^{\prime}}\operatorname{dt}(\chi|_{K_{i}})[L^{\prime}_{i,\chi}]+\sum_{x\in D_{F}}pt_{x}[\mathrm{F}^{*}T_{x}^{*}X]$ is a 2-cycle. Later, we consider the difference $FCCj_{!}\mathcal{F}-FCCj_{!}\Lambda$ . This cycle is a 2-cycle, so the intersection number with the 0-section is defined.

Remark 5.9.

In this remark, we consider the equal characteristic case. Let $X$ be a smooth scheme over a perfect field $k$ of characteristic $p>0$ . Let $D=\cup_{i\in I}D_{i}$ be a divisor with simple normal crossings and let $j\colon U=X-D\to X$ be the open immersion. Let $\mathcal{F}$ be a locally constant sheaf of $\Lambda$ -modules of rank 1 on $U$ . Then the characteristic cycle $CCj_{!}\mathcal{F}$ is defined as a cycle on the cotangent bundle $T^{*}X$ . If the dimension of $X$ is 2, we have

CCj_{!}\mathcal{F}=[T_{X}^{*}X]+\sum_{i\in I}\operatorname{dt}(\chi|_{K_{i}})[L^{\prime\prime}_{i,\chi}]+\sum_{x\in D_{F}}t_{x}[T_{x}^{*}X]

by [21, Theorem 6.1]. Here, $L^{\prime\prime}_{i,\chi}$ denotes the vector bundle defined by the characteristic form in the sense of [21] and $t_{x}$ is defined in [21] by the same form as (5.4).

Let $\mathrm{F}\colon X\to X$ be the Frobenius. If we put

FCCj_{!}\mathcal{F}=-\Big{(}\frac{1}{p}[\mathrm{F}^{*}T_{X}^{*}X]+\sum_{i\in I}\operatorname{dt}(\chi|_{K_{i}})[\mathrm{F}^{*}L^{\prime\prime}_{i,\chi}]+\sum_{x\in D_{F}}pt_{x}[\mathrm{F}^{*}T_{x}^{*}X]\Big{)}

as a cycle on $FT^{*}X\cong\mathrm{F}^{*}T^{*}X$ . Then we have

\mathrm{F}_{*}FCCj_{!}\mathcal{F}=-p\cdot CCj_{!}\mathcal{F}

where $\mathrm{F}_{*}$ denotes the pushforward by the projection $\mathrm{F}^{*}T^{*}X\to T^{*}X$ .

The rationality of the characteristic form (Theorem 1.3) implies the integrality of the coefficients of the fibers in the F-characteristic cycle.

Lemma 5.10.

The coefficients $pt_{x}$ of the fibers $[\mathrm{F}^{*}T^{*}_{x}X]$ in the F-characteristic cycle (5.3) are integers. If $(X,U,\chi)$ is clean at $x\in D_{F}$ , we have $t_{x}\geq 0$ .

Proof.

In the definition of $t_{x}$ (5.4), the terms other than $\operatorname{sw}(\chi|_{K_{i}})\operatorname{ord}^{\prime}(\chi;x,D_{i})$ are integers. By Definition 5.2, we see that $p\cdot\operatorname{ord}^{\prime}(\chi;x,D_{i})$ are integers.

If $(X,U,\chi)$ is clean at $x$ , we have

t_{x}=\#I_{x}-1+\sum_{i\in I_{W,x}}\operatorname{sw}(\chi|_{K_{i}})\cdot\operatorname{ord}^{\prime}(\chi;x,D_{i})+\sum_{i\in I_{\textup{II},x}}(1-\#I_{x})

by (5.4). Since we have $\operatorname{ord}^{\prime}(\chi;x,D_{i})\geq 0$ , we have $t_{x}\geq 0$ unless $\operatorname{ord}^{\prime}(\chi;x,D_{i})=0$ and $\#I_{x}=\#I_{\textup{II},x}=2$ . If $\operatorname{ord}^{\prime}(\chi;x,D_{i})=0$ and $\#I_{x}=\#I_{\textup{II},x}=2$ , we have $\operatorname{rsw}(\chi)_{x}=0$ and this contradicts the assumption. ∎

Remark 5.11.

The author conjectures that the terms $\operatorname{sw}(\chi|_{K_{i}})\operatorname{ord}^{\prime}(\chi;x,D_{i})$ are also integers and thus $t_{x}$ are integers. We can check that $\operatorname{sw}(\chi|_{K_{i}})\operatorname{ord}^{\prime}(\chi;x,D_{i})$ is an integer in the following cases:
1. The character $\chi|_{K_{i}}$ is of type I.
2. The character $\chi|_{K_{i}}$ is defined by a Kummer equation of degree $p$ .
Indeed, in the case 1, $\operatorname{ord}(\chi;x,D_{i})$ is an integer. In the case 2, if the character $\chi|_{K_{i}}$ is of type II, the Swan conductor $\operatorname{sw}(\chi|_{K_{i}})$ is divisible by $p$ .

The author also conjectures $t_{x}\geq 0$ even if $(X,U,\chi)$ is not clean at $x$ . In the equal characteristic case, this follows from the fact that $j_{!}\mathcal{F}[2]$ is perverse by [14, Proposition 5.14.1].

Let $\mathrm{F}\colon X_{F}\to X_{F}$ be the Frobenius. We define

\tau_{D}\colon F\Omega^{1}_{X}\to\mathrm{F}^{*}\Omega^{1}_{X}(\log D)|_{X_{F}}

by the composition of the maps

F\Omega^{1}_{X}\to F\Omega^{1}_{X}/\mathcal{O}_{X_{F}}\cdot w(p)\cong\mathrm{F}^{*}\Omega^{1}_{X_{F}}\to\mathrm{F}^{*}\Omega^{1}_{X}(\log D)|_{X_{F}}

where the middle isomorphism is the map [16, (4-1)]. We also define a morphism of vector bundles over $D_{F}$ by

\tau_{D}\colon FT^{*}X|_{D_{F}}\to\mathrm{F}^{*}T^{*}X(\log D)|_{D_{F}}.

Let

\tau_{D}^{!}\colon\operatorname{CH}_{2}(FT^{*}X|_{D_{F}})\to\operatorname{CH}_{2}(T^{*}X(\log D)|_{D_{F}})

be the Gysin homomorphism for $\tau_{D}$ .

Lemma 5.12 (cf. [21, Lemma 4.3(i)]).

Assume $\dim X=2$ . Let $i$ be an element of $I_{\textup{I},\chi}$ . Let $\mathrm{F}\colon D_{i}\to D_{i}$ be the Frobenius. Then,

(1)

$\dim\tau_{D}^{-1}(\mathrm{F}^{*}L_{i,\chi})=2$ .
(2)

$L^{\prime}_{i,\chi}=\mathrm{F}^{*}T^{*}_{D_{i}}X$ .

(3)

We have

\tau_{D}^{!}([\mathrm{F}^{*}L_{i,\chi}])=[L^{\prime}_{i,\chi}]+\sum_{x\in D_{i}}p(\operatorname{ord}^{\prime}(\chi;x,D_{i})-\operatorname{ord}(\chi;x,D_{i}))[\mathrm{F}^{*}T^{*}_{x}X]+\sum_{x\in Z_{\textup{II},\chi}\cap D_{i}}p[\mathrm{F}^{*}T^{*}_{x}X]

in $Z_{2}(\tau_{D}^{-1}(\mathrm{F}^{*}L_{i,\chi}))$ .

Proof.

We may assume $I=\{1,2\}$ and $i=1$ . Let $x$ be a closed point of $D$ and $(\pi_{1},\pi_{2})$ be a local coordinate at $x$ such that $\pi_{i}$ is a local equation of $D_{i}$ for $i\in I_{x}$ . Then $F\Omega^{1}_{X,x}$ is a free $O_{X,x}$ -module with base $(w\pi_{1},w\pi_{2})$ . Its dual base is denoted by $(\partial^{\prime}/\partial^{\prime}\pi_{1},\partial^{\prime}/\partial^{\prime}\pi_{2})$ . Let $\mathcal{I},\mathcal{I}^{\prime},\mathcal{J}$ be the defining ideal sheaves of $L_{1,\chi}\subset T^{*}X(\log D)|_{D_{i}}$ , $L^{\prime}_{i,\chi}\subset FT^{*}X|_{D_{i}}$ and $\tau_{D}^{-1}(\mathrm{F}^{*}L_{i,\chi})\subset FT^{*}X|_{D_{i}}$ respectively.

First, we consider the case $I_{x}=\{1\}$ . Then $\Omega^{1}_{X}(\log D)_{x}$ is a free $O_{X,x}$ -module with base $(d\log\pi_{1},d\pi_{2})$ . Its dual base is denoted by $(\partial/\partial\log\pi_{1},\partial/\partial\pi_{2})$ . If we put

\operatorname{rsw}(\chi)_{x}=(\alpha_{1}d\log\pi_{1}+\alpha_{2}d\pi_{2})/\pi_{1}^{n_{1}}

where $\alpha_{1},\alpha_{2}\in\mathcal{O}_{X,x}$ and $n_{1}=\operatorname{sw}(\chi|_{K_{1}})$ , then

\mathcal{I}_{x}=(\pi_{2}^{-\operatorname{ord}(\chi;x,D_{1})}(\alpha_{2}\partial/\partial\log\pi_{1}-\alpha_{1}\partial/\partial\pi_{2})),

and

\mathcal{J}_{x}=(\pi_{2}^{-p\operatorname{ord}(\chi;x,D_{1})}(-\alpha_{1}^{p}\partial/\partial\pi_{2})).

Since $\chi|_{K_{1}}$ is of type I,

\operatorname{char}(\chi)_{x}=\alpha_{1}^{p}w\pi_{1}/\pi_{1}^{p(n_{1}+1)}

by Proposition 3.2 and thus we have $\operatorname{ord}^{\prime}(\chi;x,D_{i})=\operatorname{ord}_{\pi_{2}}(\alpha_{1})$ . Hence we have

\mathcal{J}_{x}=(\pi_{2}^{p\operatorname{ord}^{\prime}(\chi;x,D_{1})-p\operatorname{ord}(\chi;x,D_{1})}\partial/\partial\pi_{2}),

\mathcal{I}^{\prime}_{x}=(\pi_{2}^{-p\operatorname{ord}^{\prime}(\chi;x,D_{1})}(-\alpha_{1}^{p}\partial^{\prime}/\partial^{\prime}\pi_{2}))=(\partial^{\prime}/\partial^{\prime}\pi_{2}).

Hence the assertion follows.

Second, we consider the case $I_{x}=\{1,2\}$ . Then $\Omega^{1}_{X}(\log D)_{x}$ is a free $O_{X,x}$ -module with base $(d\log\pi_{1},d\log\pi_{2})$ . Its dual base is denoted by $(\partial/\partial\log\pi_{1},\partial/\partial\log\pi_{2})$ . If we put

\operatorname{rsw}(\chi)_{x}=(\alpha_{1}d\log\pi_{1}+\alpha_{2}d\pi_{2})/\pi_{1}^{n_{1}}\pi_{1}^{n_{2}}

where $\alpha_{1},\alpha_{2}\in\mathcal{O}_{X,x}$ and $n_{i}=\operatorname{sw}(\chi|_{K_{i}})$ , then

\mathcal{I}_{x}=(\pi_{2}^{-\operatorname{ord}(\chi;x,D_{1})}(\alpha_{2}\partial/\partial\log\pi_{1}-\alpha_{1}\partial/\partial\log\pi_{2})),

and

\mathcal{J}_{x}=(\pi_{2}^{-p\operatorname{ord}(\chi;x,D_{1})}(-\alpha_{1}^{p}\pi_{2}^{p}\partial/\partial\pi_{2})).

Since $i=1$ is an element of $I_{\textup{I},\chi}$ ,

\operatorname{char}(\chi)_{x}=\alpha_{1}^{p}\pi_{2}^{p(1-\delta)}w\pi_{1}/\pi_{1}^{p(n_{1}+1)}\pi_{2}^{p(n_{2}+\delta)}

by Proposition 3.2 where $\delta$ is 1 if $\chi|_{K_{2}}$ is tamely ramified or of type I and 0 if $\chi|_{K_{2}}$ is type II. Hence we have $\operatorname{ord}^{\prime}(\chi;x,D_{i})=\operatorname{ord}_{\pi_{2}}(\alpha_{1})+1-\delta$ and

\mathcal{J}_{x}=(\pi_{2}^{p\operatorname{ord}^{\prime}(\chi;x,D_{1})-p\operatorname{ord}(\chi;x,D_{1})+p\delta}\partial/\partial\pi_{2}),

\mathcal{I}^{\prime}_{x}=(\partial^{\prime}/\partial^{\prime}\pi_{2}).

Hence the assertion follows. ∎

Lemma 5.13 (cf. [21, Lemma 4.4, Lemma 4.5]).

Assume $\dim X=2$ . Let $i$ be an element of $I_{\textup{II},\chi}$ . Let $q^{\prime}_{i}\colon\tau_{D}^{-1}(\mathrm{F}^{*}L_{i,\chi})\to D_{i}$ be the canonical projection. Let $\mathrm{F}\colon D_{i}\to D_{i}$ be the Frobenius. Then,

(1)

$\tau_{D}^{-1}(\mathrm{F}^{*}L_{i,\chi})=FT^{*}X|_{D_{i}}$ .

(2)

We have

\tau_{D}^{!}([\mathrm{F}^{*}L_{i,\chi}])={q^{\prime}_{i}}^{*}(c_{1}(\mathrm{F}^{*}T^{*}X(\log D)|_{D_{i}})\cap[D_{i}]-c_{1}(\mathcal{O}_{X}(-pR_{\chi})|_{D_{i}})\cap[D_{i}]-\sum_{x\in D_{i}}p\operatorname{ord}(\chi;x,D_{i})[x])

in $\operatorname{CH}_{2}(FT^{*}X|_{D_{i}})$ .

(3)

We have

[L^{\prime}_{i,\chi}]={q^{\prime}_{i}}^{*}(c_{1}(FT^{*}X|_{D_{i}})\cap[D_{i}]-c_{1}(\mathcal{O}_{X}(-pR^{\prime}_{\chi})|_{D_{i}})\cap[D_{i}]-\sum_{x\in D_{i}}p\operatorname{ord}^{\prime}(\chi;x,D_{i})[x])

in $\operatorname{CH}_{2}(FT^{*}X|_{D_{i}})$ .

(4)

We have

[\mathrm{F}^{*}T^{*}_{D_{i}}X]={q^{\prime}_{i}}^{*}(c_{1}(\mathcal{O}_{X}(-pR_{\chi})|_{D_{i}})\cap[D_{i}]+\sum_{x\in D_{i}}p(\operatorname{ord}(\chi;x,D_{i})-\#I_{x}+1)[x])

in $\operatorname{CH}_{2}(FT^{*}X|_{D_{i}})$ .

Proof.

(1) We use the same notation as in the proof of Lemma 5.12. Since $\chi|_{K_{1}}$ is of type II, the refined Swan conductor is the image of the characteristic form by Proposition 3.4 and thus $\alpha_{1}=0$ . Hence $\mathcal{I}^{\prime}=0$ and we have $\tau_{D}^{-1}(\mathrm{F}^{*}L_{i,\chi})=FT^{*}X|_{D_{i}}$ .

(2) By applying the excess intersection formula to the cartesian diagram

we have

[\mathrm{F}^{*}L_{i,\chi}]=q_{i}^{*}(c_{1}(\mathrm{F}^{*}T^{*}X(\log D)|_{D_{i}})\cap[D_{i}]-c_{1}(\mathrm{F}^{*}L_{i,\chi})\cap[D_{i}])

in $\operatorname{CH}_{2}(\mathrm{F}^{*}T^{*}X(\log D)|_{D_{i}})$ where the map $q_{i}\colon\mathrm{F}^{*}T^{*}X(\log D)|_{D_{i}}\to D_{i}$ is the canonical projection. Since the sub vector bundle $L_{i,\chi}$ of $T^{*}X(\log D)|_{D_{i}}$ is defined by the image of the injection

\times\operatorname{rsw}(\chi)|_{D_{i}}\colon\mathcal{O}_{X}(-R_{\chi})\otimes_{\mathcal{O}_{X}}\mathcal{O}_{D_{i}}(\sum_{x\in D_{i}}\operatorname{ord}(\chi;x,D_{i})[x])\to\Omega^{1}_{X}(\log D)|_{D_{i}},

the assertion holds.

(3) By applying the excess intersection formula to the cartesian diagram

we have

[L^{\prime}_{i,\chi}]={q^{\prime}_{i}}^{*}(c_{1}(FT^{*}X|_{D_{i}})\cap[D_{i}]-c_{1}(L^{\prime}_{i,\chi})\cap[D_{i}])

in $\operatorname{CH}_{2}(FT^{*}X|_{D_{i}})$ . Since the sub vector bundle $L^{\prime}_{i,\chi}$ of $FT^{*}X|_{D_{i}}$ is defined by the image of the injection

\times\operatorname{char}(\chi)|_{D_{i}}\colon\mathcal{O}_{X}(-pR^{\prime}_{\chi})\otimes_{\mathcal{O}_{X}}\mathcal{O}_{D_{i}}(\sum_{x\in D_{i}}p\operatorname{ord}^{\prime}(\chi;x,D_{i})[x])\to F\Omega^{1}_{X}|_{D_{i}},

the assertion holds.

(4) By applying the excess intersection formula to the cartesian diagram

we have

[\mathrm{F}^{*}T^{*}_{D_{i}}X]={q^{\prime}_{i}}^{*}(c_{1}(\mathrm{F}^{*}T^{*}D_{i})\cap[D_{i}])

in $\operatorname{CH}_{2}(FT^{*}X|_{D_{i}})$ since the sequence

0\to\mathrm{F}^{*}T^{*}_{D_{i}}X\to FT^{*}X|_{D_{i}}\to FT^{*}D_{i}\to 0

is exact by [15, (2-12)] and $FT^{*}D_{i}\cong\mathrm{F}^{*}T^{*}D_{i}$ by [15, (2-4)]. Since $D_{i}$ is a scheme over $F$ , the computation in [21, Lemma 4.5(iii)] implies that we have an equality

c_{1}(T^{*}D_{i})\cap[D_{i}]=c_{1}(\mathcal{O}_{D_{i}}(-(R_{\chi}\cap D_{i}))\cap[D_{i}]+\sum_{x\in D_{i}}(\operatorname{ord}(\chi;x,D_{i})-\#I_{x}+1)[x]

in $\operatorname{CH}_{0}(D_{i})$ . Since $\mathrm{F}^{*}\Omega^{1}_{D_{i}}=(\Omega^{1}_{D_{i}})^{\otimes p}$ , we have

c_{1}(\mathrm{F}^{*}T^{*}D_{i})\cap[D_{i}]=c_{1}(\mathcal{O}_{X}(-pR_{\chi})|_{D_{i}})\cap[D_{i}]+\sum_{x\in D_{i}}p(\operatorname{ord}(\chi;x,D_{i})-\#I_{x}+1)[x]

in $\operatorname{CH}_{0}(D_{i})$ . ∎

Lemma 5.14.

Assume $D_{i}$ is contained in the closed fiber $D_{F}$ . Let $\mathrm{F}\colon D_{i}\to D_{i}$ be the Frobenius. Then we have

c_{1}(\mathrm{F}^{*}T^{*}X(\log D)|_{D_{i}})\cap[D_{i}]=c_{1}(FT^{*}X|_{D_{i}})\cap[D_{i}]+\sum_{j\in I}c_{1}(\mathcal{O}_{X}(pD_{j})|_{D_{i}})\cap[D_{i}].

in $\operatorname{CH}_{0}(D_{F})$ .

Proof.

Let $D^{\prime}_{i}$ be the closed subscheme consisting of the closed points $x$ of $D_{i}$ such that $\#I_{x}=2$ . By the two exact sequences

0\to\mathrm{F}^{*}N_{D_{i}/X}\to F\Omega^{1}_{X}|_{D_{i}}\to\mathrm{F}^{*}\Omega^{1}_{D_{i}}\to 0,

0\to\mathrm{F}^{*}\Omega^{1}_{D_{i}}(\log D^{\prime}_{i})\to\mathrm{F}^{*}\Omega^{1}_{X}(\log D)|_{D_{i}}\to\mathcal{O}_{D_{i}}\to 0

of locally free $D_{i}$ -modules, we have

c_{1}(\mathrm{F}^{*}T^{*}X(\log D)|_{D_{i}})\cap[D_{i}]-c_{1}(FT^{*}X|_{D_{F}})\cap[D_{i}]\\ =c_{1}(\mathrm{F}^{*}\Omega^{1}_{D_{i}}(\log D^{\prime}_{i}))\cap[D_{i}]-c_{1}(\mathrm{F}^{*}\Omega^{1}_{D_{i}})\cap[D_{i}]-c_{1}(\mathrm{F}^{*}N_{D_{i}/X})\cap[D_{i}].

Applying [21, (4.10)] to the scheme $D_{i}$ , we obtain

c_{1}(\Omega^{1}_{D_{i}}(\log D^{\prime}_{i}))\cap[D_{i}]-c_{1}(\Omega^{1}_{D_{i}})\cap[D_{i}]=\sum_{j\neq i\in I}c_{1}(\mathcal{O}_{D_{i}}(D_{j}\cap D_{i}))\cap[D_{i}].

Since we have $\mathrm{F}^{*}N_{D_{i}/X}=\mathrm{F}^{*}\mathcal{O}_{X}(-D_{i})|_{D_{i}}$ , the assertion follows. ∎

Theorem 5.15.

Assume $\dim X=2$ and $X$ is proper over $\mathcal{O}_{K}$ . Then we have

(FCCj_{!}\mathcal{F}-FCCj_{!}\Lambda,FT^{*}_{X}X|_{X_{F}})_{FT^{*}X|_{X_{F}}}=p\cdot(\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\mathcal{F})-\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\Lambda)).

Proof.

We do some computations used later. By Lemma 5.12, we have

\tau_{D}^{!}(\sum_{i\in I_{\textup{I},\chi}}\operatorname{sw}(\chi|_{K_{i}})[\mathrm{F}^{*}L_{i,\chi}])=\sum_{i\in I_{\textup{I},\chi}}\operatorname{sw}(\chi|_{K_{i}})\Big{(}[L^{\prime}_{i,\chi}]\\ +\sum_{x\in D_{i}}p(\operatorname{ord}^{\prime}(\chi;x,D_{i})-\operatorname{ord}(\chi;x,D_{i}))[\mathrm{F}^{*}T^{*}_{x}X]+\sum_{x\in Z_{\textup{II},\chi}\cap D_{i}}p[\mathrm{F}^{*}T^{*}_{x}X]\Big{)}.

(5.5)

We note that if $i\in I_{\textup{I},\chi}$ , we have

{q^{\prime}_{i}}^{*}(c_{1}(\mathcal{O}_{X}(Z_{\textup{II},\chi})|_{D_{i}})\cap[D_{i}])=\sum_{x\in Z_{\textup{II},\chi}\cap D_{i}}[\mathrm{F}^{*}T^{*}_{x}X].

By Lemma 5.13 (2) and (3) and Lemma 5.14, we have

\tau_{D}^{!}(\sum_{i\in I_{\textup{II},\chi}}\operatorname{sw}(\chi|_{K_{i}})[\mathrm{F}^{*}L_{i,\chi}])=\sum_{i\in I_{\textup{II},\chi}}\operatorname{sw}(\chi|_{K_{i}})\Big{(}[L^{\prime}_{i,\chi}]\\ +\sum_{x\in D_{i}}p(\operatorname{ord}^{\prime}(\chi;x,D_{i})-\operatorname{ord}(\chi;x,D_{i}))[\mathrm{F}^{*}T^{*}_{x}X]+{q^{\prime}_{i}}^{*}(c_{1}(\mathcal{O}_{X}(pZ_{\textup{II},\chi})|_{D_{i}})\cap[D_{i}]\Big{)}.

(5.6)

Since we have

\sum_{i\in I}\operatorname{sw}(\chi|_{K_{i}}){q^{\prime}_{i}}^{*}(c_{1}(\mathcal{O}_{X}(pZ_{\textup{II},\chi})|_{D_{i}})\cap[D_{i}])=\sum_{i\in I_{\textup{II},\chi}}{q^{\prime}_{i}}^{*}(c_{1}(\mathcal{O}_{X}(pR_{\chi})|_{D_{i}})\cap[D_{i}]),

we have

\sum_{i\in I}\operatorname{sw}(\chi|_{K_{i}}){q^{\prime}_{i}}^{*}(c_{1}(\mathcal{O}_{X}(pZ_{\textup{II},\chi})|_{D_{i}})\cap[D_{i}])=-[\mathrm{F}^{*}T^{*}_{D_{i}}X]+\sum_{x\in D_{i}}p(\operatorname{ord}(\chi;x,D_{i})-\#I_{x}+1)[\mathrm{F}^{*}T_{x}^{*}X])

by Lemma 5.13 (4). The sum of equalities (5.5) and (5.6) shows the equality

\tau_{D}^{!}(\sum_{i\in I}\operatorname{sw}(\chi|_{K_{i}})[\mathrm{F}^{*}L_{i,\chi}])=-[\mathrm{F}^{*}T^{*}_{D_{i}}X]+\sum_{i\in I_{\chi}}\operatorname{sw}(\chi|_{K_{i}})\Big{(}[L^{\prime}_{i,\chi}]\\ +\sum_{x\in D_{i}}p(\operatorname{ord}^{\prime}(\chi;x,D_{i})-\operatorname{ord}(\chi;x,D_{i}))[\mathrm{F}^{*}T^{*}_{x}X]\Big{)}+\sum_{x\in D_{i}}p(\operatorname{ord}(\chi;x,D_{i})-\#I_{x}+1)[\mathrm{F}^{*}T_{x}^{*}X].

(5.7)

We have

\begin{split}(\mathrm{F}^{*}L_{i,\chi},T^{*}_{X}X(\log D)|_{D_{F}})_{\mathrm{F}^{*}T^{*}X(\log D)|_{D_{F}}}&=c_{1}\Big{(}\mathrm{F}^{*}\big{(}T^{*}X(\log D)|_{D_{F}}/L_{i,\chi}\big{)}\Big{)}\cap[D_{F}]\\ &=p\cdot(c_{1}(T^{*}X(\log D)|_{D_{F}}/L_{i,\chi})\cap[D_{F}])\\ &=p\cdot(L_{i,\chi},T^{*}_{X}X(\log D)|_{D_{F}})_{T^{*}X(\log D)|_{D_{F}}}.\end{split}

(5.8)

First, we assume that $s_{x}=0$ for every $x\in D_{F}$ . Then it suffices to show that we have

FCCj_{!}\mathcal{F}-FCCj_{!}\Lambda=-\tau_{D}^{!}(\mathrm{F}^{*}(CC^{\log}j_{!}\mathcal{F}-[T^{*}_{X}X(\log D)|_{D_{F}}]))

(5.9)

in $\operatorname{CH}_{2}(T^{*}X(\log D)|_{D_{F}})$ by Theorem 5.5 and (5.8). This equality holds by (5.7) and the definition of $t_{x}$ (5.4).

Next, we consider the general case. By the definition of the logarithmic characteristic cycle, we have

CC^{\log}j_{!}\mathcal{F}-[T^{*}_{X}X(\log D)|_{D_{F}}]-\sum_{x\in D_{F}}s_{x}[T_{x}^{*}X(\log D)]=\sum_{i\in I}\operatorname{sw}(\chi|_{K_{i}})[L_{i,\chi}].

Then the equality (5.7) shows that we have

\begin{split}&\tau_{D}^{!}(-\mathrm{F}^{*}(CC^{\log}j_{!}\mathcal{F}-[T^{*}_{X}X(\log D)|_{D_{F}}]-\sum_{x\in D_{F}}s_{x}[T^{*}_{x}X(\log D)]))\\ =&\tau_{D}^{!}(-\sum_{i\in I}\operatorname{sw}(\chi|_{K_{i}})\mathrm{F}^{*}[L_{i,\chi}])\\ =&FCCj_{!}\mathcal{F}-FCCj_{!}\Lambda+\sum_{x\in D_{F}}ps_{x}[\mathrm{F}^{*}T^{*}_{x}X]\end{split}

(5.10)

by the definition of $t_{x}$ . By Theorem 5.5, we have

(-(CC^{\log}j_{!}\mathcal{F}-[T^{*}_{X}X(\log D)|_{D_{F}}]-\sum_{x\in D_{F}}s_{x}[T_{x}^{*}X(\log D)]),T^{*}_{X}X(\log D)|_{D_{F}})_{T^{*}X(\log D)|_{D_{F}}}\\ =\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\mathcal{F})-\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\Lambda)+\sum_{x\in D_{F}}s_{x}

(5.11)

since the intersection number $(T_{x}^{*}X(\log D),T^{*}_{X}X(\log D)|_{D_{F}})$ is 1. By (5.8), we have

(-\mathrm{F}^{*}(CC^{\log}j_{!}\mathcal{F}-[T^{*}_{X}X(\log D)|_{D_{F}}]-\sum_{x\in D_{F}}s_{x}[T_{x}^{*}X(\log D)]),T^{*}_{X}X(\log D)|_{D_{F}})_{\mathrm{F}^{*}T^{*}X(\log D)|_{D_{F}}}\\ =p\cdot(\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\mathcal{F})-\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\Lambda)+\sum_{x\in D_{F}}s_{x}).

(5.12)

By (5.10) and (5.12), we have

(FCCj_{!}\mathcal{F}-FCCj_{!}\Lambda+\sum_{x\in D_{F}}ps_{x}[\mathrm{F}^{*}T_{x}^{*}X],FT^{*}_{X}X|_{D_{F}})_{FT^{*}X|_{D_{F}}}\\ =p\cdot(\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\mathcal{F})-\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\Lambda)+\sum_{x\in D_{F}}s_{x}).

Since the intersection number $(\mathrm{F}^{*}T_{x}^{*}X,FT^{*}_{X}X|_{D_{F}})$ is 1, the assertion follows. ∎

5.3 Example

In this subsection, we give an example of the F-characteristic cycle.

Let $p>2$ be a prime number and let $\zeta_{p}$ be a primitive $p$ -th root. Let $K$ be a complete discrete valuation field tamely ramified over $\mathbf{Q}_{p}(\zeta_{p})$ with valuation ring $\mathcal{O}_{K}$ and with residue field $F$ . Let $e=\operatorname{ord}_{K}p$ be the absolute ramification index and put $e^{\prime}=pe/(p-1)$ . We fix a uniformizer $\pi$ and write $p=u\pi^{e}$ with some $u\in\mathcal{O}_{K}^{\times}$ . Let $a,b,c$ be integers satisfying $0<a,b<p$ , $(p,c)=1$ and $a+b+c=0$ . We put $X=\mathbf{P}^{1}_{\mathcal{O}_{K}}$ . Let $U$ be the open subscheme $\operatorname{Spec}K[x^{\pm 1},(1-x)^{-1}]$ of $X$ and let $j\colon U\to X$ be the open immersion. Let $\mathcal{K}$ be the Kummer sheaf defined by $t^{p}=(-1)^{c}x^{a}(1-x)^{b}$ on $U$ . For convenience, we change the coordinate by $y=x+a/c$ . Let $D,E_{1},E_{2},E_{3}$ be divisors defined by $(\pi=0),(y-a/c=0),(y+b/c=0),(y=\infty)$ respectively. Then $D\cup E_{1}\cup E_{2}\cup E_{3}$ is a divisor with simple normal crossings and $U$ is the complement of this divisor. Let $z_{0}$ be the closed point $\{\pi=y=0\}$ and let $z_{i}$ be the closed point $E_{i}\cap D$ for $i=1,2,3$ . Let $M$ be the local field at the generic point of $D$ .

We compute the F-characteristic cycle $FCCj_{!}\mathcal{K}$ of $j_{!}\mathcal{K}$ . Applying Theorem 5.15, we compute the Swan conductor of $\mathrm{H}^{1}(\mathbf{P}^{1}_{\overline{K}},j_{!}\mathcal{K})$ . This cohomology group realizes the Jacobi sum Hecke character as in [5]. Coleman-McCallum [5], Miki [11] and Tsushima [18] computed the conductor or explicitly the ramified component of the Jacobi sum Hecke character in more general cases by different methods.

Remark 5.16.

We note that the Swan conductor can be calculated easier by computing the logarithmic characteristic cycle and applying Kato-Saito’s conductor formula (Theorem 5.5) because we need to compute the logarithmic characteristic cycle for the computation of the F-characteristic cycle. The subject of this article is the non-logarithmic theory, so we compute the Swan conductor using the F-characteristic cycle.

We have $\operatorname{sw}(\chi|_{M})=\operatorname{dt}(\chi|_{M})=e^{\prime}$ and the character $\chi|_{M}$ is of type II. We have

\operatorname{rsw}\chi=\frac{-cy\cdot dy}{(1-\zeta_{p})^{p}\cdot(y-a/c)^{a}(y+b/c)^{b}}

(5.13)

and

\operatorname{char}\chi=\frac{-c^{p}y^{p}\cdot wy}{(1-\zeta_{p})^{p^{2}}\cdot(y-a/c)^{ap}(y+b/c)^{bp}}

(5.14)

on the complement of $E_{3}$ . The character $\chi$ is not clean and not non-degenerate only at $z_{0}$ and we have $\operatorname{ord}(\chi;z_{0},D)=\operatorname{ord}^{\prime}(\chi;z_{0},D)=1$ .

We prove an elementary lemma used later.

Lemma 5.17.

Let $d$ be a rational number satisfying $v_{p}(d)\geq 0$ . We put $r=v_{p}(d^{p-1}-1)$ . Then there exists an integer $l$ such that $p^{r}$ divides $1-dl^{p}$ . There does not exist any integer $l$ such that $p^{r+1}$ divides $1-dl^{p}$ .

Proof.

Since $p^{r}$ divides $d^{p-1}-1=(d-1)(d^{p-2}+\cdots d+1)$ , we see that $p^{r}$ divides $d-1$ or $d(d^{p-3}+\cdots+1)+1$ . Therefore, we may take $l=1$ or $l=-(d^{p-3}+\cdots+1)$ because we have $dl^{p}\equiv d^{p}l^{p}\equiv 1\mod p^{r}$ .

If there exists an integer $l$ such that $p^{r+1}$ divides $1-dl^{p}$ , then we have $v_{p}(d^{p-1}-1)\geq r+1$ and this is a contradiction. ∎

Now we compute the F-characteristic cycle of $j_{!}\mathcal{K}$ . We have to divide into two cases.

Case 1 : we assume $v_{p}((a^{a}b^{b}c^{c})^{p-1}-1)=1$ .

Lemma 5.18.

We put $Y_{n}=\operatorname{Spec}\mathcal{O}_{K}[y_{n}]$ for a natural number $1\leq n\leq e/2$ . Let $M_{n}$ be the local field at the generic point of the closed fiber. Let $\chi_{n}$ be the Kummer character defined around $(y_{n}=0)$ by $t^{p}=(-1)^{a}(y_{n}\pi^{n}-a/c)^{a}(y_{n}\pi^{n}+b/c)^{b}$ . Then we have the following properties.

$1$ . We have $\operatorname{sw}(\chi_{n}|_{M_{n}})=e^{\prime}-2n$ .

$2$ . If $n<e/2$ , the character $\chi_{n}$ is not clean at $y_{n}=0$ .

$3$ . If $n=e/2$ , the character $\chi_{n}$ is clean.

Proof.

By Lemma 5.17, we may take an integer $l$ such that $p$ divides $1-a^{a}b^{b}c^{c}l^{p}$ . Then we have

	$\displaystyle(lt)^{p}$	$\displaystyle=(-1)^{a}l^{p}(y_{n}\pi^{n}-a/c)^{a}(y_{n}\pi^{n}+b/c)^{b}$
		$\displaystyle=l^{p}\Big{(}a^{a}b^{b}c^{c}+\frac{a^{a-1}b^{b-1}}{2c^{a+b-3}}y_{n}^{2}\pi^{2n}+\cdots\Big{)}$
		$\displaystyle=1+pm+\frac{a^{a-1}b^{b-1}l^{p}}{2c^{a+b-3}}y_{n}^{2}\pi^{2n}+\cdots$
		$\displaystyle=1+mu\pi^{e}+\frac{a^{a-1}b^{b-1}l^{p}}{2c^{a+b-3}}y_{n}^{2}\pi^{2n}+\cdots$

for some rational number $m$ such that $v_{p}(m)=0$ and the omitted part is divided by $\pi^{3n}$ . Hence, the assertion 1 follows. Around the closed point $\{\pi=y_{n}=0\}$ , the refined Swan conductor is

\operatorname{rsw}(\chi)=\frac{-a^{a-1}b^{b-1}c^{3-a-b}(ny_{n}^{2}\cdot d\log\pi+y_{n}\cdot dy_{n})}{(1-\zeta_{p})^{p}\cdot\pi^{-2n}\cdot(y_{n}\pi^{n}-a/c)^{a}(y_{n}\pi^{n}+b/c)^{b}}

if $n<e/2$ and

\operatorname{rsw}(\chi)=\frac{-((2^{-1}a^{a-1}b^{b-1}c^{3-a-b}l^{p}y_{n}^{2}+mu)e\cdot d\log\pi+a^{a-1}b^{b-1}c^{3-a-b}l^{p}y_{n}\cdot dy_{n})}{(1-\zeta_{p})^{p}\cdot\pi^{-2n}\cdot(y_{n}\pi^{n}-a/c)^{a}(y_{n}\pi^{n}+b/c)^{b}l^{p}}

if $n=e/2$ by [8, Corollary 8.2.3]. Since we assume $e$ is prime to $p$ , the assertion follows. ∎

We compute the coefficient $s_{z_{0}}$ of the fiber in the logarithmic characteristic cycle. We may consider locally around $z_{0}$ . We define the successive blowups as follows.

Let $X_{1}\to X$ be the blowup at the closed point $\{\pi=y=0\}$ . The scheme $X_{1}$ is a union of two open subschemes $U_{1}=\operatorname{Spec}\mathcal{O}_{K}[y,x_{1}]/(yx_{1}-\pi)$ and $Y_{1}=\operatorname{Spec}\mathcal{O}_{K}[y_{1}]$ where $y_{1}\pi=y$ . Then we can check that the character $\chi$ is clean on $U_{1}$ . By Lemma 5.18, $\chi$ is not clean at $y_{1}=0$ . Let $X_{2}\to X_{1}$ be the blowup at the closed point $\{\pi=y_{1}=0\}$ . Repeating this process, we get the successive blowups $X_{e/2}\to\cdots\to X_{1}\to X_{0}=X$ at non-clean closed points. The character $\chi$ is clean on $X_{e/2}$ by Lemma 5.18. Hence the coefficient $s_{z_{0}}$ is equal to $e^{\prime}-\sum_{1\leq i\leq e/2}2=e^{\prime}-e$ by [7, Remark 5.8]. Hence we have

FCCj_{!}\mathcal{K}-FCCj_{!}\Lambda=-e^{\prime}[L^{\prime}]+[\mathrm{F}^{*}T^{*}_{X_{F}}X]-p(e^{\prime}-e+1)[\mathrm{F}^{*}T^{*}_{z_{0}}X]+p[\mathrm{F}^{*}T_{z_{1}}^{*}X]+p[\mathrm{F}^{*}T_{z_{2}}^{*}X]+p[\mathrm{F}^{*}T_{z_{3}}^{*}X]

where $L^{\prime}$ is defined by the characteristic form (5.14).

We compute the intersection number with the 0-section. We have

\displaystyle([\mathrm{F}^{*}T^{*}_{X_{F}}X],FT^{*}_{X}X|_{X_{F}})_{FT^{*}X|_{X_{F}}}=c_{1}(\mathrm{F}^{*}\Omega^{1}_{\mathbf{P}^{1}_{F}})\cap[\mathbf{P}^{1}_{F}]=-2p

Since $L^{\prime}$ is defined by the image of the injection

\times\operatorname{char}\chi\colon\mathcal{O}_{X}(-p(e^{\prime}D+E_{1}+E_{2}+E_{3}))\otimes_{\mathcal{O}_{X}}\mathcal{O}_{D}(p[z_{0}]),

we have

	$\displaystyle([L^{\prime}],FT^{}_{X}X\|_{X_{F}})_{FT^{}X\|_{X_{F}}}$	$\displaystyle=c_{1}(F\Omega^{1}_{X}\|_{X_{F}})\cap[X_{F}]+pe^{\prime}(D,D)_{X}+3p-p$
		$\displaystyle=c_{1}(\mathrm{F}^{}\Omega^{1}_{X_{F}})\cap[X_{F}]+c_{1}(\mathrm{F}^{}N_{X_{F}}X)\cap[X_{F}]+2p$
		$\displaystyle=-2p+0+2p=0$

Hence we have

(FCCj_{!}\mathcal{K}-FCCj_{!}\Lambda,FT^{*}_{X}X|_{X_{F}})_{FT^{*}X|_{X_{F}}}=-p(e^{\prime}-e)

Since we have $\mathrm{H}^{i}(X,j_{!}\mathcal{K})=0$ for $i\neq 1$ and $\operatorname{Sw}_{K}(X_{\overline{K}},j_{!}\Lambda)=0$ , the Swan conductor of $\mathrm{H}^{1}(\mathbf{P}^{1}_{\overline{K}},j_{!}\mathcal{K})$ is $e^{\prime}-e=e/(p-1)$ by Theorem 5.15.

Case 2 : we assume $v_{p}((a^{a}b^{b}c^{c})^{p-1}-1)\geq 2$ .

Lemma 5.19.

We put $Y_{n}=\operatorname{Spec}\mathcal{O}_{K}[y_{n}]$ for a natural number $1\leq n\leq(e^{\prime}-1)/2$ . Let $\chi_{n}$ be the Kummer character defined around $(y_{n}=0)$ by $t^{p}=(-1)^{a}(y_{n}\pi^{n}-a/c)^{a}(y_{n}\pi^{n}+b/c)^{b}$ . Then we have the following claims.

$1$ . We have $\operatorname{sw}(\chi_{n}|_{M_{n}})=e^{\prime}-2n$ .

$2$ . The character $\chi_{n}$ is not clean at $y_{n}=0$ .

Proof.

We can prove the assertions in the same way as Lemma 5.18. ∎

We compute the coefficient $s_{z_{0}}$ of the fiber in the logarithmic characteristic cycle. We may consider locally around $z_{0}$ . Take the successive blowups $X_{(e^{\prime}-1)/2}\to\cdots\to X_{0}=X$ at non-clean closed points in the same way as in Case 1. Contrary to Case 1, the character $\chi$ is still not clean on $X_{(e^{\prime}-1)/2}$ .

The scheme $X_{(e^{\prime}-1)/2}$ contains the open subscheme $Y_{(e^{\prime}-1)/2}=\operatorname{Spec}\mathcal{O}_{K}[y_{(e^{\prime}-1)/2}]$ . We put $y^{\prime}=y_{(e^{\prime}-1)/2}$ . Let $W\to X_{(e^{\prime}-1)/2}$ be the blowup at the closed point $\pi=y^{\prime}=0$ . The scheme $W$ is the union of two open subschemes $U=\operatorname{Spec}\mathcal{O}_{K}[y^{\prime},w]/(y^{\prime}w-\pi)$ and $V=\operatorname{Spec}\mathcal{O}_{K}[y^{\prime\prime}]$ where $y^{\prime\prime}\pi=y^{\prime}$ . The character $\chi$ is unramified on $V$ . On $U$ , the character $\chi$ is defined by

t^{p}=(-1)^{a}(y^{\prime(e^{\prime}+1)/2}w^{(e^{\prime}-1)/2}-a/c)^{a}(y^{\prime(e^{\prime}+1)/2}w^{(e^{\prime}-1)/2}+b/c)^{b}.

We can check $\chi$ is not clean at the closed point $\{y^{\prime}=w=0\}$ . Further, let $W^{\prime}\to W$ be the blowup at the closed point $\{y^{\prime}=w=0\}$ and $U^{\prime}$ be the open subscheme $\operatorname{Spec}\mathcal{O}_{K}[y^{\prime},w,w^{\prime}]/(y^{\prime}w^{\prime}-w,y^{\prime}w-\pi)$ . Then the the character $\chi$ is defined by

t^{p}=(-1)^{a}(y^{\prime e^{\prime}}w^{\prime(e^{\prime}-1)/2}-a/c)^{a}(y^{\prime e^{\prime}}w^{\prime(e^{\prime}-1)/2}+b/c)^{b}

and the refined Swan conductor of $\chi$ is

\displaystyle\operatorname{rsw}(\chi)

\displaystyle=\frac{-2^{-1}a^{a-1}b^{b-1}c^{3-a-b}(e^{\prime}-1)\cdot d\log w^{\prime}}{(1-\zeta_{p})^{p}y^{\prime-2e^{\prime}}w^{\prime-(e^{\prime}-1)}\cdot(y^{\prime e^{\prime}}w^{\prime(e^{\prime}-1)/2}-a/c)^{a}(y^{\prime e^{\prime}}w^{\prime(e^{\prime}-1)/2}+b/c)^{b}}

by [8, Corollary 8.2.3]. Hence the character $\chi$ is clean on $W^{\prime}$ .

We see that the coefficient $s_{z_{0}}$ is equal to $e^{\prime}-(\sum_{1\leq i\leq(e^{\prime}-1)/2}2+1)=0$ by [7, Remark 5.8]. Hence we have

FCCj_{!}\mathcal{K}-FCCj_{!}\Lambda=-e^{\prime}[L^{\prime}]+[\mathrm{F}^{*}T^{*}_{X_{F}}X]-p[\mathrm{F}^{*}T^{*}_{z_{0}}X]+p[\mathrm{F}^{*}T_{z_{1}}^{*}X]+p[\mathrm{F}^{*}T_{z_{2}}^{*}X]+p[\mathrm{F}^{*}T_{z_{3}}^{*}X].

Computing similarly as Case 1, we have

(FCCj_{!}\mathcal{K}-FCCj_{!}\Lambda,FT^{*}_{X}X|_{X_{F}})_{FT^{*}X|_{X_{F}}}=0

and the Swan conductor of $\mathrm{H}^{1}(\mathbf{P}^{1}_{\overline{K}},j_{!}\mathcal{K})$ is 0 by Theorem 5.15.

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Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-Ku, Tokyo 153-8914, Japan

Email address: [email protected]

	$\displaystyle([L^{\prime}],FT^{}_{X}X\|_{X_{F}})_{FT^{}X\|_{X_{F}}}$	$\displaystyle=c_{1}(F\Omega^{1}_{X}\|_{X_{F}})\cap[X_{F}]+pe^{\prime}(D,D)_{X}+3p-p$
		$\displaystyle=c_{1}(\mathrm{F}^{}\Omega^{1}_{X_{F}})\cap[X_{F}]+c_{1}(\mathrm{F}^{}N_{X_{F}}X)\cap[X_{F}]+2p$
		$\displaystyle=-2p+0+2p=0$

F-characteristic cycle of a rank one sheaf on an arithmetic surface

Abstract

Introduction

Theorem 0.1 (rationality, Theorem 1.3).

Theorem 0.2 (integrality, Theorem 1.5, cf. (5.1)).

Theorem 0.3 (Theorem 5.15).

1 Characteristic form

1.1 Cotangent complex and FW-differential

Proposition 1.1.

Proof.

Proposition 1.2 ([16, Corollary 4.12]).

1.2 Characteristic form

Theorem 1.3.

Remark 1.4.

Theorem 1.5.

2 Refined Swan conductor

Lemma 2.1 (cf. [17, Lemma 1.2.3.1]).

Proof.

Proposition 2.2 (cf. [17, Theorem 4.3.3]).

Proof (Saito).

Remark 2.3.

Proposition 2.4.

Proof.

Theorem 2.5 ([6, Theorem 7.1, 7.3]).

3 Comparison

Proposition 3.1 ([17, Proposition 1.1.8.2, 1.1.10]).

Proposition 3.2.

Lemma 3.3.

Proof (Saito).

Proof of Proposition 3.2.

Proposition 3.4.

Proof (Saito).

4 Proof of the rationality and the integrality

Proof of Theorem 1.3.

Lemma 4.1.

Proof.

Proof of Theorem 1.5.

(a) The case where r=1r=1 and χ|K1\chi|_{K_{1}} is of type I.

(b) The case where r=1r=1 and χ|K1\chi|_{K_{1}} is of type II.

(c) The case where r=2r=2 and χ|K1\chi|_{K_{1}} or χ|K2\chi|_{K_{2}} is tamely ramified.

(d) The case where r=2r=2 and χ|K1\chi|_{K_{1}} and χ|K2\chi|_{K_{2}} are both of type I.

(e) The case where r=2r=2 and χ|K1\chi|_{K_{1}} is of type II and χ|K2\chi|_{K_{2}} is of type I, or χ|K1\chi|_{K_{1}} is of type I and χ|K2\chi|_{K_{2}} is of type II.

(f) The case where r=2r=2 and χ|K1\chi|_{K_{1}} and χ|K2\chi|_{K_{2}} are both of type II.

5 F-Characteristic cycle

5.1 Refined Swan conductor and characteristic form of a rank one sheaf

Definition 5.1 ([7, Definition 4.2]).

Definition 5.2.

Remark 5.3.

5.2 F-characteristic cycle

Definition 5.4 ([6, (3.4.4)]).

Theorem 5.5 (Conductor formula, [8, Corollary 7.5.3, Theorem 8.3.7]).

Proof.

Definition 5.6.

Lemma 5.7.

Proof.

Remark 5.8.

Remark 5.9.

Lemma 5.10.

Proof.

Remark 5.11.

Lemma 5.12 (cf. [21, Lemma 4.3(i)]).

Proof.

Lemma 5.13 (cf. [21, Lemma 4.4, Lemma 4.5]).

Proof.

Lemma 5.14.

Proof.

Theorem 5.15.

Proof.

5.3 Example

Remark 5.16.

Lemma 5.17.

Proof.

Case 1 : we assume vp​((aa​bb​cc)p−1−1)=1v_{p}((a^{a}b^{b}c^{c})^{p-1}-1)=1.

Lemma 5.18.

Proof.

Case 2 : we assume vp​((aa​bb​cc)p−1−1)≥2v_{p}((a^{a}b^{b}c^{c})^{p-1}-1)\geq 2.

Lemma 5.19.

Proof.

References

F-characteristic cycle of a rank one sheaf
on an arithmetic surface

(a) The case where $r=1$ and $\chi|_{K_{1}}$ is of type I.

(b) The case where $r=1$ and $\chi|_{K_{1}}$ is of type II.

(c) The case where $r=2$ and $\chi|_{K_{1}}$ or $\chi|_{K_{2}}$ is tamely ramified.

(d) The case where $r=2$ and $\chi|_{K_{1}}$ and $\chi|_{K_{2}}$ are both of type I.

(e) The case where $r=2$ and $\chi|_{K_{1}}$ is of type II and $\chi|_{K_{2}}$ is of type I, or $\chi|_{K_{1}}$ is of type I and $\chi|_{K_{2}}$ is of type II.

(f) The case where $r=2$ and $\chi|_{K_{1}}$ and $\chi|_{K_{2}}$ are both of type II.

Case 1 : we assume $v_{p}((a^{a}b^{b}c^{c})^{p-1}-1)=1$ .

Case 2 : we assume $v_{p}((a^{a}b^{b}c^{c})^{p-1}-1)\geq 2$ .