Derived characterizations
for rational pairs
à la Schwede-Takagi and Kollár-Kovács

A ‘pair’ in the style of [ST08] is $(Y,\mathcal{I}^{c})$ where $Y$ is a quasi-compact separated normal irreducible scheme $Y$ that is essentially of finite type over a field $k$ of characteristic zero¹¹1We only impose $Y$ to be normal and irreducible for our work, and this not required for other notions in [ST08]., $\mathcal{I}$ is an ideal sheaf on $Y$, and $c$ is a nonnegative real number. Recall that a log resolution for such datum $(Y,\mathcal{I}^{c})$ is a proper birational morphism $f\colon\widetilde{Y}\to Y$ from a regular scheme such that $\mathcal{I}\cdot\mathcal{O}_{\widetilde{Y}}=\mathcal{O}_{\widetilde{Y}}(-G)$ is a line bundle and $\operatorname{exc}(f)\cup\operatorname{supp}(G)$ is a simple normal crossings divisor, where $\operatorname{exc}(f)$ denotes the exceptional locus of $f$. Finally, a pair $(Y,\mathcal{I}^{c})$ has rational singularities à la Schwede–Takagi if the natural map $\mathcal{O}_{Y}\to\mathbb{R}f_{\ast}\mathcal{O}_{\widetilde{Y}}(\lfloor c\cdot G\rfloor)$ is an isomorphism. See Section 2.2 for details.

This brings attention to our first result.

A gives a simpler criterion to check whether $(Y,D)$ has rational singularities à la Schwede–Takagi as opposed to asking for the natural map $\mathcal{O}_{Y}\to\mathbb{R}f_{\ast}\mathcal{O}_{\widetilde{Y}}(\lfloor c\cdot G\rfloor)$ to be an isomorphism. Specifically, it says we only need to require that $\mathcal{O}_{Y}$ be a direct summand of an object which is a finite direct sum of shifts of $\mathbb{R}f_{\ast}\mathcal{O}_{\widetilde{Y}}(\lfloor c\cdot G\rfloor)$ (e.g. an object of the form $\bigoplus_{n\in\mathbb{Z}}\mathbb{R}f_{\ast}\mathcal{O}_{\widetilde{Y}}(\lfloor c\cdot G\rfloor)^{\oplus r_{n}}[n])$.

This gives us a weaker characterization than [ST08, Theorem 3.11] where rational pairs in the sense of loc. cit. were shown to be equivalent to the natural map $\mathcal{O}_{Y}\to\mathbb{R}f_{\ast}\mathcal{O}_{\widetilde{Y}}(\lfloor c\cdot G\rfloor)$ splitting, yet our statement doesn’t directly work with said map. We observe that our constraint for $f$ being ‘locally projective’ is mild in practice (e.g. such $f$ always exist for varieties over a field of characteristic zero).

This component of our work is motivated by a notion of ‘pairs’ in the flavor of [Kol13] but we work in slightly more general setting than loc. cit. Specifically, the class of schemes considered below are those which are quasi-compact separated quasi-excellent of characteristic zero and admit a dualizing complex. A ‘pair’ in this setting will be any such scheme $Y$ which is normal and irreducible, with a choice of Weil divisor $D$ on $Y$ whose coefficients are all one. We fix such a datum in this subsection, and denote it by $(Y,D)$.

There is a notion of thrifty resolution, related to that which appeared in Section 1.1, for the pair $(Y,D)$. This is a proper birational morphism $f\colon\widetilde{Y}\to Y$ from a regular scheme $\widetilde{Y}$ such that the strict transform $D_{Y}$ of $D$ is a simple normal crossing (snc) divisor for which a few technical conditions regarding stratum of the strict transform and exceptional locus must satisfy. These morphisms always exist by [Tem18]. See Section 2.3 for details.

We now propose the following definition for rational pairs in this generality. The motivation for doing so is that our next result, and its argument, are applicable in a more general setting than only that of varieties over a field.

A key difference between rational pairs in the sense of Kollár–Kovács and Schwede–Takagi is that simple normal crossing pairs are of the former, whereas not necessarily of the latter. This brings attention to our next result.

C gives a simpler criterion to check whether $(Y,D)$ has rational singularities à la Kollár–Kovács as opposed to asking for the natural map $\mathcal{O}_{Y}(-D)\to\mathbb{R}f_{\ast}\mathcal{O}_{\widetilde{Y}}(-D_{Y})$ to be an isomorphism. Specifically, it says we only need to require that $\mathcal{O}_{Y}(-D)$ be a direct summand of an object which is a finite direct sum of shifts of $\mathbb{R}f_{\ast}\mathcal{O}_{\widetilde{Y}}(-D_{Y})$ (e.g. an object of the form $\bigoplus_{n\in\mathbb{Z}}\mathbb{R}f_{\ast}\mathcal{O}_{\widetilde{Y}}(-D_{Y})^{\oplus r_{n}}[n])$.

There were a few interesting steps toward progressing on C. Specifically, we realized it is possible to mimic classical results in the literature for our generality. One being a variation of [Kov00, Theorem 1], [Bha12, Theorem 2.12], and [Mur21, Theorem 9.5] in the context of B (see Proposition 3.5). The other being a cohomological vanishing statement for thrifty resolutions in this general setting (see Lemma 3.4).

Let $X$ be a scheme. The following triangulated categories are of interest to our work:

1. (1)

   $D(X):=D(\operatorname{Mod}(X))$ is the derived category of $\mathcal{O}_{X}$-modules.

2. (2)

   $D_{\operatorname{qc}}(X)$ is the (strictly full) subcategory of $D(X)$ consisting of complexes with quasi-coherent cohomology.

3. (3)

   $D_{\operatorname{coh}}^{b}(X)$ is the (strictly full) subcategory of $D(X)$ consisting of complexes having bounded and coherent cohomology.

If $X=\operatorname{Spec}A$ is affine, then we might at times abuse notation and write $D_{\operatorname{qc}}(A)$ for $D_{\operatorname{qc}}(X)$; similar conventions will occur for the other categories. There is a triangulated equivalence of $D_{\operatorname{qc}}(X)$ with $D(\operatorname{Qcoh}(X))$ [Sta24, Tag 09T4], and $D^{b}_{\operatorname{coh}}(X)$ with $D^{b}(\operatorname{coh}(X))$ [Sta24, Tag 0FDB]. We freely use this throughout.

This section gives a brisk recap on generation for triangulated categories. See [BVdB03, Rou08, ABIM10] for further background. Let $\mathcal{T}$ be a triangulated category with shift functor $[1]\colon\mathcal{T}\to\mathcal{T}$. Suppose $\mathcal{S}$ is a subcategory of $\mathcal{T}$. We say $\mathcal{S}$ is thick if it is a triangulated subcategory of $\mathcal{T}$ which is closed under direct summands. The smallest thick subcategory containing $\mathcal{S}$ in $\mathcal{T}$ is denoted by $\langle\mathcal{S}\rangle$; if $\mathcal{S}$ consists of a single object $G$, then $\langle\mathcal{S}\rangle$ will be written as $\langle G\rangle$. Denote by $\operatorname{add}(\mathcal{S})$ for the smallest strictly full²²2This means closed under isomorphisms in the ambient category. subcategory of $\mathcal{T}$ containing $\mathcal{S}$ which is closed under shifts, finite coproducts, and direct summands. Inductively, let

• •

  $\langle\mathcal{S}\rangle_{0}$ consists of all objects in $\mathcal{T}$ isomorphic to the zero objects

• •

  $\langle\mathcal{S}\rangle_{1}:=\operatorname{add}(\mathcal{S})$.

• •

  $\langle\mathcal{S}\rangle_{n}:=\operatorname{add}\{\operatorname{cone}(\phi):\phi\in\operatorname{Hom}_{\mathcal{T}}(\langle\mathcal{S}\rangle_{n-1},\langle\mathcal{S}\rangle_{1})\}$

We denote $\langle\mathcal{S}\rangle_{n}$ by $\langle G\rangle_{n}$ if $\mathcal{S}$ consists of a single object $G$. At times, one might be interested in the cases where there is an object $G$ such that $\langle G\rangle=D^{b}_{\operatorname{coh}}(X)$, and further, when there is an $n\geq 0$ such that $\langle G\rangle_{n}=D^{b}_{\operatorname{coh}}(X)$. Any such object is respectively called a classical or strong generator.

We highlight some motivation for applications these notions beyond the interests of our work. Specifically, for the case $\mathcal{T}=D^{b}_{\operatorname{coh}}(X)$ where $X$ is a Noetherian scheme. It has connections to the closedness for the singular locus of $X$ [IT19, DL24a] and relations to geometric invariants such as Krull dimension [Ola23, DLM24b]. Moreover, we know in large generality that classical and/or strong generators exist for suitable $X$ [Aok21, ELS20], and at times can be explicitly written by hand [BIL⁺23, DLT23, Lan24, DL24b].

Let $\mathcal{C}$ be an additive category. We say that $\mathcal{C}$ is a Krull-Schmidt category if every object of $\mathcal{C}$ is isomorphic to a finite coproduct of objects having local endomorphism rings. Moreover, an object of $\mathcal{C}$ is said to be indecomposable if it is not isomorphic to a coproduct of two nonzero objects. We refer the reader to [Ati56, WW76] for further detail.

If $\mathcal{C}$ is a Krull-Schmidt category, then every object is isomorphic to a finite coproduct of indecomposables, which is unique up to permutations [Kra15, Theorem 4.2]. The case where $\mathcal{C}$ satisfies a linearity condition over a Noetherian complete local ring yield instances of Krull-Schmidt categories (see [LC07, Corollary B]).

We recap the notion of ‘pairs’ in [ST08]. A pair à la Schwede–Takagi is a tuple $(Y,\mathcal{I}^{c})$ where $Y$ is a quasi-compact separated normal irreducible scheme $Y$ that is essentially of finite type over a field $k$ of characteristic zero, $\mathcal{I}$ is an ideal sheaf on $Y$, and $c$ is a nonnegative real number. Loc. cit. relaxes conditions of normality and irreducibility on $Y$ for other notions that are introduced in their work. Our work will leverage these conditions later when reducing arguments to a problem in local algebra. See [ST08, KLM97, KM08] for details.

A log resolution for $(Y,\mathcal{I}^{c})$ is a proper birational morphism $f\colon\widetilde{Y}\to Y$ from a regular scheme such that $\mathcal{I}\cdot\mathcal{O}_{\widetilde{Y}}=\mathcal{O}_{\widetilde{Y}}(-G)$ is a line bundle and $\operatorname{exc}(f)\cup\operatorname{supp}(G)$ is a simple normal crossings divisor. Here $\operatorname{exc}(f)$ denotes the exceptional locus of $f$, which is defined as the closed subset $\widetilde{Y}\setminus U$ for $U$ the largest open subscheme for which $f$ is an isomorphism over.

We recall the definition for rational pairs in the sense of [ST08]. We say $(Y,\mathcal{I}^{c})$ has rational singularities à la Schwede–Takagi if the natural map $\mathcal{O}_{Y}\to\mathbb{R}f_{\ast}\mathcal{O}_{\widetilde{Y}}(\lfloor c\cdot G\rfloor)$ is an isomorphism for $f\colon\widetilde{Y}\to Y$ a log resolution of $(X,\mathcal{I}^{c})$ with $\mathcal{I}\cdot\mathcal{O}_{\widetilde{Y}}=\mathcal{O}_{\widetilde{Y}}(-G)$.

We briefly state a notion of ‘pairs’ motivated by that of [Kol13]. Specifically, we will present the ideas in general setting than loc. cit. due to the contents of our work. To set the stage:

A pair à la Kollár–Kovács be any scheme $Y$ in Convention 2.2 which is normal and irreducible, with a choice of Weil divisor $D$ on $Y$ whose coefficients are all one. We denote such datum by $(Y,D)$.

The following class of morphisms will be needed for defining rational singularities in this setting. A thrifty resolution for the pair $(Y,D)$ is a proper birational morphism $f\colon\widetilde{Y}\to Y$ from a regular scheme $\widetilde{Y}$ such that the strict transform $D_{Y}$ of $D$ is a simple normal crossing (snc) divisor, $\operatorname{exc}(f)$ does not contain any stratum³³3A stratum, in the sense of [Kol13, Definition 1.7], is an irreducible component of $\bigcap_{i\in I}D_{i}$ where $I$ is a subset of $J$ and $D_{Y}=\sum_{j\in J}D_{j}$. of $(Y,D_{Y})$, and $f(\operatorname{exc}(f))$ does not contain any stratum of the snc locus⁴⁴4This is the largest open subscheme $U$ of $Y$ such that $(U,D|_{U})$ is a snc pair in the sense of [Kol13, Definition 1.7]. of $(Y,D_{Y})$.

It follows from [Tem18, Theorem 1.1.6] that thrifty resolutions always exist. See [Kol13, Definition 2.79] or [Tem18, $\S 1.1.5$] for details. We recall the following notion in Section 1.2, which was first introduced for varieties in [Kol13, Definition 2.80].

We briefly recall dualizing complexes for schemes. Let $X$ be a Noetherian scheme. See [Sta24, Tag 0DWE] or [Har66] for details. Suppose $R$ is a Noetherian ring. An object $\omega^{\bullet}_{R}$ in $D(R)$ is called a dualizing complex if the following conditions are satisfied:

1. (1)

   $\omega^{\bullet}_{R}$ has finite injective dimension

2. (2)

   $H^{i}(\omega^{\bullet}_{R})$ is a finite $R$-module for all $i$

3. (3)

   the natural map $R\to\mathbb{R}\operatorname{Hom}_{R}(\omega^{\bullet}_{R},\omega^{\bullet}_{R})$ is an isomorphism in $D(R)$.

This extends naturally to schemes as follows to schemes. An object $K$ of $D_{\operatorname{qc}}(X)$ is called a dualizing complex if $j^{\ast}K$ is a dualizing complex⁵⁵5This means, by abuse of language, $j^{\ast}K$ is isomorphic to the image of a dualizing complex in $D(H^{0}(U,\mathcal{O}_{U}))$ under the sheafification functor. for $D_{\operatorname{qc}}(U)$ where $j\colon U\to X$ is an open immersion from an affine scheme. There many cases where these objects exist, e.g. Noetherian complete local rings [Sta24, Tag 0BFR]. This includes any scheme which is of finite type and separated over a Noetherian scheme that admits a dualizing complex [Sta24, Tag 0AU3].

Let $K$ be a dualizing complex for $X$. Then $K$ belongs to $D_{\operatorname{coh}}^{b}(X)$ and the functor $\operatorname{\mathbb{R}\mathcal{H}\!\mathit{om}}_{\mathcal{O}_{X}}(-,K)\colon D_{\operatorname{coh}}^{\#}(X)\to D_{\operatorname{coh}}^{\#}(X)$ is an anti-equivalence of triangulated categories for $\#$ in $\{\textit{blank},+,b\}$. See [Sta24, Tag 0A89]. Suppose further that $K^{\prime}$ is another dualizing complex in $D_{\operatorname{qc}}(X)$. There is an invertible object⁶⁶6This means there is an $L^{\prime}$ in $D_{\operatorname{qc}}(X)$ such that $L\otimes^{\mathbb{L}}L^{\prime}$ is isomorphic to $\mathcal{O}_{X}$ in $D_{\operatorname{qc}}(X)$. $L$ in $D_{\operatorname{qc}}(X)$ such that $K$ is isomorphic to $K^{\prime}\otimes^{\mathbb{L}}L$ in $D_{\operatorname{qc}}(X)$. [Sta24, Tag 0ATP]. We often will say ‘let $\omega^{\bullet}_{X}$ be a dualizing complex for $X$’ when we mean some object which is a dualizing complex as these are unique up to tensoring by an invertible object.

Let $\omega^{\bullet}_{X}$ be a dualizing complex for $X$. We say that $H^{n}(\omega_{X}^{\bullet})$ is the dualizing module (or dualizing sheaf) for $X$ with $n$ the smallest integer such that $H^{n}(\omega^{\bullet}_{X})$ is nonzero⁷⁷7This exists by [Sta24, Tag 0AWF].. If $X$ is Cohen-Macaulay, then $\omega^{\bullet}_{X}$ is concentrated in one degree. See [Sta24, Tag 0AWT]. If $X$ is Gorenstein, then $\mathcal{O}_{X}[0]$ is a dualizing complex for $D_{\operatorname{qc}}(X)$. See [Sta24, Tag 0BFQ].

There are many useful relations among dualizing complexes, dualizing modules, derived sheaf hom functors, and derived pushforward functors. This falls under the term of ‘duality’. See [Sta24, Tag 0DWE] for a collection of such results. We have tried our best in what follows in the next section to appropriately make reference to where ‘duality’ is being used.

The following stems from a similar, yet mildly different, vein as [Kol13, Theorem 2.74]. A key difference is that we only impose $f_{\ast}E$ belonging to $\langle\mathbb{R}f_{\ast}E\rangle_{1}$ as opposed requiring the natural map $f_{\ast}E\to\mathbb{R}f_{\ast}E$ split as in loc. cit.

We record the following elementary lemma for sake of convenience.

We now start with the first flavor of our main results. Briefly, let us recall notions from Section 2.2. Consider a pair (à la Schwede–Takagi) $(Y,\mathcal{I}^{c})$ where $Y$ is a quasi-compact separated normal irreducible scheme $Y$ that is essentially of finite type over a field $k$ of characteristic zero, $\mathcal{I}$ is an ideal sheaf on $Y$, and $c$ is a nonnegative real number. Suppose $f\colon\widetilde{Y}\to Y$ is a log resolution of $(Y,\mathcal{I}^{c})$ such that $\mathcal{I}\cdot\mathcal{O}_{\widetilde{Y}}=\mathcal{O}_{\widetilde{Y}}(-G)$.

Next, we discuss the last flavor of our main results. We briefly remind ourselves of notions from Section 2.3. Our 2.2 is to work with quasi-compact separated quasi-excellent schemes of characteristic zero that admit a dualizing complex. Consider a pair (à la Kollár–Kovács) where $Y$ is a scheme in our convention which is normal and irreducible and $D$ is Weil divisor $D$ (on $Y$) whose coefficients are all one. Let $f\colon\widetilde{Y}\to Y$ be a thrifty resolution of $(Y,D)$.

We start with a (straightforward) generalization of a result for varieties.

Any pair of thrifty resolutions for $(Y,D)$ can be dominated by a third thrifty resolution of $(Y,D)$. This tells us Definition 2.3 is independent of the choice of thrifty resolution. The following is a variation of [Kov00, Theorem 1], [Bha12, Theorem 2.12] and [Mur21, Theorem 9.5] in our setting. Its proof follows in a similar vein, so we only give a sketch.

It follows from [Tem18, Theorem 1.1.6] that we can find a thrifty resolution of $(Y,D)$ which is locally projective. We now end our work with the following.