Spectral Excision and Descent for Almost Perfect Complexes

In our proof of 1.3, we will make use of the notion of a universal descent morphism of $\operatorname{\mathbb{E}}_{\infty}$-rings in the sense of [7, D.3.1.1], which was introduced originally by Akhil Mathew in [9, 3.18]. We note that the notion of universal descent morphisms makes sense more generally for $\operatorname{\mathbb{E}}_{2}$-rings. However, it is in the commutative setting that the class of universal descent morphisms has the descent property of [7, D.3.5.8], which will play an important role in our discussion of almost perfect complexes. For this reason, we focus our attention to the case of $\operatorname{\mathbb{E}}_{\infty}$-rings (unlike Milnor’s excision, which works for associative rings).

Under the additional assumption that the underlying map of commutative rings $\pi_{0}A^{\prime}\rightarrow\pi_{0}B^{\prime}$ is surjective, $\operatorname{APerf}(\sigma)$ is already known to be a pullback diagram by virtue of [7, 16.2.0.2] and [7, 16.2.3.1].

In the special case where $\sigma$ is a pullback square of connective $\operatorname{\mathbb{E}}_{\infty}$-rings for which the map $\pi_{0}B\rightarrow\pi_{0}B^{\prime}$ is a surjection whose kernel is a nilpotent ideal of $\pi_{0}B$, 1.3 can be deduced from [7, 16.2.0.2] and [7, 2.7.3.2].

By restricting the equivalence of 1.3 to the full subcategories spanned by the dualizable objects (see the proof of 2.15), we immediately deduce that the canonical map $\operatorname{Perf}(A)\rightarrow\operatorname{Perf}(A^{\prime})\times_{\operatorname{Perf}(B^{\prime})}\operatorname{Perf}(B)$ is an equivalence of $\infty$-categories. We remark that this equivalence is proven in [9, 2.23] under the additional assumption that the map $\pi_{0}A^{\prime}\rightarrow\pi_{0}B^{\prime}$ is a surjection.

According to [4, 1.17] of Markus Land and Georg Tamme, $\operatorname{Perf}(\sigma)$ is a pullback diagram of $\infty$-categories if $\sigma$ is a pullback square of $\operatorname{\mathbb{E}}_{1}$-rings for which the functor $\operatorname{Perf}(B\odot^{B^{\prime}}_{A}A^{\prime})\rightarrow\operatorname{Perf}(B^{\prime})$ is conservative (see [4, 1.3] for more details). In the case of connective $\operatorname{\mathbb{E}}_{\infty}$-rings, the condition appearing in [4, 1.17] is satisfied if the map $\pi_{0}B\rightarrow\pi_{0}B^{\prime}$ is surjective (see [4, p.912] and [4, 1.3]). Consequently, 1.7, which is an immediate consequence of our more general result 1.3, can be deduced from [4, 1.17]. We note that our proof of 1.7, which is very succinct, does not involve the construction of the $\operatorname{\mathbb{E}}_{1}$-ring $B\odot^{B^{\prime}}_{A}A^{\prime}$ appearing in the statement of [4, 1.17].

If $f:A\rightarrow B$ is a morphism of noetherian simplicial commutative rings for which the induced map $\pi_{0}f$ is a finitely presented $v$-cover of ordinary commutative rings, it then follows from [3, 3.3.1] of Halpern-Leistner and Preygel that almost perfect complexes satisfy (not necessarily universal) descent for the morphism $f$; see 2.13. Our descent result 1.11 can be regarded as a generalization of their work (in the affine case): it holds without the locally noetherian assumption appearing in [3, 3.3.1] and is valid more generally for connective $\operatorname{\mathbb{E}}_{\infty}$-rings, rather than merely for simplicial commutative rings. Moreover, the $\operatorname{APerf}$-descent for the morphism $f$ is universal (that is, $\operatorname{APerf}$ satisfies descent for arbitrary base change of the map $f$).

Let $f:A\rightarrow B$ be a morphism of noetherian connective $\operatorname{\mathbb{E}}_{\infty}$-rings for which the induced map $\pi_{0}f$ is a finitely presented $v$-cover of ordinary commutative rings. Then [3, 3.3.6] of Halpern-Leistner and Preygel shows that the functor $\operatorname{Mod}^{\operatorname{acn}}$ satisfies (not necessarily universal) descent for the map $f$ (see 2.9). Using a slight variant of the proof of 1.11, we can remove the noetherian assumption on $A$ and $B$ to show that $f$ is of universal $\operatorname{Mod}^{\operatorname{acn}}$-descent; see 3.8.

As a consequence of 1.11, we will see in 3.9 that $f$ is also of universal $\operatorname{Perf}$-descent. In particular, the functor $\operatorname{Perf}:\operatorname{CAlg}^{\operatorname{cn}}\rightarrow\operatorname{\mathcal{C}at}_{\infty}$ satisfies descent with respect to the Grothendieck topology on $(\operatorname{CAlg}^{\operatorname{cn}})^{\operatorname{op}}$ which is characterized by the following property: a sieve $\operatorname{\mathcal{C}}\subseteq(\operatorname{CAlg}^{\operatorname{cn}})^{\operatorname{op}}_{/A}$ is a covering if and only if it contains a finite collection of maps $\{A\rightarrow A_{i}\}_{1\leq i\leq n}$ for which the induced map $f:A\rightarrow\prod A_{i}$ is a $v$-cover such that the underlying map of commutative rings $\pi_{0}f$ exhibits $\pi_{0}(\prod A_{i})$ as a finitely presented $\pi_{0}A$-algebra. In contrast with [2, 11.2] for ordinary perfect schemes, the sheaf $\operatorname{Perf}$ is not hypercomplete (that is, it is not true in general that if $U_{\bullet}:\Delta^{\operatorname{op}}\rightarrow(\operatorname{CAlg}^{\operatorname{cn}})^{\operatorname{op}}$ is a hypercovering with respect to the above Grothendieck topology in the sense of [7, A.5.7.1], then the composition $\Delta\stackrel{{\scriptstyle U_{\bullet}}}{{\longrightarrow}}\operatorname{CAlg}^{\operatorname{cn}}\stackrel{{\scriptstyle\operatorname{Perf}}}{{\longrightarrow}}\operatorname{\mathcal{C}at}_{\infty}$ is a limit diagram). To see this, we note that for every connective $\operatorname{\mathbb{E}}_{\infty}$-ring $R$, the truncation map $R\rightarrow\pi_{0}R$ is a $v$-cover, so that the constant cosimplicial $\operatorname{\mathbb{E}}_{\infty}$-ring with value $\pi_{0}R$ is a hypercovering of $R$ with respect to the above Grothendieck topology. Consequently, if the functor $\operatorname{Perf}$ is hypercomplete, then the natural map $\operatorname{Perf}(R)\rightarrow\operatorname{Perf}(\pi_{0}R)$ is an equivalence of $\infty$-categories, which is false in general.

Fix integers $a\leq b$. Since the functor $\operatorname{Perf}$ is a sheaf for the Grothendieck topology appearing in 1.14 by virtue of 3.9, so is the subfunctor $\operatorname{Perf}^{[a,b]}$ which assigns to each connective $\operatorname{\mathbb{E}}_{\infty}$-ring $R$ the full subcategory $\operatorname{Perf}^{[a,b]}(R)\subseteq\operatorname{Perf}(R)$ spanned by those perfect complexes whose Tor-amplitude is contained in $[a,b]$; see, for example, [8, 7.2.4.21]. As Akhil Mathew pointed out, the functor $\operatorname{Perf}^{[a,b]}:\operatorname{CAlg}^{\operatorname{cn}}\rightarrow\operatorname{\mathcal{C}at}_{\infty}$ does not take values in the $\infty$-category of $(b-a+1)$-categories in the sense of [6, 2.3.4.1], unlike in the case of ordinary perfect schemes of [2, 11.2]. Consequently, the fact that $\operatorname{Perf}^{[a,b]}$ is a sheaf does not guarantee that it is a hypercomplete sheaf (see [6, 6.5.2.9]). In fact, for a connective $\operatorname{\mathbb{E}}_{\infty}$-ring $R$, the canonical map $\operatorname{Perf}^{[a,b]}(R)\rightarrow\operatorname{Perf}^{[a,b]}(\pi_{0}R)$ is not an equivalence in general, so that $\operatorname{Perf}^{[a,b]}$ is not hypercomplete.

Alternatively, we can define a $v$-cover of affine spectral Deligne-Mumford stacks $\operatorname{Spec}(f):\operatorname{Spec}B\rightarrow\operatorname{Spec}A$ as in 2.1, by virtue of the universal property of $0$-truncations of spectral Deligne-Mumford stacks (see [7, 1.4.6.3]).

The class of universal descent morphisms, which will play a central role in our proofs of 1.3 and 1.11, allows us to do descent theory in the spectral setting. In particular, if $f:A\rightarrow B$ is a universal descent morphism of $\operatorname{\mathbb{E}}_{\infty}$-rings, it then follows from [7, D.3.5.8] that the canonical map $\operatorname{Mod}_{A}\rightarrow\lim\operatorname{Mod}_{B^{\bullet}}$ is an equivalence of symmetric monoidal $\infty$-categories, where $B^{\bullet}$ denotes the Čech nerve of $f$ formed in the $\infty$-category $(\operatorname{CAlg}^{\operatorname{cn}})^{\operatorname{op}}$.

Let $f:A\rightarrow B$ be a morphism of $\operatorname{\mathbb{E}}_{\infty}$-rings. Let $B^{\bullet}$ denote the Čech nerve of $f$ (so that $B^{n}$ is given by the $(n+1)$-fold tensor product $B\otimes_{A}\cdots\otimes_{A}B$ for each $n\geq 0$). We denote by $\operatorname{Tot}^{\bullet}(B/A)$ the Tot-tower of $B^{\bullet}$: that is, the sequence of $A$-modules

Here $\operatorname{Tot}^{n}(B/A)$ denotes the limit of the diagram $\{B^{m}\}$ taken over the full subcategory $\Delta_{\leq n}\subseteq\Delta$ spanned by those objects $[m]\in\Delta$ such that $m\leq n$. We note that if $f:A\rightarrow B$ is a universal descent morphism, then $A$ is a retract of $\operatorname{Tot}^{n}(B/A)$ for some integer $n\geq 0$ (see the proof of [7, D.3.2.1]). We will use this fact to deduce some descending properties of universal descent morphisms; see 2.11.

In the situation of 2.11, it follows from 2.15 that the $A$-module $M$ is perfect if and only if the $B$-module $M\otimes_{A}B$ is perfect (see [9, 3.28]).

The proof of 2.15 shows that every morphism which is either of universal $\operatorname{Mod}^{\operatorname{acn}}$-descent or of universal $\operatorname{APerf}$-descent is of universal $\operatorname{Perf}$-descent.

The converse of 2.15 is not necessarily true. For example, if $R=\operatorname{Sym}^{\ast}_{\mathbb{Q}}(\Sigma^{2}\mathbb{Q})$ denotes the free $\operatorname{\mathbb{E}}_{\infty}$-algebra over $\mathbb{Q}$ on a single generator $t$ of degree $2$, then we will see from 3.9 that the truncation map $R\rightarrow\pi_{0}R$ is of universal $\operatorname{Perf}$-descent. However, the extension of scalars functor $\operatorname{Mod}_{R}\rightarrow\operatorname{Mod}_{\pi_{0}R}$ is not conservative (since the image of the localization $R[t^{-1}]$ vanishes), so that the truncation map is not a universal descent morphism (see 2.7).

Arguing as in the proof of 1.11 (using $\operatorname{Mod}^{\operatorname{acn}}$ in place of $\operatorname{APerf}$), we deduce that in the situation of 1.11, $f$ is of universal $\operatorname{Mod}^{\operatorname{acn}}$-descent.

In the special case where $X$ is representable by an affine spectral Deligne-Mumford stack $\operatorname{Spec}A$, the full subcategory $\operatorname{Perf}(X)\subseteq\operatorname{QCoh}(X)$ corresponds to the full subcategory $\operatorname{Perf}(A)\subseteq\operatorname{Mod}_{A}$ under the equivalence of $\infty$-categories $\operatorname{QCoh}(X)\simeq\operatorname{Mod}_{A}$.

We note that $\operatorname{QCoh}(X)$ admits a symmetric monoidal structure of [7, p.497]. According to [7, 6.2.6.2], an object $F\in\operatorname{QCoh}(X)$ is dualizable as an object of the symmetric monoidal $\infty$-category $\operatorname{QCoh}(X)$ if and only if it belongs to $\operatorname{Perf}(X)$.