The Stacks project

115.21 The stack of coherent sheaves in the non-flat case

In Quot, Theorem 99.5.12 the assumption that $f : X \to B$ is flat is not necessary. In this section we modify the method of proof based on ideas from derived algebraic geometry to get around the flatness hypothesis. An entirely different method is used in Quot, Section 99.6 to get exactly the same result; this is why the method from this section is obsolete.

The only step in the proof of Quot, Theorem 99.5.12 which uses flatness is in the application of Quot, Lemma 99.5.11. The lemma is used to construct an obstruction theory as in Artin's Axioms, Section 98.24. The proof of the lemma relies on Deformation Theory, Lemmas 91.12.1 and 91.12.5 from Deformation Theory, Section 91.12. This is how the assumption that $f$ is flat comes about. Before we go on, note that results (2) and (3) of Deformation Theory, Lemmas 91.12.1 do hold without the assumption that $f$ is flat as they rely on Deformation Theory, Lemmas 91.11.7 and 91.11.4 which do not have any flatness assumptions.

Before we give the details we give some motivation for the construction from derived algebraic geometry, since we think it will clarify what follows. Let $A$ be a finite type algebra over the locally Noetherian base $S$. Denote $X \otimes ^\mathbf {L} A$ a “derived base change” of $X$ to $A$ and denote $i : X_ A \to X \otimes ^\mathbf {L} A$ the canonical inclusion morphism. The object $X \otimes ^\mathbf {L} A$ does not (yet) have a definition in the Stacks project; we may think of it as the algebraic space $X_ A$ endowed with a simplicial sheaf of rings $\mathcal{O}_{X \otimes ^\mathbf {L} A}$ whose homology sheaves are

\[ H_ i(\mathcal{O}_{X \otimes ^\mathbf {L} A}) = \text{Tor}^{\mathcal{O}_ S}_ i(\mathcal{O}_ X, A). \]

The morphism $X \otimes ^\mathbf {L} A \to \mathop{\mathrm{Spec}}(A)$ is flat (the terms of the simplicial sheaf of rings being $A$-flat), so the usual material for deformations of flat modules applies to it. Thus we see that we get an obstruction theory using the groups

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{X \otimes ^\mathbf {L} A}(i_*\mathcal{F}, i_*\mathcal{F} \otimes _ A M) \]

where $i = 0, 1, 2$ for inf auts, inf defs, obstructions. Note that a flat deformation of $i_*\mathcal{F}$ to $X \otimes ^\mathbf {L} A'$ is automatically of the form $i'_*\mathcal{F}'$ where $\mathcal{F}'$ is a flat deformation of $\mathcal{F}$. By adjunction of the functors $Li^*$ and $i_* = Ri_*$ these ext groups are equal to

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(Li^*(i_*\mathcal{F}), \mathcal{F} \otimes _ A M) \]

Thus we obtain obstruction groups of exactly the same form as in the proof of Quot, Lemma 99.5.11 with the only change being that one replaces the first occurrence of $\mathcal{F}$ by the complex $Li^*(i_*\mathcal{F})$.

Below we prove the non-flat version of the lemma by a “direct” construction of $E(\mathcal{F}) = Li^*(i_*\mathcal{F})$ and direct proof of its relationship to the deformation theory of $\mathcal{F}$. In fact, it suffices to construct $\tau _{\geq -2}E(\mathcal{F})$, as we are only interested in the ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(Li^*(i_*\mathcal{F}), \mathcal{F} \otimes _ A M)$ for $i = 0, 1, 2$. We can even identify the cohomology sheaves

\[ H^ i(E(\mathcal{F})) = \left\{ \begin{matrix} 0 & \text{if }i > 0 \\ \mathcal{F} & \text{if } i = 0 \\ 0 & \text{if } i = -1 \\ \text{Tor}_1^{\mathcal{O}_ S}(\mathcal{O}_ X, A) \otimes _{\mathcal{O}_ X} \mathcal{F} & \text{if } i = -2 \end{matrix} \right. \]

This observation will guide our construction of $E(\mathcal{F})$ in the remarks below.

Remark 115.21.1 (Direct construction). Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $U$ be another algebraic space over $B$. Denote $q : X \times _ B U \to U$ the second projection. Consider the distinguished triangle

\[ Lq^*L_{U/B} \to L_{X \times _ B U/B} \to E \to Lq^*L_{U/B}[1] \]

of Cotangent, Section 92.28. For any sheaf $\mathcal{F}$ of $\mathcal{O}_{X \times _ B U}$-modules we have the Atiyah class

\[ \mathcal{F} \to L_{X \times _ B U/B} \otimes _{\mathcal{O}_{X \times _ B U}}^\mathbf {L} \mathcal{F}[1] \]

see Cotangent, Section 92.19. We can compose this with the map to $E$ and choose a distinguished triangle

\[ E(\mathcal{F}) \to \mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_{X \times _ B U}}^\mathbf {L} E[1] \to E(\mathcal{F})[1] \]

in $D(\mathcal{O}_{X \times _ B U})$. By construction the Atiyah class lifts to a map

\[ e_\mathcal {F} : E(\mathcal{F}) \longrightarrow Lq^*L_{U/B} \otimes _{\mathcal{O}_{X \times _ B U}}^\mathbf {L} \mathcal{F}[1] \]

fitting into a morphism of distinguished triangles

\[ \xymatrix{ \mathcal{F} \otimes ^\mathbf {L} Lq^*L_{U/B}[1] \ar[r] & \mathcal{F} \otimes ^\mathbf {L} L_{X \times _ B U/B}[1] \ar[r] & \mathcal{F} \otimes ^\mathbf {L} E[1] \\ E(\mathcal{F}) \ar[r] \ar[u]^{e_\mathcal {F}} & \mathcal{F} \ar[r] \ar[u]^{Atiyah} & \mathcal{F} \otimes ^\mathbf {L} E[1] \ar[u]^{=} } \]

Given $S, B, X, f, U, \mathcal{F}$ we fix a choice of $E(\mathcal{F})$ and $e_\mathcal {F}$.

Remark 115.21.2 (Construction of obstruction class). With notation as in Remark 115.21.1 let $i : U \to U'$ be a first order thickening of $U$ over $B$. Let $\mathcal{I} \subset \mathcal{O}_{U'}$ be the quasi-coherent sheaf of ideals cutting out $B$ in $B'$. The fundamental triangle

\[ Li^*L_{U'/B} \to L_{U/B} \to L_{U/U'} \to Li^*L_{U'/B}[1] \]

together with the map $L_{U/U'} \to \mathcal{I}[1]$ determine a map $e_{U'} : L_{U/B} \to \mathcal{I}[1]$. Combined with the map $e_\mathcal {F}$ of the previous remark we obtain

\[ (\text{id}_\mathcal {F} \otimes Lq^*e_{U'}) \cup e_\mathcal {F} : E(\mathcal{F}) \longrightarrow \mathcal{F} \otimes _{\mathcal{O}_{X \times _ B U}} q^*\mathcal{I}[2] \]

(we have also composed with the map from the derived tensor product to the usual tensor product). In other words, we obtain an element

\[ \xi _{U'} \in \mathop{\mathrm{Ext}}\nolimits ^2_{\mathcal{O}_{X \times _ B U}}( E(\mathcal{F}), \mathcal{F} \otimes _{\mathcal{O}_{X \times _ B U}} q^*\mathcal{I}) \]

Lemma 115.21.3. In the situation of Remark 115.21.2 assume that $\mathcal{F}$ is flat over $U$. Then the vanishing of the class $\xi _{U'}$ is a necessary and sufficient condition for the existence of a $\mathcal{O}_{X \times _ B U'}$-module $\mathcal{F}'$ flat over $U'$ with $i^*\mathcal{F}' \cong \mathcal{F}$.

Proof (sketch). We will use the criterion of Deformation Theory, Lemma 91.11.8. We will abbreviate $\mathcal{O} = \mathcal{O}_{X \times _ B U}$ and $\mathcal{O}' = \mathcal{O}_{X \times _ B U'}$. Consider the short exact sequence

\[ 0 \to \mathcal{I} \to \mathcal{O}_{U'} \to \mathcal{O}_ U \to 0. \]

Let $\mathcal{J} \subset \mathcal{O}'$ be the quasi-coherent sheaf of ideals cutting out $X \times _ B U$. By the above we obtain an exact sequence

\[ \text{Tor}_1^{\mathcal{O}_ B}(\mathcal{O}_ X, \mathcal{O}_ U) \to q^*\mathcal{I} \to \mathcal{J} \to 0 \]

where the $\text{Tor}_1^{\mathcal{O}_ B}(\mathcal{O}_ X, \mathcal{O}_ U)$ is an abbreviation for

\[ \text{Tor}_1^{h^{-1}\mathcal{O}_ B}(p^{-1}\mathcal{O}_ X, q^{-1}\mathcal{O}_ U) \otimes _{(p^{-1}\mathcal{O}_ X\otimes _{h^{-1}\mathcal{O}_ B}q^{-1}\mathcal{O}_ U)} \mathcal{O}. \]

Tensoring with $\mathcal{F}$ we obtain the exact sequence

\[ \mathcal{F} \otimes _\mathcal {O} \text{Tor}_1^{\mathcal{O}_ B}(\mathcal{O}_ X, \mathcal{O}_ U) \to \mathcal{F} \otimes _\mathcal {O} q^*\mathcal{I} \to \mathcal{F} \otimes _\mathcal {O} \mathcal{J} \to 0 \]

(Note that the roles of the letters $\mathcal{I}$ and $\mathcal{J}$ are reversed relative to the notation in Deformation Theory, Lemma 91.11.8.) Condition (1) of the lemma is that the last map above is an isomorphism, i.e., that the first map is zero. The vanishing of this map may be checked on stalks at geometric points $\overline{z} = (\overline{x}, \overline{u}) : \mathop{\mathrm{Spec}}(k) \to X \times _ B U$. Set $R = \mathcal{O}_{B, \overline{b}}$, $A = \mathcal{O}_{X, \overline{x}}$, $B = \mathcal{O}_{U, \overline{u}}$, and $C = \mathcal{O}_{\overline{z}}$. By Cotangent, Lemma 92.28.2 and the defining triangle for $E(\mathcal{F})$ we see that

\[ H^{-2}(E(\mathcal{F}))_{\overline{z}} = \mathcal{F}_{\overline{z}} \otimes \text{Tor}_1^ R(A, B) \]

The map $\xi _{U'}$ therefore induces a map

\[ \mathcal{F}_{\overline{z}} \otimes \text{Tor}_1^ R(A, B) \longrightarrow \mathcal{F}_{\overline{z}} \otimes _ B \mathcal{I}_{\overline{u}} \]

We claim this map is the same as the stalk of the map described above (proof omitted; this is a purely ring theoretic statement). Thus we see that condition (1) of Deformation Theory, Lemma 91.11.8 is equivalent to the vanishing $H^{-2}(\xi _{U'}) : H^{-2}(E(\mathcal{F})) \to \mathcal{F} \otimes \mathcal{I}$.

To finish the proof we show that, assuming that condition (1) is satisfied, condition (2) is equivalent to the vanishing of $\xi _{U'}$. In the rest of the proof we write $\mathcal{F} \otimes \mathcal{I}$ to denote $\mathcal{F} \otimes _\mathcal {O} q^*\mathcal{I} = \mathcal{F} \otimes _\mathcal {O} \mathcal{J}$. A consideration of the spectral sequence

\[ \mathop{\mathrm{Ext}}\nolimits ^ i(H^{-j}(E(\mathcal{F})), \mathcal{F} \otimes \mathcal{I}) \Rightarrow \mathop{\mathrm{Ext}}\nolimits ^{i + j}(E(\mathcal{F}), \mathcal{F} \otimes \mathcal{I}) \]

using that $H^0(E(\mathcal{F})) = \mathcal{F}$ and $H^{-1}(E(\mathcal{F})) = 0$ shows that there is an exact sequence

\[ 0 \to \mathop{\mathrm{Ext}}\nolimits ^2(\mathcal{F}, \mathcal{F} \otimes \mathcal{I}) \to \mathop{\mathrm{Ext}}\nolimits ^2(E(\mathcal{F}), \mathcal{F} \otimes \mathcal{I}) \to \mathop{\mathrm{Hom}}\nolimits (H^{-2}(E(\mathcal{F})), \mathcal{F} \otimes \mathcal{I}) \]

Thus our element $\xi _{U'}$ is an element of $\mathop{\mathrm{Ext}}\nolimits ^2(\mathcal{F}, \mathcal{F} \otimes \mathcal{I})$. The proof is finished by showing this element agrees with the element of Deformation Theory, Lemma 91.11.8 a verification we omit. $\square$

Lemma 115.21.4. In Quot, Situation 99.5.1 assume that $S$ is a locally Noetherian scheme and $S = B$. Let $\mathcal{X} = \textit{Coh}_{X/B}$. Then we have openness of versality for $\mathcal{X}$ (see Artin's Axioms, Definition 98.13.1).

Proof (sketch). Let $U \to S$ be of finite type morphism of schemes, $x$ an object of $\mathcal{X}$ over $U$ and $u_0 \in U$ a finite type point such that $x$ is versal at $u_0$. After shrinking $U$ we may assume that $u_0$ is a closed point (Morphisms, Lemma 29.16.1) and $U = \mathop{\mathrm{Spec}}(A)$ with $U \to S$ mapping into an affine open $\mathop{\mathrm{Spec}}(\Lambda )$ of $S$. We will use Artin's Axioms, Lemma 98.24.4 to prove the lemma. Let $\mathcal{F}$ be the coherent module on $X_ A = \mathop{\mathrm{Spec}}(A) \times _ S X$ flat over $A$ corresponding to the given object $x$.

Choose $E(\mathcal{F})$ and $e_\mathcal {F}$ as in Remark 115.21.1. The description of the cohomology sheaves of $E(\mathcal{F})$ shows that

\[ \mathop{\mathrm{Ext}}\nolimits ^1(E(\mathcal{F}), \mathcal{F} \otimes _ A M) = \mathop{\mathrm{Ext}}\nolimits ^1(\mathcal{F}, \mathcal{F} \otimes _ A M) \]

for any $A$-module $M$. Using this and using Deformation Theory, Lemma 91.11.7 we have an isomorphism of functors

\[ T_ x(M) = \mathop{\mathrm{Ext}}\nolimits ^1_{X_ A}(E(\mathcal{F}), \mathcal{F} \otimes _ A M) \]

By Lemma 115.21.3 given any surjection $A' \to A$ of $\Lambda $-algebras with square zero kernel $I$ we have an obstruction class

\[ \xi _{A'} \in \mathop{\mathrm{Ext}}\nolimits ^2_{X_ A}(E(\mathcal{F}), \mathcal{F} \otimes _ A I) \]

Apply Derived Categories of Spaces, Lemma 75.23.3 to the computation of the Ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(E(\mathcal{F}), \mathcal{F} \otimes _ A M)$ for $i \leq m$ with $m = 2$. We omit the verification that $E(\mathcal{F})$ is in $D^-_{\textit{Coh}}$; hint: use Cotangent, Lemma 92.5.4. We find a perfect object $K \in D(A)$ and functorial isomorphisms

\[ H^ i(K \otimes _ A^\mathbf {L} M) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{X_ A}(E(\mathcal{F}), \mathcal{F} \otimes _ A M) \]

for $i \leq m$ compatible with boundary maps. This object $K$, together with the displayed identifications above gives us a datum as in Artin's Axioms, Situation 98.24.2. Finally, condition (iv) of Artin's Axioms, Lemma 98.24.3 holds by a variant of Deformation Theory, Lemma 91.12.5 whose formulation and proof we omit. Thus Artin's Axioms, Lemma 98.24.4 applies and the lemma is proved. $\square$

Theorem 115.21.5. Let $S$ be a scheme. Let $f : X \to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation and separated. Then $\textit{Coh}_{X/B}$ is an algebraic stack over $S$.

Proof. This theorem is a copy of Quot, Theorem 99.6.1. The reason we have this copy here is that with the material in this section we get a second proof (as discussed at the beginning of this section). Namely, we argue exactly as in the proof of Quot, Theorem 99.5.12 except that we substitute Lemma 115.21.4 for Quot, Lemma 99.5.11. $\square$

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