The Stacks project

113.18 Deformations and obstructions of flat modules

In this section we sketch a construction of a deformation theory for the stack of coherent sheaves for any algebraic space $X$ over a ring $\Lambda $. This material is obsolete due to the improved discussion in Quot, Section 97.6.

Our setup will be the following. We assume given

  1. a ring $\Lambda $,

  2. an algebraic space $X$ over $\Lambda $,

  3. a $\Lambda $-algebra $A$, set $X_ A = X \times _{\mathop{\mathrm{Spec}}(\Lambda )} \mathop{\mathrm{Spec}}(A)$, and

  4. a finitely presented $\mathcal{O}_{X_ A}$-module $\mathcal{F}$ flat over $A$.

In this situation we will consider all possible surjections

\[ 0 \to I \to A' \to A \to 0 \]

where $A'$ is a $\Lambda $-algebra whose kernel $I$ is an ideal of square zero in $A'$. Given $A'$ we obtain a first order thickening $X_ A \to X_{A'}$ of algebraic spaces over $\mathop{\mathrm{Spec}}(\Lambda )$. For each of these we consider the problem of lifting $\mathcal{F}$ to a finitely presented module $\mathcal{F}'$ on $X_{A'}$ flat over $A'$. We would like to replicate the results of Deformation Theory, Lemma 89.12.1 in this setting.

To be more precise let $\textit{Lift}(\mathcal{F}, A')$ denote the category of pairs $(\mathcal{F}', \alpha )$ where $\mathcal{F}'$ is a finitely presented module on $X_{A'}$ flat over $A'$ and $\alpha : \mathcal{F}'|_{X_ A} \to \mathcal{F}$ is an isomorphism. Morphisms $(\mathcal{F}'_1, \alpha _1) \to (\mathcal{F}'_2, \alpha _2)$ are isomorphisms $\mathcal{F}'_1 \to \mathcal{F}'_2$ which are compatible with $\alpha _1$ and $\alpha _2$. The set of isomorphism classes of $\textit{Lift}(\mathcal{F}, A')$ is denoted $\text{Lift}(\mathcal{F}, A')$.

Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ X \otimes _\Lambda A$-modules on $X_{\acute{e}tale}$ flat over $A$. We introduce the category $\textit{Lift}(\mathcal{G}, A')$ of pairs $(\mathcal{G}', \beta )$ where $\mathcal{G}'$ is a sheaf of $\mathcal{O}_ X \otimes _\Lambda A'$-modules flat over $A'$ and $\beta $ is an isomorphism $\mathcal{G}' \otimes _{A'} A \to \mathcal{G}$.

Lemma 113.18.1. Notation and assumptions as above. Let $p : X_ A \to X$ denote the projection. Given $A'$ denote $p' : X_{A'} \to X$ the projection. The functor $p'_*$ induces an equivalence of categories between

  1. the category $\textit{Lift}(\mathcal{F}, A')$, and

  2. the category $\textit{Lift}(p_*\mathcal{F}, A')$.

Proof. FIXME. $\square$

Let $\mathcal{H}$ be a sheaf of $\mathcal{O} \otimes _\Lambda A$-modules on $\mathcal{C}_{X/\Lambda }$ flat over $A$. We introduce the category $\textit{Lift}_\mathcal {O}(\mathcal{H}, A')$ whose objects are pairs $(\mathcal{H}', \gamma )$ where $\mathcal{H}'$ is a sheaf of $\mathcal{O} \otimes _\Lambda A'$-modules flat over $A'$ and $\gamma : \mathcal{H}' \otimes _ A A' \to \mathcal{H}$ is an isomorphism of $\mathcal{O} \otimes _\Lambda A$-modules.

Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ X \otimes _\Lambda A$-modules on $X_{\acute{e}tale}$ flat over $A$. Consider the morphisms $i$ and $\pi $ of Cotangent, Equation (90.27.1.1). Denote $\underline{\mathcal{G}} = \pi ^{-1}(\mathcal{G})$. It is simply given by the rule $(U \to \mathbf{A}) \mapsto \mathcal{G}(U)$ hence it is a sheaf of $\underline{\mathcal{O}}_ X \otimes _\Lambda A$-modules. Denote $i_*\underline{\mathcal{G}}$ the same sheaf but viewed as a sheaf of $\mathcal{O} \otimes _\Lambda A$-modules.

Lemma 113.18.2. Notation and assumptions as above. The functor $\pi _!$ induces an equivalence of categories between

  1. the category $\textit{Lift}_\mathcal {O}(i_*\underline{\mathcal{G}}, A')$, and

  2. the category $\textit{Lift}(\mathcal{G}, A')$.

Proof. FIXME. $\square$

Lemma 113.18.3. Notation and assumptions as in Lemma 113.18.2. Consider the object

\[ L = L(\Lambda , X, A, \mathcal{G}) = L\pi _!(Li^*(i_*(\underline{\mathcal{G}}))) \]

of $D(\mathcal{O}_ X \otimes _\Lambda A)$. Given a surjection $A' \to A$ of $\Lambda $-algebras with square zero kernel $I$ we have

  1. The category $\textit{Lift}(\mathcal{G}, A')$ is nonempty if and only if a certain class $\xi \in \mathop{\mathrm{Ext}}\nolimits ^2_{\mathcal{O}_ X \otimes A}(L, \mathcal{G} \otimes _ A I)$ is zero.

  2. If $\textit{Lift}(\mathcal{G}, A')$ is nonempty, then $\text{Lift}(\mathcal{G}, A')$ is principal homogeneous under $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X \otimes A}(L, \mathcal{G} \otimes _ A I)$.

  3. Given a lift $\mathcal{G}'$, the set of automorphisms of $\mathcal{G}'$ which pull back to $\text{id}_\mathcal {G}$ is canonically isomorphic to $\mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_ X \otimes A}(L, \mathcal{G} \otimes _ A I)$.

Proof. FIXME. $\square$

Finally, we put everything together as follows.

Proposition 113.18.4. With $\Lambda $, $X$, $A$, $\mathcal{F}$ as above. There exists a canonical object $L = L(\Lambda , X, A, \mathcal{F})$ of $D(X_ A)$ such that given a surjection $A' \to A$ of $\Lambda $-algebras with square zero kernel $I$ we have

  1. The category $\textit{Lift}(\mathcal{F}, A')$ is nonempty if and only if a certain class $\xi \in \mathop{\mathrm{Ext}}\nolimits ^2_{X_ A}(L, \mathcal{F} \otimes _ A I)$ is zero.

  2. If $\textit{Lift}(\mathcal{F}, A')$ is nonempty, then $\text{Lift}(\mathcal{F}, A')$ is principal homogeneous under $\mathop{\mathrm{Ext}}\nolimits ^1_{X_ A}(L, \mathcal{F} \otimes _ A I)$.

  3. Given a lift $\mathcal{F}'$, the set of automorphisms of $\mathcal{F}'$ which pull back to $\text{id}_\mathcal {F}$ is canonically isomorphic to $\mathop{\mathrm{Ext}}\nolimits ^0_{X_ A}(L, \mathcal{F} \otimes _ A I)$.

Proof. FIXME. $\square$

Lemma 113.18.5. In the situation of Proposition 113.18.4, if $X \to \mathop{\mathrm{Spec}}(\Lambda )$ is locally of finite type and $\Lambda $ is Noetherian, then $L$ is pseudo-coherent.

Proof. FIXME. $\square$


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