The Stacks project

Lemma 115.26.1. Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is the directed colimit of its finite type quasi-coherent submodules.

Lemma 115.26.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The map $\{ \mathop{\mathrm{Spec}}(k) \to X \text{ monomorphism}\} \to |X|$ is injective.

Lemma 115.26.5. Let $X$ be a locally ringed space. A direct summand of a finite free $\mathcal{O}_ X$-module is finite locally free.

Lemma 115.26.6. Let $R$ be a ring. Let $E$ be an $R$-module. The following are equivalent

1. $E$ is an injective $R$-module, and

2. given an ideal $I \subset R$ and a module map $\varphi : I \to E$ there exists an extension of $\varphi $ to an $R$-module map $R \to E$.

Lemma 115.26.7. Let $R$ be a local ring.

1. If $(M, N, \varphi , \psi )$ is a $2$-periodic complex such that $M$, $N$ have finite length. Then $e_ R(M, N, \varphi , \psi ) = \text{length}_ R(M) - \text{length}_ R(N)$.

2. If $(M, \varphi , \psi )$ is a $(2, 1)$-periodic complex such that $M$ has finite length. Then $e_ R(M, \varphi , \psi ) = 0$.

3. Suppose that we have a short exact sequence of $2$-periodic complexes

   \[ 0 \to (M_1, N_1, \varphi _1, \psi _1) \to (M_2, N_2, \varphi _2, \psi _2) \to (M_3, N_3, \varphi _3, \psi _3) \to 0 \]

   If two out of three have cohomology modules of finite length so does the third and we have

   \[ e_ R(M_2, N_2, \varphi _2, \psi _2) = e_ R(M_1, N_1, \varphi _1, \psi _1) + e_ R(M_3, N_3, \varphi _3, \psi _3). \]

Lemma 115.26.8. Let $A$ be a ring and let $I$ be an $A$-module.

1. The set of extensions of rings $0 \to I \to A' \to A \to 0$ where $I$ is an ideal of square zero is canonically bijective to $\mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{A/\mathbf{Z}}, I)$.

2. Given a ring map $A \to B$, a $B$-module $N$, an $A$-module map $c : I \to N$, and given extensions of rings with square zero kernels:

   1. $0 \to I \to A' \to A \to 0$ corresponding to $\alpha \in \mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{A/\mathbf{Z}}, I)$, and

   2. $0 \to N \to B' \to B \to 0$ corresponding to $\beta \in \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/\mathbf{Z}}, N)$

   then there is a map $A' \to B'$ fitting into Deformation Theory, Equation (91.2.0.1) if and only if $\beta $ and $\alpha $ map to the same element of $\mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{A/\mathbf{Z}}, N)$.

Lemma 115.26.9. Let $(S, \mathcal{O}_ S)$ be a ringed space and let $\mathcal{J}$ be an $\mathcal{O}_ S$-module.

1. The set of extensions of sheaves of rings $0 \to \mathcal{J} \to \mathcal{O}_{S'} \to \mathcal{O}_ S \to 0$ where $\mathcal{J}$ is an ideal of square zero is canonically bijective to $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ S}(\mathop{N\! L}\nolimits _{S/\mathbf{Z}}, \mathcal{J})$.

2. Given a morphism of ringed spaces $f : (X, \mathcal{O}_ X) \to (S, \mathcal{O}_ S)$, an $\mathcal{O}_ X$-module $\mathcal{G}$, an $f$-map $c : \mathcal{J} \to \mathcal{G}$, and given extensions of sheaves of rings with square zero kernels:

   1. $0 \to \mathcal{J} \to \mathcal{O}_{S'} \to \mathcal{O}_ S \to 0$ corresponding to $\alpha \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ S}(\mathop{N\! L}\nolimits _{S/\mathbf{Z}}, \mathcal{J})$,

   2. $0 \to \mathcal{G} \to \mathcal{O}_{X'} \to \mathcal{O}_ X \to 0$ corresponding to $\beta \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/\mathbf{Z}}, \mathcal{G})$

   then there is a morphism $X' \to S'$ fitting into Deformation Theory, Equation (91.7.0.1) if and only if $\beta $ and $\alpha $ map to the same element of $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*\mathop{N\! L}\nolimits _{S/\mathbf{Z}}, \mathcal{G})$.

Lemma 115.26.10. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$ be a ringed topos and let $\mathcal{J}$ be an $\mathcal{O}_\mathcal {B}$-module.

1. The set of extensions of sheaves of rings $0 \to \mathcal{J} \to \mathcal{O}_{\mathcal{B}'} \to \mathcal{O}_\mathcal {B} \to 0$ where $\mathcal{J}$ is an ideal of square zero is canonically bijective to $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_\mathcal {B}}( \mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {B}/\mathbf{Z}}, \mathcal{J})$.

2. Given a morphism of ringed topoi $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$, an $\mathcal{O}$-module $\mathcal{G}$, an $f^{-1}\mathcal{O}_\mathcal {B}$-module map $c : f^{-1}\mathcal{J} \to \mathcal{G}$, and given extensions of sheaves of rings with square zero kernels:

   1. $0 \to \mathcal{J} \to \mathcal{O}_{\mathcal{B}'} \to \mathcal{O}_\mathcal {B} \to 0$ corresponding to $\alpha \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_\mathcal {B}}( \mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {B}/\mathbf{Z}}, \mathcal{J})$,

   2. $0 \to \mathcal{G} \to \mathcal{O}' \to \mathcal{O} \to 0$ corresponding to $\beta \in \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathop{N\! L}\nolimits _{\mathcal{O}/\mathbf{Z}}, \mathcal{G})$

   then there is a morphism $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}, \mathcal{O}_{\mathcal{B}'})$ fitting into Deformation Theory, Equation (91.13.0.1) if and only if $\beta $ and $\alpha $ map to the same element of $\mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}( Lf^*\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {B}/\mathbf{Z}}, \mathcal{G})$.

Lemma 115.26.12. Deformation Problems, Examples 93.4.1, 93.5.1, 93.6.1, and 93.7.1 satisfy the Rim-Schlessinger condition (RS).

Lemma 115.26.13. We have the following canonical $k$-vector space identifications:

1. In Deformation Problems, Example 93.4.1 if $x_0 = (k, V)$, then $T_{x_0}\mathcal{F} = (0)$ and $\text{Inf}_{x_0}(\mathcal{F}) = \text{End}_ k(V)$ are finite dimensional.

2. In Deformation Problems, Example 93.5.1 if $x_0 = (k, V, \rho _0)$, then $T_{x_0}\mathcal{F} = \mathop{\mathrm{Ext}}\nolimits ^1_{k[\Gamma ]}(V, V) = H^1(\Gamma , \text{End}_ k(V))$ and $\text{Inf}_{x_0}(\mathcal{F}) = H^0(\Gamma , \text{End}_ k(V))$ are finite dimensional if $\Gamma $ is finitely generated.

3. In Deformation Problems, Example 93.6.1 if $x_0 = (k, V, \rho _0)$, then $T_{x_0}\mathcal{F} = H^1_{cont}(\Gamma , \text{End}_ k(V))$ and $\text{Inf}_{x_0}(\mathcal{F}) = H^0_{cont}(\Gamma , \text{End}_ k(V))$ are finite dimensional if $\Gamma $ is topologically finitely generated.

4. In Deformation Problems, Example 93.7.1 if $x_0 = (k, P)$, then $T_{x_0}\mathcal{F}$ and $\text{Inf}_{x_0}(\mathcal{F}) = \text{Der}_ k(P, P)$ are finite dimensional if $P$ is finitely generated over $k$.