The Stacks project

114.25 Duplicate and split out references

This section is a place where we collect duplicates and references which used to say several things at the same time but are now split into their constituent parts.

Lemma 114.25.1. 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.

Proof. This is a duplicate of Properties, Lemma 28.22.3. $\square$

Lemma 114.25.2. 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.

Proof. This is a duplicate of Properties of Spaces, Lemma 65.4.11. $\square$

Theorem 114.25.3. Let $S = \mathop{\mathrm{Spec}}(K)$ with $K$ a field. Let $\overline{s}$ be a geometric point of $S$. Let $G = \text{Gal}_{\kappa (s)}$ denote the absolute Galois group. Then there is an equivalence of categories $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \to G\textit{-Sets}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$.

Proof. This is a duplicate of √Čtale Cohomology, Theorem 59.56.3. $\square$

Remark 114.25.4. You got here because of a duplicate tag. Please see Formal Deformation Theory, Section 89.12 for the actual content.

Lemma 114.25.5. Let $X$ be a locally ringed space. A direct summand of a finite free $\mathcal{O}_ X$-module is finite locally free.

Proof. This is a duplicate of Modules, Lemma 17.14.6. $\square$

Lemma 114.25.6. 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 $R \to E$.

Proof. This is Baer's criterion, see Injectives, Lemma 19.2.6. $\square$

Lemma 114.25.7. 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

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

Proof. This follows from Chow Homology, Lemmas 42.2.3 and 42.2.4. $\square$

Lemma 114.25.8. 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 (90.2.0.1) if and only if $\beta $ and $\alpha $ map to the same element of $\mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{A/\mathbf{Z}}, N)$.

Proof. This follows from Deformation Theory, Lemmas 90.2.3 and 90.2.5. $\square$

Lemma 114.25.9. 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 (90.7.0.1) if and only if $\beta $ and $\alpha $ map to the same element of $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*\mathop{N\! L}\nolimits _{S/\mathbf{Z}}, \mathcal{G})$.

Proof. This follows from Deformation Theory, Lemmas 90.7.4 and 90.7.6. $\square$

Lemma 114.25.10. 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 (90.13.0.1) if and only if $\beta $ and $\alpha $ map to the same element of $\mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}( Lf^*\mathop{N\! L}\nolimits _{\mathcal{O}_\mathcal {B}/\mathbf{Z}}, \mathcal{G})$.

Proof. This follows from Deformation Theory, Lemmas 90.13.4 and 90.13.6. $\square$

Remark 114.25.11. This tag used to point to a section describing several examples of deformation problems. These now each have their own section. See Deformation Problems, Sections 92.4, 92.5, 92.6, and 92.7.

Lemma 114.25.13. We have the following canonical $k$-vector space identifications:

  1. In Deformation Problems, Example 92.4.1 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 92.5.1 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 92.6.1 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 92.7.1 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 $k$.


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