Lemma 113.24.1. 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.
113.24 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.
Proof. This is a duplicate of Properties, Lemma 28.22.3. $\square$
Lemma 113.24.2. 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 64.4.11. $\square$
Theorem 113.24.3. 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 Étale Cohomology, Theorem 58.55.3. $\square$
Remark 113.24.4. You got here because of a duplicate tag. Please see Formal Deformation Theory, Section 88.12 for the actual content.
Lemma 113.24.5. 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 113.24.6. Let $R$ be a ring. Let $E$ be an $R$-module. The following are equivalent
$E$ is an injective $R$-module, and
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 113.24.7. Let $R$ be a local ring.
If $(M, N, \varphi , \psi )$ is a $2$-periodic complex such that $M$, $N$ have finite length. Then $e_ R(M, N, \varphi , \psi ) = \text{length}_ R(M) - \text{length}_ R(N)$.
If $(M, \varphi , \psi )$ is a $(2, 1)$-periodic complex such that $M$ has finite length. Then $e_ R(M, \varphi , \psi ) = 0$.
Suppose that we have a short exact sequence of $2$-periodic complexes
If two out of three have cohomology modules of finite length so does the third and we have
\[ 0 \to (M_1, N_1, \varphi _1, \psi _1) \to (M_2, N_2, \varphi _2, \psi _2) \to (M_3, N_3, \varphi _3, \psi _3) \to 0 \]
\[ e_ R(M_2, N_2, \varphi _2, \psi _2) = e_ R(M_1, N_1, \varphi _1, \psi _1) + e_ R(M_3, N_3, \varphi _3, \psi _3). \]
Proof. This follows from Chow Homology, Lemmas 42.2.3 and 42.2.4. $\square$
Remark 113.24.8. 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 91.4, 91.5, 91.6, and 91.7.
Lemma 113.24.9. Deformation Problems, Examples 91.4.1, 91.5.1, 91.6.1, and 91.7.1 satisfy the Rim-Schlessinger condition (RS).
Proof. This follows from Deformation Problems, Lemmas 91.4.2, 91.5.2, 91.6.2, and 91.7.2. $\square$
Lemma 113.24.10. We have the following canonical $k$-vector space identifications:
In Deformation Problems, Example 91.4.1 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.
In Deformation Problems, Example 91.5.1 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.
In Deformation Problems, Example 91.6.1 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.
In Deformation Problems, Example 91.7.1 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 $k$.
Proof. This follows from Deformation Problems, Lemmas 91.4.3, 91.5.3, 91.6.3, and 91.7.3. $\square$
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