The Stacks project

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.

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.

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 √Čtale 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

  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 113.24.7. 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. $\square$

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

  1. 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.

  2. 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.

  3. 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.

  4. 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 $k$.

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