varieties-lemma-degree-tensor-product

Let $k$ be a field, let $X$ be a proper scheme of dimension $\leq 1$ over $k$, and let $\mathcal{E}$, $\mathcal{V}$ be locally free $\mathcal{O}_X$-modules of constant finite rank. Then $$ \deg(\mathcal{E} \otimes \mathcal{V}) = \text{rank}(\mathcal{E}) \deg(\mathcal{V}) + \text{rank}(\mathcal{V}) \deg(\mathcal{E}) $$


varieties-lemma-degree-effective-Cartier-divisor

Let $k$ be a field, let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $D$ be an effective Cartier divisor on $X$. Then $D$ is finite over $\Spec(k)$ of degree $\deg(D) = \dim_k \Gamma(D, \mathcal{O}_D)$. For a locally free sheaf $\mathcal{E}$ of rank $n$ we have $$ \deg(\mathcal{E}(D)) = n\deg(D) + \deg(\mathcal{E}) $$ where $\mathcal{E}(D) = \mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{O}_X(D)$.


varieties-lemma-divisible

Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced and connected. Let $\kappa = H^0(X, \mathcal{O}_X)$. Then $\kappa/k$ is a finite extension of fields and $w = [\kappa : k]$ divides \begin{enumerate} \item $\deg(\mathcal{E})$ for all locally free $\mathcal{O}_X$-modules $\mathcal{E}$, \item $[\kappa(x) : k]$ for all closed points $x \in X$, and \item $\deg(D)$ for all closed subschemes $D \subset X$ of dimension zero. \end{enumerate}


varieties-lemma-degree-pullback-map-proper-curves

Let $k$ be a field. Let $f : X \to Y$ be a nonconstant morphism of proper curves over $k$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_Y$-module. Then $$ \deg(f^*\mathcal{E}) = \deg(X/Y) \deg(\mathcal{E}) $$


varieties-lemma-check-invertible-sheaf-trivial

Let $k$ be a field. Let $X$ be a proper curve over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. \begin{enumerate} \item If $\mathcal{L}$ has a nonzero section, then $\deg(\mathcal{L}) \geq 0$. \item If $\mathcal{L}$ has a nonzero section $s$ which vanishes at a point, then $\deg(\mathcal{L}) > 0$. \item If $\mathcal{L}$ and $\mathcal{L}^{-1}$ have nonzero sections, then $\mathcal{L} \cong \mathcal{O}_X$. \item If $\deg(\mathcal{L}) \leq 0$ and $\mathcal{L}$ has a nonzero section, then $\mathcal{L} \cong \mathcal{O}_X$. \item If $\mathcal{N} \to \mathcal{L}$ is a nonzero map of invertible $\mathcal{O}_X$-modules, then $\deg(\mathcal{L}) \geq \deg(\mathcal{N})$ and if equality holds then it is an isomorphism. \end{enumerate}


varieties-lemma-no-sections-dual-nef

Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and equidimensional of dimension $1$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. If $\deg(\mathcal{L}|_C) \leq 0$ for all irreducible components $C$ of $X$, then either $H^0(X, \mathcal{L}) = 0$ or $\mathcal{L} \cong \mathcal{O}_X$.


varieties-lemma-ample-curve

Let $k$ be a field. Let $X$ be a proper curve over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then $\mathcal{L}$ is ample if and only if $\deg(\mathcal{L}) > 0$.


varieties-lemma-ampleness-in-terms-of-degrees-components

Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $C_i \subset X$, $i = 1, \ldots, t$ be the irreducible components of dimension $1$. The following are equivalent: \begin{enumerate} \item $\mathcal{L}$ is ample, and \item $\deg(\mathcal{L}|_{C_i}) > 0$ for $i = 1, \ldots, t$. \end{enumerate}


varieties-lemma-regular-point-on-curve

Let $k$ be a field. Let $X$ be a curve over $k$. Let $x \in X$ be a closed point. We think of $x$ as a (reduced) closed subscheme of $X$ with sheaf of ideals $\mathcal{I}$. The following are equivalent \begin{enumerate} \item $\mathcal{O}_{X, x}$ is regular, \item $\mathcal{O}_{X, x}$ is normal, \item $\mathcal{O}_{X, x}$ is a discrete valuation ring, \item $\mathcal{I}$ is an invertible $\mathcal{O}_X$-module, \item $x$ is an effective Cartier divisor on $X$. \end{enumerate} If $k$ is perfect, these are also equivalent to \begin{enumerate} \item[(6)] $X \to \Spec(k)$ is smooth at $x$. \end{enumerate}


varieties-lemma-general-degree-g-line-bundle

Let $k$ be an algebraically closed field. Let $X$ be a proper curve over $k$. Then there exist \begin{enumerate} \item an invertible $\mathcal{O}_X$-module $\mathcal{L}$ with $\dim_k H^0(X, \mathcal{L}) = 1$ and $H^1(X, \mathcal{L}) = 0$, and \item an invertible $\mathcal{O}_X$-module $\mathcal{N}$ with $\dim_k H^0(X, \mathcal{N}) = 0$ and $H^1(X, \mathcal{N}) = 0$. \end{enumerate}


varieties-lemma-vanishing-degree-2g-and-1-line-bundle

Let $k$ be an algebraically closed field. Let $X$ be a proper curve over $k$. Set $g = \dim_k H^1(X, \mathcal{O}_X)$. For every invertible $\mathcal{O}_X$-module $\mathcal{L}$ with $\deg(\mathcal{L}) \geq 2g - 1$ we have $H^1(X, \mathcal{L}) = 0$.


varieties-lemma-numerical-polynomial-from-euler

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $\mathcal{L}_1, \ldots, \mathcal{L}_r$ be invertible $\mathcal{O}_X$-modules. The map $$ (n_1, \ldots, n_r) \longmapsto \chi(X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}) $$ is a numerical polynomial in $n_1, \ldots, n_r$ of total degree at most the dimension of the support of $\mathcal{F}$.


varieties-lemma-numerical-polynomial-leading-term

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $\mathcal{L}_1, \ldots, \mathcal{L}_r$ be invertible $\mathcal{O}_X$-modules. Let $d = \dim(\text{Supp}(\mathcal{F}))$. Let $Z_i \subset X$ be the irreducible components of $\text{Supp}(\mathcal{F})$ of dimension $d$. Let $\xi_i \in Z_i$ be the generic point and set $m_i = \text{length}_{\mathcal{O}_{X, \xi_i}}(\mathcal{F}_{\xi_i})$. Then $$ \chi(X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}) - \sum\nolimits_i m_i\ \chi(Z_i, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}|_{Z_i}) $$ is a numerical polynomial in $n_1, \ldots, n_r$ of total degree $< d$.


varieties-lemma-intersection-number-integer

In the situation of Definition \ref{definition-intersection-number} the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z)$ is an integer.


varieties-lemma-intersection-number-additive

In the situation of Definition \ref{definition-intersection-number} the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z)$ is additive: if $\mathcal{L}_i = \mathcal{L}_i' \otimes \mathcal{L}_i''$, then we have $$ (\mathcal{L}_1 \cdots \mathcal{L}_i \cdots \mathcal{L}_d \cdot Z) = (\mathcal{L}_1 \cdots \mathcal{L}_i' \cdots \mathcal{L}_d \cdot Z) + (\mathcal{L}_1 \cdots \mathcal{L}_i'' \cdots \mathcal{L}_d \cdot Z) $$


varieties-lemma-intersection-number-in-terms-of-components

In the situation of Definition \ref{definition-intersection-number} let $Z_i \subset Z$ be the irreducible components of dimension $d$. Let $m_i = \text{length}_{\mathcal{O}_{X, \xi_i}}(\mathcal{O}_{Z, \xi_i})$ where $\xi_i \in Z_i$ is the generic point. Then $$ (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) = \sum m_i(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z_i) $$


varieties-lemma-intersection-number-and-pullback

Let $k$ be a field. Let $f : Y \to X$ be a morphism of proper schemes over $k$. Let $Z \subset Y$ be an integral closed subscheme of dimension $d$ and let $\mathcal{L}_1, \ldots, \mathcal{L}_d$ be invertible $\mathcal{O}_X$-modules. Then $$ (f^*\mathcal{L}_1 \cdots f^*\mathcal{L}_d \cdot Z) = \deg(f|_Z : Z \to f(Z)) (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot f(Z)) $$ where $\deg(Z \to f(Z))$ is as in Morphisms, Definition \ref{morphisms-definition-degree} or $0$ if $\dim(f(Z)) < d$.


varieties-lemma-numerical-intersection-effective-Cartier-divisor

Let $k$ be a field. Let $X$ be proper over $k$. Let $Z \subset X$ be a closed subscheme of dimension $d$. Let $\mathcal{L}_1, \ldots, \mathcal{L}_d$ be invertible $\mathcal{O}_X$-modules. Assume there exists an effective Cartier divisor $D \subset Z$ such that $\mathcal{L}_1|_Z \cong \mathcal{O}_Z(D)$. Then $$ (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) = (\mathcal{L}_2 \cdots \mathcal{L}_d \cdot D) $$


varieties-lemma-ample-positive

Let $k$ be a field. Let $X$ be proper over $k$. Let $Z \subset X$ be a closed subscheme of dimension $d$. If $\mathcal{L}_1, \ldots, \mathcal{L}_d$ are ample, then $(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z)$ is positive.


varieties-lemma-degree-finite-morphism-in-terms-degrees

Let $k$ be a field. Let $f : Y \to X$ be a finite dominant morphism of proper varieties over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Then $$ \deg_{f^*\mathcal{L}}(Y) = \deg(f) \deg_\mathcal{L}(X) $$ where $\deg(f)$ is as in Morphisms, Definition \ref{morphisms-definition-degree}.


varieties-lemma-intersection-numbers-and-degrees-on-curves

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $Z \subset X$ be a closed subscheme of dimension $\leq 1$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then $$ (\mathcal{L} \cdot Z) = \deg(\mathcal{L}|_Z) $$ where $\deg(\mathcal{L}|_Z)$ is as in Definition \ref{definition-degree-invertible-sheaf}. If $\mathcal{L}$ is ample, then $\deg_\mathcal{L}(Z) = \deg(\mathcal{L}|_Z)$.


varieties-lemma-generate-over-complement

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Let $Z \subset X$ be a closed subscheme. Then there exists an integer $n_0$ such that for all $n \geq n_0$ the kernel $V_n$ of $\Gamma(X, \mathcal{L}^{\otimes n}) \to \Gamma(Z, \mathcal{L}^{\otimes n}|_Z)$ generates $\mathcal{L}^{\otimes n}|_{X \setminus Z}$ and the canonical morphism $$ X \setminus Z \longrightarrow \mathbf{P}(V_n) $$ is an immersion of schemes over $k$.


varieties-lemma-bertini

In Situation \ref{situation-family-divisors} assume \begin{enumerate} \item $X$ is smooth over $k$, \item the image of $\psi : V \to \Gamma(X, \mathcal{L})$ generates $\mathcal{L}$, \item the corresponding morphism $\varphi_{\mathcal{L}, \psi} : X \to \mathbf{P}(V)$ is an immersion. \end{enumerate} Then for general $v \in V \otimes_k k'$ the scheme $H_v$ is smooth over $k'$.


varieties-lemma-vanishin-h0-negative

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. If $\text{Ass}(\mathcal{F})$ does not contain any closed points, then $\Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = 0$ for $n \ll 0$.


varieties-lemma-vanishin-h1-negative

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Assume that for $x \in X$ closed we have $\text{depth}(\mathcal{F}_x) \geq 2$. Then $H^1(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes m}) = 0$ for $m \ll 0$.


varieties-lemma-connectedness-ample-divisor

Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. Assume \begin{enumerate} \item $s$ is a regular section (Divisors, Definition \ref{divisors-definition-regular-section}), \item for every closed point $x \in X$ we have $\text{depth}(\mathcal{O}_{X, x}) \geq 2$, and \item $X$ is connected. \end{enumerate} Then the zero scheme $Z(s)$ of $s$ is connected.


varieties-proposition-units-general

Let $k$ be a field. Let $X$ be a scheme over $k$. Assume that $X$ is locally of finite type over $k$, connected, reduced, and has finitely many irreducible components. Then $\mathcal{O}(X)^*/k^*$ is a finitely generated abelian group if in addition to the conditions above at least one of the following conditions is satisfied: \begin{enumerate} \item the integral closure of $k$ in $\Gamma(X, \mathcal{O}_X)$ is $k$, \item $X$ has a $k$-rational point, or \item $X$ is geometrically integral. \end{enumerate}


varieties-proposition-unique-base-field

Let $X$ be a scheme. Let $a : X \to \Spec(k_1)$ and $b : X \to \Spec(k_2)$ be morphisms from $X$ to spectra of fields. Assume $a, b$ are locally of finite type, and $X$ is reduced, and connected. Then we have $k_1' = k_2'$, where $k_i' \subset \Gamma(X, \mathcal{O}_X)$ is the integral closure of $k_i$ in $\Gamma(X, \mathcal{O}_X)$.


varieties-proposition-dim-1-noetherian-separated-has-ample

Let $X$ be a Noetherian separated scheme of dimension $1$. Then $X$ has an ample invertible sheaf.


varieties-proposition-finite-set-of-points-of-codim-1-in-affine

Let $X$ be a separated scheme such that every quasi-compact open has a finite number of irreducible components. Let $x_1, \ldots, x_r \in X$ be points such that $\mathcal{O}_{X, x_i}$ is Noetherian of dimension $\leq 1$. Then there exists an affine open subscheme of $X$ containing all of $x_1, \ldots, x_r$.


varieties-proposition-asymptotic-riemann-roch

Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $d$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Then $$ \dim_k \Gamma(X, \mathcal{L}^{\otimes n}) \sim c n^d + l.o.t. $$ where $c = \deg_\mathcal{L}(X)/d!$ is a positive constant.


cotangent-theorem-quillen-spectral-sequence

Let $A \to B$ be a surjective ring map. Consider the sheaf $\Omega = \Omega_{\mathcal{O}/A} \otimes_\mathcal{O} \underline{B}$ of $\underline{B}$-modules on $\mathcal{C}_{B/A}$, see Section \ref{section-compute-L-pi-shriek}. Then there is a spectral sequence with $E_1$-page $$ E_1^{p, q} = H_{- p - q}(\mathcal{C}_{B/A}, \text{Sym}^p_{\underline{B}}(\Omega)) \Rightarrow \text{Tor}^A_{- p - q}(B, B) $$ with $d_r$ of bidegree $(r, -r + 1)$. Moreover, $H_i(\mathcal{C}_{B/A}, \text{Sym}^k_{\underline{B}}(\Omega)) = 0$ for $i < k$.


cotangent-lemma-colimit-cotangent-complex

Let $A_i \to B_i$ be a system of ring maps over a directed index set $I$. Then $\colim L_{B_i/A_i} = L_{\colim B_i/\colim A_i}$.


cotangent-lemma-identify-pi-shriek

With notation as above let $P_\bullet$ be a simplicial $A$-algebra endowed with an augmentation $\epsilon : P_\bullet \to B$. Assume each $P_n$ is a polynomial algebra over $A$ and $\epsilon$ is a trivial Kan fibration on underlying simplicial sets. Then $$ L\pi_!(\mathcal{F}) = \mathcal{F}(P_\bullet, \epsilon) $$ in $D(\textit{Ab})$, resp.\ $D(B)$ functorially in $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, resp.\ $\textit{Mod}(\underline{B})$.


cotangent-lemma-pi-shriek-standard

Let $A \to B$ be a ring map. Let $\epsilon : P_\bullet \to B$ be the standard resolution of $B$ over $A$. Let $\pi$ be as in (\ref{equation-pi}). Then $$ L\pi_!(\mathcal{F}) = \mathcal{F}(P_\bullet, \epsilon) $$ in $D(\textit{Ab})$, resp.\ $D(B)$ functorially in $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, resp.\ $\textit{Mod}(\underline{B})$.


cotangent-lemma-compute-cotangent-complex

Let $A \to B$ be a ring map.  Let $\pi$ and $i$ be as in (\ref{equation-pi}). There is a canonical isomorphism $$ L_{B/A} = L\pi_!(Li^*\Omega_{\mathcal{O}/A}) = L\pi_!(i^*\Omega_{\mathcal{O}/A}) = L\pi_!(\Omega_{\mathcal{O}/A} \otimes_\mathcal{O} \underline{B}) $$ in $D(B)$.


cotangent-lemma-pi-lower-shriek-constant-sheaf

If $A \to B$ is a ring map, then $L\pi_!(\pi^{-1}M) = M$ with $\pi$ as in (\ref{equation-pi}).


cotangent-lemma-identify-H0

If $A \to B$ is a ring map, then $H^0(L_{B/A}) = \Omega_{B/A}$.


cotangent-lemma-pi-lower-shriek-polynomial-algebra

If $B$ is a polynomial algebra over the ring $A$, then with $\pi$ as in (\ref{equation-pi}) we have that $\pi_!$ is exact and $\pi_!\mathcal{F} = \mathcal{F}(B \to B)$.


cotangent-lemma-cotangent-complex-polynomial-algebra

If $B$ is a polynomial algebra over the ring $A$, then $L_{B/A}$ is quasi-isomorphic to $\Omega_{B/A}[0]$.


cotangent-lemma-polynomial

Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map. Let $\mathcal{A}$ be the category of $A$-algebra maps $C \to B$. Let $n \geq 0$ and let $P_\bullet$ be a simplicial object of $\mathcal{A}$ such that \begin{enumerate} \item $P_\bullet \to B$ is a trivial Kan fibration of simplicial sets, \item $P_k$ is finite type over $A$ for $k \leq n$, \item $P_\bullet = \text{cosk}_n \text{sk}_n P_\bullet$ as simplicial objects of $\mathcal{A}$. \end{enumerate} Then $P_{n + 1}$ is a finite type $A$-algebra.


cotangent-lemma-pi-shriek-finite

Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map. Let $\pi$, $\underline{B}$ be as in (\ref{equation-pi}). If $\mathcal{F}$ is an $\underline{B}$-module such that $\mathcal{F}(P, \alpha)$ is a finite $B$-module for all $\alpha : P = A[x_1, \ldots, x_n] \to B$, then the cohomology modules of $L\pi_!(\mathcal{F})$ are finite $B$-modules.


cotangent-lemma-cotangent-finite

Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map. Then $H^n(L_{B/A})$ is a finite $B$-module for all $n \in \mathbf{Z}$.


cotangent-lemma-O-homology-B-homology

Let $A \to B$ be a ring map. Let $\pi$, $\mathcal{O}$, $\underline{B}$ be as in (\ref{equation-pi}). For any $\mathcal{O}$-module $\mathcal{F}$ we have $$ L\pi_!(\mathcal{F}) = L\pi_!(Li^*\mathcal{F}) = L\pi_!(\mathcal{F} \otimes_\mathcal{O}^\mathbf{L} \underline{B}) $$ in $D(\textit{Ab})$.


cotangent-lemma-apply-O-B-comparison

Let $A \to B$ be a ring map. Let $\pi$, $\mathcal{O}$, $\underline{B}$ be as in (\ref{equation-pi}). We have $$ L\pi_!(\mathcal{O}) = L\pi_!(\underline{B}) = B \quad\text{and}\quad L_{B/A} = L\pi_!(\Omega_{\mathcal{O}/A} \otimes_\mathcal{O} \underline{B}) = L\pi_!(\Omega_{\mathcal{O}/A}) $$ in $D(\textit{Ab})$.


cotangent-lemma-special-case-triangle

Let $A \to B \to C$ be ring maps. If $B$ is a polynomial algebra over $A$, then there is a distinguished triangle  $L_{B/A} \otimes_B^\mathbf{L} C \to L_{C/A} \to L_{C/B} \to L_{B/A} \otimes_B^\mathbf{L} C[1]$ in $D(C)$.


cotangent-lemma-flat-base-change

Assume (\ref{equation-commutative-square}) induces a quasi-isomorphism $B \otimes_A^\mathbf{L} A' = B'$. Then, with notation as in (\ref{equation-double-square}) and $\mathcal{F}' \in \textit{Ab}(\mathcal{C}')$, we have $L\pi_!(g^{-1}\mathcal{F}') = L\pi'_!(\mathcal{F}')$.


cotangent-lemma-flat-base-change-cotangent-complex

If (\ref{equation-commutative-square}) induces a quasi-isomorphism $B \otimes_A^\mathbf{L} A' = B'$, then the functoriality map induces an isomorphism $$ L_{B/A} \otimes_B^\mathbf{L} B' \longrightarrow L_{B'/A'} $$


cotangent-lemma-cotangent-complex-product

Let $A \to B$ and $A \to C$ be ring maps. Then the map $L_{B \times C/A} \to L_{B/A} \oplus L_{C/A}$ is an isomorphism in $D(B \times C)$.


cotangent-lemma-triangle-ses

With notation as in (\ref{equation-three-maps}) set $$ \begin{matrix} \Omega_1 = \Omega_{\mathcal{O}/A} \otimes_\mathcal{O} \underline{B} \text{ on }\mathcal{C}_{B/A} \\ \Omega_2 = \Omega_{\mathcal{O}/A} \otimes_\mathcal{O} \underline{C} \text{ on }\mathcal{C}_{C/A} \\ \Omega_3 = \Omega_{\mathcal{O}/B} \otimes_\mathcal{O} \underline{C} \text{ on }\mathcal{C}_{C/B} \end{matrix} $$ Then we have a canonical short exact sequence of sheaves of $\underline{C}$-modules $$ 0 \to g_1^{-1}\Omega_1 \otimes_{\underline{B}} \underline{C} \to g_2^{-1}\Omega_2 \to g_3^{-1}\Omega_3 \to 0 $$ on $\mathcal{C}_{C/B/A}$.


cotangent-lemma-polynomial-on-top

With notation as in (\ref{equation-three-maps}) suppose that $C$ is a polynomial algebra over $B$. Then $L\pi_!(g_3^{-1}\mathcal{F}) = L\pi_{3, !}\mathcal{F} = \pi_{3, !}\mathcal{F}$ for any abelian sheaf $\mathcal{F}$ on $\mathcal{C}_{C/B}$


cotangent-lemma-triangle-compute-lower-shriek

With notation as in (\ref{equation-three-maps}) we have $Lg_{i, !} \circ g_i^{-1} = \text{id}$ for $i = 1, 2, 3$ and hence also $L\pi_! \circ g_i^{-1} = L\pi_{i, !}$ for $i = 1, 2, 3$.


cotangent-lemma-localize-at-bottom

Let $A \to A' \to B$ be ring maps such that $B = B \otimes_A^\mathbf{L} A'$. Then $L_{B/A} = L_{B/A'}$ in $D(B)$.


cotangent-lemma-derived-diagonal

Let $A \to B$ be a ring map such that $B = B \otimes_A^\mathbf{L} B$. Then $L_{B/A} = 0$ in $D(B)$.


cotangent-lemma-bootstrap

Let $A \to B$ be a ring map such that $\text{Tor}^A_i(B, B) = 0$ for $i > 0$ and such that $L_{B/B \otimes_A B} = 0$. Then $L_{B/A} = 0$ in $D(B)$.


cotangent-lemma-when-zero

The cotangent complex $L_{B/A}$ is zero in each of the following cases: \begin{enumerate} \item $A \to B$ and $B \otimes_A B \to B$ are flat, i.e., $A \to B$ is weakly \'etale (More on Algebra, Definition \ref{more-algebra-definition-weakly-etale}), \item $A \to B$ is a flat epimorphism of rings, \item $B = S^{-1}A$ for some multiplicative subset $S \subset A$, \item $A \to B$ is unramified and flat, \item $A \to B$ is \'etale, \item $A \to B$ is a filtered colimit of ring maps for which the cotangent complex vanishes, \item $B$ is a henselization of a local ring of $A$, \item $B$ is a strict henselization of a local ring of $A$, and \item add more here. \end{enumerate}


cotangent-lemma-localize-on-top

Let $A \to B \to C$ be ring maps such that $L_{C/B} = 0$. Then $L_{C/A} = L_{B/A} \otimes_B^\mathbf{L} C$.


cotangent-lemma-localize

Let $A \to B$ be ring maps and $S \subset A$, $T \subset B$ multiplicative subsets such that $S$ maps into $T$. Then $L_{T^{-1}B/S^{-1}A} = L_{B/A} \otimes_B T^{-1}B$ in $D(T^{-1}B)$.


cotangent-lemma-cotangent-complex-henselization

Let $A \to B$ be a local ring homomorphism of local rings. Let $A^h \to B^h$, resp.\ $A^{sh} \to B^{sh}$ be the induced maps of henselizations, resp.\ strict henselizations. Then $$ L_{B^h/A^h} = L_{B^h/A} = L_{B/A} \otimes_B^\mathbf{L} B^h \quad\text{resp.}\quad L_{B^{sh}/A^{sh}} = L_{B^{sh}/A} = L_{B/A} \otimes_B^\mathbf{L} B^{sh} $$ in $D(B^h)$, resp.\ $D(B^{sh})$.


cotangent-lemma-when-projective

If $A \to B$ is a smooth ring map, then $L_{B/A} = \Omega_{B/A}[0]$.


cotangent-lemma-frobenius-homotopy

Let $A \to B$ be a ring map with $p = 0$ in $A$. Let $P_\bullet$ be the standard resolution of $B$ over $A$. The map $P_\bullet \to P_\bullet$ induced by the diagram $$ \xymatrix{ B \ar[r]_{F_B} & B \\ A \ar[u] \ar[r]^{F_A} & A \ar[u] } $$ discussed in Section \ref{section-functoriality} is homotopic to the Frobenius endomorphism $P_\bullet \to P_\bullet$ given by Frobenius on each $P_n$.


cotangent-lemma-frobenius-acts-as-zero

Let $p$ be a prime number. Let $A \to B$ be a ring homomorphism and assume that $p = 0$ in $A$. The map $L_{B/A} \to L_{B/A}$ of Section \ref{section-functoriality} induced by the Frobenius maps $F_A$ and $F_B$ is homotopic to zero.


cotangent-lemma-perfect-zero

Let $p$ be a prime number. Let $A \to B$ be a ring homomorphism and assume that $p = 0$ in $A$. If $A$ and $B$ are perfect, then $L_{B/A}$ is zero in $D(B)$.


cotangent-lemma-surjection

\begin{slogan} The cohomology of the cotangent complex of a surjective ring map is trivial in degree zero; it is the kernel modulo its square in degree $-1$. \end{slogan} Let $A \to B$ be a surjective ring map with kernel $I$. Then $H^0(L_{B/A}) = 0$ and $H^{-1}(L_{B/A}) = I/I^2$. This isomorphism comes from the map (\ref{equation-comparison-map}) for the object $(A \to B)$ of $\mathcal{C}_{B/A}$.


cotangent-lemma-relation-with-naive-cotangent-complex

Let $A \to B$ be a ring map. Then $\tau_{\geq -1}L_{B/A}$ is canonically quasi-isomorphic to the naive cotangent complex.


cotangent-lemma-vanishing-symmetric-powers

Notation and assumptions as in Cohomology on Sites, Example \ref{sites-cohomology-example-category-to-point}. Assume $\mathcal{C}$ has a cosimplicial object as in Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}. Let $\mathcal{F}$ be a flat $\underline{B}$-module such that $H_0(\mathcal{C}, \mathcal{F}) = 0$. Then $H_l(\mathcal{C}, \text{Sym}_{\underline{B}}^k(\mathcal{F})) = 0$ for $l < k$.


cotangent-lemma-map-tors-zero

Let $A$ be a ring. Let $P = A[E]$ be a polynomial ring. Set $I = (e; e \in E) \subset P$. The maps $\text{Tor}_i^P(A, I^{n + 1}) \to \text{Tor}_i^P(A, I^n)$ are zero for all $i$ and $n$.


cotangent-lemma-polynomial-ring-unique

Let $P = A[S]$ be a polynomial ring over $A$. Let $M$ be a $(P, P)$-bimodule over $A$. Given $m_s \in M$ for $s \in S$, there exists a unique $A$-biderivation $\lambda : P \to M$ mapping $s$ to $m_s$ for $s \in S$.


cotangent-lemma-compare-higher

In the situation above denote $L$ the complex (\ref{equation-lichtenbaum-schlessinger}). There is a canonical map $L_{B/A} \to L$ in $D(A)$ which induces an isomorphism $\tau_{\geq -2}L_{B/A} \to L$ in $D(B)$.


cotangent-lemma-special-case

Let $A = \mathbf{Z}[x_1, \ldots, x_n] \to B = \mathbf{Z}$ be the ring map which sends $x_i$ to $0$ for $i = 1, \ldots, n$. Let $I = (x_1, \ldots, x_n) \subset A$. Then $L_{B/A}$ is quasi-isomorphic to $I/I^2[1]$.


cotangent-lemma-mod-Koszul-regular-ideal

Let $A \to B$ be a surjective ring map whose kernel $I$ is Koszul. Then $L_{B/A}$ is quasi-isomorphic to $I/I^2[1]$.


cotangent-lemma-tensor-product-tor-independent

If $A$ and $B$ are Tor independent $R$-algebras, then the object $E$ in (\ref{equation-tensor-product}) is zero. In this case we have $$ L_{A \otimes_R B/R} = L_{A/R} \otimes_A^\mathbf{L} (A \otimes_R B) \oplus L_{B/R} \otimes_B^\mathbf{L} (A \otimes_R B) $$ which is represented by the complex $L_{A/R} \otimes_R B \oplus L_{B/R} \otimes_R A $ of $A \otimes_R B$-modules.


cotangent-lemma-tensor-product

Let $R$ be a ring and let $A$, $B$ be $R$-algebras. The object $E$ in (\ref{equation-tensor-product}) satisfies $$ H^i(E) = \left\{ \begin{matrix} 0 & \text{if} & i \geq -1 \\ \text{Tor}_1^R(A, B) & \text{if} & i = -2 \end{matrix} \right. $$


cotangent-lemma-pullback-cotangent-morphism-topoi

Let $f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ be a morphism of topoi. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. Then $f^{-1}L_{\mathcal{B}/\mathcal{A}} = L_{f^{-1}\mathcal{B}/f^{-1}\mathcal{A}}$.


cotangent-lemma-compute-L-morphism-sheaves-rings

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. Then $H^i(L_{\mathcal{B}/\mathcal{A}})$ is the sheaf associated to the presheaf $U \mapsto H^i(L_{\mathcal{B}(U)/\mathcal{A}(U)})$.


cotangent-lemma-H0-L-morphism-sheaves-rings

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. Then $H^0(L_{\mathcal{B}/\mathcal{A}}) = \Omega_{\mathcal{B}/\mathcal{A}}$.


cotangent-lemma-compute-L-product-sheaves-rings

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ and $\mathcal{A} \to \mathcal{B}'$ be homomorphisms of sheaves of rings on $\mathcal{C}$. Then $$ L_{\mathcal{B} \times \mathcal{B}'/\mathcal{A}} \longrightarrow L_{\mathcal{B}/\mathcal{A}} \oplus L_{\mathcal{B}'/\mathcal{A}} $$ is an isomorphism in $D(\mathcal{B} \times \mathcal{B}')$.


cotangent-lemma-triangle-sheaves-rings

Let $\mathcal{D}$ be a site. Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ be homomorphisms of sheaves of rings on $\mathcal{D}$. There is a canonical distinguished triangle $$ L_{\mathcal{B}/\mathcal{A}} \otimes_\mathcal{B}^\mathbf{L} \mathcal{C} \to L_{\mathcal{C}/\mathcal{A}} \to L_{\mathcal{C}/\mathcal{B}} \to L_{\mathcal{B}/\mathcal{A}} \otimes_\mathcal{B}^\mathbf{L} \mathcal{C}[1] $$ in $D(\mathcal{C})$.


cotangent-lemma-stalk-cotangent-complex

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. If $p$ is a point of $\mathcal{C}$, then $(L_{\mathcal{B}/\mathcal{A}})_p = L_{\mathcal{B}_p/\mathcal{A}_p}$.


cotangent-lemma-compare-cotangent-complex-with-naive

Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. There is a canonical map $L_{\mathcal{B}/\mathcal{A}} \to \NL_{\mathcal{B}/\mathcal{A}}$ which identifies the naive cotangent complex with the truncation $\tau_{\geq -1}L_{\mathcal{B}/\mathcal{A}}$.


cotangent-lemma-H0-L-morphism-ringed-spaces

Let $f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$ be a morphism of ringed spaces. Then $H^0(L_{X/S}) = \Omega_{X/S}$.


cotangent-lemma-triangle-ringed-spaces

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Then there is a canonical distinguished triangle $$ Lf^* L_{Y/Z} \to L_{X/Z} \to L_{X/Y} \to Lf^*L_{Y/Z}[1] $$ in $D(\mathcal{O}_X)$.


cotangent-lemma-compare-cotangent-complex-with-naive-ringed-spaces

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. There is a canonical map $L_{X/Y} \to \NL_{X/Y}$ which identifies the naive cotangent complex with the truncation $\tau_{\geq -1}L_{X/Y}$.


cotangent-lemma-H0-L-morphism-ringed-topoi

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$ be a morphism of ringed topoi. Then $H^0(L_f) = \Omega_f$.


cotangent-lemma-triangle-ringed-topoi

Let $f : (\Sh(\mathcal{C}_1), \mathcal{O}_1) \to (\Sh(\mathcal{C}_2), \mathcal{O}_2)$ and $g : (\Sh(\mathcal{C}_2), \mathcal{O}_2) \to (\Sh(\mathcal{C}_3), \mathcal{O}_3)$ be morphisms of ringed topoi. Then there is a canonical distinguished triangle $$ Lf^* L_g \to L_{g \circ f} \to L_f \to Lf^*L_g[1] $$ in $D(\mathcal{O}_1)$.


cotangent-lemma-compare-cotangent-complex-with-naive-ringed-topoi

Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$ be a morphism of ringed topoi. There is a canonical map $L_f \to \NL_f$ which identifies the naive cotangent complex with the truncation $\tau_{\geq -1}L_f$.


cotangent-lemma-morphism-affine-schemes

Let $f : X \to Y$ be a morphism of schemes. Let $U = \Spec(A) \subset X$ and $V = \Spec(B) \subset Y$ be affine opens such that $f(U) \subset V$. There is a canonical map $$ \widetilde{L_{B/A}} \longrightarrow L_{X/Y}|_U $$ of complexes which is an isomorphism in $D(\mathcal{O}_U)$. This map is compatible with restricting to smaller affine opens of $X$ and $Y$.


cotangent-lemma-scheme-over-ring

Let $\Lambda$ be a ring. Let $X$ be a scheme over $\Lambda$. Then $$ L_{X/\Spec(\Lambda)} = L_{\mathcal{O}_X/\underline{\Lambda}} $$ where $\underline{\Lambda}$ is the constant sheaf with value $\Lambda$ on $X$.


cotangent-lemma-category-fibred

In the situation above the category $\mathcal{C}_{X/\Lambda}$ is fibred over $X_{Zar}$.


cotangent-lemma-compare-spaces-schemes

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ and $Y$ representable by schemes $X_0$ and $Y_0$. Then there is a canonical identification $L_{X/Y} = \epsilon^*L_{X_0/Y_0}$ in $D(\mathcal{O}_X)$ where $\epsilon$ is as in Derived Categories of Spaces, Section \ref{spaces-perfect-section-derived-quasi-coherent-etale} and $L_{X_0/Y_0}$ is as in  Definition \ref{definition-cotangent-morphism-schemes}.


cotangent-lemma-space-over-ring

Let $\Lambda$ be a ring. Let $X$ be an algebraic space over $\Lambda$. Then $$ L_{X/\Spec(\Lambda)} = L_{\mathcal{O}_X/\underline{\Lambda}} $$ where $\underline{\Lambda}$ is the constant sheaf with value $\Lambda$ on $X_\etale$.


cotangent-lemma-category-fibred-space

In the situation above the category $\mathcal{C}_{X/\Lambda}$ is fibred over $X_\etale$.


cotangent-lemma-fibre-product

Let $S$ be a scheme. Let $X \to B$ and $Y \to B$ be morphisms of algebraic spaces over $S$. The object $E$ in (\ref{equation-fibre-product}) satisfies $H^i(E) = 0$ for $i = 0, -1$ and for a geometric point $(\overline{x}, \overline{y}) : \Spec(k) \to X \times_B Y$ we have $$ H^{-2}(E)_{(\overline{x}, \overline{y})} = \text{Tor}_1^R(A, B) \otimes_{A \otimes_R B} C $$ where $R = \mathcal{O}_{B, \overline{b}}$, $A = \mathcal{O}_{X, \overline{x}}$, $B = \mathcal{O}_{Y, \overline{y}}$, and $C = \mathcal{O}_{X \times_B Y, (\overline{x}, \overline{y})}$.


cotangent-proposition-polynomial

Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map. There exists a simplicial $A$-algebra $P_\bullet$ with an augmentation $\epsilon : P_\bullet \to B$ such that each $P_n$ is a polynomial algebra of finite type over $A$ and such that $\epsilon$ is a trivial Kan fibration of simplicial sets.


cotangent-proposition-triangle

Let $A \to B \to C$ be ring maps. There is a canonical distinguished triangle $$ L_{B/A} \otimes_B^\mathbf{L} C \to L_{C/A} \to L_{C/B} \to L_{B/A} \otimes_B^\mathbf{L} C[1] $$ in $D(C)$.


spaces-cohomology-theorem-formal-functions

In Situation \ref{situation-formal-functions}. Fix $p \geq 0$. The system of maps $$ H^p(X, \mathcal{F})/I^nH^p(X, \mathcal{F}) \longrightarrow H^p(X, \mathcal{F}/I^n\mathcal{F}) $$ define an isomorphism of limits $$ H^p(X, \mathcal{F})^\wedge \longrightarrow \lim_n H^p(X, \mathcal{F}/I^n\mathcal{F}) $$ where the left hand side is the completion of the $A$-module $H^p(X, \mathcal{F})$ with respect to the ideal $I$, see Algebra, Section \ref{algebra-section-completion}. Moreover, this is in fact a homeomorphism for the limit topologies.


spaces-cohomology-lemma-higher-direct-image

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is quasi-compact and quasi-separated, then $R^if_*$ transforms quasi-coherent $\mathcal{O}_X$-modules into quasi-coherent $\mathcal{O}_Y$-modules.


spaces-cohomology-lemma-quasi-coherence-higher-direct-images-application

Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$. For any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ and any affine object $V$ of $Y_\etale$ we have $$ H^q(V \times_Y X, \mathcal{F}) = H^0(V, R^qf_*\mathcal{F}) $$ for all $q \in \mathbf{Z}$.


spaces-cohomology-lemma-finite-higher-direct-image-zero

Let $S$ be a scheme. Let $f : X \to Y$ be an integral (for example finite) morphism of algebraic spaces. Then $f_* : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$ is an exact functor and $R^pf_* = 0$ for $p > 0$.


spaces-cohomology-lemma-stalk-push-finite

Let $S$ be a scheme. Let $f : X \to Y$ be a finite morphism of algebraic spaces over $S$. Let $\overline{y}$ be a geometric point of $Y$ with lifts $\overline{x}_1, \ldots, \overline{x}_n$ in $X$. Then $$ (f_*\mathcal{F})_{\overline{y}} = \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{F}_{\overline{x}_i} $$ for any sheaf $\mathcal{F}$ on $X_\etale$.


spaces-cohomology-lemma-finite-rings

Let $S$ be a scheme. Let $\pi : X \to Y$ be a finite morphism of algebraic spaces over $S$. Let $\mathcal{A}$ be a sheaf of rings on $X_\etale$. Let $\mathcal{B}$ be a sheaf of rings on $Y_\etale$. Let $\varphi : \mathcal{B} \to \pi_*\mathcal{A}$ be a homomorphism of sheaves of rings so that we obtain a morphism of ringed topoi $$ f = (\pi, \varphi) : (\Sh(X_\etale), \mathcal{A}) \longrightarrow (\Sh(Y_\etale), \mathcal{B}). $$ For a sheaf of $\mathcal{A}$-modules $\mathcal{F}$ and a sheaf of $\mathcal{B}$-modules $\mathcal{G}$ the canonical map $$ \mathcal{G} \otimes_\mathcal{B} f_*\mathcal{F} \longrightarrow f_*(f^*\mathcal{G} \otimes_\mathcal{A} \mathcal{F}). $$ is an isomorphism.


spaces-cohomology-lemma-projection-formula-finite

With $S$, $X$, $Y$, $\pi$, $\mathcal{A}$, $\mathcal{B}$, $\varphi$, and $f$ as in Lemma \ref{lemma-finite-rings} we have $$ K \otimes_\mathcal{B}^\mathbf{L} Rf_*M = Rf_*(Lf^*K \otimes_\mathcal{A}^\mathbf{L} M) $$ in $D(\mathcal{B})$ for any $K \in D(\mathcal{B})$ and $M \in D(\mathcal{A})$.


spaces-cohomology-lemma-colimits

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is quasi-compact and quasi-separated, then $$ \colim_i H^p(X, \mathcal{F}_i) \longrightarrow H^p(X, \colim_i \mathcal{F}_i) $$ is an isomorphism for every filtered diagram of abelian sheaves on $X_\etale$.


spaces-cohomology-lemma-colimit-cohomology

\begin{slogan} Higher direct images of qcqs morphisms commute with filtered colimits of sheaves. \end{slogan} Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\mathcal{F} = \colim \mathcal{F}_i$ be a filtered colimit of abelian sheaves on $X_\etale$. Then for any $p \geq 0$ we have $$ R^pf_*\mathcal{F} = \colim R^pf_*\mathcal{F}_i. $$


spaces-cohomology-lemma-finite-presentation-quasi-compact-colimit

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $I$ be a directed set and let $(\mathcal{F}_i, \varphi_{ii'})$ be a system over $I$ of quasi-coherent $\mathcal{O}_X$-modules. Let $\mathcal{G}$ be an $\mathcal{O}_X$-module of finite presentation. Then we have $$ \colim_i \Hom_X(\mathcal{G}, \mathcal{F}_i) = \Hom_X(\mathcal{G}, \colim_i \mathcal{F}_i). $$


spaces-cohomology-lemma-product-is-tensor-product

Let $S$ be a scheme. Let $f_i : U_i \to X$ be \'etale morphisms of algebraic spaces over $S$. Then there are isomorphisms $$ f_{1, !}\underline{\mathbf{Z}} \otimes_{\mathbf{Z}} f_{2, !}\underline{\mathbf{Z}} \longrightarrow f_{12, !}\underline{\mathbf{Z}} $$ where $f_{12} : U_1 \times_X U_2 \to X$ is the structure morphism and $$ (f_1 \amalg f_2)_! \underline{\mathbf{Z}} \longrightarrow f_{1, !}\underline{\mathbf{Z}} \oplus f_{2, !}\underline{\mathbf{Z}} $$


spaces-cohomology-lemma-alternating-cech-to-cohomology

Let $S$ be a scheme. Let $f : U \to X$ be a surjective \'etale morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be an object of $\textit{Ab}(X_\etale)$. There exists a canonical map $$ \check{\mathcal{C}}^\bullet_{alt}(f, \mathcal{F}) \longrightarrow R\Gamma(X, \mathcal{F}) $$ in $D(\textit{Ab})$. Moreover, there is a spectral sequence with $E_1$-page $$ E_1^{p, q} = \Ext_{\textit{Ab}(X_\etale)}^q(K^p, \mathcal{F}) $$ converging to $H^{p + q}(X, \mathcal{F})$ where $K^p = \wedge^{p + 1}f_!\underline{\mathbf{Z}}$.


spaces-cohomology-lemma-compute

Let $S$ be a scheme. Let $f : U \to X$ be a surjective, \'etale, and separated morphism of algebraic spaces over $S$. For $p \geq 0$ set $$ W_p = U \times_X \ldots \times_X U \setminus \text{all diagonals} $$ where the fibre product has $p + 1$ factors. There is a free action of $S_{p + 1}$ on $W_p$ over $X$ and $$ \Hom(K^p, \mathcal{F}) = S_{p + 1}\text{-anti-invariant elements of } \mathcal{F}(W_p) $$ functorially in $\mathcal{F}$ where $K^p = \wedge^{p + 1}f_!\underline{\mathbf{Z}}$.


spaces-cohomology-lemma-twist

Let $S$ be a scheme. Let $W$ be an algebraic space over $S$. Let $G$ be a finite group acting freely on $W$. Let $U = W/G$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quotient}. Let $\chi : G \to \{+1, -1\}$ be a character. Then there exists a rank 1 locally free sheaf of $\mathbf{Z}$-modules $\underline{\mathbf{Z}}(\chi)$ on $U_\etale$ such that for every abelian sheaf $\mathcal{F}$ on $U_\etale$ we have $$ H^0(W, \mathcal{F}|_W)^\chi = H^0(U, \mathcal{F} \otimes_{\mathbf{Z}} \underline{\mathbf{Z}}(\chi)) $$


spaces-cohomology-lemma-alternating-spectral-sequence

Let $S$ be a scheme. Let $f : U \to X$ be a surjective, \'etale, and separated morphism of algebraic spaces over $S$. For $p \geq 0$ set $$ W_p = U \times_X \ldots \times_X U \setminus \text{all diagonals} $$ (with $p + 1$ factors) as in Lemma \ref{lemma-compute}. Let $\chi_p : S_{p + 1} \to \{+1, -1\}$ be the sign character. Let $U_p = W_p/S_{p + 1}$ and $\underline{\mathbf{Z}}(\chi_p)$ be as in Lemma \ref{lemma-twist}. Then the spectral sequence of Lemma \ref{lemma-alternating-cech-to-cohomology} has $E_1$-page $$ E_1^{p, q} = H^q(U_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)) $$ and converges to $H^{p + q}(X, \mathcal{F})$.


spaces-cohomology-lemma-quasi-coherent-twist

With $S$, $W$, $G$, $U$, $\chi$ as in Lemma \ref{lemma-twist}. If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_U$-module, then so is $\mathcal{F} \otimes_{\mathbf{Z}} \underline{\mathbf{Z}}(\chi)$.


spaces-cohomology-lemma-vanishing-quasi-separated

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and quasi-separated. Then we can choose \begin{enumerate} \item an affine scheme $U$, \item a surjective \'etale morphism $f : U \to X$, \item an integer $d$ bounding the degrees of the fibres of $U \to X$, \item for every $p = 0, 1, \ldots, d$ a surjective \'etale morphism $V_p \to U_p$ from an affine scheme $V_p$ where $U_p$ is as in Lemma \ref{lemma-alternating-spectral-sequence}, and \item an integer $d_p$ bounding the degree of the fibres of $V_p \to U_p$. \end{enumerate} Moreover, whenever we have (1) -- (5), then for any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ we have $H^q(X, \mathcal{F}) = 0$ for $q \geq \max(d_p + p)$.


spaces-cohomology-lemma-vanishing-higher-direct-images

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that \begin{enumerate} \item $f$ is quasi-compact and quasi-separated, and \item $Y$ is quasi-compact. \end{enumerate} Then there exists an integer $n(X \to Y)$ such that for any algebraic space $Y'$, any morphism $Y' \to Y$ and any quasi-coherent sheaf $\mathcal{F}'$ on $X' = Y' \times_Y X$ the higher direct images $R^if'_*\mathcal{F}'$ are zero for $i \geq n(X \to Y)$.


spaces-cohomology-lemma-affine-vanishing-higher-direct-images

Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Then $R^if_*\mathcal{F} = 0$ for $i > 0$ and any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$.


spaces-cohomology-lemma-relative-affine-cohomology

Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $H^i(X, \mathcal{F}) = H^i(Y, f_*\mathcal{F})$ for all $i \geq 0$.


spaces-cohomology-lemma-etale-localization-sheaf-with-support

Let $S$ be a scheme. Let $f : X \to Y$ be an \'etale morphism of algebraic spaces over $S$. Let $Z \subset Y$ be a closed subspace such that $f^{-1}(Z) \to Z$ is an isomorphism of algebraic spaces. Let $\mathcal{F}$ be an abelian sheaf on $X$. Then $$ \mathcal{H}^q_Z(\mathcal{F}) = \mathcal{H}^q_{f^{-1}(Z)}(f^{-1}\mathcal{F}) $$ as abelian sheaves on $Z = f^{-1}(Z)$ and we have $H^q_Z(Y, \mathcal{F}) = H^q_{f^{-1}(Z)}(X, f^{-1}\mathcal{F})$.


spaces-cohomology-lemma-complexes-with-support-on-closed

Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. The map $Ri_* = i_* : D(Z_\etale) \to D(X_\etale)$ induces an equivalence $D(Z_\etale) \to D_{|Z|}(X_\etale)$ with quasi-inverse $$ i^{-1}|_{D_Z(X_\etale)} = R\mathcal{H}_Z|_{D_{|Z|}(X_\etale)} $$


spaces-cohomology-lemma-vanishing-above-dimension

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Assume $\dim(X) \leq d$ for some integer $d$. Let $\mathcal{F}$ be a quasi-coherent sheaf $\mathcal{F}$ on $X$. \begin{enumerate} \item $H^q(X, \mathcal{F}) = 0$ for $q > d$, \item $H^d(X, \mathcal{F}) \to H^d(U, \mathcal{F})$ is surjective for any quasi-compact open $U \subset X$, \item $H^q_Z(X, \mathcal{F}) = 0$ for $q > d$ for any closed subspace $Z \subset X$ whose complement is quasi-compact. \end{enumerate}


spaces-cohomology-lemma-affine-base-change

Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. In this case $f_*\mathcal{F} \cong Rf_*\mathcal{F}$ is a quasi-coherent sheaf, and for every diagram (\ref{equation-base-change-diagram}) we have $$ g^*f_*\mathcal{F} = f'_*(g')^*\mathcal{F}. $$


spaces-cohomology-lemma-flat-base-change-cohomology

Let $S$ be a scheme. Consider a cartesian diagram of algebraic spaces $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module with pullback $\mathcal{F}' = (g')^*\mathcal{F}$. Assume that $g$ is flat and that $f$ is quasi-compact and quasi-separated. For any $i \geq 0$ \begin{enumerate} \item the base change map of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-base-change-map-flat-case} is an isomorphism $$ g^*R^if_*\mathcal{F} \longrightarrow R^if'_*\mathcal{F}', $$ \item if $Y = \Spec(A)$ and $Y' = \Spec(B)$, then $H^i(X, \mathcal{F}) \otimes_A B = H^i(X', \mathcal{F}')$. \end{enumerate}


spaces-cohomology-lemma-coherent-Noetherian

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is coherent, \item $\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_X$-module, \item $\mathcal{F}$ is a finitely presented $\mathcal{O}_X$-module, \item for any \'etale morphism $\varphi : U \to X$ where $U$ is a scheme the pullback $\varphi^*\mathcal{F}$ is a coherent module on $U$, and \item there exists a surjective \'etale morphism $\varphi : U \to X$ where $U$ is a scheme such that the pullback $\varphi^*\mathcal{F}$ is a coherent module on $U$. \end{enumerate} In particular $\mathcal{O}_X$ is coherent, any invertible $\mathcal{O}_X$-module is coherent, and more generally any finite locally free $\mathcal{O}_X$-module is coherent.


spaces-cohomology-lemma-coherent-abelian-Noetherian

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. The category of coherent $\mathcal{O}_X$-modules is abelian. More precisely, the kernel and cokernel of a map of coherent $\mathcal{O}_X$-modules are coherent. Any extension of coherent sheaves is coherent.


spaces-cohomology-lemma-coherent-Noetherian-quasi-coherent-sub-quotient

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Any quasi-coherent submodule of $\mathcal{F}$ is coherent. Any quasi-coherent quotient module of $\mathcal{F}$ is coherent.


spaces-cohomology-lemma-coherent-support-closed

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{F}$ and $\mathcal{G}$ the quasi-coherent $\mathcal{O}_Z$-module such that $i_*\mathcal{G} = \mathcal{F}$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-scheme-theoretic-support}. Then $\mathcal{G}$ is a coherent $\mathcal{O}_Z$-module.


spaces-cohomology-lemma-i-star-equivalence

Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of locally Noetherian algebraic spaces over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent sheaf of ideals cutting out $Z$. The functor $i_*$ induces an equivalence between the category of coherent $\mathcal{O}_X$-modules annihilated by $\mathcal{I}$ and the category of coherent $\mathcal{O}_Z$-modules.


spaces-cohomology-lemma-finite-pushforward-coherent

Let $S$ be a scheme. Let $f : X \to Y$ be a finite morphism of algebraic spaces over $S$ with $Y$ locally Noetherian. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Assume $f$ is finite and $Y$ locally Noetherian. Then $R^pf_*\mathcal{F} = 0$ for $p > 0$ and $f_*\mathcal{F}$ is coherent.


spaces-cohomology-lemma-acc-coherent

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The ascending chain condition holds for quasi-coherent submodules of $\mathcal{F}$. In other words, given any sequence $$ \mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots \subset \mathcal{F} $$ of quasi-coherent submodules, then $\mathcal{F}_n = \mathcal{F}_{n + 1} = \ldots $ for some $n \geq 0$.


spaces-cohomology-lemma-power-ideal-kills-sheaf

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals corresponding to a closed subspace $Z \subset X$. Then there is some $n \geq 0$ such that $\mathcal{I}^n\mathcal{F} = 0$ if and only if $\text{Supp}(\mathcal{F}) \subset Z$ (set theoretically).


spaces-cohomology-lemma-Artin-Rees

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $\mathcal{G} \subset \mathcal{F}$ be a quasi-coherent subsheaf. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Then there exists a $c \geq 0$ such that for all $n \geq c$ we have $$ \mathcal{I}^{n - c}(\mathcal{I}^c\mathcal{F} \cap \mathcal{G}) = \mathcal{I}^n\mathcal{F} \cap \mathcal{G} $$


spaces-cohomology-lemma-homs-over-open

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G}$ be a coherent $\mathcal{O}_X$-module. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Denote $Z \subset X$ the corresponding closed subspace and set $U = X \setminus Z$. There is a canonical isomorphism $$ \colim_n \Hom_{\mathcal{O}_X}(\mathcal{I}^n\mathcal{G}, \mathcal{F}) \longrightarrow \Hom_{\mathcal{O}_U}(\mathcal{G}|_U, \mathcal{F}|_U). $$ In particular we have an isomorphism $$ \colim_n \Hom_{\mathcal{O}_X}(\mathcal{I}^n, \mathcal{F}) \longrightarrow \Gamma(U, \mathcal{F}). $$


spaces-cohomology-lemma-prepare-filter-support

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Suppose that $\text{Supp}(\mathcal{F}) = Z \cup Z'$ with $Z$, $Z'$ closed. Then there exists a short exact sequence of coherent sheaves $$ 0 \to \mathcal{G}' \to \mathcal{F} \to \mathcal{G} \to 0 $$ with $\text{Supp}(\mathcal{G}') \subset Z'$ and $\text{Supp}(\mathcal{G}) \subset Z$.


spaces-cohomology-lemma-prepare-filter-irreducible

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Assume that the scheme theoretic support of $\mathcal{F}$ is a reduced $Z \subset X$ with $|Z|$ irreducible. Then there exist an integer $r > 0$, a nonzero sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Z$, and an injective map of coherent sheaves $$ i_*\left(\mathcal{I}^{\oplus r}\right) \to \mathcal{F} $$ whose cokernel is supported on a proper closed subspace of $Z$.


spaces-cohomology-lemma-coherent-filter

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent sheaf on $X$. There exists a filtration $$ 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F} $$ by coherent subsheaves such that for each $j = 1, \ldots, m$ there exists a reduced closed subspace $Z_j \subset X$ with $|Z_j|$ irreducible and a sheaf of ideals $\mathcal{I}_j \subset \mathcal{O}_{Z_j}$ such that $$ \mathcal{F}_j/\mathcal{F}_{j - 1} \cong (Z_j \to X)_* \mathcal{I}_j $$


spaces-cohomology-lemma-property-initial

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{P}$ be a property of coherent sheaves on $X$. Assume \begin{enumerate} \item For any short exact sequence of coherent sheaves $$ 0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0 $$ if $\mathcal{F}_i$, $i = 1, 2$ have property $\mathcal{P}$ then so does $\mathcal{F}$. \item For every reduced closed subspace $Z \subset X$ with $|Z|$ irreducible and every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Z$ we have $\mathcal{P}$ for $i_*\mathcal{I}$. \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.


spaces-cohomology-lemma-property-higher-rank-cohomological

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{P}$ be a property of coherent sheaves on $X$. Assume \begin{enumerate} \item For any short exact sequence of coherent sheaves $$ 0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0 $$ if $\mathcal{F}_i$, $i = 1, 2$ have property $\mathcal{P}$ then so does $\mathcal{F}$. \item If $\mathcal{P}$ holds for $\mathcal{F}^{\oplus r}$ for some $r \geq 1$, then it holds for $\mathcal{F}$. \item For every reduced closed subspace $i : Z \to X$ with $|Z|$ irreducible there exists a coherent sheaf $\mathcal{G}$ on $Z$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z$, \item for every nonzero quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_Z$ there exists a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{I}\mathcal{G}$ such that $\text{Supp}(\mathcal{G}/\mathcal{G}')$ is proper closed in $|Z|$ and such that $\mathcal{P}$ holds for $i_*\mathcal{G}'$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.


spaces-cohomology-lemma-property-higher-rank-cohomological-variant

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{P}$ be a property of coherent sheaves on $X$. Assume \begin{enumerate} \item For any short exact sequence of coherent sheaves on $X$ if two out of three have property $\mathcal{P}$ so does the third. \item If $\mathcal{P}$ holds  for $\mathcal{F}^{\oplus r}$ for some $r \geq 1$, then it holds for $\mathcal{F}$. \item For every reduced closed subspace $i : Z \to X$ with $|Z|$ irreducible there exists a coherent sheaf $\mathcal{G}$ on $X$ whose scheme theoretic support is $Z$ such that $\mathcal{P}$ holds for $\mathcal{G}$. \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.


spaces-cohomology-lemma-directed-colimit-coherent

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Every quasi-coherent $\mathcal{O}_X$-module is the filtered colimit of its coherent submodules.


spaces-cohomology-lemma-direct-colimit-finite-presentation

Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$ with $Y$ Noetherian. Then every quasi-coherent $\mathcal{O}_X$-module is a filtered colimit of finitely presented $\mathcal{O}_X$-modules.


spaces-cohomology-lemma-vanishing-compute

In Situation \ref{situation-vanishing} for an $A$-module $M$ we have $p_*(M \otimes_A \mathcal{O}_X) = \widetilde{M}$ and $\Gamma(X, M \otimes_A \mathcal{O}_X) = M$.


spaces-cohomology-lemma-vanishing-base-change

In Situation \ref{situation-vanishing}. \begin{enumerate} \item Given an affine morphism $X' \to X$ of algebraic spaces, we have $H^1(X', \mathcal{F}') = 0$ for every quasi-coherent $\mathcal{O}_{X'}$-module $\mathcal{F}'$. \item Given an $A$-algebra $A'$ setting $X' = X \times_{\Spec(A)} \Spec(A')$ the morphism $X' \to X$ is affine and $\Gamma(X', \mathcal{O}_{X'}) = A'$. \end{enumerate}


spaces-cohomology-lemma-vanishing-separate-closed

In Situation \ref{situation-vanishing}. Let $Z_0, Z_1 \subset |X|$ be disjoint closed subsets. Then there exists an $a \in A$ such that $Z_0 \subset V(a)$ and $Z_1 \subset V(a - 1)$.


spaces-cohomology-lemma-vanishing-injective

In Situation \ref{situation-vanishing} the morphism $p : X \to \Spec(A)$ is universally injective.


spaces-cohomology-lemma-vanishing-separated

In Situation \ref{situation-vanishing} the morphism $p : X \to \Spec(A)$ is separated.


spaces-cohomology-lemma-Noetherian-h1-zero-invertible

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume that for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ there exists an $n \geq 1$ such that $H^1(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = 0$. Then $X$ is a scheme and $\mathcal{L}$ is ample on $X$.


spaces-cohomology-lemma-finite-morphism-Noetherian

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Assume $f$ is finite, surjective and $X$ locally Noetherian. Let $i : Z \to X$ be a closed immersion. Denote $i' : Z' \to Y$ the inverse image of $Z$ (Morphisms of Spaces, Section \ref{spaces-morphisms-section-closed-immersions}) and $f' : Z' \to Z$ the induced morphism. Then $\mathcal{G} = f'_*\mathcal{O}_{Z'}$ is a coherent $\mathcal{O}_Z$-module whose support is $Z$.


spaces-cohomology-lemma-affine-morphism-projection-ideal

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $Y$. Let $\mathcal{I}$ be a quasi-coherent sheaf of ideals on $X$. If $f$ is affine then $\mathcal{I}f_*\mathcal{F} = f_*(f^{-1}\mathcal{I}\mathcal{F})$ (with notation as explained in the proof).


spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ finite, \item $f$ surjective, \item $Y$ affine, and \item $X$ Noetherian. \end{enumerate} Then $X$ is affine.


spaces-cohomology-lemma-weak-chow

Let $A$ be a ring. Let $X$ be an algebraic space over $\Spec(A)$ whose structure morphism $X \to \Spec(A)$ is separated of finite type. Then there exists a proper surjective morphism $X' \to X$ where $X'$ is a scheme which is H-quasi-projective over $\Spec(A)$.


spaces-cohomology-lemma-check-separated-dvr

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $Y$ is locally Noetherian, \item $f$ is locally of finite type and quasi-separated, \item for every commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & Y } $$ where $A$ is a discrete valuation ring and $K$ its fraction field, there is at most one dotted arrow making the diagram commute. \end{enumerate} Then $f$ is separated.


spaces-cohomology-lemma-check-proper-dvr

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $Y$ is locally Noetherian, \item $f$ is of finite type and quasi-separated, \item for every commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & Y } $$ where $A$ is a discrete valuation ring and $K$ its fraction field, there is a unique dotted arrow making the diagram commute. \end{enumerate} Then $f$ is proper.


spaces-cohomology-lemma-kill-by-twisting

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ X \ar[r]_i \ar[rd]_f & \mathbf{P}^n_Y \ar[d] \\ & Y } $$ of algebraic spaces over $S$. Assume $i$ is a closed immersion and $Y$ Noetherian. Set $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_Y}(1)$. Let $\mathcal{F}$ be a coherent module on $X$. Then there exists an integer $d_0$ such that for all $d \geq d_0$ we have $R^pf_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$.


spaces-cohomology-lemma-proper-pushforward-coherent

Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$ with $Y$ locally Noetherian. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $R^if_*\mathcal{F}$ is a coherent $\mathcal{O}_Y$-module for all $i \geq 0$.


spaces-cohomology-lemma-proper-over-affine-cohomology-finite

Let $A$ be a Noetherian ring. Let $f : X \to \Spec(A)$ be a proper morphism of algebraic spaces. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $H^i(X, \mathcal{F})$ is finite $A$-module for all $i \geq 0$.


spaces-cohomology-lemma-graded-finiteness

Let $A$ be a Noetherian ring. Let $B$ be a finitely generated graded $A$-algebra. Let $f : X \to \Spec(A)$ be a proper morphism of algebraic spaces. Set $\mathcal{B} = f^*\widetilde B$. Let $\mathcal{F}$ be a quasi-coherent graded $\mathcal{B}$-module of finite type. For every $p \geq 0$ the graded $B$-module $H^p(X, \mathcal{F})$ is a finite $B$-module.


spaces-cohomology-lemma-vanshing-gives-ample

Let $R$ be a Noetherian ring. Let $X$ be a proper algebraic space over $R$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $X$ is a scheme and $\mathcal{L}$ is ample on $X$, \item for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ there exists an $n_0 \geq 0$ such that $H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for all $n \geq n_0$ and $p > 0$, and \item for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ there exists an $n \geq 1$ such that $H^1(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$. \end{enumerate}


spaces-cohomology-lemma-cohomology-powers-ideal-times-F

In Situation \ref{situation-formal-functions}. Set $B = \bigoplus_{n \geq 0} I^n$. Then for every $p \geq 0$ the graded $B$-module $\bigoplus_{n \geq 0} H^p(X, I^n\mathcal{F})$ is a finite $B$-module.


spaces-cohomology-lemma-cohomology-powers-ideal-application

In Situation \ref{situation-formal-functions}. For every $p \geq 0$ there exists an integer $c \geq 0$ such that \begin{enumerate} \item the multiplication map $I^{n - c} \otimes H^p(X, I^c\mathcal{F}) \to H^p(X, I^n\mathcal{F})$ is surjective for all $n \geq c$, and \item the image of $H^p(X, I^{n + m}\mathcal{F}) \to H^p(X, I^n\mathcal{F})$ is contained in the submodule $I^{m - c} H^p(X, I^n\mathcal{F})$ for all $n \geq 0$, $m \geq c$. \end{enumerate}


spaces-cohomology-lemma-ML-cohomology-powers-ideal

In Situation \ref{situation-formal-functions}. Fix $p \geq 0$. \begin{enumerate} \item There exists a $c_1 \geq 0$ such that for all $n \geq c_1$ we have $$ \Ker( H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F}) ) \subset I^{n - c_1}H^p(X, \mathcal{F}). $$ \item The inverse system $$ \left(H^p(X, \mathcal{F}/I^n\mathcal{F})\right)_{n \in \mathbf{N}} $$ satisfies the Mittag-Leffler condition (see Homology, Definition \ref{homology-definition-Mittag-Leffler}). \item In fact for any $p$ and $n$ there exists a $c_2(n) \geq n$ such that $$ \Im(H^p(X, \mathcal{F}/I^k\mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F})) = \Im(H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F})) $$ for all $k \geq c_2(n)$. \end{enumerate}


spaces-cohomology-lemma-spell-out-theorem-formal-functions

Let $A$ be a ring. Let $I \subset A$ be an ideal. Assume $A$ is Noetherian and complete with respect to $I$. Let $f : X \to \Spec(A)$ be a proper morphism of algebraic spaces. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then $$ H^p(X, \mathcal{F}) = \lim_n H^p(X, \mathcal{F}/I^n\mathcal{F}) $$ for all $p \geq 0$.


spaces-cohomology-lemma-formal-functions-stalk

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ and let $\mathcal{F}$ be a quasi-coherent sheaf on $Y$. Assume \begin{enumerate} \item $Y$ locally Noetherian, \item $f$ proper, and \item $\mathcal{F}$ coherent. \end{enumerate} Let $\overline{y}$ be a geometric point of $Y$. Consider the ``infinitesimal neighbourhoods'' $$ \xymatrix{ X_n = \Spec(\mathcal{O}_{Y, \overline{y}}/\mathfrak m_{\overline{y}}^n) \times_Y X \ar[r]_-{i_n} \ar[d]_{f_n} & X \ar[d]^f \\ \Spec(\mathcal{O}_{Y, \overline{y}}/\mathfrak m_{\overline{y}}^n) \ar[r]^-{c_n} & Y } $$ of the fibre $X_1 = X_{\overline{y}}$ and set $\mathcal{F}_n = i_n^*\mathcal{F}$. Then we have $$ \left(R^pf_*\mathcal{F}\right)_{\overline{y}}^\wedge \cong \lim_n H^p(X_n, \mathcal{F}_n) $$ as $\mathcal{O}_{Y, \overline{y}}^\wedge$-modules.


spaces-cohomology-lemma-higher-direct-images-zero-finite-fibre

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{y}$ be a geometric point of $Y$. Assume \begin{enumerate} \item $Y$ locally Noetherian, \item $f$ is proper, and \item $X_{\overline{y}}$ has discrete underlying topological space. \end{enumerate} Then for any coherent sheaf $\mathcal{F}$ on $X$ we have $(R^pf_*\mathcal{F})_{\overline{y}} = 0$ for all $p > 0$.


spaces-cohomology-lemma-higher-direct-images-zero-above-dimension-fibre

\begin{slogan} For proper maps, stalks of higher direct images are trivial in degrees larger than the dimension of the fibre. \end{slogan} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{y}$ be a geometric point of $Y$. Assume \begin{enumerate} \item $Y$ locally Noetherian, \item $f$ is proper, and \item $\dim(X_{\overline{y}}) = d$. \end{enumerate} Then for any coherent sheaf $\mathcal{F}$ on $X$ we have $(R^pf_*\mathcal{F})_{\overline{y}} = 0$ for all $p > d$.


spaces-cohomology-lemma-characterize-finite

(For a more general version see More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-characterize-finite}). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $Y$ is locally Noetherian. The following are equivalent \begin{enumerate} \item $f$ is finite, and \item $f$ is proper and $|X_k|$ is a discrete space for every morphism $\Spec(k) \to Y$ where $k$ is a field. \end{enumerate}


spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood

(For a more general version see More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood}). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{y}$ be a geometric point of $Y$. Assume \begin{enumerate} \item $Y$ is locally Noetherian, \item $f$ is proper, and \item $|X_{\overline{y}}|$ is finite. \end{enumerate} Then there exists an open neighbourhood $V \subset Y$ of $\overline{y}$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.


spaces-cohomology-proposition-vanishing

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and separated. Let $U$ be an affine scheme, and let $f : U \to X$ be a surjective \'etale morphism. Let $d$ be an upper bound for the size of the fibres of $|U| \to |X|$. Then for any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ we have $H^q(X, \mathcal{F}) = 0$ for $q \geq d$.


spaces-cohomology-proposition-vanishing-affine

\begin{slogan} Serre's criterion for affineness in the setting of algebraic spaces. \end{slogan} A quasi-compact and quasi-separated algebraic space is affine if and only if all higher cohomology groups of quasi-coherent sheaves vanish. More precisely, any algebraic space as in Situation \ref{situation-vanishing} is an affine scheme.


artin-theorem-contractions

\begin{reference} \cite[Theorem 3.1]{ArtinII} \end{reference} Let $S$ be a locally Noetherian scheme such that $\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s \in S$. Let $X'$ be an algebraic space locally of finite type over $S$. Let $T' \subset |X'|$ be a closed subset. Let $W$ be a locally Noetherian formal algebraic space over $S$ with $W_{red}$ locally of finite type over $S$. Finally, we let $$ g : X'_{/T'} \longrightarrow W $$ be a formal modification, see Algebraization of Formal Spaces, Definition \ref{restricted-definition-formal-modification}. If $X'$ and $W$ are separated\footnote{See Remark \ref{remark-separated-needed}.} over $S$, then there exists a proper morphism $f : X' \to X$ of algebraic spaces over $S$, a closed subset $T \subset |X|$, and an isomorphism $a : X_{/T} \to W$ of formal algebraic spaces such that \begin{enumerate} \item $T'$ is the inverse image of $T$ by $|f| : |X'| \to |X|$, \item $f : X' \to X$ maps $X' \setminus T'$ isomorphically to $X \setminus T$, and \item $g = a \circ f_{/T}$ where $f_{/T} : X'_{/T'} \to X_{/T}$ is the induced morphism. \end{enumerate} In other words, $(f : X' \to X, T, a)$ is a solution as defined earlier in this section.


artin-lemma-formally-smooth-on-deformation-categories

Let $S$ be a locally Noetherian scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume either \begin{enumerate} \item $F$ is formally smooth on objects (Criteria for Representability, Section \ref{criteria-section-formally-smooth}), \item $F$ is representable by algebraic spaces and formally smooth, or \item $F$ is representable by algebraic spaces and smooth. \end{enumerate} Then for every finite type field $k$ over $S$ and object $x_0$ of $\mathcal{X}$ over $k$ the functor (\ref{equation-functoriality}) is smooth in the sense of Formal Deformation Theory, Definition \ref{formal-defos-definition-smooth-morphism}.


artin-lemma-fibre-product-deformation-categories

Let $S$ be a locally Noetherian scheme. Let $$ \xymatrix{ \mathcal{W} \ar[d] \ar[r] & \mathcal{Z} \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{Y} } $$ be a $2$-fibre product of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $k$ be a finite type field over $S$ and $w_0$ an object of $\mathcal{W}$ over $k$. Let $x_0, z_0, y_0$ be the images of $w_0$ under the morphisms in the diagram. Then $$ \xymatrix{ \mathcal{F}_{\mathcal{W}, k, w_0} \ar[d] \ar[r] & \mathcal{F}_{\mathcal{Z}, k, z_0} \ar[d] \\ \mathcal{F}_{\mathcal{X}, k, x_0} \ar[r] & \mathcal{F}_{\mathcal{Y}, k, y_0} } $$ is a fibre product of predeformation categories.


artin-lemma-pushout

\begin{slogan} Algebraic stacks satisfy the (strong) Rim-Schlessinger condition \end{slogan} Let $S$ be a scheme. Let $$ \xymatrix{ X \ar[r] \ar[d] & X' \ar[d] \\ Y \ar[r] & Y' } $$ be a pushout in the category of schemes over $S$ where $X \to X'$ is a thickening and $X \to Y$ is affine, see More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}. Let $\mathcal{Z}$ be an algebraic stack over $S$. Then the functor of fibre categories $$ \mathcal{Z}_{Y'} \longrightarrow \mathcal{Z}_Y \times_{\mathcal{Z}_X} \mathcal{Z}_{X'} $$ is an equivalence of categories.


artin-lemma-algebraic-stack-RS

Let $\mathcal{X}$ be an algebraic stack over a locally Noetherian base $S$. Then $\mathcal{X}$ satisfies (RS).


artin-lemma-fibre-product-RS

Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ satisfy (RS), then so does $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$.


artin-lemma-deformation-category

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$ satisfying (RS). For any field $k$ of finite type over $S$ and any object $x_0$ of $\mathcal{X}$ lying over $k$ the predeformation category $p : \mathcal{F}_{\mathcal{X}, k, x_0} \to \mathcal{C}_\Lambda$ (\ref{equation-predeformation-category}) is a deformation category, see Formal Deformation Theory, Definition \ref{formal-defos-definition-deformation-category}.


artin-lemma-change-of-field

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Let $k$ be a field of finite type over $S$ and let $l/k$ be a finite extension. Let $x_0$ be an object of $\mathcal{F}$ lying over $\Spec(k)$. Denote $x_{l, 0}$ the restriction of $x_0$ to $\Spec(l)$. Then there is a canonical functor $$ (\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k} \longrightarrow \mathcal{F}_{\mathcal{X}, l, x_{l, 0}} $$ of categories cofibred in groupoids over $\mathcal{C}_{\Lambda, l}$. If $\mathcal{X}$ satisfies (RS), then this functor is an equivalence.


artin-lemma-finite-dimension

Let $S$ be a locally Noetherian scheme. Assume \begin{enumerate} \item $\mathcal{X}$ is an algebraic stack, \item $U$ is a scheme locally of finite type over $S$, and \item $(\Sch/U)_{fppf} \to \mathcal{X}$ is a smooth surjective morphism. \end{enumerate} Then, for any $\mathcal{F} = \mathcal{F}_{\mathcal{X}, k, x_0}$ as in Section \ref{section-predeformation-categories} the tangent space $T\mathcal{F}$ and infinitesimal automorphism space $\text{Inf}(\mathcal{F})$ have finite dimension over $k$


artin-lemma-fibre-product-tangent-spaces

Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{Z}$ satisfy (RS). Let $k$ be a field of finite type over $S$ and let $w_0$ be an object of $\mathcal{W} = \mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ over $k$. Denote $x_0, y_0, z_0$ the objects of $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ you get from $w_0$. Then there is a $6$-term exact sequence $$ \xymatrix{ 0 \ar[r] & \text{Inf}(\mathcal{F}_{\mathcal{W}, k, w_0}) \ar[r] & \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) \oplus \text{Inf}(\mathcal{F}_{\mathcal{Z}, k, z_0}) \ar[r] & \text{Inf}(\mathcal{F}_{\mathcal{Y}, k, y_0}) \ar[lld] \\  & T\mathcal{F}_{\mathcal{W}, k, w_0} \ar[r] & T\mathcal{F}_{\mathcal{X}, k, x_0} \oplus T\mathcal{F}_{\mathcal{Z}, k, z_0} \ar[r] & T\mathcal{F}_{\mathcal{Y}, k, y_0} } $$ of $k$-vector spaces.


artin-lemma-smooth-lift-formal

Let $S$ be a locally Noetherian scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\eta = (R, \eta_n, g_n)$ be a formal object of $\mathcal{Y}$ and let $\xi_1$ be an object of $\mathcal{X}$ with $F(\xi_1) \cong \eta_1$. If $F$ is formally smooth on objects (see Criteria for Representability, Section \ref{criteria-section-formally-smooth}), then there exists a formal object $\xi = (R, \xi_n, f_n)$ of $\mathcal{X}$ such that $F(\xi) \cong \eta$.


artin-lemma-effective

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. The functor (\ref{equation-approximation}) is an equivalence.


artin-lemma-fibre-product-effective

Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If the functor (\ref{equation-approximation}) is an equivalence for  $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$, then it is  an equivalence for $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$.


artin-lemma-approximate

Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $x$ be an object of $\mathcal{X}$ lying over $\Spec(R)$ where $R$ is a Noetherian complete local ring with residue field $k$ of finite type over $S$. Let $s \in S$ be the image of $\Spec(k) \to S$. Assume that (a) $\mathcal{O}_{S, s}$ is a G-ring and (b) $p$ is limit preserving on objects. Then for every integer $N \geq 1$ there exist \begin{enumerate} \item a finite type $S$-algebra $A$, \item a maximal ideal $\mathfrak m_A \subset A$, \item an object $x_A$ of $\mathcal{X}$ over $\Spec(A)$, \item an $S$-isomorphism $R/\mathfrak m_R^N \cong A/\mathfrak m_A^N$, \item an isomorphism $x|_{\Spec(R/\mathfrak m_R^N)} \cong x_A|_{\Spec(A/\mathfrak m_A^N)}$ compatible with (4), and \item an isomorphism $\text{Gr}_{\mathfrak m_R}(R) \cong \text{Gr}_{\mathfrak m_A}(A)$ of graded $k$-algebras. \end{enumerate}


artin-lemma-fibre-product-limit-preserving

Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item If $\mathcal{X} \to (\Sch/S)_{fppf}$ and $\mathcal{Z} \to (\Sch/S)_{fppf}$ are limit preserving on objects and $\mathcal{Y}$ is limit preserving, then $\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to (\Sch/S)_{fppf}$ is limit preserving on objects. \item If $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ are limit preserving, then so is $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$. \end{enumerate}


artin-lemma-limit-preserving-algebraic-space

Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Then the following are equivalent \begin{enumerate} \item $\mathcal{X}$ is a stack in setoids and $\mathcal{X} \to (\Sch/S)_{fppf}$ is limit preserving on objects, \item $\mathcal{X}$ is a stack in setoids and limit preserving, \item $\mathcal{X}$ is representable by an algebraic space locally of finite presentation. \end{enumerate}


artin-lemma-diagonal

Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces and $\mathcal{X}$ is limit preserving. Then $\Delta$ is locally of finite type.


artin-lemma-versality-matches

With notation as in Definition \ref{definition-versal}. Let $R = \mathcal{O}_{U, u_0}^\wedge$. Let $\xi$ be the formal object of $\mathcal{X}$ over $R$ associated to $x|_{\Spec(R)}$, see (\ref{equation-approximation}). Then $$ x\text{ is versal at }u_0 \Leftrightarrow \xi\text{ is versal} $$


artin-lemma-versal-change-of-field

Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition \ref{definition-versal}. Let $l$ be a field and let $u_{l, 0} : \Spec(l) \to U$ be a morphism with image $u_0$ such that $l/k = \kappa(u_0)$ is finite. Set $x_{l, 0} = x_0|_{\Spec(l)}$. If $\mathcal{X}$ satisfies (RS) and $x$ is versal at $u_0$, then $$ \mathcal{F}_{(\Sch/U)_{fppf}, l, u_{l, 0}} \longrightarrow \mathcal{F}_{\mathcal{X}, l, x_{l, 0}} $$ is smooth.


artin-lemma-base-change-versal

Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition \ref{definition-versal}. Assume \begin{enumerate} \item $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces, \item $\Delta$ is locally of finite type (for example if $\mathcal{X}$ is limit preserving), and \item $\mathcal{X}$ has (RS). \end{enumerate} Let $V$ be a scheme locally of finite type over $S$ and let $y$ be an object of $\mathcal{X}$ over $V$. Form the $2$-fibre product $$ \xymatrix{ \mathcal{Z} \ar[r] \ar[d] & (\Sch/U)_{fppf} \ar[d]^x \\ (\Sch/V)_{fppf} \ar[r]^y & \mathcal{X} } $$ Let $Z$ be the algebraic space representing $\mathcal{Z}$ and let $z_0 \in |Z|$ be a finite type point lying over $u_0$. If $x$ is versal at $u_0$, then the morphism $Z \to V$ is smooth at $z_0$.


artin-lemma-approximate-versal

Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\xi = (R, \xi_n, f_n)$ be a formal object of $\mathcal{X}$ with $\xi_1$ lying over $\Spec(k) \to S$ with image $s \in S$. Assume \begin{enumerate} \item $\xi$ is versal, \item $\xi$ is effective, \item $\mathcal{O}_{S, s}$ is a G-ring, and \item $p : \mathcal{X} \to (\Sch/S)_{fppf}$ is limit preserving on objects. \end{enumerate} Then there exist a morphism of finite type $U \to S$, a finite type point $u_0 \in U$ with residue field $k$, and an object $x$ of $\mathcal{X}$ over $U$ such that $x$ is versal at $u_0$ and such that $x|_{\Spec(\mathcal{O}_{U, u_0}/\mathfrak m_{u_0}^n)} \cong \xi_n$.


artin-lemma-versal-smooth

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Let $U$ be a scheme locally of finite type over $S$. Let $x$ be an object of $\mathcal{X}$ over $U$. Assume that $x$ is versal at every finite type point of $U$ and that $\mathcal{X}$ satisfies (RS). Then $x : (\Sch/U)_{fppf} \to \mathcal{X}$ satisfies (\ref{equation-smooth}).


artin-lemma-composition-smooth

Let $S$ be a locally Noetherian scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be composable $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $f$ and $g$ satisfy (\ref{equation-smooth}) so does $g \circ f$.


artin-lemma-base-change-smooth

Let $S$ be a locally Noetherian scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ and $\mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $f$ satisfies (\ref{equation-smooth}) so does the projection $\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}$.


artin-lemma-smooth-smooth

Let $S$ be a locally Noetherian scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $f$ is formally smooth on objects, then $f$ satisfies (\ref{equation-smooth}). If $f$ is representable by algebraic spaces and smooth, then $f$ satisfies (\ref{equation-smooth}).


artin-lemma-implies-smooth

Let $S$ be a locally Noetherian scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume \begin{enumerate} \item $f$ is representable by algebraic spaces, \item $f$ satisfies (\ref{equation-smooth}), \item $\mathcal{X} \to (\Sch/S)_{fppf}$ is limit preserving on objects, and \item $\mathcal{Y}$ is limit preserving. \end{enumerate} Then $f$ is smooth.


artin-lemma-get-smooth

Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $k$ be a finite type field over $S$ and let $x_0$ be an object of $\mathcal{X}$ over $\Spec(k)$ with image $s \in S$. Assume \begin{enumerate} \item $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces, \item $\mathcal{X}$ satisfies axioms [1], [2], [3] (see Section \ref{section-axioms}), \item every formal object of $\mathcal{X}$ is effective, \item openness of versality holds for $\mathcal{X}$, and \item $\mathcal{O}_{S, s}$ is a G-ring. \end{enumerate} Then there exist a morphism of finite type $U \to S$ and an object $x$ of $\mathcal{X}$ over $U$ such that $$ x : (\Sch/U)_{fppf} \longrightarrow \mathcal{X} $$ is smooth and such that there exists a finite type point $u_0 \in U$ whose residue field is $k$ and such that $x|_{u_0} \cong x_0$.


artin-lemma-diagonal-representable

Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Assume that \begin{enumerate} \item $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces, \item $\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3] (see Section \ref{section-axioms}), \item every formal object of $\mathcal{X}$ is effective, \item $\mathcal{X}$ satisfies openness of versality, and \item $\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$. \end{enumerate} Then $\mathcal{X}$ is an algebraic stack.


artin-lemma-fibre-product-RS-star

Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ satisfy (RS*), then so does $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$.


artin-lemma-single-point

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$ having (RS*). Let $x$ be an object of $\mathcal{X}$ over an affine scheme $U$ of finite type over $S$. Let $u \in U$ be a finite type point such that $x$ is not versal at $u$. Then there exists a morphism $x \to y$ of $\mathcal{X}$ lying over $U \to T$ satisfying \begin{enumerate} \item the morphism $U \to T$ is a first order thickening, \item we have a short exact sequence $$ 0 \to \kappa(u) \to \mathcal{O}_T \to \mathcal{O}_U \to 0 $$ \item there does {\bf not} exist a pair $(W, \alpha)$ consisting of an open neighbourhood $W \subset T$ of $u$ and a morphism $\beta : y|_W \to x$ such that the composition $$ x|_{U \cap W} \xrightarrow{\text{restriction of }x \to y} y|_W \xrightarrow{\beta} x $$ is the canonical morphism $x|_{U \cap W} \to x$. \end{enumerate}


artin-lemma-generalization-versality

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Assume \begin{enumerate} \item $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces, \item $\mathcal{X}$ has (RS*), \item $\mathcal{X}$ is limit preserving. \end{enumerate} Let $x$ be an object of $\mathcal{X}$ over a scheme $U$ of finite type over $S$. Let $u \leadsto u_0$ be a specialization of finite type points of $U$ such that $x$ is versal at $u_0$. Then $x$ is versal at $u$.


artin-lemma-infinite-sequence-pre

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$ having (RS*). Let $x$ be an object of $\mathcal{X}$ over an affine scheme $U$ of finite type over $S$. Let $u_n \in U$, $n \geq 1$ be finite type points such that (a) there are no specializations $u_n \leadsto u_m$ for $n \not = m$, and (b) $x$ is not versal at $u_n$ for all $n$. Then there exist morphisms $$ x \to x_1 \to x_2 \to \ldots \quad\text{in }\mathcal{X}\text{ lying over }\quad U \to U_1 \to U_2 \to \ldots $$ over $S$ such that \begin{enumerate} \item for each $n$ the morphism $U \to U_n$ is a first order thickening, \item for each $n$ we have a short exact sequence $$ 0 \to \kappa(u_n) \to \mathcal{O}_{U_n} \to \mathcal{O}_{U_{n - 1}} \to 0 $$ with $U_0 = U$ for $n = 1$, \item for each $n$ there does {\bf not} exist a pair $(W, \alpha)$ consisting of an open neighbourhood $W \subset U_n$ of $u_n$ and a morphism $\alpha : x_n|_W \to x$ such that the composition $$ x|_{U \cap W} \xrightarrow{\text{restriction of }x \to x_n} x_n|_W \xrightarrow{\alpha} x $$ is the canonical morphism $x|_{U \cap W} \to x$. \end{enumerate}


artin-lemma-SGE-implies-openness-versality

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Assume \begin{enumerate} \item $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces, \item $\mathcal{X}$ has (RS*), \item $\mathcal{X}$ is limit preserving, \item systems $(\xi_n)$ as in Remark \ref{remark-strong-effectiveness} where $\Ker(R_m \to R_n)$ is an ideal of square zero for all $m \geq n$ are effective. \end{enumerate} Then $\mathcal{X}$ satisfies openness of versality.


artin-lemma-functoriality

Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $$ \xymatrix{ B' \ar[r] & B \\ A' \ar[u] \ar[r] & A \ar[u] } $$ be a commutative diagram of $S$-algebras. Let $x$ be an object of $\mathcal{X}$ over $\Spec(A)$, let $y$ be an object of $\mathcal{Y}$ over $\Spec(B)$, and let $\phi : f(x)|_{\Spec(B)} \to y$ be a morphism of $\mathcal{Y}$ over $\Spec(B)$. Then there is a canonical functor $$ \textit{Lift}(x, A') \longrightarrow \textit{Lift}(y, B') $$ of categories of lifts induced by $f$ and $\phi$. The construction is compatible with compositions of $1$-morphisms of categories fibred in groupoids in an obvious manner.


artin-lemma-properties-lift-RS-star

Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $\mathcal{X}$ satisfies condition (RS*). Let $A$ be an $S$-algebra and let $x$ be an object of $\mathcal{X}$ over $\Spec(A)$. \begin{enumerate} \item There exists an $A$-linear functor $\text{Inf}_x : \text{Mod}_A \to \text{Mod}_A$ such that given a deformation situation $(x, A' \to A)$ and a lift $x'$ there is an isomorphism $\text{Inf}_x(I) \to \text{Inf}(x'/x)$ where $I = \Ker(A' \to A)$. \item There exists an $A$-linear functor $T_x : \text{Mod}_A \to \text{Mod}_A$ such that \begin{enumerate} \item given $M$ in $\text{Mod}_A$ there is a bijection $T_x(M) \to \text{Lift}(x, A[M])$, \item given a deformation situation $(x, A' \to A)$ there is an action $$ T_x(I) \times \text{Lift}(x, A') \to \text{Lift}(x, A') $$ where $I = \Ker(A' \to A)$. It is simply transitive if $\text{Lift}(x, A') \not = \emptyset$. \end{enumerate} \end{enumerate}


artin-lemma-ses-inf-and-T

Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{Z}$ satisfy (RS*). Let $A$ be an $S$-algebra and let $w$ be an object of $\mathcal{W} = \mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ over $A$. Denote $x, y, z$ the objects of $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ you get from $w$. For any $A$-module $M$ there is a $6$-term exact sequence $$ \xymatrix{ 0 \ar[r] & \text{Inf}_w(M) \ar[r] & \text{Inf}_x(M) \oplus \text{Inf}_z(M) \ar[r] & \text{Inf}_y(M) \ar[lld] \\  & T_w(M) \ar[r] & T_x(M) \oplus T_z(M) \ar[r] & T_y(M) } $$ of $A$-modules.


artin-lemma-get-openness-obstruction-theory

\begin{reference} This is \cite[Theorem 4.4]{Hall-coherent} \end{reference} Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. Assume \begin{enumerate} \item $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces, \item $\mathcal{X}$ has (RS*), \item $\mathcal{X}$ is limit preserving, \item there exists an obstruction theory\footnote{Analyzing the proof the reader sees that in fact it suffices to check the functoriality (ii) of obstruction classes in Definition \ref{definition-obstruction-theory} for maps $(y, B' \to B) \to (x, A' \to A)$ with $B = A$ and $y = x$.}, \item for an object $x$ of $\mathcal{X}$ over $\Spec(A)$ and $A$-modules $M_n$, $n \geq 1$ we have \begin{enumerate} \item $T_x(\prod M_n) = \prod T_x(M_n)$, \item $\mathcal{O}_x(\prod M_n) \to \prod \mathcal{O}_x(M_n)$ is injective. \end{enumerate} \end{enumerate} Then $\mathcal{X}$ satisfies openness of versality.


artin-lemma-compute-ext-into-field

Let $A \to k$ be a ring map with $k$ a field. Let $E \in D^-(A)$. Then $\Ext^i_A(E, k) = \Hom_k(H^{-i}(E \otimes^\mathbf{L} k), k)$.


artin-lemma-construct-essential-surjection

Let $\Lambda \to A \to k$ be finite type ring maps of Noetherian rings with $k = \kappa(\mathfrak p)$ for some prime $\mathfrak p$ of $A$. Let $\xi : E \to \NL_{A/\Lambda}$ be morphism of $D^{-}(A)$ such that $H^{-1}(\xi \otimes^{\mathbf{L}} k)$ is not surjective. Then there exists a surjection $A' \to A$ of $\Lambda$-algebras such that \begin{enumerate} \item[(a)] $I = \Ker(A' \to A)$ has square zero and is isomorphic to $k$ as an $A$-module, \item[(b)] $\Omega_{A'/\Lambda} \otimes k = \Omega_{A/\Lambda} \otimes k$, and \item[(c)] $E \to \NL_{A/A'}$ is zero. \end{enumerate}


artin-lemma-characterize-versal

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$ satisfying (RS*). Let $U = \Spec(A)$ be an affine scheme of finite type over $S$ which maps into an affine open $\Spec(\Lambda)$. Let $x$ be an object of $\mathcal{X}$ over $U$. Let $\xi : E \to \NL_{A/\Lambda}$ be a morphism of $D^{-}(A)$. Assume \begin{enumerate} \item[(i)] for every deformation situation $(x, A' \to A)$ we have: $x$ lifts to $\Spec(A')$ if and only if $E \to \NL_{A/\Lambda} \to \NL_{A/A'}$ is zero, and \item[(ii)] there is an isomorphism of functors $T_x(-) \to \Ext^0_A(E, -)$ such that $E \to \NL_{A/\Lambda} \to \Omega^1_{A/\Lambda}$ corresponds to the canonical element (see Remark \ref{remark-canonical-element}). \end{enumerate} Let $u_0 \in U$ be a finite type point with residue field $k = \kappa(u_0)$. Consider the following statements \begin{enumerate} \item $x$ is versal at $u_0$, and \item $\xi : E \to \NL_{A/\Lambda}$ induces a surjection $H^{-1}(E \otimes_A^{\mathbf{L}} k) \to H^{-1}(\NL_{A/\Lambda} \otimes_A^{\mathbf{L}} k)$ and an injection $H^0(E \otimes_A^{\mathbf{L}} k) \to H^0(\NL_{A/\Lambda} \otimes_A^{\mathbf{L}} k)$. \end{enumerate} Then we always have (2) $\Rightarrow$ (1) and we have (1) $\Rightarrow$ (2) if $u_0$ is a closed point.


artin-lemma-openness

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$ satisfying (RS*). Let $U = \Spec(A)$ be an affine scheme of finite type over $S$ which maps into an affine open $\Spec(\Lambda)$. Let $x$ be an object of $\mathcal{X}$ over $U$. Let $\xi : E \to \NL_{A/\Lambda}$ be a morphism of $D^{-}(A)$. Assume \begin{enumerate} \item[(i)] for every deformation situation $(x, A' \to A)$ we have: $x$ lifts to $\Spec(A')$ if and only if $E \to \NL_{A/\Lambda} \to \NL_{A/A'}$ is zero, \item[(ii)] there is an isomorphism of functors $T_x(-) \to \Ext^0_A(E, -)$ such that $E \to \NL_{A/\Lambda} \to \Omega^1_{A/\Lambda}$ corresponds to the canonical element (see Remark \ref{remark-canonical-element}), \item[(iii)] the cohomology groups of $E$ are finite $A$-modules. \end{enumerate} If $x$ is versal at a closed point $u_0 \in U$, then there exists an open neighbourhood $u_0 \in U' \subset U$ such that $x$ is versal at every finite type point of $U'$.


artin-lemma-dual-obstruction

In Situation \ref{situation-dual}. Assume furthermore that \begin{enumerate} \item[(iv)] given a short exact sequence of deformation situations as in Remark \ref{remark-short-exact-sequence-thickenings} and a lift $x'_2 \in \text{Lift}(x, A_2')$ then $o_x(A_3') \in H^2(K^\bullet \otimes_A^\mathbf{L} I_3)$ equals $\partial\theta$ where $\theta \in H^1(K^\bullet \otimes_A^\mathbf{L} I_1)$ is the element corresponding to $x'_2|_{\Spec(A_1')}$ via $A_1' = A[I_1]$ and the given map $T_x(-) \to H^1(K^\bullet \otimes_A^\mathbf{L} -)$. \end{enumerate} In this case there exists an element $\xi \in H^1(K^\bullet \otimes_A^\mathbf{L} \NL_{A/\Lambda})$ such that \begin{enumerate} \item for every deformation situation $(x, A' \to A)$ we have $\xi_{A'} = o_x(A')$, and \item $\xi_{can}$ matches the canonical element of Remark \ref{remark-canonical-element} via the given transformation $T_x(-) \to H^1(K^\bullet \otimes_A^\mathbf{L} -)$. \end{enumerate}


artin-lemma-dual-openness

In Situation \ref{situation-dual} assume that (iv) of Lemma \ref{lemma-dual-obstruction} holds and that $K^\bullet$ is a perfect object of $D(A)$. In this case, if $x$ is versal at a closed point $u_0 \in U$ then there exists an open neighbourhood $u_0 \in U' \subset U$ such that $x$ is versal at every finite type point of $U'$.


artin-lemma-canonical-extension

Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. Restricting along the inclusion functor $(\textit{Noetherian}/S)_\tau \to (\Sch/S)_\tau$ defines an equivalence of categories between \begin{enumerate} \item the category of limit preserving sheaves on $(\Sch/S)_\tau$ and \item the category of limit preserving sheaves on $(\textit{Noetherian}/S)_\tau$ \end{enumerate}


artin-lemma-representable-limit-preserving

Let $X$ be an object of $(\textit{Noetherian}/S)_\tau$. If the functor of points $h_X : (\textit{Noetherian}/S)_\tau^{opp} \to \textit{Sets}$ is limit preserving, then $X$ is locally of finite presentation over $S$.


artin-lemma-representable

Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. Let $F', G' : (\Sch/S)_\tau^{opp} \to \textit{Sets}$ be limit preserving and sheaves. Let $a' : F' \to G'$ be a transformation of functors. Denote $a : F \to G$ the restriction of $a' : F' \to G'$ to $(\textit{Noetherian}/S)_\tau$. The following are equivalent \begin{enumerate} \item $a'$ is representable (as a transformation of functors, see Categories, Definition \ref{categories-definition-representable-morphism}), and \item for every object $V$ of $(\textit{Noetherian}/S)_\tau$ and every map $V \to G$ the fibre product $F \times_G V : (\textit{Noetherian}/S)_\tau^{opp} \to \textit{Sets}$ is a representable functor, and \item same as in (2) but only for $V$ affine finite type over $S$ mapping into an affine open of $S$. \end{enumerate}


artin-lemma-functor

In Situation \ref{situation-contractions} the rule $F$ that sends a locally Noetherian scheme $V$ over $S$ to the set of triples $(Z, u', \hat x)$ satisfying the compatibility condition and which sends a a morphism $\varphi : V_2 \to V_1$ of locally Noetherian schemes over $S$ to the map $$ F(\varphi) : F(V_1) \longrightarrow F(V_2) $$ sending an element $(Z_1, u'_1, \hat x_1)$ of $F(V_1)$ to $(Z_2, u'_2, \hat x_2)$ in $F(V_2)$ given by \begin{enumerate} \item $Z_2 \subset V_2$ is the inverse image of $Z_1$ by $\varphi$, \item $u'_2$ is the composition of $u'_1$ and $\varphi|_{V_2 \setminus Z_2} : V_2 \setminus Z_2 \to V_1 \setminus Z_1$, \item $\hat x_2$ is the composition of $\hat x_1$ and $\varphi_{/Z_2} : V_{2, /Z_2} \to V_{1, /Z_1}$ \end{enumerate} is a contravariant functor.


artin-lemma-solution

In Situation \ref{situation-contractions} if there exists a solution $(f : X' \to X, T, a)$ then there is a functorial bijection $F(V) = \Mor_S(V, X)$ on the category of locally Noetherian schemes $V$ over $S$.


artin-lemma-functor-is-solution

In Situation \ref{situation-contractions} if there exists an algebraic space $X$ locally of finite type over $S$ and a functorial bijection $F(V) = \Mor_S(V, X)$ on the category of locally Noetherian schemes $V$ over $S$, then $X$ is a solution.


artin-lemma-closed-immersion

In Situation \ref{situation-contractions} assume given a closed subset $Z \subset S$ such that \begin{enumerate} \item the inverse image of $Z$ in $X'$ is $T'$, \item $U' \to S \setminus Z$ is a closed immersion, \item $W \to S_{/Z}$ is a closed immersion. \end{enumerate} Then there exists a solution $(f : X' \to X, T, a)$ and moreover $X \to S$ is a closed immersion.


artin-lemma-diagonal-contractions

In Situation \ref{situation-contractions} assume $X' \to S$ and $W \to S$ are separated. Then the diagonal $\Delta : F \to F \times F$ is representable by closed immersions.


artin-lemma-sheaf

In Situation \ref{situation-contractions} the functor $F$ satisfies the sheaf property for all \'etale coverings of locally Noetherian schemes over $S$.


artin-lemma-limit-preserving

In Situation \ref{situation-contractions} the functor $F$ is limit preserving: for any directed limit $V = \lim V_\lambda$ of Noetherian affine schemes over $S$ we have $F(V) = \colim F(V_\lambda)$.


artin-lemma-rs

In Situation \ref{situation-contractions} the functor $F$ satisfies the Rim-Schlessinger condition (RS).


artin-lemma-finite-dim

In Situation \ref{situation-contractions} the tangent spaces of the functor $F$ are finite dimensional.


artin-lemma-formal-object-effective

In Situation \ref{situation-contractions} assume $X' \to S$ is separated. Then every formal object for $F$ is effective.


artin-lemma-openness-smoothness

Let $S$ be a locally Noetherian scheme. Let $V$ be a scheme locally of finite type over $S$. Let $Z \subset V$ be closed. Let $W$ be a locally Noetherian formal algebraic space over $S$ such that $W_{red}$ is locally of finite type over $S$. Let $g : V_{/Z} \to W$ be an adic morphism of formal algebraic spaces over $S$. Let $v \in V$ be a closed point such that $g$ is versal at $v$ (as in Section \ref{section-axioms-functors}). Then after replacing $V$ by an open neighbourhood of $v$ the morphism $g$ is smooth (see proof).


artin-lemma-openness-versality

In Situation \ref{situation-contractions} the functor $F$ satisfies openness of versality.


artin-proposition-spaces-diagonal-representable

Let $S$ be a locally Noetherian scheme. Let $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Assume that \begin{enumerate} \item $\Delta : F \to F \times F$ is representable by algebraic spaces, \item $F$ satisfies axioms [-1], [0], [1], [2], [3], [4], [5] (see Section \ref{section-axioms-functors}), and \item $\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$. \end{enumerate} Then $F$ is an algebraic space.


artin-proposition-spaces-diagonal-representable-noetherian

Let $S$ be a locally Noetherian scheme. Let $F : (\textit{Noetherian}/S)_\etale^{opp} \to \textit{Sets}$ be a functor. Assume that \begin{enumerate} \item $\Delta : F \to F \times F$ is representable (as a transformation of functors, see Categories, Definition \ref{categories-definition-representable-morphism}), \item $F$ satisfies axioms [-1], [0], [1], [2], [3], [4], [5] (see above), and \item $\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$. \end{enumerate} Then there exists a unique algebraic space $F' : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ whose restriction to $(\textit{Noetherian}/S)_\etale$ is $F$ (see proof for elucidation).