derived-definition-injective-resolution

Let $\mathcal{A}$ be an abelian category. Let $A \in \Ob(\mathcal{A})$. An {\it injective resolution of $A$} is a complex $I^\bullet$ together with a map $A \to I^0$ such that: \begin{enumerate} \item We have $I^n = 0$ for $n < 0$. \item Each $I^n$ is an injective object of $\mathcal{A}$. \item The map $A \to I^0$ is an isomorphism onto $\Ker(d^0)$. \item We have $H^i(I^\bullet) = 0$ for $i > 0$. \end{enumerate} Hence $A[0] \to I^\bullet$ is a quasi-isomorphism. In other words the complex $$ \ldots \to 0 \to A \to I^0 \to I^1 \to \ldots $$ is acyclic. Let $K^\bullet$ be a complex in $\mathcal{A}$. An {\it injective resolution of $K^\bullet$} is a complex $I^\bullet$ together with a map $\alpha : K^\bullet \to I^\bullet$ of complexes such that \begin{enumerate} \item We have $I^n = 0$ for $n \ll 0$, i.e., $I^\bullet$ is bounded below. \item Each $I^n$ is an injective object of $\mathcal{A}$. \item The map $\alpha : K^\bullet \to I^\bullet$ is a quasi-isomorphism. \end{enumerate}


derived-definition-projective-resolution

Let $\mathcal{A}$ be an abelian category. Let $A \in \Ob(\mathcal{A})$. An {\it projective resolution of $A$} is a complex $P^\bullet$ together with a map $P^0 \to A$ such that: \begin{enumerate} \item We have $P^n = 0$ for $n > 0$. \item Each $P^n$ is an projective object of $\mathcal{A}$. \item The map $P^0 \to A$ induces an isomorphism $\Coker(d^{-1}) \to A$. \item We have $H^i(P^\bullet) = 0$ for $i < 0$. \end{enumerate} Hence $P^\bullet \to A[0]$ is a quasi-isomorphism. In other words the complex $$ \ldots \to P^{-1} \to P^0 \to A \to 0 \to \ldots $$ is acyclic. Let $K^\bullet$ be a complex in $\mathcal{A}$. An {\it projective resolution of $K^\bullet$} is a complex $P^\bullet$ together with a map $\alpha : P^\bullet \to K^\bullet$ of complexes such that \begin{enumerate} \item We have $P^n = 0$ for $n \gg 0$, i.e., $P^\bullet$ is bounded above. \item Each $P^n$ is an projective object of $\mathcal{A}$. \item The map $\alpha : P^\bullet \to K^\bullet$ is a quasi-isomorphism. \end{enumerate}


derived-definition-cartan-eilenberg

Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$ be a bounded below complex. A {\it Cartan-Eilenberg resolution} of $K^\bullet$ is given by a double complex $I^{\bullet, \bullet}$ and a morphism of complexes $\epsilon : K^\bullet \to I^{\bullet, 0}$ with the following properties: \begin{enumerate} \item There exists a $i \ll 0$ such that $I^{p, q} = 0$ for all $p < i$ and all $q$. \item We have $I^{p, q} = 0$ if $q < 0$. \item The complex $I^{p, \bullet}$ is an injective resolution of $K^p$. \item The complex $\Ker(d_1^{p, \bullet})$ is an injective resolution of $\Ker(d_K^p)$. \item The complex $\Im(d_1^{p, \bullet})$ is an injective resolution of $\Im(d_K^p)$. \item The complex $H^p_I(I^{\bullet, \bullet})$ is an injective resolution of $H^p(K^\bullet)$. \end{enumerate}


derived-definition-localization-functor

Let $\mathcal{A}$ be an abelian category with enough injectives. A {\it resolution functor}\footnote{This is likely nonstandard terminology.} for $\mathcal{A}$ is given by the following data: \begin{enumerate} \item for all $K^\bullet \in \Ob(K^{+}(\mathcal{A}))$ a bounded below complex of injectives $j(K^\bullet)$, and \item for all $K^\bullet \in \Ob(K^{+}(\mathcal{A}))$ a quasi-isomorphism $i_{K^\bullet} : K^\bullet \to j(K^\bullet)$. \end{enumerate}


derived-definition-filtered-complexes-notation

Let $\mathcal{A}$ be an abelian category. We say an object $I$ of $\text{Fil}^f(\mathcal{A})$ is {\it filtered injective} if each $\text{gr}^p(I)$ is an injective object of $\mathcal{A}$.


derived-definition-ext

Let $\mathcal{A}$ be an abelian category. Let $i \in \mathbf{Z}$. Let $X, Y$ be objects of $D(\mathcal{A})$. The {\it $i$th extension group} of $X$ by $Y$ is the group $$ \Ext^i_\mathcal{A}(X, Y) = \Hom_{D(\mathcal{A})}(X, Y[i]) = \Hom_{D(\mathcal{A})}(X[-i], Y). $$ If $A, B \in \Ob(\mathcal{A})$ we set $\Ext^i_\mathcal{A}(A, B) = \text{Ext}^i_\mathcal{A}(A[0], B[0])$.


derived-definition-yoneda-extension

Let $\mathcal{A}$ be an abelian category. Let $A, B \in \Ob(\mathcal{A})$. A degree $i$ {\it Yoneda extension} of $B$ by $A$ is an exact sequence $$ E : 0 \to A \to Z_{i - 1} \to Z_{i - 2} \to \ldots \to Z_0 \to B \to 0 $$ in $\mathcal{A}$. We say two Yoneda extensions $E$ and $E'$ of the same degree are {\it equivalent} if there exists a commutative diagram $$ \xymatrix{ 0 \ar[r] & A \ar[r] & Z_{i - 1} \ar[r] & \ldots \ar[r] & Z_0 \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & A \ar[r] \ar[u]^{\text{id}} \ar[d]_{\text{id}} & Z''_{i - 1} \ar[r] \ar[u] \ar[d] & \ldots \ar[r] & Z''_0 \ar[r] \ar[u] \ar[d] & B \ar[r] \ar[u]_{\text{id}} \ar[d]^{\text{id}} & 0 \\ 0 \ar[r] & A \ar[r] & Z'_{i - 1} \ar[r] & \ldots \ar[r] & Z'_0 \ar[r] & B \ar[r] & 0 } $$ where the middle row is a Yoneda extension as well.


derived-definition-K-zero

Let $\mathcal{D}$ be a triangulated category. We denote $K_0(\mathcal{D})$ the {\it zeroth $K$-group of $\mathcal{D}$}. It is the abelian group constructed as follows. Take the free abelian group on the objects on $\mathcal{D}$ and for every distinguished triangle $X \to Y \to Z$ impose the relation $[Y] - [X] - [Z] = 0$.


derived-definition-K-injective

Let $\mathcal{A}$ be an abelian category. A complex $I^\bullet$ is {\it K-injective} if for every acyclic complex $M^\bullet$ we have $\Hom_{K(\mathcal{A})}(M^\bullet, I^\bullet) = 0$.


derived-definition-derived-colimit

Let $\mathcal{D}$ be a triangulated category. Let $(K_n, f_n)$ be a system of objects of $\mathcal{D}$. We say an object $K$ is a {\it derived colimit}, or a {\it homotopy colimit} of the system $(K_n)$ if the direct sum $\bigoplus K_n$ exists and there is a distinguished triangle $$ \bigoplus K_n \to \bigoplus K_n \to K \to \bigoplus K_n[1] $$ where the map $\bigoplus K_n \to \bigoplus K_n$ is given by $1 - f_n$ in degree $n$. If this is the case, then we sometimes indicate this by the notation $K = \text{hocolim} K_n$.


derived-definition-derived-limit

Let $\mathcal{D}$ be a triangulated category. Let $(K_n, f_n)$ be an inverse system of objects of $\mathcal{D}$. We say an object $K$ is a {\it derived limit}, or a {\it homotopy limit} of the system $(K_n)$ if the product $\prod K_n$ exists and there is a distinguished triangle $$ K \to \prod K_n \to \prod K_n \to K[1] $$ where the map $\prod K_n \to \prod K_n$ is given by $(k_n) \mapsto (k_n - f_{n+1}(k_{n + 1}))$. If this is the case, then we sometimes indicate this by the notation $K = R\lim K_n$.


derived-definition-generators

Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. \begin{enumerate} \item We say $E$ is a {\it classical generator} of $\mathcal{D}$ if the smallest strictly full, saturated, triangulated subcategory of $\mathcal{D}$ containing $E$ is equal to $\mathcal{D}$, in other words, if $\langle E \rangle = \mathcal{D}$. \item We say $E$ is a {\it strong generator} of $\mathcal{D}$ if $\langle E \rangle_n = \mathcal{D}$ for some $n \geq 1$. \item We say $E$ is a {\it weak generator} or a {\it generator} of $\mathcal{D}$ if for any nonzero object $K$ of $\mathcal{D}$ there exists an integer $n$ and a nonzero map $E \to K[n]$. \end{enumerate}


derived-definition-compact-object

Let $\mathcal{D}$ be an additive category with arbitrary direct sums. A {\it compact object} of $\mathcal{D}$ is an object $K$ such that the map $$ \bigoplus\nolimits_{i \in I} \Hom_{\mathcal{D}}(K, E_i) \longrightarrow \Hom_{\mathcal{D}}(K, \bigoplus\nolimits_{i \in I} E_i) $$ is bijective for any set $I$ and objects $E_i \in \Ob(\mathcal{D})$ parametrized by $i \in I$.


derived-definition-compactly-generated

Let $\mathcal{D}$ be a triangulated category with arbitrary direct sums. We say $\mathcal{D}$ is {\it compactly generated} if there exists a set $E_i$, $i \in I$ of compact objects such that $\bigoplus E_i$ generates $\mathcal{D}$.


derived-definition-orthogonal

Let $\mathcal{D}$ be an additive category. Let $\mathcal{A} \subset \mathcal{D}$ be a full subcategory. The {\it right orthogonal} $\mathcal{A}^\perp$ of $\mathcal{A}$ is the full subcategory consisting of the objects $X$ of $\mathcal{D}$ such that $\Hom(A, X) = 0$ for all $A \in \Ob(\mathcal{A})$. The {\it left orthogonal} ${}^\perp\mathcal{A}$ of $\mathcal{A}$ is the full subcategory consisting of the objects $X$ of $\mathcal{D}$ such that $\Hom(X, A) = 0$ for all $A \in \Ob(\mathcal{A})$.


derived-definition-admissible

Let $\mathcal{D}$ be a triangulated category. A {\it right admissible} subcategory of $\mathcal{D}$ is a strictly full triangulated subcategory satisfying the equivalent conditions of Lemma \ref{lemma-right-adjoint}. A {\it left admissible} subcategory of $\mathcal{D}$ is a strictly full triangulated subcategory satisfying the equivalent conditions of Lemma \ref{lemma-left-adjoint}. A {\it two-sided admissible} subcategory is one which is both right and left admissible.


derived-definition-postnikov-system

Let $\mathcal{D}$ be a triangulated category. Let $$ X_n \to X_{n - 1} \to \ldots \to X_0 $$ be a complex in $\mathcal{D}$. A {\it Postnikov system} is defined inductively as follows. \begin{enumerate} \item If $n = 0$, then it is an isomorphism $Y_0 \to X_0$. \item If $n = 1$, then it is a choice of a distinguished triangle $$ Y_1 \to X_1 \to Y_0 \to Y_1[1] $$ where $X_1 \to Y_0$ composed with $Y_0 \to X_0$ is the given morphism $X_1 \to X_0$. \item If $n > 1$, then it is a choice of a Postnikov system for $X_{n - 1} \to \ldots \to X_0$ and a choice of a distinguished triangle $$ Y_n \to X_n \to Y_{n - 1} \to Y_n[1] $$ where the morphism $X_n \to Y_{n - 1}$ composed with $Y_{n - 1} \to X_{n - 1}$ is the given morphism $X_n \to X_{n - 1}$. \end{enumerate} Given a morphism \begin{equation} \label{equation-map-complexes} \vcenter{ \xymatrix{ X_n \ar[r] \ar[d] & X_{n - 1} \ar[r] \ar[d] & \ldots \ar[r] & X_0 \ar[d] \\ X'_n \ar[r] & X'_{n - 1} \ar[r] & \ldots \ar[r] & X'_0 } } \end{equation} between complexes of the same length in $\mathcal{D}$ there is an obvious notion of a {\it morphism of Postnikov systems}.


cohomology-definition-torsor

Let $X$ be a topological space. Let $\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $X$. A {\it torsor}, or more precisely a {\it $\mathcal{G}$-torsor}, is a sheaf of sets $\mathcal{F}$ on $X$ endowed with an action $\mathcal{G} \times \mathcal{F} \to \mathcal{F}$ such that \begin{enumerate} \item whenever $\mathcal{F}(U)$ is nonempty the action $\mathcal{G}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ is simply transitive, and \item for every $x \in X$ the stalk $\mathcal{F}_x$ is nonempty. \end{enumerate} A {\it morphism of $\mathcal{G}$-torsors} $\mathcal{F} \to \mathcal{F}'$ is simply a morphism of sheaves of sets compatible with the $\mathcal{G}$-actions. The {\it trivial $\mathcal{G}$-torsor} is the sheaf $\mathcal{G}$ endowed with the obvious left $\mathcal{G}$-action.


cohomology-definition-cech-complex

Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Let $\mathcal{F}$ be an abelian presheaf on $X$. The complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ is the {\it {\v C}ech complex} associated to $\mathcal{F}$ and the open covering $\mathcal{U}$. Its cohomology groups $H^i(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}))$ are called the {\it {\v C}ech cohomology groups} associated to $\mathcal{F}$ and the covering $\mathcal{U}$. They are denoted $\check H^i(\mathcal{U}, \mathcal{F})$.


cohomology-definition-flasque

Let $X$ be a topological space. We say a presheaf of sets $\mathcal{F}$ is {\it flasque} or {\it flabby} if for every $U \subset V$ open in $X$ the restriction map $\mathcal{F}(V) \to \mathcal{F}(U)$ is surjective.


cohomology-definition-alternating-cech-complex

Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Let $\mathcal{F}$ be an abelian presheaf on $X$. The complex $\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})$ is the {\it alternating {\v C}ech complex} associated to $\mathcal{F}$ and the open covering $\mathcal{U}$.


cohomology-definition-ordered-cech-complex

Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Assume given a total ordering on $I$. Let $\mathcal{F}$ be an abelian presheaf on $X$. The complex $\check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{F})$ is the {\it ordered {\v C}ech complex} associated to $\mathcal{F}$, the open covering $\mathcal{U}$ and the given total ordering on $I$.


cohomology-definition-covering-locally-finite

Let $X$ be a topological space. An open covering $X = \bigcup_{i \in I} U_i$ is said to be {\it locally finite} if for every $x \in X$ there exists an open neighbourhood $W$ of $x$ such that $\{i \in I \mid W \cap U_i \not = \emptyset\}$ is finite.


cohomology-definition-K-flat

Let $(X, \mathcal{O}_X)$ be a ringed space. A complex $\mathcal{K}^\bullet$ of $\mathcal{O}_X$-modules is called {\it K-flat} if for every acyclic complex $\mathcal{F}^\bullet$ of $\mathcal{O}_X$-modules the complex $$ \text{Tot}(\mathcal{F}^\bullet \otimes_{\mathcal{O}_X} \mathcal{K}^\bullet) $$ is acyclic.


cohomology-definition-derived-tor

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}^\bullet$ be an object of $D(\mathcal{O}_X)$. The {\it derived tensor product} $$ - \otimes_{\mathcal{O}_X}^{\mathbf{L}} \mathcal{F}^\bullet : D(\mathcal{O}_X) \longrightarrow D(\mathcal{O}_X) $$ is the exact functor of triangulated categories described above.


cohomology-definition-tor

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_X$-modules. The {\it Tor}'s of $\mathcal{F}$ and $\mathcal{G}$ are define by the formula $$ \text{Tor}_p^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) = H^{-p}(\mathcal{F} \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}) $$ with derived tensor product as defined above.


cohomology-definition-strictly-perfect

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{E}^\bullet$ be a complex of $\mathcal{O}_X$-modules. We say $\mathcal{E}^\bullet$ is {\it strictly perfect} if $\mathcal{E}^i$ is zero for all but finitely many $i$ and $\mathcal{E}^i$ is a direct summand of a finite free $\mathcal{O}_X$-module for all $i$.


cohomology-definition-pseudo-coherent

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{E}^\bullet$ be a complex of $\mathcal{O}_X$-modules. Let $m \in \mathbf{Z}$. \begin{enumerate} \item We say $\mathcal{E}^\bullet$ is {\it $m$-pseudo-coherent} if there exists an open covering $X = \bigcup U_i$ and for each $i$ a morphism of complexes $\alpha_i : \mathcal{E}_i^\bullet \to \mathcal{E}^\bullet|_{U_i}$ where $\mathcal{E}_i$ is strictly perfect on $U_i$ and $H^j(\alpha_i)$ is an isomorphism for $j > m$ and $H^m(\alpha_i)$ is surjective. \item We say $\mathcal{E}^\bullet$ is {\it pseudo-coherent} if it is $m$-pseudo-coherent for all $m$. \item We say an object $E$ of $D(\mathcal{O}_X)$ is {\it $m$-pseudo-coherent} (resp.\ {\it pseudo-coherent}) if and only if it can be represented by a $m$-pseudo-coherent (resp.\ pseudo-coherent) complex of $\mathcal{O}_X$-modules. \end{enumerate}


cohomology-definition-tor-amplitude

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $a, b \in \mathbf{Z}$ with $a \leq b$. \begin{enumerate} \item We say $E$ has {\it tor-amplitude in $[a, b]$} if $H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}) = 0$ for all $\mathcal{O}_X$-modules $\mathcal{F}$ and all $i \not \in [a, b]$. \item We say $E$ has {\it finite tor dimension} if it has tor-amplitude in $[a, b]$ for some $a, b$. \item We say $E$ {\it locally has finite tor dimension} if there exists an open covering $X = \bigcup U_i$ such that $E|_{U_i}$ has finite tor dimension for all $i$. \end{enumerate} An $\mathcal{O}_X$-module $\mathcal{F}$ has {\it tor dimension $\leq d$} if $\mathcal{F}[0]$ viewed as an object of $D(\mathcal{O}_X)$ has tor-amplitude in $[-d, 0]$.


cohomology-definition-perfect

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{E}^\bullet$ be a complex of $\mathcal{O}_X$-modules. We say $\mathcal{E}^\bullet$ is {\it perfect} if there exists an open covering $X = \bigcup U_i$ such that for each $i$ there exists a morphism of complexes $\mathcal{E}_i^\bullet \to \mathcal{E}^\bullet|_{U_i}$ which is a quasi-isomorphism with $\mathcal{E}_i^\bullet$ a strictly perfect complex of $\mathcal{O}_{U_i}$-modules. An object $E$ of $D(\mathcal{O}_X)$ is {\it perfect} if it can be represented by a perfect complex of $\mathcal{O}_X$-modules.


stacks-introduction-definition-smooth

We say a morphism $S \to \mathcal{M}_{1, 1}$ is {\it smooth} if for every morphism $S' \to \mathcal{M}_{1, 1}$ the projection morphism $$ S \times_{\mathcal{M}_{1, 1}} S' \longrightarrow S' $$ is smooth.


stacks-introduction-definition-algebraic-stack

We say $\mathcal{M}_{1, 1}$ is an {\it algebraic stack} if and only if \begin{enumerate} \item We have descent for objects for the \'etale topology on $\Sch$. \item The key fact holds. \item there exists a surjective and smooth morphism $S \to \mathcal{M}_{1, 1}$. \end{enumerate}


restricted-definition-rig-smooth-homomorphism

Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Let $B$ be an object of (\ref{equation-C-prime}). We say $B$ is {\it rig-smooth over $(A, I)$} if there exists an integer $c \geq 0$ such that $I^c$ annihilates $\Ext^1_B(\NL_{B/A}^\wedge, N)$ for every $B$-module $N$.


restricted-definition-rig-etale-homomorphism

Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Let $B$ be an object of (\ref{equation-C-prime}). We say $B$ is {\it rig-\'etale over $(A, I)$} if there exists an integer $c \geq 0$ such that for all $a \in I^c$ multiplication by $a$ on $\NL_{B/A}^\wedge$ is zero in $D(B)$.


restricted-definition-flat

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is {\it flat} if for every commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and \'etale, the morphism $U \to V$ corresponds to a flat map of adic Noetherian topological rings.


restricted-definition-rig-closed

Let $A$ be a Noetherian adic topological ring. Let $\mathfrak q \subset A$ be a prime ideal. We say $\mathfrak q$ is {\it rig-closed} if the equivalent conditions of Lemma \ref{lemma-rig-point} are satisfied.


restricted-definition-completed-principal-localization

Let $A$ be an adic topological ring which has a finitely generated ideal of definition. Let $f \in A$. The {\it completed principal localization} $A_{\{f\}}$ of $A$ is the completion of $A_f = A[1/f]$ of the principal localization of $A$ at $f$ with respect to any ideal of definition of $A$.


restricted-definition-naively-rig-flat

Let $\varphi : A \to B$ be a continuous ring homomorphism between adic Noetherian topological rings, i.e., $\varphi$ is an arrow of $\textit{WAdm}^{Noeth}$. We say $\varphi$ is {\it naively rig-flat} if $\varphi$ is adic, topologically of finite type, and satisfies the equivalent conditions of Lemma \ref{lemma-naively-rig-flat-continuous}.


restricted-definition-rig-flat-continuous-homomorphism

Let $\varphi : A \to B$ be a continuous ring homomorphism between adic Noetherian topological rings, i.e., $\varphi$ is an arrow of $\textit{WAdm}^{Noeth}$. We say $\varphi$ is {\it rig-flat} if $\varphi$ is adic, topologically of finite type, and for all $f \in A$ the induced map $$ A_{\{f\}} \longrightarrow B_{\{f\}} $$ is naively rig-flat (Definition \ref{definition-naively-rig-flat}).


restricted-definition-rig-flat

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is {\it rig-flat} if for every commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and \'etale, the morphism $U \to V$ corresponds to a rig-flat map of adic Noetherian topological rings.


restricted-definition-rig-smooth-continuous-homomorphism

Let $\varphi : A \to B$ be a continuous ring homomorphism between adic Noetherian topological rings, i.e., $\varphi$ is an arrow of $\textit{WAdm}^{Noeth}$. We say $\varphi$ is {\it rig-smooth} if the equivalent conditions of Lemma \ref{lemma-rig-smooth-continuous} hold.


restricted-definition-rig-smooth

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is {\it rig-smooth} if for every commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and \'etale, the morphism $U \to V$ corresponds to a rig-smooth map of adic Noetherian topological rings.


restricted-definition-rig-etale-continuous-homomorphism

Let $\varphi : A \to B$ be a continuous ring homomorphism between adic Noetherian topological rings, i.e., $\varphi$ is an arrow of $\textit{WAdm}^{Noeth}$. We say $\varphi$ is {\it rig-etale} if the equivalent conditions of Lemma \ref{lemma-rig-etale-continuous} hold.


restricted-definition-rig-etale

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is {\it rig-\'etale} if for every commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and \'etale, the morphism $U \to V$ corresponds to a rig-\'etale map of adic Noetherian topological rings.


restricted-definition-rig-surjective

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. Assume that $X$ and $Y$ are locally Noetherian and that $f$ is locally of finite type. We say $f$ is {\it rig-surjective} if for every solid diagram $$ \xymatrix{ \text{Spf}(R') \ar@{..>}[r] \ar@{..>}[d] & X \ar[d]^f \\ \text{Spf}(R) \ar[r]^-p & Y } $$ where $R$ is a complete discrete valuation ring and where $p$ is an adic morphism there exists an extension of complete discrete valuation rings $R \subset R'$ and a morphism $\text{Spf}(R') \to X$ making the displayed diagram commute.


restricted-definition-formal-modification

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is a {\it formal modification} if \begin{enumerate} \item $f$ is a proper morphism (Formal Spaces, Definition \ref{formal-spaces-definition-proper}), \item $f$ is rig-\'etale, \item $f$ is rig-surjective, \item $\Delta_f : X \to X \times_Y X$ is rig-surjective. \end{enumerate}


bootstrap-definition-morphism-representable-by-spaces

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F$, $G$ be presheaves on $\Sch_{fppf}/S$. We say a morphism $a : F \to G$ is {\it representable by algebraic spaces} if for every $U \in \Ob((\Sch/S)_{fppf})$ and any $\xi : U \to G$ the fiber product $U \times_{\xi, G} F$ is an algebraic space.


bootstrap-definition-property-transformation

Let $S$ be a scheme. Let $a : F \to G$ be a map of presheaves on $(\Sch/S)_{fppf}$ which is representable by algebraic spaces. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which \begin{enumerate} \item is preserved under any base change, and \item is fppf local on the base, see Descent on Spaces, Definition \ref{spaces-descent-definition-property-morphisms-local}. \end{enumerate} In this case we say that $a$ has {\it property $\mathcal{P}$} if for every scheme $U$ and $\xi : U \to G$ the resulting morphism of algebraic spaces $U \times_G F \to U$ has property $\mathcal{P}$.


spaces-perfect-definition-supported-on

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $T \subset |X|$ be a closed subset. We say $E$ is {\it supported on $T$} if the cohomology sheaves $H^i(E)$ are supported on $T$.


spaces-perfect-definition-derived-quasi-coherent

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\it derived category of $\mathcal{O}_X$-modules with quasi-coherent cohomology sheaves} is denoted $D_\QCoh(\mathcal{O}_X)$.


spaces-perfect-definition-proper-over-base

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $T \subset |X|$ be a closed subset. We say {\it $T$ is proper over $Y$} if the equivalent conditions of Lemma \ref{lemma-closed-proper-over-base} are satisfied.


spaces-perfect-definition-elementary-distinguished-square

Let $S$ be a scheme. A commutative diagram $$ \xymatrix{ U \times_W V \ar[r] \ar[d] & V \ar[d]^f \\ U \ar[r]^j & W } $$ of algebraic spaces over $S$ is called an {\it elementary distinguished square} if \begin{enumerate} \item $U$ is an open subspace of $W$ and $j$ is the inclusion morphism, \item $f$ is \'etale, and \item setting $T = W \setminus U$ (with reduced induced subspace structure) the morphism $f^{-1}(T) \to T$ is an isomorphism. \end{enumerate} We will indicate this by saying: ``Let $(U \subset W, f : V \to W)$ be an elementary distinguished square.''


spaces-perfect-definition-approximation-holds

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider triples $(T, E, m)$ where \begin{enumerate} \item $T \subset |X|$ is a closed subset, \item $E$ is an object of $D_\QCoh(\mathcal{O}_X)$, and \item $m \in \mathbf{Z}$. \end{enumerate} We say {\it approximation holds for the triple} $(T, E, m)$ if there exists a perfect object $P$ of $D(\mathcal{O}_X)$ supported on $T$ and a map $\alpha : P \to E$ which induces isomorphisms $H^i(P) \to H^i(E)$ for $i > m$ and a surjection $H^m(P) \to H^m(E)$.


spaces-perfect-definition-approximation

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say {\it approximation by perfect complexes holds} on $X$ if for any closed subset $T \subset |X|$ such that the morphism $X \setminus T \to X$ is quasi-compact there exists an integer $r$ such that for every triple $(T, E, m)$ as in Definition \ref{definition-approximation-holds} with \begin{enumerate} \item $E$ is $(m - r)$-pseudo-coherent, and \item $H^i(E)$ is supported on $T$ for $i \geq m - r$ \end{enumerate} approximation holds.


spaces-perfect-definition-tor-independent

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X$, $Y$ be algebraic spaces over $B$. We say $X$ and $Y$ are {\it Tor independent over $B$} if and only if for every commutative diagram $$ \xymatrix{ \Spec(k) \ar[d]_{\overline{y}} \ar[dr]_{\overline{b}} \ar[r]_-{\overline{x}} & X \ar[d] \\ Y \ar[r] & B } $$ of geometric points the rings $\mathcal{O}_{X, \overline{x}}$ and $\mathcal{O}_{Y, \overline{y}}$ are Tor independent over $\mathcal{O}_{B, \overline{b}}$ (see More on Algebra, Definition \ref{more-algebra-definition-tor-independent}).


dualizing-definition-essential

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item An injection $A \subset B$ of $\mathcal{A}$ is {\it essential}, or we say that $B$ is an {\it essential extension of} $A$, if every nonzero subobject $B' \subset B$ has nonzero intersection with $A$. \item A surjection $f : A \to B$ of $\mathcal{A}$ is {\it essential} if for every proper subobject $A' \subset A$ we have $f(A') \not = B$. \end{enumerate}


dualizing-definition-projective-cover

Let $R$ be a ring. A surjection $P \to M$ of $R$-modules is said to be a {\it projective cover}, or sometimes a {\it projective envelope}, if $P$ is a projective $R$-module and $P \to M$ is an essential surjection.


dualizing-definition-injective-hull

Let $R$ be a ring. A injection $M \to I$ of $R$-modules is said to be an {\it injective hull} if $I$ is a injective $R$-module and $M \to I$ is an essential injection.


dualizing-definition-indecomposable

An object $X$ of an additive category is called {\it indecomposable} if it is nonzero and if $X = Y \oplus Z$, then either $Y = 0$ or $Z = 0$.


dualizing-definition-dualizing

Let $A$ be a Noetherian ring. A {\it dualizing complex} is a complex of $A$-modules $\omega_A^\bullet$ such that \begin{enumerate} \item $\omega_A^\bullet$ has finite injective dimension, \item $H^i(\omega_A^\bullet)$ is a finite $A$-module for all $i$, and \item $A \to R\Hom_A(\omega_A^\bullet, \omega_A^\bullet)$ is a quasi-isomorphism. \end{enumerate}


dualizing-definition-gorenstein

Gorenstein rings. \begin{enumerate} \item Let $A$ be a Noetherian local ring. We say $A$ is {\it Gorenstein} if $A[0]$ is a dualizing complex for $A$. \item Let $A$ be a Noetherian ring. We say $A$ is {\it Gorenstein} if $A_\mathfrak p$ is Gorenstein for every prime $\mathfrak p$ of $A$. \end{enumerate}


dualizing-definition-relative-dualizing-complex

Let $R \to A$ be a flat ring map of finite presentation. A {\it relative dualizing complex} is an object $K \in D(A)$ such that \begin{enumerate} \item $K$ is $R$-perfect (More on Algebra, Definition \ref{more-algebra-definition-relatively-perfect}), and \item $R\Hom_{A \otimes_R A}(A, K \otimes_A^\mathbf{L} (A \otimes_R A))$ is isomorphic to $A$. \end{enumerate}


properties-definition-integral

Let $X$ be a scheme. We say $X$ is {\it integral} if it is nonempty and for every nonempty affine open $\Spec(R) = U \subset X$ the ring $R$ is an integral domain.


properties-definition-property-local

Let $P$ be a property of rings. We say that $P$ is {\it local} if the following hold: \begin{enumerate} \item For any ring $R$, and any $f \in R$ we have $P(R) \Rightarrow P(R_f)$. \item For any ring $R$, and $f_i \in R$ such that $(f_1, \ldots, f_n) = R$ then $\forall i, P(R_{f_i}) \Rightarrow P(R)$. \end{enumerate}


properties-definition-locally-P

Let $P$ be a property of rings. Let $X$ be a scheme. We say $X$ is {\it locally $P$} if for any $x \in X$ there exists an affine open neighbourhood $U$ of $x$ in $X$ such that $\mathcal{O}_X(U)$ has property $P$.


properties-definition-noetherian

Let $X$ be a scheme. \begin{enumerate} \item We say $X$ is {\it locally Noetherian} if every $x \in X$ has an affine open neighbourhood $\Spec(R) = U \subset X$ such that the ring $R$ is Noetherian. \item We say $X$ is {\it Noetherian} if $X$ is locally Noetherian and quasi-compact. \end{enumerate}


properties-definition-jacobson

A scheme $S$ is said to be {\it Jacobson} if its underlying topological space is Jacobson.


properties-definition-normal

A scheme $X$ is {\it normal} if and only if for all $x \in X$ the local ring $\mathcal{O}_{X, x}$ is a normal domain.


properties-definition-Cohen-Macaulay

Let $X$ be a scheme. We say $X$ is {\it Cohen-Macaulay} if for every $x \in X$ there exists an affine open neighbourhood $U \subset X$ of $x$ such that the ring $\mathcal{O}_X(U)$ is Noetherian and Cohen-Macaulay.


properties-definition-regular

Let $X$ be a scheme. We say $X$ is {\it regular}, or {\it nonsingular} if for every $x \in X$ there exists an affine open neighbourhood $U \subset X$ of $x$ such that the ring $\mathcal{O}_X(U)$ is Noetherian and regular.


properties-definition-dimension

Let $X$ be a scheme. \begin{enumerate} \item The {\it dimension} of $X$ is just the dimension of $X$ as a topological spaces, see Topology, Definition \ref{topology-definition-Krull}. \item For $x \in X$ we denote $\dim_x(X)$ the dimension of the underlying topological space of $X$ at $x$ as in Topology, Definition \ref{topology-definition-Krull}. We say $\dim_x(X)$ is the {\it dimension of $X$ at $x$}. \end{enumerate}


properties-definition-catenary

Let $S$ be a scheme. We say $S$ is {\it catenary} if the underlying topological space of $S$ is catenary.


properties-definition-Rk

Let $X$ be a locally Noetherian scheme. Let $k \geq 0$. \begin{enumerate} \item We say $X$ is {\it regular in codimension $k$}, or we say $X$ has property {\it $(R_k)$} if for every $x \in X$ we have $$ \dim(\mathcal{O}_{X, x}) \leq k \Rightarrow \mathcal{O}_{X, x}\text{ is regular} $$ \item We say $X$ has property {\it $(S_k)$} if for every $x \in X$ we have $\text{depth}(\mathcal{O}_{X, x}) \geq \min(k, \dim(\mathcal{O}_{X, x}))$. \end{enumerate}


properties-definition-nagata

Let $X$ be a scheme. \begin{enumerate} \item Assume $X$ integral. We say $X$ is {\it Japanese} if for every $x \in X$ there exists an affine open neighbourhood $x \in U \subset X$ such that the ring $\mathcal{O}_X(U)$ is Japanese (see Algebra, Definition \ref{algebra-definition-N}). \item We say $X$ is {\it universally Japanese} if for every $x \in X$ there exists an affine open neighbourhood $x \in U \subset X$ such that the ring $\mathcal{O}_X(U)$ is universally Japanese (see Algebra, Definition \ref{algebra-definition-nagata}). \item We say $X$ is {\it Nagata} if for every $x \in X$ there exists an affine open neighbourhood $x \in U \subset X$ such that the ring $\mathcal{O}_X(U)$ is Nagata (see Algebra, Definition \ref{algebra-definition-nagata}). \end{enumerate}


properties-definition-singular-locus

Let $X$ be a locally Noetherian scheme. The {\it regular locus} $\text{Reg}(X)$ of $X$ is the set of $x \in X$ such that $\mathcal{O}_{X, x}$ is a regular local ring. The {\it singular locus} $\text{Sing}(X)$ is the complement $X \setminus \text{Reg}(X)$, i.e., the set of points $x \in X$ such that $\mathcal{O}_{X, x}$ is not a regular local ring.


properties-definition-unibranch

\begin{reference} \cite[Chapter IV (6.15.1)]{EGA4} \end{reference} Let $X$ be a scheme. Let $x \in X$. We say $X$ is {\it unibranch at $x$} if the local ring $\mathcal{O}_{X, x}$ is unibranch. We say $X$ is {\it geometrically unibranch at $x$} if the local ring $\mathcal{O}_{X, x}$ is geometrically unibranch. We say $X$ is {\it unibranch} if $X$ is unibranch at all of its points. We say $X$ is {\it geometrically unibranch} if $X$ is geometrically unibranch at all of its points.


properties-definition-number-of-branches

Let $X$ be a scheme. Let $x \in X$. The {\it number of branches of $X$ at $x$} is the number of branches of the local ring $\mathcal{O}_{X, x}$ as defined in More on Algebra, Definition \ref{more-algebra-definition-number-of-branches}. The {\it number of geometric branches of $X$ at $x$} is the number of geometric branches of the local ring $\mathcal{O}_{X, x}$ as defined in More on Algebra, Definition \ref{more-algebra-definition-number-of-branches}.


properties-definition-quasi-affine

A scheme $X$ is called {\it quasi-affine} if it is quasi-compact and isomorphic to an open subscheme of an affine scheme.


properties-definition-locally-projective

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. We say $\mathcal{F}$ is {\it locally projective} if for every affine open $U \subset X$ the $\mathcal{O}_X(U)$-module $\mathcal{F}(U)$ is projective.


properties-definition-kappa-generated

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\kappa$ be an infinite cardinal. We say a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is {\it $\kappa$-generated} if there exists an open covering $X = \bigcup U_i$ such that $\mathcal{F}|_{U_i}$ is generated by a subset $R_i \subset \mathcal{F}(U_i)$ whose cardinality is at most $\kappa$.


properties-definition-subsheaf-sections-annihilated-by-ideal

Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The subsheaf $\mathcal{F}' \subset \mathcal{F}$ defined in Lemma \ref{lemma-sections-annihilated-by-ideal} above is called the {\it subsheaf of sections annihilated by $\mathcal{I}$}.


properties-definition-subsheaf-sections-supported-on-closed

Let $X$ be a scheme. Let $T \subset X$ be a closed subset whose complement is retrocompact in $X$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The quasi-coherent subsheaf $\mathcal{F}' \subset \mathcal{F}$ defined in Lemma \ref{lemma-sections-supported-on-closed-subset} is called the {\it subsheaf of sections supported on $T$}.


properties-definition-ample

\begin{reference} \cite[II Definition 4.5.3]{EGA} \end{reference} Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. We say $\mathcal{L}$ is {\it ample} if \begin{enumerate} \item $X$ is quasi-compact, and \item for every $x \in X$ there exists an $n \geq 1$ and $s \in \Gamma(X, \mathcal{L}^{\otimes n})$ such that $x \in X_s$ and $X_s$ is affine. \end{enumerate}


criteria-definition-algebraic

Let $S$ be a scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$. We say that $F$ is {\it algebraic} if for every scheme $T$ and every object $\xi$ of $\mathcal{Y}$ over $T$ the $2$-fibre product $$ (\Sch/T)_{fppf} \times_{\xi, \mathcal{Y}} \mathcal{X} $$ is an algebraic stack over $S$.


coherent-definition-depth

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $k \geq 0$ be an integer. \begin{enumerate} \item We say $\mathcal{F}$ has {\it depth $k$ at a point} $x$ of $X$ if $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) = k$. \item We say $X$ has {\it depth $k$ at a point} $x$ of $X$ if $\text{depth}(\mathcal{O}_{X, x}) = k$. \item We say $\mathcal{F}$ has property {\it $(S_k)$} if $$ \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) \geq \min(k, \dim(\text{Supp}(\mathcal{F}_x))) $$ for all $x \in X$. \item We say $X$ has property {\it $(S_k)$} if $\mathcal{O}_X$ has property $(S_k)$. \end{enumerate}


coherent-definition-Cohen-Macaulay

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. We say $\mathcal{F}$ is {\it Cohen-Macaulay} if and only if $(S_k)$ holds for all $k \geq 0$.


coherent-definition-proper-over-base

Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $Z \subset X$ be a closed subset. We say {\it $Z$ is proper over $S$} if the equivalent conditions of Lemma \ref{lemma-closed-proper-over-base} are satisfied.


formal-defos-definition-CLambda

Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k$ be a finite ring map where $k$ is a field. We define {\it $\mathcal{C}_\Lambda$} to be the category with \begin{enumerate} \item objects are pairs $(A, \varphi)$ where $A$ is an Artinian local $\Lambda$-algebra and where $\varphi : A/\mathfrak m_A \to k$ is a $\Lambda$-algebra isomorphism, and \item morphisms $f : (B, \psi) \to (A, \varphi)$ are local $\Lambda$-algebra homomorphisms such that $\varphi \circ (f \bmod \mathfrak m) = \psi$. \end{enumerate} We say we are in the {\it classical case} if $\Lambda$ is a Noetherian complete local ring and $k$ is its residue field.


formal-defos-definition-small-extension

Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$.  We say $f$ is a {\it small extension} if it is surjective and $\Ker(f)$ is a nonzero principal ideal which is annihilated by $\mathfrak{m}_B$.


formal-defos-definition-tangent-space-ring

Let $R \to S$ be a local homomorphism of local rings. The {\it relative cotangent space}\footnote{Caution: We will see later that in our general setting the tangent space of an object $A \in \mathcal{C}_\Lambda$ over $\Lambda$ should not be defined simply as the $k$-linear dual of the relative cotangent space. In fact, the correct definition of the relative cotangent space is $\Omega_{S/R} \otimes_S S/\mathfrak m_S$.} of $R$ over $S$ is the $S/\mathfrak m_S$-vector space $\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)$.


formal-defos-definition-essential-surjection

Let $f: B \to A$ be a ring map in $\mathcal{C}_\Lambda$.  We say $f$ is an {\it essential surjection} if it has the following properties: \begin{enumerate} \item $f$ is surjective. \item If $g: C \to B$ is a ring map in $\mathcal{C}_\Lambda$ such that $f \circ g$ is surjective, then $g$ is surjective. \end{enumerate}


formal-defos-definition-completion-CLambda

Let $\Lambda$ be a Noetherian ring and let $\Lambda \to k$ be a finite ring map where $k$ is a field. We define {\it $\widehat{\mathcal{C}}_\Lambda$} to be the category with \begin{enumerate} \item objects are pairs $(R, \varphi)$ where $R$ is a Noetherian complete local $\Lambda$-algebra and where $\varphi : R/\mathfrak m_R \to k$ is a $\Lambda$-algebra isomorphism, and \item morphisms $f : (S, \psi) \to (R, \varphi)$ are local $\Lambda$-algebra homomorphisms such that $\varphi \circ (f \bmod \mathfrak m) = \psi$. \end{enumerate}


formal-defos-definition-category-cofibred-groupoids

Let $\mathcal{C}$ be a category.  A {\it category cofibered in groupoids over $\mathcal{C}$} is a category $\mathcal{F}$ equipped with a functor $p: \mathcal{F} \to \mathcal{C}$ such that $\mathcal{F}^{opp}$ is a category fibered in groupoids over $\mathcal{C}^{opp}$ via $p^{opp}: \mathcal{F}^{opp} \to \mathcal{C}^{opp}$.


formal-defos-definition-prorepresentable

Let $F : \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. We say $F$ is {\it prorepresentable} if there exists an isomorphism $F \cong \underline{R}|_{\mathcal{C}_\Lambda}$ of functors for some $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$.


formal-defos-definition-predeformation-category

A {\it predeformation category} $\mathcal{F}$ is a category cofibered in groupoids over $\mathcal{C}_\Lambda$ such that $\mathcal{F}(k)$ is equivalent to a category with a single object and a single morphism, i.e., $\mathcal{F}(k)$ contains at least one object and there is a unique morphism between any two objects. A {\it morphism of predeformation categories} is a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$.


formal-defos-definition-formal-objects

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. The {\it category $\widehat{\mathcal{F}}$ of formal objects of  $\mathcal{F}$} is the category with the following objects and morphisms. \begin{enumerate} \item A {\it formal object $\xi = (R, \xi_n, f_n)$ of $\mathcal{F}$} consists of an object $R$ of $\widehat{\mathcal{C}}_\Lambda$, and a collection indexed by $n \in \mathbf{N}$ of objects $\xi_n$ of $\mathcal{F}(R/\mathfrak m_R^n)$ and morphisms $f_n : \xi_{n + 1} \to \xi_n$ lying over the projection $R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n$. \item Let $\xi = (R, \xi_n, f_n)$ and $\eta = (S, \eta_n, g_n)$ be formal objects of $\mathcal{F}$.  A {\it morphism $a : \xi \to \eta$ of formal objects} consists of a map $a_0 : R \to S$ in $\widehat{\mathcal{C}}_\Lambda$ and a collection $a_n : \xi_n \to \eta_n$ of morphisms of $\mathcal{F}$ lying over $R/\mathfrak m_R^n \to S/\mathfrak m_S^n$, such that for every $n$ the diagram $$ \xymatrix{ \xi_{n + 1} \ar[r]_{f_n} \ar[d]_{a_{n + 1}} & \xi_n \ar[d]^{a_n} \\ \eta_{n + 1} \ar[r]^{g_n} & \eta_n } $$ commutes. \end{enumerate}


formal-defos-definition-completion

Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids. The category cofibered in groupoids $\widehat{p} : \widehat{\mathcal  F} \to \widehat{\mathcal{C}}_\Lambda$ is called the {\it completion of $\mathcal{F}$}.


formal-defos-definition-smooth-morphism

Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$.  We say  $\varphi$ is {\it smooth} if it satisfies the following condition: Let $B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$.  Let $y \in \Ob(\mathcal{G}(B)), x \in \Ob(\mathcal{F}(A))$, and $y \to \varphi(x)$ be a morphism lying over $B \to A$.  Then there exists $x' \in \Ob(\mathcal{F}(B))$, a morphism $x' \to x$ lying over $B \to A$, and a morphism $\varphi(x') \to y$ lying over $\text{id}: B \to B$, such that the diagram $$ \xymatrix{ \varphi(x') \ar[r] \ar[dr] & y \ar[d] \\ & \varphi(x) } $$ commutes.


formal-defos-definition-versal

Let $\mathcal{F}$ be a category cofibered in groupoids.  Let $\xi$ be a formal object of $\mathcal{F}$ lying over $R \in \Ob(\widehat{\mathcal{C}}_\Lambda)$. We say $\xi$ is {\it versal} if the corresponding morphism $\underline{\xi}: \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$ of Remark \ref{remark-formal-objects-yoneda} is smooth.


formal-defos-definition-cofibered-groupoid-projection-smooth

Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids. We say $\mathcal{F}$ is {\it smooth} or {\it unobstructed} if its structure morphism $p$ is smooth in the sense of Definition \ref{definition-smooth-morphism}.


formal-defos-definition-S1-S2

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$. We define {\it conditions (S1) and (S2)} on $\mathcal{F}$ as follows: \begin{enumerate} \item[(S1)] Every diagram in $\mathcal{F}$ $$ \vcenter{ \xymatrix{            & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{            & A_2 \ar[d] \\ A_1 \ar[r] & A } } $$ in $\mathcal{C}_\Lambda$ with $A_2 \to A$ surjective can be completed to a commutative diagram $$ \vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r]      & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r]      & A. } } $$ \item[(S2)] The condition of (S1) holds for diagrams in $\mathcal{F}$ lying over a diagram in $\mathcal{C}_\Lambda$ of the form $$ \xymatrix{           & k[\epsilon] \ar[d] \\ A  \ar[r] & k. } $$ Moreover, if we have two commutative diagrams in $\mathcal{F}$ $$ \vcenter{ \xymatrix{ y \ar[r]_c \ar[d]_a & x_\epsilon \ar[d]^e \\ x \ar[r]^d          & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r]_{c'} \ar[d]_{a'} & x_\epsilon \ar[d]^e \\ x \ar[r]^d                 & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\ A  \ar[r] & k } } $$ then there exists a morphism $b : y \to y'$ in $\mathcal{F}(A \times_k k[\epsilon])$ such that $a = a' \circ b$. \end{enumerate}


formal-defos-definition-linear

Let $L: \text{Mod}^{fg}_R \to \text{Mod}_R$, resp.\ $L: \text{Mod}_R \to \text{Mod}_R$ be a functor.  We say that $L$ is {\it $R$-linear} if for every pair of objects $M, N$ of $\text{Mod}^{fg}_R$, resp.\ $\text{Mod}_R$ the map $$ L : \Hom_R(M, N) \longrightarrow \Hom_R(L(M), L(N)) $$ is a map of $R$-modules.


formal-defos-definition-tangent-space-over-R

Let $\mathcal{C}$ be a category as in Lemma \ref{lemma-tangent-space-functor}. Let $F : \mathcal{C} \to \textit{Sets}$ be a functor such that $F(R)$ is a one element set. The {\it tangent space $TF$ of $F$} is $F(R[\epsilon])$.


formal-defos-definition-tangent-space

Let $\mathcal{F}$ be a predeformation category. The {\it tangent space $T \mathcal{F}$ of $\mathcal{F}$} is the set $\overline{\mathcal{F}}(k[\epsilon])$ of isomorphism classes of objects in the fiber category $\mathcal F(k[\epsilon])$.


formal-defos-definition-differential

Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism predeformation categories. The {\it differential $d \varphi : T \mathcal{F} \to T \mathcal{G}$ of $\varphi$} is the map obtained by evaluating the morphism of functors $\overline{\varphi}: \overline{\mathcal{F}} \to \overline{\mathcal{G}}$ at $A = k[\epsilon]$.


formal-defos-definition-minimal-versal

Let $\mathcal{F}$ be a predeformation category. We say a versal formal object $\xi$ of $\mathcal{F}$ is {\it minimal}\footnote{This may be nonstandard terminology. Many authors tie this notion in with properties of tangent spaces. We will make the link in Section \ref{section-miniversal-objects-existence}.} if for any morphism of formal objects $\xi' \to \xi$ the underlying map on rings is surjective. Sometimes a minimal versal formal object is called {\it miniversal}.


formal-defos-definition-RS

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$.  We say that $\mathcal{F}$ satisfies {\it condition (RS)} if for every diagram in $\mathcal{F}$ $$ \vcenter{ \xymatrix{            & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{            & A_2 \ar[d] \\ A_1 \ar[r] & A } } $$ in $\mathcal{C}_\Lambda$ with $A_2 \to A$ surjective, there exists a fiber product $x_1 \times_x x_2$ in $\mathcal{F}$ such that the diagram $$ \vcenter{ \xymatrix{ x_1 \times_x x_2 \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r]      & x } } \quad\text{lies over}\quad \vcenter{ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r]      & A. } } $$


formal-defos-definition-deformation-category

A {\it deformation category} is a predeformation category $\mathcal{F}$ satisfying (RS). A morphism of deformation categories is a morphism of categories over $\mathcal{C}_\Lambda$.


formal-defos-definition-lifts

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $f: A' \to A$ be a map in $\mathcal{C}_\Lambda$. Let $x \in \mathcal{F}(A)$. The category $\textit{Lift}(x, f)$ of lifts of $x$ along $f$ is the category with the following objects and morphisms. \begin{enumerate} \item Objects: A {\it lift of $x$ along $f$} is a morphism $x' \to x$ lying over $f$. \item Morphisms: A {\it morphism of lifts} from $a_1 : x'_1 \to x$ to $a_2 : x'_2 \to x$ is a morphism $b : x'_1 \to x'_2$ in $\mathcal{F}(A')$ such that $a_2 = a_1 \circ b$. \end{enumerate} The set $\text{Lift}(x, f)$ of lifts of $x$ along $f$ is the set of isomorphism classes of $\textit{Lift}(x, f)$.


formal-defos-definition-relative-infinitesimal-auts

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$. Let $x' \to x$ be a morphism in $\mathcal{F}$ lying over $A' \to A$. The kernel $$ \text{Inf}(x'/x) = \Ker(\text{Aut}_{A'}(x') \to \text{Aut}_A(x)) $$ is the {\it group of infinitesimal automorphisms of $x'$ over $x$}.


formal-defos-definition-infinitesimal-auts

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$. Let $x_0 \in \Ob(\mathcal{F}(k))$. Assume a choice of pushforward $x_0 \to x_0'$ of $x_0$ along the map $k \to k[\epsilon], a \mapsto a$ has been made. Then there is a unique map $x'_0 \to x_0$ such that $x_0 \to x_0' \to x_0$ is the identity on $x_0$. Then $$ \text{Inf}_{x_0}(\mathcal F) = \text{Inf}(x'_0/x_0) $$ is the {\it group of infinitesimal automorphisms of $x_0$}


formal-defos-definition-automorphism-functor

Let $p : \mathcal{F} \to \mathcal{C}$ be a category cofibered in groupoids over an arbitrary base category $\mathcal{C}$. Assume a choice of pushforwards has been made. Let $x \in \Ob(\mathcal{F})$ and let $U = p(x)$. Let $U/\mathcal{C}$ denote the category of objects under $U$. The {\it automorphism functor of $x$} is the functor $\mathit{Aut}(x) : U/\mathcal{C} \to \textit{Sets}$ sending an object $f : U \to V$ to $\text{Aut}_V(f_*x)$ and sending a morphism $$ \xymatrix{ V' \ar[rr] &                    & V\\           & U \ar[ul]^{f'}  \ar[ur]_f & } $$ to the homomorphism $\text{Aut}_{V'}(f'_*x) \to \text{Aut}_V(f_*x)$ coming from the unique morphism $f'_*x \to f_*x$ lying over $V' \to V$ and compatible with $x \to f'_*x$ and $x \to f_*x$.


formal-defos-definition-groupoid-in-functors

Let $\mathcal{C}$ be a category. The {\it category of groupoids in functors on $\mathcal{C}$} is the category with the following objects and morphisms. \begin{enumerate} \item Objects: A {\it groupoid in functors on $\mathcal{C}$} is a quintuple $(U, R, s, t, c)$ where $U, R : \mathcal{C} \to \textit{Sets}$ are functors and $s, t : R \to U$ and $c : R \times_{s, U, t} R \to R$ are morphisms with the following property: For any object $T$ of $\mathcal{C}$, the quintuple $$ (U(T), R(T), s, t, c) $$ is a groupoid category. \item Morphisms: A {\it morphism $(U, R, s, t, c) \to (U', R', s', t', c')$ of groupoids in functors on $\mathcal{C}$} consists of morphisms $U \to U'$ and $R \to R'$ with the following property: For any object $T$ of $\mathcal{C}$, the induced maps $U(T) \to U'(T)$ and $R(T) \to R'(T)$ define a functor between groupoid categories $$ (U(T), R(T), s, t, c) \to (U'(T), R'(T), s', t', c'). $$ \end{enumerate}


formal-defos-definition-representable

Let $\mathcal{C}$ be a category. A groupoid in functors on $\mathcal{C}$ is {\it representable} if it is isomorphic to one of the form $(\underline{U}, \underline{R}, s, t, c)$ where $U$ and $R$ are objects of $\mathcal{C}$ and the pushout $R \amalg_{s, U, t} R$ exists.


formal-defos-definition-restricting-groupoids-in-functors

Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\mathcal{C}$. Let $\mathcal{C}'$ be a subcategory of $\mathcal{C}$. The {\it restriction $(U, R, s, t, c)|_{\mathcal{C}'}$ of $(U, R, s, t, c)$ to $\mathcal{C}'$} is the groupoid in functors on $\mathcal{C}'$ given by $(U|_{\mathcal{C}'}, R|_{\mathcal C'}, s|_{\mathcal{C}'}, t|_{\mathcal{C}'}, c|_{\mathcal{C}'})$.


formal-defos-definition-quotient

Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\mathcal{C}$. \begin{enumerate} \item The assignment $T \mapsto  (U(T), R(T), s, t, c)$ determines a functor $\mathcal{C} \to \textit{Groupoids}$. The {\it quotient category cofibered in groupoids $[U/R] \to \mathcal{C}$} is the category cofibered in groupoids over $\mathcal{C}$ associated to this functor (as in Remarks \ref{remarks-cofibered-groupoids} (\ref{item-construction-associated-cofibered-groupoid})). \item The {\it quotient morphism $U \to [U/R]$} is the morphism of categories cofibered in groupoids over $\mathcal{C}$ defined by the rules \begin{enumerate} \item $x \in U(T)$ maps to the object $(T, x) \in \Ob([U/R](T))$, and \item $x \in U(T)$ and $f : T \to T'$ give rise to the morphism $(f, \text{id}_{U(f)(x)}): (T, x) \to (T, U(f)(x))$ lying over $f : T \to T'$. \end{enumerate} \end{enumerate}


formal-defos-definition-prorepresentable-groupoid-in-functors

A groupoid in functors on $\mathcal{C}_\Lambda$ is {\it prorepresentable} if it is isomorphic to $(\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda}$ for some representable groupoid in functors $(\underline{R_0}, \underline{R_1}, s, t, c)$ on the category $\widehat{\mathcal{C}}_\Lambda$.


formal-defos-definition-completion-groupoid-in-functors

Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$. The {\it completion $(U, R, s, t, c)^{\wedge}$ of $(U, R, s, t, c)$} is the groupoid in functors $(\widehat{U}, \widehat{R}, \widehat{s}, \widehat{t}, \widehat{c})$ on $\widehat{\mathcal{C}}_\Lambda$ described above.


formal-defos-definition-smooth-groupoid-in-functors

Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}_\Lambda$. We say $(U, R, s, t, c)$ is {\it smooth} if $s, t: R \to U$ are smooth.


formal-defos-definition-presentation

Let $\mathcal{F}$ be a category cofibered in groupoids over a category $\mathcal{C}$. Let $(U, R, s, t, c)$ be a groupoid in functors on $\mathcal{C}$. A {\it presentation of $\mathcal{F}$ by $(U, R, s, t, c)$} is an equivalence $\varphi : [U/R] \to \mathcal{F}$ of categories cofibered in groupoids over $\mathcal{C}$.


formal-defos-definition-minimal-groupoid-in-functors

Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$. \begin{enumerate} \item We say $(U, R, s, t, c)$ is {\it normalized} if the groupoid $(U(k[\epsilon]), R(k[\epsilon]), s, t, c)$ is totally disconnected, i.e., there are no morphisms between distinct objects. \item We say $(U, R, s, t, c)$ is {\it minimal} if the $U \to [U/R]$ is given by a minimal versal formal object of $[U/R]$. \end{enumerate}


adequate-definition-module-valued-functor

Let $A$ be a ring. A {\it module-valued functor} is a functor $F : \textit{Alg}_A \to \textit{Ab}$ such that \begin{enumerate} \item for every object $B$ of $\textit{Alg}_A$ the group $F(B)$ is endowed with the structure of a $B$-module, and \item for any morphism $B \to B'$ of $\textit{Alg}_A$ the map $F(B) \to F(B')$ is $B$-linear. \end{enumerate} A {\it morphism of module-valued functors} is a transformation of functors $\varphi : F \to G$ such that $F(B) \to G(B)$ is $B$-linear for all $B \in \Ob(\textit{Alg}_A)$.


adequate-definition-adequate-functor

Let $A$ be a ring. A module-valued functor $F$ on $\textit{Alg}_A$ is called \begin{enumerate} \item {\it adequate} if there exists a map of $A$-modules $M \to N$ such that $F$ is isomorphic to $\Ker(\underline{M} \to \underline{N})$. \item {\it linearly adequate} if $F$ is isomorphic to the kernel of a map $\underline{A^{\oplus n}} \to \underline{A^{\oplus m}}$. \end{enumerate}


adequate-definition-adequate

A sheaf of $\mathcal{O}$-modules $\mathcal{F}$ on $(\Sch/S)_\tau$ is {\it adequate} if there exists a $\tau$-covering $\{\Spec(A_i) \to S\}_{i \in I}$ such that $F_{\mathcal{F}, A_i}$ is adequate for all $i \in I$.


adequate-definition-category-adequate-modules

Let $S$ be a scheme. The category of adequate $\mathcal{O}$-modules on $(\Sch/S)_\tau$ is denoted {\it $\textit{Adeq}(\mathcal{O})$} or {\it $\textit{Adeq}((\Sch/S)_\tau, \mathcal{O})$}. If we want to think just about the abelian category of adequate modules without choosing a topology we simply write {\it $\textit{Adeq}(S)$}.


adequate-definition-pure

Let $A$ be a ring. \begin{enumerate} \item An $A$-module $P$ is said to be {\it pure projective} if for every universally exact sequence $0 \to K \to M \to N \to 0$ of $A$-module the sequence $0 \to \Hom_A(P, K) \to \Hom_A(P, M) \to \Hom_A(P, N) \to 0$ is exact. \item An $A$-module $I$ is said to be {\it pure injective} if for every universally exact sequence $0 \to K \to M \to N \to 0$ of $A$-module the sequence $0 \to \Hom_A(N, I) \to \Hom_A(M, I) \to \Hom_A(K, I) \to 0$ is exact. \end{enumerate}


adequate-definition-pure-resolution

Let $A$ be a ring. Let $M$ be an $A$-module. \begin{enumerate} \item A {\it pure projective resolution} $P_\bullet \to M$ is a universally exact sequence $$ \ldots \to P_1 \to P_0 \to M \to 0 $$ with each $P_i$ pure projective. \item A {\it pure injective resolution} $M \to I^\bullet$ is a universally exact sequence $$ 0 \to M \to I^0 \to I^1 \to \ldots $$ with each $I^i$ pure injective. \end{enumerate}


adequate-definition-pure-ext

Let $A$ be a ring and let $M$, $N$ be $A$-modules. The $i$th {\it pure extension module} $\text{Pext}^i_A(M, N)$ is the $i$th cohomology module of the complex $\Hom_A(M, I^\bullet)$ where $I^\bullet$ is a pure injective resolution of $N$.


spaces-topologies-definition-zariski-covering

Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. A {\it Zariski covering of $X$} is a family of morphisms $\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces over $S$ such that each $f_i$ is an open immersion and such that $$ |X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|), $$ i.e., the morphisms are jointly surjective.


spaces-topologies-definition-etale-covering

Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. An {\it \'etale covering of $X$} is a family of morphisms $\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces over $S$ such that each $f_i$ is \'etale and such that $$ |X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|), $$ i.e., the morphisms are jointly surjective.


spaces-topologies-definition-big-etale-site

Let $S$ be a scheme. A big \'etale site {\it $(\textit{Spaces}/S)_\etale$} is any site constructed as follows: \begin{enumerate} \item Choose a big \'etale site $(\Sch/S)_\etale$ as in Topologies, Section \ref{topologies-section-etale}. \item As underlying category take the category $\textit{Spaces}/S$ of algebraic spaces over $S$ (see discussion in Section \ref{section-procedure} why this is a set). \item Choose any set of coverings as in Sets, Lemma \ref{sets-lemma-coverings-site} starting with the category $\textit{Spaces}/S$ and the class of \'etale coverings of Definition \ref{definition-etale-covering}. \end{enumerate}


spaces-topologies-definition-big-small-etale

Let $S$ be a scheme. Let $(\textit{Spaces}/S)_\etale$ be as in Definition \ref{definition-big-etale-site}. Let $X$ be an algebraic space over $S$, i.e., an object of $(\textit{Spaces}/S)_\etale$. Then the big \'etale site {\it $(\textit{Spaces}/X)_\etale$} of $X$ is the localization of the site $(\textit{Spaces}/S)_\etale$ at $X$ introduced in Sites, Section \ref{sites-section-localize}.


spaces-topologies-definition-restriction-small-etale

In the situation of Lemma \ref{lemma-at-the-bottom-etale} the functor $i_X^{-1} = \pi_{X, *}$ is often called the {\it restriction to the small \'etale site}, and for a sheaf $\mathcal{F}$ on the big \'etale site we often denote $\mathcal{F}|_{X_\etale}$ this restriction.


spaces-topologies-definition-smooth-covering

Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. A {\it smooth covering of $X$} is a family of morphisms $\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces over $S$ such that each $f_i$ is smooth and such that $$ |X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|), $$ i.e., the morphisms are jointly surjective.


spaces-topologies-definition-syntomic-covering

Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. A {\it syntomic covering of $X$} is a family of morphisms $\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces over $S$ such that each $f_i$ is syntomic and such that $$ |X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|), $$ i.e., the morphisms are jointly surjective.


spaces-topologies-definition-fppf-covering

Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. An {\it fppf covering of $X$} is a family of morphisms $\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces over $S$ such that each $f_i$ is flat and locally of finite presentation and such that $$ |X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|), $$ i.e., the morphisms are jointly surjective.


spaces-topologies-definition-big-fppf-site

Let $S$ be a scheme. A big fppf site {\it $(\textit{Spaces}/S)_{fppf}$} is any site constructed as follows: \begin{enumerate} \item Choose a big fppf site $(\Sch/S)_{fppf}$ as in Topologies, Section \ref{topologies-section-fppf}. \item As underlying category take the category $\textit{Spaces}/S$ of algebraic spaces over $S$ (see discussion in Section \ref{section-procedure} why this is a set). \item Choose any set of coverings as in Sets, Lemma \ref{sets-lemma-coverings-site} starting with the category $\textit{Spaces}/S$ and the class of fppf coverings of Definition \ref{definition-fppf-covering}. \end{enumerate}


spaces-topologies-definition-big-small-fppf

Let $S$ be a scheme. Let $(\textit{Spaces}/S)_{fppf}$ be as in Definition \ref{definition-big-fppf-site}. Let $X$ be an algebraic space over $S$, i.e., an object of $(\textit{Spaces}/S)_{fppf}$. Then the big fppf site {\it $(\textit{Spaces}/X)_{fppf}$} of $X$ is the localization of the site $(\textit{Spaces}/S)_{fppf}$ at $X$ introduced in Sites, Section \ref{sites-section-localize}.


spaces-topologies-definition-ph-covering

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. A {\it ph covering of $X$} is a family of morphisms $\{X_i \to X\}_{i \in I}$ of algebraic spaces over $S$ such that $f_i$ is locally of finite type and such that for every $U \to X$ with $U$ affine there exists a standard ph covering $\{U_j \to U\}_{j = 1, \ldots, m}$ refining the family $\{X_i \times_X U \to U\}_{i \in I}$.


spaces-topologies-definition-big-ph-site

Let $S$ be a scheme. A big ph site {\it $(\textit{Spaces}/S)_{ph}$} is any site constructed as follows: \begin{enumerate} \item Choose a big ph site $(\Sch/S)_{ph}$ as in Topologies, Section \ref{topologies-section-ph}. \item As underlying category take the category $\textit{Spaces}/S$ of algebraic spaces over $S$ (see discussion in Section \ref{section-procedure} why this is a set). \item Choose any set of coverings as in Sets, Lemma \ref{sets-lemma-coverings-site} starting with the category $\textit{Spaces}/S$ and the class of ph coverings of Definition \ref{definition-ph-covering}. \end{enumerate}


spaces-topologies-definition-big-small-ph

Let $S$ be a scheme. Let $(\textit{Spaces}/S)_{ph}$ be as in Definition \ref{definition-big-ph-site}. Let $X$ be an algebraic space over $S$, i.e., an object of $(\textit{Spaces}/S)_{ph}$. Then the big ph site {\it $(\textit{Spaces}/X)_{ph}$} of $X$ is the localization of the site $(\textit{Spaces}/S)_{ph}$ at $X$ introduced in Sites, Section \ref{sites-section-localize}.


spaces-topologies-definition-fpqc-covering

Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. An {\it fpqc covering of $X$} is a family of morphisms $\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces such that each $f_i$ is flat and such that for every affine scheme $Z$ and morphism $h : Z \to X$ there exists a standard fpqc covering $\{g_j : Z_j \to Z\}_{j = 1, \ldots, m}$ which refines the family $\{X_i \times_X Z \to Z\}_{i \in I}$.


proetale-definition-w-local

A spectral space $X$ is {\it w-local} if the set of closed points $X_0$ is closed and every point of $X$ specializes to a unique closed point. A continuous map $f : X \to Y$ of w-local spaces is {\it w-local} if it is spectral and maps any closed point of $X$ to a closed point of $Y$.


proetale-definition-local-isomorphism

Let $\varphi : A \to B$ be a ring map. \begin{enumerate} \item We say $A \to B$ is a {\it local isomorphism} if for every prime $\mathfrak q \subset B$ there exists a $g \in B$, $g \not \in \mathfrak q$ such that $A \to B_g$ induces an open immersion $\Spec(B_g) \to \Spec(A)$. \item We say $A \to B$ {\it identifies local rings} if for every prime $\mathfrak q \subset B$ the canonical map $A_{\varphi^{-1}(\mathfrak q)} \to B_\mathfrak q$ is an isomorphism. \end{enumerate}


proetale-definition-ind-zariski

A ring map $A \to B$ is said to be {\it ind-Zariski} if $B$ can be written as a filtered colimit $B = \colim B_i$ with each $A \to B_i$ a local isomorphism.


proetale-definition-ind-etale

A ring map $A \to B$ is said to be {\it ind-\'etale} if $B$ can be written as a filtered colimit of \'etale $A$-algebras.


proetale-definition-w-contractible

Let $A$ be a ring. We say $A$ is {\it w-contractible} if every faithfully flat weakly \'etale ring map $A \to B$ has a section.


proetale-definition-fpqc-covering

Let $T$ be a scheme. A {\it pro-\'etale covering of $T$} is a family of morphisms $\{f_i : T_i \to T\}_{i \in I}$ of schemes such that each $f_i$ is weakly-\'etale and such that for every affine open $U \subset T$ there exists $n \geq 0$, a map $a : \{1, \ldots, n\} \to I$ and affine opens $V_j \subset T_{a(j)}$, $j = 1, \ldots, n$ with $\bigcup_{j = 1}^n f_{a(j)}(V_j) = U$.


proetale-definition-standard-proetale

Let $T$ be an affine scheme. A {\it standard pro-\'etale covering} of $T$ is a family $\{f_i : T_i \to T\}_{i = 1, \ldots, n}$ where each $T_j$ is affine, each $f_i$ is weakly \'etale, and $T = \bigcup f_i(T_i)$.


proetale-definition-big-proetale-site

A {\it big pro-\'etale site} is any site $\Sch_\proetale$ as in Sites, Definition \ref{sites-definition-site} constructed as follows: \begin{enumerate} \item Choose any set of schemes $S_0$, and any set of pro-\'etale coverings $\text{Cov}_0$ among these schemes. \item Change the function $Bound$ of Sets, Equation (\ref{sets-equation-bound}) into $$ Bound(\kappa) = \max\{\kappa^{2^{2^{2^\kappa}}}, \kappa^{\aleph_0}, \kappa^+\}. $$ \item As underlying category take any category $\Sch_\alpha$ constructed as in Sets, Lemma \ref{sets-lemma-construct-category} starting with the set $S_0$ and the function $Bound$. \item Choose any set of coverings as in Sets, Lemma \ref{sets-lemma-coverings-site} starting with the category $\Sch_\alpha$ and the class of pro-\'etale coverings, and the set $\text{Cov}_0$ chosen above. \end{enumerate}


proetale-definition-big-small-proetale

Let $S$ be a scheme. Let $\Sch_\proetale$ be a big pro-\'etale site containing $S$. \begin{enumerate} \item The {\it big pro-\'etale site of $S$}, denoted $(\Sch/S)_\proetale$, is the site $\Sch_\proetale/S$ introduced in Sites, Section \ref{sites-section-localize}. \item The {\it small pro-\'etale site of $S$}, which we denote $S_\proetale$, is the full subcategory of $(\Sch/S)_\proetale$ whose objects are those $U/S$ such that $U \to S$ is weakly \'etale. A covering of $S_\proetale$ is any covering $\{U_i \to U\}$ of $(\Sch/S)_\proetale$ with $U \in \Ob(S_\proetale)$. \item The {\it big affine pro-\'etale site of $S$}, denoted $(\textit{Aff}/S)_\proetale$, is the full subcategory of $(\Sch/S)_\proetale$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_\proetale$ is any covering $\{U_i \to U\}$ of $(\Sch/S)_\proetale$ which is a standard pro-\'etale covering. \end{enumerate}


proetale-definition-restriction-small-proetale

In the situation of Lemma \ref{lemma-at-the-bottom} the functor $i_S^{-1} = \pi_{S, *}$ is often called the {\it restriction to the small pro-\'etale site}, and for a sheaf $\mathcal{F}$ on the big pro-\'etale site we denote $\mathcal{F}|_{S_\proetale}$ this restriction.


proetale-definition-extension-zero

Let $j : U \to X$ be a weakly \'etale morphism of schemes. \begin{enumerate} \item The restriction functor $j^{-1} : \Sh(X_\proetale) \to \Sh(U_\proetale)$ has a left adjoint $j_!^{Sh} : \Sh(X_\proetale) \to \Sh(U_\proetale)$. \item The restriction functor $j^{-1} : \textit{Ab}(X_\proetale) \to \textit{Ab}(U_\proetale)$ has a left adjoint which is denoted $j_! : \textit{Ab}(U_\proetale) \to \textit{Ab}(X_\proetale)$ and called {\it extension by zero}. \item Let $\Lambda$ be a ring. The functor $j^{-1} : \textit{Mod}(X_\proetale, \Lambda) \to \textit{Mod}(U_\proetale, \Lambda)$ has a left adjoint $j_! : \textit{Mod}(U_\proetale, \Lambda) \to \textit{Mod}(X_\proetale, \Lambda)$ and called {\it extension by zero}. \end{enumerate}


proetale-definition-constructible

Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. A sheaf of $\Lambda$-modules on $X_\proetale$ is {\it constructible} if for every affine open $U \subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \coprod_i U_i$ such that $\mathcal{F}|_{U_i}$ is of finite type and locally constant for all $i$.


proetale-definition-adic

Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $X$ be a scheme. Let $\mathcal{F}$ be a sheaf of $\Lambda$-modules on $X_\proetale$. \begin{enumerate} \item We say $\mathcal{F}$ is a {\it constructible $\Lambda$-sheaf} if $\mathcal{F} = \lim \mathcal{F}/I^n\mathcal{F}$ and each $\mathcal{F}/I^n\mathcal{F}$ is a constructible sheaf of $\Lambda/I^n$-modules. \item If $\mathcal{F}$ is a constructible $\Lambda$-sheaf, then we say $\mathcal{F}$ is {\it lisse} if each $\mathcal{F}/I^n\mathcal{F}$ is locally constant. \item We say $\mathcal{F}$ is {\it adic lisse}\footnote{This may be nonstandard notation.} if there exists a $I$-adically complete $\Lambda$-module $M$ with $M/IM$ finite such that $\mathcal{F}$ is locally isomorphic to $$ \underline{M}^\wedge = \lim \underline{M/I^nM}. $$ \item We say $\mathcal{F}$ is {\it adic constructible}\footnote{This may be nonstandard notation.} if for every affine open $U \subset X$ there exists a decomposition $U = \coprod U_i$ into constructible locally closed subschemes such that $\mathcal{F}|_{U_i}$ is adic lisse. \end{enumerate}


proetale-definition-Dbc

Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $X$ be a scheme. An object $K$ of $D(X_\proetale, \Lambda)$ is called {\it constructible} if \begin{enumerate} \item $K$ is derived complete with respect to $I$, \item $K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}$ has constructible cohomology sheaves and locally has finite tor dimension. \end{enumerate} We denote $D_{cons}(X, \Lambda)$ the full subcategory of constructible $K$ in $D(X_\proetale, \Lambda)$.


proetale-definition-adic-constructible

Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $K \in D(X_\proetale, \Lambda)$. \begin{enumerate} \item We say $K$ is {\it adic lisse}\footnote{This may be nonstandard notation} if there exists a finite complex of finite projective $\Lambda^\wedge$-modules $M^\bullet$ such that $K$ is locally isomorphic to $$ \underline{M^a}^\wedge \to \ldots \to \underline{M^b}^\wedge $$ \item We say $K$ is {\it adic constructible}\footnote{This may be nonstandard notation.} if for every affine open $U \subset X$ there exists a decomposition $U = \coprod U_i$ into constructible locally closed subschemes such that $K|_{U_i}$ is adic lisse. \end{enumerate}


formal-spaces-definition-toplogy-tensor-product

Let $R$ be a topological ring. Let $M$ and $N$ be linearly topologized $R$-modules. The {\it tensor product} of $M$ and $N$ is the (usual) tensor product $M \otimes_R N$ endowed with the linear topology defined by declaring $$ \Im(M_\mu \otimes_R N + M \otimes_R N_\nu \longrightarrow M \otimes_R N) $$ to be a fundamental system of open submodules, where $M_\mu \subset M$ and $N_\nu \subset N$ run through fundamental systems of open submodules in $M$ and $N$. The {\it completed tensor product} $$ M \widehat{\otimes}_R N = \lim M \otimes_R N/(M_\mu \otimes_R N + M \otimes_R N_\nu) = \lim M/M_\mu \otimes_R N/N_\nu $$ is the completion of the tensor product.


formal-spaces-definition-weakly-admissible

Let $A$ be a linearly topologized ring. \begin{enumerate} \item An element $f \in A$ is called {\it topologically nilpotent} if $f^n \to 0$ as $n \to \infty$. \item A {\it weak ideal of definition} for $A$ is an open ideal $I \subset A$ consisting entirely of topologically nilpotent elements. \item We say $A$ is {\it weakly pre-admissible} if $A$ has a weak ideal of definition. \item We say $A$ is {\it weakly admissible} if $A$ is weakly pre-admissible and complete\footnote{By our conventions this includes separated.}. \end{enumerate}


formal-spaces-definition-taut

Let $\varphi : A \to B$ be a continuous map of linearly topologized rings. We say $\varphi$ is {\it taut}\footnote{This is nonstandard notation. The definition generalizes to modules, by saying a linearly topologized $A$-module $M$ is $A$-taut if for every open ideal $I \subset A$ the closure of $IM$ in $M$ is open and these closures form a fundamental system of neighbourhoods of $0$ in $M$.} if for every open ideal $I \subset A$ the closure of the ideal $\varphi(I)B$ is open and these closures form a fundamental system of open ideals.


formal-spaces-definition-affine-formal-algebraic-space

Let $S$ be a scheme. We say a sheaf $X$ on $(\Sch/S)_{fppf}$ is an {\it affine formal algebraic space} if there exist \begin{enumerate} \item a directed set $\Lambda$, \item a system $(X_\lambda, f_{\lambda \mu})$ over $\Lambda$ in $(\Sch/S)_{fppf}$ where \begin{enumerate} \item each $X_\lambda$ is affine, \item each $f_{\lambda \mu} : X_\lambda \to X_\mu$ is a thickening, \end{enumerate} \end{enumerate} such that $$ X \cong \colim_{\lambda \in \Lambda} X_\lambda $$ as fppf sheaves and $X$ satisfies a set theoretic condition (see Remark \ref{remark-set-theoretic}). A {\it morphism of affine formal algebraic spaces} over $S$ is a map of sheaves.


formal-spaces-definition-types-affine-formal-algebraic-space

Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. We say $X$ is {\it McQuillan} if $X$ satisfies the equivalent conditions of Lemma \ref{lemma-mcquillan-affine-formal-algebraic-space}. Let $A$ be the weakly admissible topological ring associated to $X$. We say \begin{enumerate} \item $X$ is {\it classical} if $X$ is McQuillan and $A$ is admissible, \item $X$ is {\it adic} if $X$ is McQuillan and $A$ is adic, \item $X$ is {\it adic*} if $X$ is McQuillan, $A$ is adic, and $A$ has a finitely generated ideal of definition, and \item $X$ is {\it Noetherian} if $X$ is McQuillan and $A$ is both Noetherian and adic. \end{enumerate}


formal-spaces-definition-affine-formal-spectrum

Let $S$ be a scheme. Let $A$ be a weakly admissible topological ring over $S$, see Definition \ref{definition-weakly-admissible}\footnote{See More on Algebra, Definition \ref{more-algebra-definition-topological-ring} for the classical case and see Remark \ref{remark-mcquillan} for a discussion of differences.}. The {\it formal spectrum} of $A$ is the affine formal algebraic space $$ \text{Spf}(A) = \colim \Spec(A/I) $$ where the colimit is over the set of weak ideals of definition of $A$ and taken in the category $\Sh((\Sch/S)_{fppf})$.


formal-spaces-definition-countable

Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. We say $X$ is {\it countably indexed} if the equivalent conditions of Lemma \ref{lemma-countable-affine-formal-algebraic-space} are satisfied.


formal-spaces-definition-formal-algebraic-space

Let $S$ be a scheme. We say a sheaf $X$ on $(\Sch/S)_{fppf}$ is a {\it formal algebraic space} if there exist a family of maps $\{X_i \to X\}_{i \in I}$ of sheaves such that \begin{enumerate} \item $X_i$ is an affine formal algebraic space, \item $X_i \to X$ is representable by algebraic spaces and \'etale, \item $\coprod X_i \to X$ is surjective as a map of sheaves \end{enumerate} and $X$ satisfies a set theoretic condition (see Remark \ref{remark-set-theoretic}). A {\it morphism of formal algebraic spaces} over $S$ is a map of sheaves.


formal-spaces-definition-completion

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset. The formal algebraic space of Lemma \ref{lemma-completion-is-formal-algebraic-space} is called the {\it completion of $X$ along $T$}.


formal-spaces-definition-separated

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. We say \begin{enumerate} \item $X$ is {\it quasi-separated} if the equivalent conditions of Lemma \ref{lemma-characterize-quasi-separated} are satisfied. \item $X$ is {\it separated} if the equivalent conditions of Lemma \ref{lemma-characterize-separated} are satisfied. \end{enumerate}


formal-spaces-definition-quasi-compact

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. We say $X$ is {\it quasi-compact} if the equivalent conditions of Lemma \ref{lemma-characterize-quasi-compact} are satisfied.


formal-spaces-definition-quasi-compact-morphism

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. We say $f$ is {\it quasi-compact} if the equivalent conditions of Lemma \ref{lemma-characterize-quasi-compact-morphism} are satisfied.


formal-spaces-definition-types-formal-algebraic-spaces

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. We say $X$ is {\it locally countably indexed}, {\it locally adic*}, or {\it locally Noetherian} if the equivalent conditions of Lemma \ref{lemma-type-local} hold for the corresponding property.


formal-spaces-definition-adic-homomorphism

Let $A$ and $B$ be pre-adic topological rings. A ring homomorphism $\varphi : A \to B$ is {\it adic}\footnote{This may be nonstandard terminology.} if there exists an ideal of definition $I \subset A$ such that the topology on $B$ is the $I$-adic topology.


formal-spaces-definition-adic-morphism

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. Assume $X$ and $Y$ are locally adic*. We say $f$ is an {\it adic morphism} if $f$ is representable by algebraic spaces. See discussion above.


formal-spaces-definition-finite-type

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of formal algebraic spaces over $S$. \begin{enumerate} \item We say $f$ is {\it locally of finite type} if $f$ is representable by algebraic spaces and is locally of finite type in the sense of Bootstrap, Definition \ref{bootstrap-definition-property-transformation}. \item We say $f$ is of {\it finite type} if $f$ is locally of finite type and quasi-compact (Definition \ref{definition-quasi-compact-morphism}). \end{enumerate}


formal-spaces-definition-surjective

Let $S$ be a scheme. A morphism $f : X \to Y$ of formal algebraic spaces over $S$ is said to be {\it surjective} if it induces a surjective morphism $X_{red} \to Y_{red}$ on underlying reduced algebraic spaces.


formal-spaces-definition-monomorphism

Let $S$ be a scheme. A morphism of formal algebraic spaces over $S$ is called a {\it monomorphism} if it is an injective map of sheaves.


formal-spaces-definition-closed-immersion

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of formal algebraic spaces over $S$. We say $f$ is a {\it closed immersion} if $f$ is representable by algebraic spaces and a closed immersion in the sense of Bootstrap, Definition \ref{bootstrap-definition-property-transformation}.


formal-spaces-definition-topologically-finite-type

Let $A \to B$ be a continuous map of topological rings (More on Algebra, Definition \ref{more-algebra-definition-topological-ring}). We say $B$ is {\it topologically of finite type over} $A$ if there exists an $A$-algebra map $A[x_1, \ldots, x_n] \to B$ whose image is dense in $B$.


formal-spaces-definition-separated-morphism

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. Let $\Delta_{X/Y} : X \to X \times_Y X$ be the diagonal morphism. \begin{enumerate} \item We say $f$ is {\it separated} if $\Delta_{X/Y}$ is a closed immersion. \item We say $f$ is {\it quasi-separated} if $\Delta_{X/Y}$ is quasi-compact. \end{enumerate}


formal-spaces-definition-proper

Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of formal algebraic spaces over $S$. We say $f$ is {\it proper} if $f$ is representable by algebraic spaces and is proper in the sense of Bootstrap, Definition \ref{bootstrap-definition-property-transformation}.


formal-spaces-definition-etale-sites

Let $S$ be a scheme. Let $X$ be a formal algebraic space with reduction $X_{red}$ (Lemma \ref{lemma-reduction-formal-algebraic-space}). \begin{enumerate} \item The {\it small \'etale site} $X_\etale$ of $X$ is the site $X_{red, \etale}$ of Properties of Spaces, Definition \ref{spaces-properties-definition-etale-site}. \item The site $X_{spaces, \etale}$ is the site $X_{red, spaces, \etale}$ of Properties of Spaces, Definition \ref{spaces-properties-definition-spaces-etale-site}. \item The site $X_{affine, \etale}$ is the site $X_{red, affine, \etale}$ of Properties of Spaces, Lemma \ref{spaces-properties-lemma-alternative}. \end{enumerate}


pione-definition-G-set-continuous

Let $G$ be a topological group. A {\it $G$-set}, sometimes called a {\it discrete $G$-set}, is a set $X$ endowed with a left action $a : G \times X \to X$ such that $a$ is continuous when $X$ is given the discrete topology and $G \times X$ the product topology. A {\it morphism of $G$-sets} $f : X \to Y$ is simply any $G$-equivariant map from $X$ to $Y$. The category of $G$-sets is denoted {\it $G\textit{-Sets}$}.


pione-definition-galois-category

\begin{reference} Different from the definition in \cite[Expos\'e V, Definition 5.1]{SGA1}. Compare with \cite[Definition 7.2.1]{BS}. \end{reference} Let $\mathcal{C}$ be a category and let $F : \mathcal{C} \to \textit{Sets}$ be a functor. The pair $(\mathcal{C}, F)$ is a {\it Galois category} if \begin{enumerate} \item $\mathcal{C}$ has finite limits and finite colimits, \item \label{item-connected-components} every object of $\mathcal{C}$ is a finite (possibly empty) coproduct of connected objects, \item $F(X)$ is finite for all $X \in \Ob(\mathcal{C})$, and \item $F$ reflects isomorphisms and is exact. \end{enumerate} Here we say $X \in \Ob(\mathcal{C})$ is connected if it is not initial and for any monomorphism $Y \to X$ either $Y$ is initial or $Y \to X$ is an isomorphism.


pione-definition-fundamental-group

Let $X$ be a connected scheme. Let $\overline{x}$ be a geometric point of $X$. The {\it fundamental group} of $X$ with {\it base point} $\overline{x}$ is the group $$ \pi_1(X, \overline{x}) = \text{Aut}(F_{\overline{x}}) $$ of automorphisms of the fibre functor $F_{\overline{x}} : \textit{F\'Et}_X \to \textit{Sets}$ endowed with its canonical profinite topology from Lemma \ref{lemma-aut-inverse-limit}.


stacks-cohomology-definition-flat-base-change

Let $\mathcal{X}$ be an algebraic stack and let $\mathcal{F}$ in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. We say $\mathcal{F}$ has the {\it flat base change property}\footnote{This may be nonstandard notation.} if and only if $c_\varphi$ is an isomorphism whenever $f$ is flat.


stacks-cohomology-definition-parasitic

Let $\mathcal{X}$ be an algebraic stack. A presheaf of $\mathcal{O}_\mathcal{X}$-modules $\mathcal{F}$ is {\it parasitic} if we have $\mathcal{F}(x) = 0$ for any object $x$ of $\mathcal{X}$ which lies over a scheme $U$ such that the corresponding morphism $x : U \to \mathcal{X}$ is flat.


stacks-cohomology-definition-lisse-etale

Let $\mathcal{X}$ be an algebraic stack. \begin{enumerate} \item The {\it lisse-\'etale site} of $\mathcal{X}$ is the full subcategory $\mathcal{X}_{lisse,\etale}$\footnote{In the literature the site is denoted $\text{Lis-\'et}(\mathcal{X})$ or $\text{Lis-Et}(\mathcal{X})$ and the associated topos is denoted $\mathcal{X}_{\text{lis-\'e}t}$ or $\mathcal{X}_{\text{lis-et}}$. In the Stacks project our convention is to name the site and denote the corresponding topos by $\Sh(\mathcal{C})$.} of $\mathcal{X}$ whose objects are those $x \in \Ob(\mathcal{X})$ lying over a scheme $U$ such that $x : U \to \mathcal{X}$ is smooth. A covering of $\mathcal{X}_{lisse,\etale}$ is a family of morphisms $\{x_i \to x\}_{i \in I}$ of $\mathcal{X}_{lisse,\etale}$ which forms a covering of $\mathcal{X}_\etale$. \item The {\it flat-fppf site} of $\mathcal{X}$ is the full subcategory $\mathcal{X}_{flat,fppf}$ of $\mathcal{X}$ whose objects are those $x \in \Ob(\mathcal{X})$ lying over a scheme $U$ such that $x : U \to \mathcal{X}$ is flat. A covering of $\mathcal{X}_{flat,fppf}$ is a family of morphisms $\{x_i \to x\}_{i \in I}$ of $\mathcal{X}_{flat,fppf}$ which forms a covering of $\mathcal{X}_{fppf}$. \end{enumerate}


sites-cohomology-definition-torsor

Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $\mathcal{C}$. A {\it pseudo torsor}, or more precisely a {\it pseudo $\mathcal{G}$-torsor}, is a sheaf of sets $\mathcal{F}$ on $\mathcal{C}$ endowed with an action $\mathcal{G} \times \mathcal{F} \to \mathcal{F}$ such that \begin{enumerate} \item whenever $\mathcal{F}(U)$ is nonempty the action $\mathcal{G}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ is simply transitive. \end{enumerate} A {\it morphism of pseudo $\mathcal{G}$-torsors} $\mathcal{F} \to \mathcal{F}'$ is simply a morphism of sheaves of sets compatible with the $\mathcal{G}$-actions. A {\it torsor}, or more precisely a {\it $\mathcal{G}$-torsor}, is a pseudo $\mathcal{G}$-torsor such that in addition \begin{enumerate} \item[(2)] for every $U \in \Ob(\mathcal{C})$ there exists a covering $\{U_i \to U\}_{i \in I}$ of $U$ such that $\mathcal{F}(U_i)$ is nonempty for all $i \in I$. \end{enumerate} A {\it morphism of $\mathcal{G}$-torsors} is simply a morphism of pseudo $\mathcal{G}$-torsors. The {\it trivial $\mathcal{G}$-torsor} is the sheaf $\mathcal{G}$ endowed with the obvious left $\mathcal{G}$-action.


sites-cohomology-definition-cech-complex

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times_U \ldots \times_U U_{i_p}$ exist in $\mathcal{C}$. Let $\mathcal{F}$ be an abelian presheaf on $\mathcal{C}$. The complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ is the {\it {\v C}ech complex} associated to $\mathcal{F}$ and the family $\mathcal{U}$. Its cohomology groups $H^i(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}))$ are called the {\it {\v C}ech cohomology groups} of $\mathcal{F}$ with respect to $\mathcal{U}$. They are denoted $\check H^i(\mathcal{U}, \mathcal{F})$.


sites-cohomology-definition-limp

Let $\mathcal{C}$ be a site. We say an abelian sheaf $\mathcal{F}$ is {\it totally acyclic}\footnote{Although this terminology is is used in \cite[Vbis, Proposition 1.3.10]{SGA4} this is probably nonstandard notation. In \cite[V, Definition 4.1]{SGA4} this property is dubbed ``flasque'', but we cannot use this because it would clash with our definition of flasque sheaves on topological spaces. Please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com} if you have a better suggestion.} if for every sheaf of sets $K$ we have $H^p(K, \mathcal{F}) = 0$ for all $p \geq 1$.


sites-cohomology-definition-K-flat

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A complex $\mathcal{K}^\bullet$ of $\mathcal{O}$-modules is called {\it K-flat} if for every acyclic complex $\mathcal{F}^\bullet$ of $\mathcal{O}$-modules the complex $$ \text{Tot}(\mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{K}^\bullet) $$ is acyclic.


sites-cohomology-definition-derived-tor

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}^\bullet$ be an object of $D(\mathcal{O})$. The {\it derived tensor product} $$ - \otimes_\mathcal{O}^{\mathbf{L}} \mathcal{F}^\bullet : D(\mathcal{O}) \longrightarrow D(\mathcal{O}) $$ is the exact functor of triangulated categories described above.


sites-cohomology-definition-tor

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}$-modules. The {\it Tor}'s of $\mathcal{F}$ and $\mathcal{G}$ are defined by the formula $$ \text{Tor}_p^\mathcal{O}(\mathcal{F}, \mathcal{G}) = H^{-p}(\mathcal{F} \otimes_\mathcal{O}^\mathbf{L} \mathcal{G}) $$ with derived tensor product as defined above.


sites-cohomology-definition-covering-LC

Let $\{f_i : X_i \to X\}$ be a family of morphisms with fixed target in the category $\textit{LC}$. We say this family is a {\it qc covering}\footnote{This is nonstandard notation. We chose it to remind the reader of fpqc coverings of schemes.} if for every $x \in X$ there exist $i_1, \ldots, i_n \in I$ and quasi-compact subsets $E_j \subset X_{i_j}$ such that $\bigcup f_{i_j}(E_j)$ is a neighbourhood of $x$.


sites-cohomology-definition-simplicial-module

Let $\mathcal{C}$ be a site. Let $\mathcal{A}_\bullet$ be a simplicial sheaf of rings on $\mathcal{C}$. A {\it simplicial $\mathcal{A}_\bullet$-module} $\mathcal{F}_\bullet$ (sometimes called a {\it simplicial sheaf of $\mathcal{A}_\bullet$-modules}) is a sheaf of modules over the sheaf of rings on $\Delta \times \mathcal{C}$ associated to $\mathcal{A}_\bullet$.


sites-cohomology-definition-strictly-perfect

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet$ be a complex of $\mathcal{O}$-modules. We say $\mathcal{E}^\bullet$ is {\it strictly perfect} if $\mathcal{E}^i$ is zero for all but finitely many $i$ and $\mathcal{E}^i$ is a direct summand of a finite free $\mathcal{O}$-module for all $i$.


sites-cohomology-definition-pseudo-coherent

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet$ be a complex of $\mathcal{O}$-modules. Let $m \in \mathbf{Z}$. \begin{enumerate} \item We say $\mathcal{E}^\bullet$ is {\it $m$-pseudo-coherent} if for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}$ and for each $i$ a morphism of complexes $\alpha_i : \mathcal{E}_i^\bullet \to \mathcal{E}^\bullet|_{U_i}$ where $\mathcal{E}_i$ is a strictly perfect complex of $\mathcal{O}_{U_i}$-modules and $H^j(\alpha_i)$ is an isomorphism for $j > m$ and $H^m(\alpha_i)$ is surjective. \item We say $\mathcal{E}^\bullet$ is {\it pseudo-coherent} if it is $m$-pseudo-coherent for all $m$. \item We say an object $E$ of $D(\mathcal{O})$ is {\it $m$-pseudo-coherent} (resp.\ {\it pseudo-coherent}) if and only if it can be represented by a $m$-pseudo-coherent (resp.\ pseudo-coherent) complex of $\mathcal{O}$-modules. \end{enumerate}


sites-cohomology-definition-tor-amplitude

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$ with $a \leq b$. \begin{enumerate} \item We say $E$ has {\it tor-amplitude in $[a, b]$} if $H^i(E \otimes_\mathcal{O}^\mathbf{L} \mathcal{F}) = 0$ for all $\mathcal{O}$-modules $\mathcal{F}$ and all $i \not \in [a, b]$. \item We say $E$ has {\it finite tor dimension} if it has tor-amplitude in $[a, b]$ for some $a, b$. \item We say $E$ {\it locally has finite tor dimension} if for any object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}$ such that $E|_{U_i}$ has finite tor dimension for all $i$. \end{enumerate} An $\mathcal{O}$-module $\mathcal{F}$ has {\it tor dimension $\leq d$} if $\mathcal{F}[0]$ viewed as an object of $D(\mathcal{O})$ has tor-amplitude in $[-d, 0]$.


sites-cohomology-definition-perfect

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet$ be a complex of $\mathcal{O}$-modules. We say $\mathcal{E}^\bullet$ is {\it perfect} if for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}$ such that for each $i$ there exists a morphism of complexes $\mathcal{E}_i^\bullet \to \mathcal{E}^\bullet|_{U_i}$ which is a quasi-isomorphism with $\mathcal{E}_i^\bullet$ strictly perfect. An object $E$ of $D(\mathcal{O})$ is {\it perfect} if it can be represented by a perfect complex of $\mathcal{O}$-modules.


fields-definition-field

A {\it field} is a nonzero ring where every nonzero element is invertible. Given a field a {\it subfield} is a subring that is itself a field.


fields-definition-domain

A {\it domain} or an {\it integral domain} is a nonzero ring where $0$ is the only zerodivisor.


fields-definition-characteristic

The {\it characteristic} of a field $F$ is $0$ if $\mathbf{Z} \subset F$, or is a prime $p$ if $p = 0$ in $F$. The {\it prime subfield of $F$} is the smallest subfield of $F$ which is either $\mathbf{Q} \subset F$ if the characteristic is zero, or $\mathbf{F}_p \subset F$ if the characteristic is $p > 0$.


fields-definition-extension

If $F$ is a field contained in a field $E$, then $E$ is said to be a {\it field extension} of $F$. We shall write $E/F$ to indicate that $E$ is an extension of $F$.


fields-definition-tower

A {\it tower} of fields $E_n/E_{n - 1}/\ldots/E_0$ consists of a sequence of extensions of fields $E_n/E_{n - 1}$, $E_{n - 1}/E_{n - 2}$, $\ldots$, $E_1/E_0$.


fields-definition-generated-by

Let $k$ be a field. If $F/k$ is an extension of fields and $S \subset F$, we write $k(S)$ for the smallest subfield of $F$ containing $k$ and $S$. We will say that $S$ {\it generates the field extension} $k(S)/k$. If $S = \{\alpha\}$ is a singleton, then we write $k(\alpha)$ instead of $k(\{\alpha\})$. We say $F/k$ is a {\it finitely generated field extension} if there exists a finite subset $S \subset F$ with $F = k(S)$.


fields-definition-degree

Let $F/E$ be an extension of fields. The dimension of $F$ considered as an $E$-vector space is called the {\it degree} of the extension and is denoted $[F : E]$. If $[F : E] < \infty$ then $F$ is said to be a {\it finite} extension of $E$.


fields-definition-number-field

A field $K$ is said to be a {\it number field} if it has characteristic $0$ and the extension $\mathbf{Q} \subset K$ is finite.


fields-definition-algebraic

Consider a field extension $F/E$. An element $\alpha \in F$ is said to be {\it algebraic} over $E$ if $\alpha$ is the root of some nonzero polynomial with coefficients in $E$. If all elements of $F$ are algebraic then $F$ is said to be an {\it algebraic extension} of $E$.


fields-definition-minimal-polynomial

The polynomial $P$ above is called the {\it minimal polynomial} of $\alpha$ over $k$.


fields-definition-algebraically-closed

A field $F$ is said to be {\it algebraically closed} if every algebraic extension $E/F$ is trivial, i.e., $E = F$.


fields-definition-algebraic-closure

Let $F$ be a field. We say $F$ is {\it algebraically closed} if every algebraic extension $E/F$ is trivial, i.e., $E = F$. An {\it algebraic closure} of $F$ is a field $\overline{F}$ containing $F$ such that: \begin{enumerate} \item $\overline{F}$ is algebraic over $F$. \item $\overline{F}$ is algebraically closed. \end{enumerate}


fields-definition-relatively-prime

If $k$ is any field, we say that two polynomials in $k[x]$ are {\it relatively prime} if they generate the unit ideal in $k[x]$.


fields-definition-separable

Let $F$ be a field. Let $K/F$ be an extension of fields. \begin{enumerate} \item We say an irreducible polynomial $P$ over $F$ is {\it separable} if it is relatively prime to its derivative. \item Given $\alpha \in K$ algebraic over $F$ we say $\alpha$ is {\it separable} over $F$ if its minimal polynomial is separable over $F$. \item If $K$ is an algebraic extension of $F$, we say $K$ is {\it separable}\footnote{For nonalgebraic extensions this definition does not make sense and is not the correct one. We refer the reader to Algebra, Sections \ref{algebra-section-separability} and \ref{algebra-section-separability-continued}.} over $F$ if every element of $K$ is separable over $F$. \end{enumerate}


fields-definition-separable-degree

Let $F$ be a field. Let $P$ be an irreducible polynomial over $F$. The {\it separable degree} of $P$ is the cardinality of the set of roots of $P$ in any algebraic closure of $F$ (see discussion above). Notation $\deg_s(P)$.


fields-definition-purely-inseparable

Let $F$ be a field of characteristic $p > 0$. Let $K/F$ be an extension. \begin{enumerate} \item An element $\alpha \in K$ is {\it purely inseparable} over $F$ if there exists a power $q$ of $p$ such that $\alpha^q \in F$. \item The extension $K/F$ is said to be {\it purely inseparable} if and only if every element of $K$ is purely inseparable over $F$. \end{enumerate}


fields-definition-insep-degree

Let $E/F$ be an algebraic field extension. Let $E_{sep}$ be the subextension found in Lemma \ref{lemma-separable-first}. \begin{enumerate} \item The integer $[E_{sep} : F]$ is called the {\it separable degree} of the extension. Notation $[E : F]_s$. \item The integer $[E : E_{sep}]$ is called the {\it inseparable degree}, or the {\it degree of inseparability} of the extension. Notation $[E : F]_i$. \end{enumerate}


fields-definition-normal

Let $E/F$ be an algebraic field extension. We say $E$ is {\it normal} over $F$ if for all $\alpha \in E$ the minimal polynomial $P$ of $\alpha$ over $F$ splits completely into linear factors over $E$.


fields-definition-automorphisms

Let $E/F$ be an extension of fields. Then $\text{Aut}(E/F)$ or $\text{Aut}_F(E)$ denotes the automorphism group of $E$ as an object of the category of $F$-extensions. Elements of $\text{Aut}(E/F)$ are called {\it automorphisms of $E$ over $F$} or {\it automorphisms of $E/F$}.


fields-definition-splitting-field

Let $F$ be a field. Let $P \in F[x]$ be a nonconstant polynomial. The field extension $E/F$ constructed in Lemma \ref{lemma-splitting-field} is called the {\it splitting field of $P$ over $F$}.


fields-definition-normal-closure

Let $E/F$ be a finite extension of fields. The field extension $K/E$ constructed in Lemma \ref{lemma-normal-closure} is called the {\it normal closure $E$ over $F$}.


fields-definition-trace-norm

Let $L/K$ be a finite extension of fields. For $\alpha \in L$ we define the {\it trace} $\text{Trace}_{L/K}(\alpha) = \text{Trace}_K(\alpha : L \to L)$ and the {\it norm} $\text{Norm}_{L/K}(\alpha) = \det_K(\alpha : L \to L)$.