limits-lemma-proper-limit-of-proper-finite-presentation-noetherian

Let $f : X \to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Then there exists a directed set $I$, an inverse system $(f_i : X_i \to S_i)$ of morphisms of schemes over $I$, such that the transition morphisms $X_i \to X_{i'}$ and $S_i \to S_{i'}$ are affine, such that $f_i$ is proper, such that $S_i$ is of finite type over $\mathbf{Z}$, and such that $(X \to S) = \lim (X_i \to S_i)$.


limits-lemma-finite-type-eventually-proper

Let $S$ be a scheme. Let $X = \lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Let $Y \to X$ be a morphism of schemes over $S$. If $Y \to X$ is proper, $X_i$ quasi-compact and quasi-separated, and $Y$ locally of finite type over $S$, then $Y \to X_i$ is proper for $i$ large enough.


limits-lemma-eventually-proper-support

Assumptions and notation as in Situation \ref{situation-descent-property}. Let $\mathcal{F}_0$ be a quasi-coherent $\mathcal{O}_{X_0}$-module. Denote $\mathcal{F}$ and $\mathcal{F}_i$ the pullbacks of $\mathcal{F}_0$ to $X$ and $X_i$. Assume \begin{enumerate} \item $f_0$ is locally of finite type, \item $\mathcal{F}_0$ is of finite type, \item the scheme theoretic support of $\mathcal{F}$ is proper over $Y$. \end{enumerate} Then the scheme theoretic support of $\mathcal{F}_i$ is proper over $Y_i$ for some $i$.


limits-lemma-separate

Let $f : X \to S$ be a quasi-compact morphism of schemes. Let $g : T \to S$ be a morphism of schemes. Let $t \in T$ be a point and $Z \subset X_T$ be a closed subscheme such that $Z \cap X_t = \emptyset$. Then there exists an open neighbourhood $V \subset T$ of $t$, a commutative diagram $$ \xymatrix{ V \ar[d] \ar[r]_a & T' \ar[d]^b \\ T \ar[r]^g & S, } $$ and a closed subscheme $Z' \subset X_{T'}$ such that \begin{enumerate} \item the morphism $b : T' \to S$ is locally of finite presentation, \item with $t' = a(t)$ we have $Z' \cap X_{t'} = \emptyset$, and \item $Z \cap X_V$ maps into $Z'$ via the morphism $X_V \to X_{T'}$. \end{enumerate} Moreover, we may assume $V$ and $T'$ are affine.


limits-lemma-test-universally-closed

Let $f : X \to S$ be a quasi-compact morphism of schemes. The following are equivalent \begin{enumerate} \item $f$ is universally closed, \item for every morphism $S' \to S$ which is locally of finite presentation the base change $X_{S'} \to S'$ is closed, and \item for every $n$ the morphism $\mathbf{A}^n \times X \to \mathbf{A}^n \times S$ is closed. \end{enumerate}


limits-lemma-limited-base-change

Let $S$ be a scheme. Let $f : X \to S$ be a separated morphism of finite type. The following are equivalent: \begin{enumerate} \item The morphism $f$ is proper. \item For any morphism $S' \to S$ which is locally of finite type the base change $X_{S'} \to S'$ is closed. \item For every $n \geq 0$ the morphism $\mathbf{A}^n \times X \to \mathbf{A}^n \times S$ is closed. \end{enumerate}


limits-lemma-reach-point-closure-Noetherian

Let $f : X \to Y$ be a morphism of schemes. Assume $f$ finite type and $Y$ locally Noetherian. Let $y \in Y$ be a point in the closure of the image of $f$. Then there exists a commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d]^f \\ \Spec(A) \ar[r] & Y } $$ where $A$ is a discrete valuation ring and $K$ is its field of fractions mapping the closed point of $\Spec(A)$ to $y$. Moreover, we can assume that the image point of $\Spec(K) \to X$ is a generic point $\eta$ of an irreducible component of $X$ and that $K = \kappa(\eta)$.


limits-lemma-Noetherian-dvr-valuative-separation

Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a morphism of schemes. Assume $f$ is locally of finite type. The following are equivalent: \begin{enumerate} \item The morphism $f$ is separated. \item For any diagram (\ref{equation-valuative}) there is at most one dotted arrow. \item For all diagrams (\ref{equation-valuative}) with $A$ a discrete valuation ring there is at most one dotted arrow. \item For any irreducible component $X_0$ of $X$ with generic point $\eta \in X_0$, for any discrete valuation ring $A \subset K = \kappa(\eta)$ with fraction field $K$ and any diagram (\ref{equation-valuative}) such that the morphism $\Spec(K) \to X$ is the canonical one (see Schemes, Section \ref{schemes-section-points}) there is at most one dotted arrow. \end{enumerate}


limits-lemma-Noetherian-dvr-valuative-proper

Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a morphism of finite type. The following are equivalent: \begin{enumerate} \item The morphism $f$ is proper. \item For any diagram (\ref{equation-valuative}) there exists exactly one dotted arrow. \item For all diagrams (\ref{equation-valuative}) with $A$ a discrete valuation ring there exists exactly one dotted arrow. \item For any irreducible component $X_0$ of $X$ with generic point $\eta \in X_0$, for any discrete valuation ring $A \subset K = \kappa(\eta)$ with fraction field $K$ and any diagram (\ref{equation-valuative}) such that the morphism $\Spec(K) \to X$ is the canonical one (see Schemes, Section \ref{schemes-section-points}) there exists exactly one dotted arrow. \end{enumerate}


limits-lemma-check-universally-closed-Noetherian

Let $f : X \to S$ be a finite type morphism of schemes. Assume $S$ is locally Noetherian. Then the following are equivalent \begin{enumerate} \item $f$ is universally closed, \item for every $n$ the morphism $\mathbf{A}^n \times X \to \mathbf{A}^n \times S$ is closed, \item for any diagram (\ref{equation-valuative}) there exists some dotted arrow, \item for all diagrams (\ref{equation-valuative}) with $A$ a discrete valuation ring there exists some dotted arrow. \end{enumerate}


limits-lemma-refined-valuative-criterion-proper

Let $f : X \to S$ and $h : U \to X$ be morphisms of schemes. Assume that $S$ is locally Noetherian, that $f$ and $h$ are of finite type, that $f$ is separated, and that $h(U)$ is dense in $X$. If given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^f \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & S } $$ where $A$ is a discrete valuation ring with field of fractions $K$, there exists a dotted arrow making the diagram commute, then $f$ is proper.


limits-lemma-refined-valuative-criterion-separated

Let $f : X \to S$ and $h : U \to X$ be morphisms of schemes. Assume that $S$ is locally Noetherian, that $f$ is locally of finite type, that $h$ is of finite type, and that $h(U)$ is dense in $X$. If given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^f \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & S } $$ where $A$ is a discrete valuation ring with field of fractions $K$, there exists at most one dotted arrow making the diagram commute, then $f$ is separated.


limits-lemma-limit-dimension

Let $I$ be a directed set. Let $(f_i : X_i \to S_i)$ be an inverse system of morphisms of schemes over $I$. Assume \begin{enumerate} \item all the morphisms $S_{i'} \to S_i$ are affine, \item all the schemes $S_i$ are quasi-compact and quasi-separated, \item the morphisms $f_i$ are of finite type, and \item the morphisms $X_{i'} \to X_i \times_{S_i} S_{i'}$ are closed immersions. \end{enumerate} Let $f : X = \lim_i X_i \to S = \lim_i S_i$ be the limit. Let $d \geq 0$. If every fibre of $f$ has dimension $\leq d$, then for some $i$ every fibre of $f_i$ has dimension $\leq d$.


limits-lemma-descend-quasi-finite

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ is a quasi-finite morphism, and \item $f_0$ is locally of finite type, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is quasi-finite.


limits-lemma-descend-dimension-d

Notation and assumptions as in Situation \ref{situation-descent-property}. If \begin{enumerate} \item $f$ has relative dimension $d$, and \item $f_0$ is locally of finite presentation, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ has relative dimension $d$.


limits-lemma-approximate-given-relative-dimension

Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a morphism of finite presentation. Let $d \geq 0$ be an integer. If $Z \subset X$ be a closed subscheme such that $\dim(Z_s) \leq d$ for all $s \in S$, then there exists a closed subscheme $Z' \subset X$ such that \begin{enumerate} \item $Z \subset Z'$, \item $Z' \to X$ is of finite presentation, and \item $\dim(Z'_s) \leq d$ for all $s \in S$. \end{enumerate}


limits-lemma-top-cohomology-functor

Let $f : X \to Y$ be a morphism of schemes. Let $d \geq 0$. Assume \begin{enumerate} \item $X$ and $Y$ are quasi-compact and quasi-separated, and \item $R^if_*\mathcal{F} = 0$ for $i > d$ and every quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$. \end{enumerate} Then we have \begin{enumerate} \item[(a)] for any base change diagram $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ we have $R^if'_*\mathcal{F}' = 0$ for $i > d$ and any quasi-coherent $\mathcal{O}_{X'}$-module $\mathcal{F}'$, \item[(b)] $R^df'_*(\mathcal{F}' \otimes_{\mathcal{O}_{X'}} (f')^*\mathcal{G}') = R^df'_*\mathcal{F}' \otimes_{\mathcal{O}_{Y'}} \mathcal{G}'$ for any quasi-coherent $\mathcal{O}_{Y'}$-module $\mathcal{G}'$, \item[(c)] formation of $R^df'_*\mathcal{F}'$ commutes with arbitrary further base change (see proof for explanation). \end{enumerate}


limits-lemma-higher-direct-images-zero-above-dimension-fibre

Let $f : X \to Y$ be a morphism of schemes. Let $y \in Y$. Assume $f$ is proper and $\dim(X_y) = d$. Then \begin{enumerate} \item for $\mathcal{F} \in \QCoh(\mathcal{O}_X)$ we have $(R^if_*\mathcal{F})_y = 0$ for all $i > d$, \item there is an affine open neighbourhood $V \subset Y$ of $y$ such that $f^{-1}(V) \to V$ and $d$ satisfy the assumptions and conclusions of Lemma \ref{lemma-top-cohomology-functor}. \end{enumerate}


limits-lemma-proper-top-cohomology-finite-type

Let $f : X \to Y$ be a morphism of schemes. Let $d \geq 0$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $f$ is a proper morphism all of whose fibres have dimension $\leq d$, \item $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module of finite type. \end{enumerate} Then $R^df_*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module of finite type.


limits-lemma-glueing-near-closed-point

Let $S$ be a scheme. Let $s \in S$ be a closed point such that $U = S \setminus \{s\} \to S$ is quasi-compact. With $V = \Spec(\mathcal{O}_{S, s}) \setminus \{s\}$ there is an equivalence of categories $$ \left\{ \begin{matrix} X \to S\text{ of finite presentation} \end{matrix} \right\} \longrightarrow \left\{ \vcenter{ \xymatrix{ X' \ar[d] & Y' \ar[d] \ar[l] \ar[r] & Y \ar[d] \\ U & V \ar[l] \ar[r] & \Spec(\mathcal{O}_{S, s}) } } \right\} $$ where on the right hand side we consider commutative diagrams whose squares are cartesian and whose vertical arrows are of finite presentation.


limits-lemma-glueing-near-closed-point-modules

Let $S$ be a scheme. Let $s \in S$ be a closed point such that $U = S \setminus \{s\} \to S$ is quasi-compact. With $V = \Spec(\mathcal{O}_{S, s}) \setminus \{s\}$ there is an equivalence of categories $$ \left\{ \mathcal{O}_S\text{-modules }\mathcal{F}\text{ of finite presentation} \right\} \longrightarrow \left\{ (\mathcal{G}, \mathcal{H}, \alpha) \right\} $$ where on the right hand side we consider triples consisting of a $\mathcal{O}_U$-module $\mathcal{G}$ of finite presentation, a $\mathcal{O}_{\Spec(\mathcal{O}_{S, s})}$-module $\mathcal{H}$ of finite presentation, and an isomorphism $\alpha : \mathcal{G}|_V \to \mathcal{H}|_V$ of $\mathcal{O}_V$-modules.


limits-lemma-glueing-near-point

Let $S$ be a scheme. Let $U \subset S$ be a retrocompact open. Let $s \in S$ be a point in the complement of $U$. With $V = \Spec(\mathcal{O}_{S, s}) \cap U$ there is an equivalence of categories $$ \colim_{s \in U' \supset U\text{ open}} \left\{ \vcenter{ \xymatrix{ X \ar[d] \\ U' } } \right\} \longrightarrow \left\{ \vcenter{ \xymatrix{ X' \ar[d] & Y' \ar[d] \ar[l] \ar[r] & Y \ar[d] \\ U & V \ar[l] \ar[r] & \Spec(\mathcal{O}_{S, s}) } } \right\} $$ where on the left hand side the vertical arrow is of finite presentation and on the right hand side we consider commutative diagrams whose squares are cartesian and whose vertical arrows are of finite presentation.


limits-lemma-glueing-near-point-properties

Notation and assumptions as in Lemma \ref{lemma-glueing-near-point}. Let $U \subset U' \subset X$ be an open containing $s$. \begin{enumerate} \item  Let $f' : X \to U'$ correspond to $f : X' \to U$ and $g : Y \to \Spec(\mathcal{O}_{S, s})$ via the equivalence. If $f$ and $g$ are separated, proper, finite, \'etale, then after possibly shrinking $U'$ the morphism $f'$ has the same property. \item Let $a : X_1 \to X_2$ be a morphism of schemes of finite presentation over $U'$ with base change $a' : X'_1 \to X'_2$ over $U$ and $b : Y_1 \to Y_2$ over $\Spec(\mathcal{O}_{S, s})$. If $a'$ and $b$ are separated, proper, finite, \'etale, then after possibly shrinking $U'$ the morphism $a$ has the same property. \end{enumerate}


limits-lemma-modifications

Let $S$ be a scheme. Let $s \in S$ be a closed point such that $U = S \setminus \{s\} \to S$ is quasi-compact. With $V = \Spec(\mathcal{O}_{S, s}) \setminus \{s\}$ the base change functor $$ \left\{ \begin{matrix} f : X \to S\text{ of finite presentation} \\ f^{-1}(U) \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} g : Y \to \Spec(\mathcal{O}_{S, s})\text{ of finite presentation} \\ g^{-1}(V) \to V\text{ is an isomorphism} \end{matrix} \right\} $$ is an equivalence of categories.


limits-lemma-modifications-properties

Notation and assumptions as in Lemma \ref{lemma-modifications}. Let $f : X \to S$ correspond to $g : Y \to \Spec(\mathcal{O}_{S, s})$ via the equivalence. Then $f$ is separated, proper, finite, \'etale and add more here if and only if $g$ is so.


limits-lemma-good-diagram

In Situation \ref{situation-limit-noetherian}. Let $X \to S$ be quasi-separated and of finite type. Then there exists an $i \in I$ and a diagram \begin{equation} \label{equation-good-diagram} \vcenter{ \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ S \ar[r] & S_i } } \end{equation} such that $W \to S_i$ is of finite type and such that the induced morphism $X \to S \times_{S_i} W$ is a closed immersion.


limits-lemma-limit-from-good-diagram

In Situation \ref{situation-limit-noetherian}. Let $X \to S$ be quasi-separated and of finite type. Given $i \in I$ and a diagram $$ \vcenter{ \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ S \ar[r] & S_i } } $$ as in (\ref{equation-good-diagram}) for $i' \geq i$ let $X_{i'}$ be the scheme theoretic image of $X \to S_{i'} \times_{S_i} W$. Then $X = \lim_{i' \geq i} X_{i'}$.


limits-lemma-morphism-good-diagram

In Situation \ref{situation-limit-noetherian}. Let $f : X \to Y$ be a morphism of schemes quasi-separated and of finite type over $S$. Let $$ \vcenter{ \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ S \ar[r] & S_{i_1} } } \quad\text{and}\quad \vcenter{ \xymatrix{ Y \ar[r] \ar[d] & V \ar[d] \\ S \ar[r] & S_{i_2} } } $$ be diagrams as in (\ref{equation-good-diagram}). Let $X = \lim_{i \geq i_1} X_i$ and $Y = \lim_{i \geq i_2} Y_i$ be the corresponding limit descriptions as in Lemma \ref{lemma-limit-from-good-diagram}. Then there exists an $i_0 \geq \max(i_1, i_2)$ and a morphism $$ (f_i)_{i \geq i_0} : (X_i)_{i \geq i_0} \to (Y_i)_{i \geq i_0} $$ of inverse systems over $(S_i)_{i \geq i_0}$ such that such that $f = \lim_{i \geq i_0} f_i$. If $(g_i)_{i \geq i_0} : (X_i)_{i \geq i_0} \to (Y_i)_{i \geq i_0}$ is a second morphism of inverse systems over $(S_i)_{i \geq i_0}$ such that such that $f = \lim_{i \geq i_0} g_i$ then $f_i = g_i$ for all $i \gg i_0$.


limits-lemma-morphism-good-diagram-flat

Notation and assumptions as in Lemma \ref{lemma-morphism-good-diagram}. If $f$ is flat and of finite presentation, then there exists an $i_3 \geq i_0$ such that for $i \geq i_3$ we have $f_i$ is flat, $X_i = Y_i \times_{Y_{i_3}} X_{i_3}$, and $X = Y \times_{Y_{i_3}} X_{i_3}$.


limits-proposition-approximate

Let $S$ be a quasi-compact and quasi-separated scheme. There exist a directed set $I$ and an inverse system of schemes $(S_i, f_{ii'})$ over $I$ such that \begin{enumerate} \item the transition morphisms $f_{ii'}$ are affine \item each $S_i$ is of finite type over $\mathbf{Z}$, and \item $S = \lim_i S_i$. \end{enumerate}


limits-proposition-characterize-locally-finite-presentation

\begin{reference} \cite[IV, Proposition 8.14.2]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $f$ is locally of finite presentation. \item For any directed set $I$, and any inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$ with each $T_i$ affine, we have $$ \Mor_S(\lim_i T_i, X) = \colim_i \Mor_S(T_i, X) $$ \item For any directed set $I$, and any inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$ with each $f_{ii'}$ affine and every $T_i$ quasi-compact and quasi-separated as a scheme, we have $$ \Mor_S(\lim_i T_i, X) = \colim_i \Mor_S(T_i, X) $$ \end{enumerate}


limits-proposition-separated-closed-in-finite-presentation

Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ is of finite type and separated, and \item $S$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a separated morphism of finite presentation $f' : X' \to S$ and a closed immersion $X \to X'$ of schemes over $S$.


limits-proposition-affine

\begin{slogan} A scheme admitting a surjective integral map from an affine scheme is affine. \end{slogan} Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is surjective and integral, and assume that $X$ is affine. Then $S$ is affine.


stacks-perfect-definition-derived

Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{M}_\mathcal{X} \subset \textit{Mod}(\mathcal{O}_\mathcal{X})$ denote the category of locally quasi-coherent $\mathcal{O}_\mathcal{X}$-modules with the flat base change property. Let $\mathcal{P}_\mathcal{X} \subset \mathcal{M}_\mathcal{X}$ be the full subcategory consisting of parasitic objects. We define the {\it derived category of $\mathcal{O}_\mathcal{X}$-modules with quasi-coherent cohomology sheaves} as the Verdier quotient\footnote{This definition is different from the one in the literature, see \cite[6.3]{olsson_sheaves}, but it agrees with that definition by Lemma \ref{lemma-derived-quasi-coherent}.} $$ D_\QCoh(\mathcal{O}_\mathcal{X}) = D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})/ D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) $$


spaces-more-morphisms-definition-radicial

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\it radicial} if for any morphism $\Spec(K) \to Y$ where $K$ is a field the reduction $(\Spec(K) \times_Y X)_{red}$ is either empty or representable by the spectrum of a purely inseparable field extension of $K$.


spaces-more-morphisms-definition-conormal-sheaf

Let $i : Z \to X$ be an immersion. The {\it conormal sheaf $\mathcal{C}_{Z/X}$ of $Z$ in $X$} or the {\it conormal sheaf of $i$} is the quasi-coherent $\mathcal{O}_Z$-module $\mathcal{I}/\mathcal{I}^2$ described above.


spaces-more-morphisms-definition-conormal-algebra

Let $i : Z \to X$ be an immersion. The {\it conormal algebra $\mathcal{C}_{Z/X, *}$ of $Z$ in $X$} or the {\it conormal algebra of $i$} is the quasi-coherent sheaf of graded $\mathcal{O}_Z$-algebras $\bigoplus_{n \geq 0} \mathcal{I}^n/\mathcal{I}^{n + 1}$ described above.


spaces-more-morphisms-definition-normal-cone

Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. The {\it normal cone $C_ZX$} of $Z$ in $X$ is $$ C_ZX = \underline{\Spec}_Z(\mathcal{C}_{Z/X, *}) $$ see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-relative-spec}. The {\it normal bundle} of $Z$ in $X$ is the vector bundle $$ N_ZX = \underline{\Spec}_Z(\text{Sym}(\mathcal{C}_{Z/X})) $$


spaces-more-morphisms-definition-sheaf-differentials

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The {\it sheaf of differentials $\Omega_{X/Y}$ of $X$ over $Y$} is sheaf of differentials (Modules on Sites, Definition \ref{sites-modules-definition-sheaf-differentials}) for the morphism of ringed topoi $$ (f_{small}, f^\sharp) : (X_\etale, \mathcal{O}_X) \to (Y_\etale, \mathcal{O}_Y) $$ of Properties of Spaces, Lemma \ref{spaces-properties-lemma-morphism-ringed-topoi}. The {\it universal $Y$-derivation} will be denoted $\text{d}_{X/Y} : \mathcal{O}_X \to \Omega_{X/Y}$.


spaces-more-morphisms-definition-thickening

Thickenings. Let $S$ be a scheme. \begin{enumerate} \item We say an algebraic space $X'$ is a {\it thickening} of an algebraic space $X$ if $X$ is a closed subspace of $X'$ and the associated topological spaces are equal. \item We say $X'$ is a {\it first order thickening} of $X$ if $X$ is a closed subspace of $X'$ and the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_{X'}$ defining $X$ has square zero. \item Given two thickenings $X \subset X'$ and $Y \subset Y'$ a {\it morphism of thickenings} is a morphism $f' : X' \to Y'$ such that $f(X) \subset Y$, i.e., such that $f'|_X$ factors through the closed subspace $Y$. In this situation we set $f = f'|_X : X \to Y$ and we say that $(f, f') : (X \subset X') \to (Y \subset Y')$ is a morphism of thickenings. \item Let $B$ be an algebraic space. We similarly define {\it thickenings over $B$}, and {\it morphisms of thickenings over $B$}. This means that the spaces $X, X', Y, Y'$ above are algebraic spaces endowed with a structure morphism to $B$, and that the morphisms $X \to X'$, $Y \to Y'$ and $f' : X' \to Y'$ are morphisms over $B$. \end{enumerate}


spaces-more-morphisms-definition-first-order-infinitesimal-neighbourhood

Let $i : Z \to X$ be an immersion of algebraic spaces. The {\it first order infinitesimal neighbourhood} of $Z$ in $X$ is the first order thickening $Z \subset Z'$ over $X$ described above.


spaces-more-morphisms-definition-formally-smooth-etale-unramified

Let $S$ be a scheme. Let $a : F \to G$ be a transformation of functors $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Consider commutative solid diagrams of the form $$ \xymatrix{ F \ar[d]_a & T \ar[d]^i \ar[l] \\ G & T' \ar[l] \ar@{-->}[lu] } $$ where $T$ and $T'$ are affine schemes and $i$ is a closed immersion defined by an ideal of square zero. \begin{enumerate} \item We say $a$ is {\it formally smooth} if given any solid diagram as above there exists a dotted arrow making the diagram commute\footnote{This is just one possible definition that one can make here. Another slightly weaker condition would be to require that the dotted arrow exists fppf locally on $T'$. This weaker notion has in some sense better formal properties.}. \item We say $a$ is {\it formally \'etale} if given any solid diagram as above there exists exactly one dotted arrow making the diagram commute. \item We say $a$ is {\it formally unramified} if given any solid diagram as above there exists at most one dotted arrow making the diagram commute. \end{enumerate}


spaces-more-morphisms-definition-formally-unramified

Let $S$ be a scheme. A morphism $f : X \to Y$ of algebraic spaces over $S$ is said to be {\it formally unramified} if it is formally unramified as a transformation of functors as in Definition \ref{definition-formally-smooth-etale-unramified}.


spaces-more-morphisms-definition-universal-thickening

Let $S$ be a scheme. Let $h : Z \to X$ be a formally unramified morphism of algebraic spaces over $S$. \begin{enumerate} \item The {\it universal first order thickening} of $Z$ over $X$ is the thickening $Z \subset Z'$ constructed in Lemma \ref{lemma-universal-thickening}. \item The {\it conormal sheaf of $Z$ over $X$} is the conormal sheaf of $Z$ in its universal first order thickening $Z'$ over $X$. \end{enumerate} We often denote the conormal sheaf $\mathcal{C}_{Z/X}$ in this situation.


spaces-more-morphisms-definition-formally-etale

Let $S$ be a scheme. A morphism $f : X \to Y$ of algebraic spaces over $S$ is said to be {\it formally \'etale} if it is formally \'etale as a transformation of functors as in Definition \ref{definition-formally-smooth-etale-unramified}.


spaces-more-morphisms-definition-formally-smooth

Let $S$ be a scheme. A morphism $f : X \to Y$ of algebraic spaces over $S$ is said to be {\it formally smooth} if it is formally smooth as a transformation of functors as in Definition \ref{definition-formally-smooth-etale-unramified}.


spaces-more-morphisms-definition-netherlander

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The {\it naive cotangent complex of $f$} is the complex defined in Modules on Sites, Definition \ref{sites-modules-definition-cotangent-complex-morphism-ringed-topoi} for the morphism of ringed topoi $f_{small}$ between the small \'etale sites of $X$ and $Y$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-morphism-ringed-topoi}. Notation: $\NL_f$ or $\NL_{X/Y}$.


spaces-more-morphisms-definition-module-flat-on-fibre

Let $S$ be a scheme. Let $X \to Y \to Z$ be morphisms of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in |X|$ be a point and denote $z \in |Z|$ its image. \begin{enumerate} \item We say {\it the restriction of $\mathcal{F}$ to its fibre over $z$ is flat at $x$ over the fibre of $Y$ over $z$} if the equivalent conditions of Lemma \ref{lemma-flat-on-fibres-at-point} are satisfied. \item We say {\it the fibre of $X$ over $z$ is flat at $x$ over the fibre of $Y$ over $z$} if the equivalent conditions of Lemma \ref{lemma-flat-on-fibres-at-point} hold with $\mathcal{F} = \mathcal{O}_X$. \item We say {\it the fibre of $X$ over $z$ is flat over the fibre of $Y$ over $z$} if for all $x \in |X|$ lying over $z$ the fibre of $X$ over $z$ is flat at $x$ over the fibre of $Y$ over $z$ \end{enumerate}


spaces-more-morphisms-definition-CM

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian (Divisors on Spaces, Definition \ref{spaces-divisors-definition-locally-Noetherian-fibre}). \begin{enumerate} \item Let $x \in |X|$, and $y = f(x)$. We say that $f$ is {\it Cohen-Macaulay at $x$} if $f$ is flat at $x$ and the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-local-source-target-at-point} hold for the property $\mathcal{P}$ described in Lemma \ref{lemma-CM-local-ring-fibre}. \item We say $f$ is a {\it Cohen-Macaulay morphism} if $f$ is Cohen-Macaulay at every point of $X$. \end{enumerate}


spaces-more-morphisms-definition-gorenstein

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian (Divisors on Spaces, Definition \ref{spaces-divisors-definition-locally-Noetherian-fibre}). \begin{enumerate} \item Let $x \in |X|$, and $y = f(x)$. We say that $f$ is {\it Gorenstein at $x$} if $f$ is flat at $x$ and the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-local-source-target-at-point} hold for the property $\mathcal{P}$ described in Lemma \ref{lemma-gorenstein-local-ring-fibre}. \item We say $f$ is a {\it Gorenstein morphism} if $f$ is Gorenstein at every point of $X$. \end{enumerate}


spaces-more-morphisms-definition-geometrically-reduced-fibre

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$. We say {\it the fibre of $f : X \to Y$ at $y$ is geometrically reduced} if the equivalent conditions of Lemma \ref{lemma-geometrically-reduced-fibre} hold.


spaces-more-morphisms-definition-regular-immersion

Let $S$ be a scheme. Let $i : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say $i$ is a {\it Koszul-regular immersion} if $i$ is representable and the equivalent conditions of Lemma \ref{lemma-representable-etale-local-target} hold with $\mathcal{P}(f) =$``$f$ is a Koszul-regular immersion''. \item We say $i$ is an {\it $H_1$-regular immersion} if $i$ is representable and the equivalent conditions of Lemma \ref{lemma-representable-etale-local-target} hold with $\mathcal{P}(f) =$``$f$ is an $H_1$-regular immersion''. \item We say $i$ is a {\it quasi-regular immersion} if $i$ is representable and the equivalent conditions of Lemma \ref{lemma-representable-etale-local-target} hold with $\mathcal{P}(f) =$``$f$ is a quasi-regular immersion''. \end{enumerate}


spaces-more-morphisms-definition-relative-pseudo-coherence

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 $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Fix $m \in \mathbf{Z}$. \begin{enumerate} \item We say $E$ is {\it $m$-pseudo-coherent relative to $Y$} if the equivalent conditions of Lemma \ref{lemma-qcoh-relative-pseudo-coherence-characterize} are satisfied. \item We say $E$ is {\it pseudo-coherent relative to $Y$} if $E$ is $m$-pseudo-coherent relative to $Y$ for all $m \in \mathbf{Z}$. \item We say $\mathcal{F}$ is {\it $m$-pseudo-coherent relative to $Y$} if $\mathcal{F}$ viewed as an object of $D_\QCoh(\mathcal{O}_X)$ is $m$-pseudo-coherent relative to $Y$. \item We say $\mathcal{F}$ is {\it pseudo-coherent relative to $Y$} if $\mathcal{F}$ viewed as an object of $D_\QCoh(\mathcal{O}_X)$ is pseudo-coherent relative to $Y$. \end{enumerate}


spaces-more-morphisms-definition-pseudo-coherent

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say $f$ is {\it pseudo-coherent} if the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-local-source-target} hold with $\mathcal{P} =$``pseudo-coherent''. \item Let $x \in |X|$. We say $f$ is {\it pseudo-coherent at $x$} if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is pseudo-coherent. \end{enumerate}


spaces-more-morphisms-definition-perfect

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say $f$ is {\it perfect} if the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-local-source-target} hold with $\mathcal{P} =$``perfect''. \item Let $x \in |X|$. We say $f$ is {\it perfect at $x$} if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is perfect. \end{enumerate}


spaces-more-morphisms-definition-lci

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item We say $f$ is a {\it Koszul morphism}, or that $f$ is a {\it local complete intersection morphism} if the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-local-source-target} hold with $\mathcal{P}(f) =$``$f$ is a local complete intersection morphism''. \item Let $x \in |X|$. We say $f$ is {\it Koszul at $x$} if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is a local complete intersection morphism. \end{enumerate}


spaces-more-morphisms-definition-relatively-perfect

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. An object $E$ of $D(\mathcal{O}_X)$ is {\it perfect relative to $Y$} or {\it $Y$-perfect} if $E$ is pseudo-coherent (Cohomology on Sites, Definition \ref{sites-cohomology-definition-pseudo-coherent}) and $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_Y)$ (Cohomology on Sites, Definition \ref{sites-cohomology-definition-tor-amplitude}).


spaces-more-morphisms-definition-nodal-family

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\it at-worst-nodal of relative dimension $1$} if the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-local-source-target} hold with $\mathcal{P} =$``at-worst-nodal of relative dimension $1$''.


algebra-definition-module-finite-type

Let $R$ be a ring. Let $M$ be an $R$-module. \begin{enumerate} \item We say $M$ is a {\it finite $R$-module}, or a {\it finitely generated $R$-module} if there exist $n \in \mathbf{N}$ and $x_1, \ldots, x_n \in M$ such that every element of $M$ is a $R$-linear combination of the $x_i$. Equivalently, this means there exists a surjection $R^{\oplus n} \to M$ for some $n \in \mathbf{N}$. \item We say $M$ is a {\it finitely presented $R$-module} or an {\it $R$-module of finite presentation} if there exist integers $n, m \in \mathbf{N}$ and an exact sequence $$ R^{\oplus m} \longrightarrow R^{\oplus n} \longrightarrow M \longrightarrow 0 $$ \end{enumerate}


algebra-definition-finite-type

Let $R \to S$ be a ring map. \begin{enumerate} \item We say $R \to S$ is of {\it finite type}, or that {\it $S$ is a finite type $R$-algebra} if there exists an $n \in \mathbf{N}$ and an surjection of $R$-algebras $R[x_1, \ldots, x_n] \to S$. \item We say $R \to S$ is of {\it finite presentation} if there exist integers $n, m \in \mathbf{N}$ and polynomials $f_1, \ldots, f_m \in R[x_1, \ldots, x_n]$ and an isomorphism of $R$-algebras $R[x_1, \ldots, x_n]/(f_1, \ldots, f_m) \cong S$. \end{enumerate}


algebra-definition-finite-ring-map

Let $\varphi : R \to S$ be a ring map. We say $\varphi : R \to S$ is {\it finite} if $S$ is finite as an $R$-module.


algebra-definition-directed-system

Let $(I, \leq)$ be a preordered set. A {\it system $(M_i, \mu_{ij})$ of $R$-modules over $I$} consists of a family of $R$-modules $\{M_i\}_{i\in I}$ indexed by $I$ and a family of $R$-module maps $\{\mu_{ij} : M_i \to M_j\}_{i \leq j}$ such that for all $i \leq j \leq k$ $$ \mu_{ii} = \text{id}_{M_i}\quad \mu_{ik} = \mu_{jk}\circ \mu_{ij} $$ We say $(M_i, \mu_{ij})$ is a {\it directed system} if $I$ is a directed set.


algebra-definition-homomorphism-directed-systems

Let $(M_i, \mu_{ij})$, $(N_i, \nu_{ij})$ be systems of $R$-modules over the same preordered set $I$. A {\it homomorphism of systems} $\Phi$ from $(M_i, \mu_{ij})$ to $(N_i, \nu_{ij})$ is by definition a family of $R$-module homomorphisms $\phi_i : M_i \to N_i$ such that $\phi_j \circ \mu_{ij} = \nu_{ij} \circ \phi_i$ for all $i \leq j$.


algebra-definition-multiplicative-subset

Let $R$ be a ring, $S$ a subset of $R$. We say $S$ is a {\it multiplicative subset of $R$} if $1\in S$ and $S$ is closed under multiplication, i.e., $s, s' \in S \Rightarrow ss' \in S$.


algebra-definition-localization

This ring is called the {\it localization of $A$ with respect to $S$}.


algebra-definition-localization-module

The $S^{-1}A$-module $S^{-1}M$ is called the {\it localization} of $M$ at $S$.


algebra-definition-relation

Let $R$ be a ring. Let $M$ be an $R$-module. Let $n \geq 0$ and $x_i \in M$ for $i = 1, \ldots, n$. A {\it relation} between $x_1, \ldots, x_n$ in $M$ is a sequence of elements $f_1, \ldots, f_n \in R$ such that $\sum_{i = 1, \ldots, n} f_i x_i = 0$.


algebra-definition-bilinear

Let $R$ be a ring, $M, N, P$ be three $R$-modules. A mapping $f : M \times N \to P$ (where $M \times N$ is viewed only as Cartesian product of two $R$-modules) is said to be {\it $R$-bilinear} if for each $x \in M$ the mapping $y\mapsto f(x, y)$ of $N$ into $P$ is $R$-linear, and for each $y\in N$ the mapping $x\mapsto f(x, y)$ is also $R$-linear.


algebra-definition-bimodule

An abelian group $N$ is called an {\it $(A, B)$-bimodule} if it is both an $A$-module and a $B$-module, and the actions $A \to End(M)$ and $B \to End(M)$ are compatible in the sense that $(ax)b = a(xb)$ for all $a\in A, b\in B, x\in N$. Usually we denote it as $_AN_B$.


algebra-definition-base-change

Let $\varphi : R \to S$ be a ring map. Let $M$ be an $S$-module. Let $R \to R'$ be any ring map. The {\it base change} of $\varphi$ by $R \to R'$ is the ring map $R' \to S \otimes_R R'$. In this situation we often write $S' = S \otimes_R R'$. The {\it base change} of the $S$-module $M$ is the $S'$-module $M \otimes_R R'$.


algebra-definition-spectrum-ring

Let $R$ be a ring. \begin{enumerate} \item The {\it spectrum} of $R$ is the set of prime ideals of $R$. It is usually denoted $\Spec(R)$. \item Given a subset $T \subset R$ we let $V(T) \subset \Spec(R)$ be the set of primes containing $T$, i.e., $V(T) = \{ \mathfrak p \in \Spec(R) \mid \forall f\in T, f\in \mathfrak p\}$. \item Given an element $f \in R$ we let $D(f) \subset \Spec(R)$ be the set of primes not containing $f$. \end{enumerate}


algebra-definition-Zariski-topology

Let $R$ be a ring. The topology on $\Spec(R)$ whose closed sets are the sets $V(T)$ is called the {\it Zariski} topology. The open subsets $D(f)$ are called the {\it standard opens} of $\Spec(R)$.


algebra-definition-local-ring

A {\it local ring} is a ring with exactly one maximal ideal. The maximal ideal is often denoted $\mathfrak m_R$ in this case. We often say ``let $(R, \mathfrak m, \kappa)$ be a local ring'' to indicate that $R$ is local, $\mathfrak m$ is its unique maximal ideal and $\kappa = R/\mathfrak m$ is its residue field. A {\it local homomorphism of local rings} is a ring map $\varphi : R \to S$ such that $R$ and $S$ are local rings and such that $\varphi(\mathfrak m_R) \subset \mathfrak m_S$. If it is given that $R$ and $S$ are local rings, then the phrase ``{\it local ring map $\varphi : R \to S$}'' means that $\varphi$ is a local homomorphism of local rings.


algebra-definition-oka-family

Let $R$ be a ring. Let $\mathcal{F}$ be a set of ideals of $R$. We say $\mathcal{F}$ is an {\it Oka family} if $R \in \mathcal{F}$ and whenever $I \subset R$ is an ideal and $(I : a), (I, a) \in \mathcal{F}$ for some $a \in R$, then $I \in \mathcal{F}$.


algebra-definition-locally-nilpotent-ideal

Let $R$ be a ring. Let $I \subset R$ be an ideal. We say $I$ is {\it locally nilpotent} if for every $x \in I$ there exists an $n \in \mathbf{N}$ such that $x^n = 0$. We say $I$ is {\it nilpotent} if there exists an $n \in \mathbf{N}$ such that $I^n = 0$.


algebra-definition-ring-jacobson

Let $R$ be a ring. We say that $R$ is a {\it Jacobson ring} if every radical ideal $I$ is the intersection of the maximal ideals containing it.


algebra-definition-integral-ring-map

Let $\varphi : R \to S$ be a ring map. \begin{enumerate} \item An element $s \in S$ is {\it integral over $R$} if there exists a monic polynomial $P(x) \in R[x]$ such that $P^\varphi(s) = 0$, where $P^\varphi(x) \in S[x]$ is the image of $P$ under $\varphi : R[x] \to S[x]$. \item  The ring map $\varphi$ is {\it integral} if every $s \in S$ is integral over $R$. \end{enumerate}


algebra-definition-integral-closure

Let $R \to S$ be a ring map. The ring $S' \subset S$ of elements integral over $R$, see Lemma \ref{lemma-integral-closure-is-ring}, is called the {\it integral closure} of $R$ in $S$. If $R \subset S$ we say that $R$ is {\it integrally closed} in $S$ if $R = S'$.


algebra-definition-domain-normal

A domain $R$ is called {\it normal} if it is integrally closed in its field of fractions.


algebra-definition-almost-integral

Let $R$ be a domain. \begin{enumerate} \item An element $g$ of the fraction field of $R$ is called {\it almost integral over $R$} if there exists an element $r \in R$, $r\not = 0$ such that $rg^n \in R$ for all $n \geq 0$. \item The domain $R$ is called {\it completely normal} if every almost integral element of the fraction field of $R$ is contained in $R$. \end{enumerate}


algebra-definition-ring-normal

A ring $R$ is called {\it normal} if for every prime $\mathfrak p \subset R$ the localization $R_{\mathfrak p}$ is a normal domain (see Definition \ref{definition-domain-normal}).


algebra-definition-integral-over-ideal

Let $\varphi : R \to S$ be a ring map. Let $I \subset R$ be an ideal. We say an element $g \in S$ is {\it integral over $I$} if there exists a monic polynomial $P = x^d + \sum_{j < d} a_j x^j$ with coefficients $a_j \in I^{d-j}$ such that $P^\varphi(g) = 0$ in $S$.


algebra-definition-flat

Let $R$ be a ring. \begin{enumerate} \item An $R$-module $M$ is called {\it flat} if whenever $N_1 \to N_2 \to N_3$ is an exact sequence of $R$-modules the sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$ is exact as well. \item An $R$-module $M$ is called {\it faithfully flat} if the complex of $R$-modules $N_1 \to N_2 \to N_3$ is exact if and only if the sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$ is exact. \item A ring map $R \to S$ is called {\it flat} if $S$ is flat as an $R$-module. \item A ring map $R \to S$ is called {\it faithfully flat} if $S$ is faithfully flat as an $R$-module. \end{enumerate}


algebra-definition-support-module

Let $R$ be a ring and let $M$ be an $R$-module. The {\it support of $M$} is the set $$ \text{Supp}(M) = \{ \mathfrak p \in \Spec(R) \mid M_{\mathfrak p} \not = 0 \} $$


algebra-definition-annihilator

Let $R$ be a ring. Let $M$ be an $R$-module. \begin{enumerate} \item Given an element $m \in M$ the {\it annihilator of $m$} is the ideal $$ \text{Ann}_R(m) = \text{Ann}(m) = \{f \in R \mid fm = 0\}. $$ \item The {\it annihilator of $M$} is the ideal $$ \text{Ann}_R(M) = \text{Ann}(M) = \{f \in R \mid fm = 0\ \forall m \in M\}. $$ \end{enumerate}


algebra-definition-going-up-down

Let $\varphi : R \to S$ be a ring map. \begin{enumerate} \item We say a $\varphi : R \to S$ satisfies {\it going up} if given primes $\mathfrak p \subset \mathfrak p'$ in $R$ and a prime $\mathfrak q$ in $S$ lying over $\mathfrak p$ there exists a prime $\mathfrak q'$ of $S$ such that (a) $\mathfrak q \subset \mathfrak q'$, and (b) $\mathfrak q'$ lies over $\mathfrak p'$. \item We say a $\varphi : R \to S$ satisfies {\it going down} if given primes $\mathfrak p \subset \mathfrak p'$ in $R$ and a prime $\mathfrak q'$ in $S$ lying over $\mathfrak p'$ there exists a prime $\mathfrak q$ of $S$ such that (a) $\mathfrak q \subset \mathfrak q'$, and (b) $\mathfrak q$ lies over $\mathfrak p$. \end{enumerate}


algebra-definition-separable-field-extension

Let $k \subset K$ be a field extension. \begin{enumerate} \item We say $K$ is {\it separably generated over $k$} if there exists a transcendence basis $\{x_i; i \in I\}$ of $K/k$ such that the extension $k(x_i; i\in I) \subset K$ is a separable algebraic extension. \item We say $K$ is {\it separable over $k$} if for every subextension $k \subset K' \subset K$ with $K'$ finitely generated over $k$, the extension $k \subset K'$ is separably generated. \end{enumerate}


algebra-definition-geometrically-reduced

Let $k$ be a field. Let $S$ be a $k$-algebra. We say $S$ is {\it geometrically reduced over $k$} if for every field extension $k \subset K$ the $K$-algebra $K \otimes_k S$ is reduced.


algebra-definition-perfect

Let $k$ be a field. We say $k$ is {\it perfect} if every field extension of $k$ is separable over $k$.


algebra-definition-perfection

Let $k$ be a field. The field extension $k \subset k'$ of Lemma \ref{lemma-perfection} is called the {\it perfect closure} of $k$. Notation $k \subset k^{perf}$.


algebra-definition-geometrically-irreducible

Let $k$ be a field. Let $S$ be a $k$-algebra. We say $S$ is {\it geometrically irreducible over $k$} if for every field extension $k \subset k'$ the spectrum of $S \otimes_k k'$ is irreducible\footnote{An irreducible space is nonempty.}.


algebra-definition-geometrically-connected

Let $k$ be a field. Let $S$ be a $k$-algebra. We say $S$ is {\it geometrically connected over $k$} if for every field extension $k \subset k'$ the spectrum of $S \otimes_k k'$ is connected.


algebra-definition-geometrically-integral

Let $k$ be a field. Let $S$ be a $k$-algebra. We say $S$ is {\it geometrically integral over $k$} if for every field extension $k \subset k'$ the ring of $S \otimes_k k'$ is a domain.


algebra-definition-valuation-ring

Valuation rings. \begin{enumerate} \item Let $K$ be a field. Let $A$, $B$ be local rings contained in $K$. We say that $B$ {\it dominates} $A$ if $A \subset B$ and $\mathfrak m_A = A \cap \mathfrak m_B$. \item Let $A$ be a ring. We say $A$ is a {\it valuation ring} if $A$ is a local domain and if $A$ is maximal for the relation of domination among local rings contained in the fraction field of $A$. \item Let $A$ be a valuation ring with fraction field $K$. If $R \subset K$ is a subring of $K$, then we say $A$ is {\it centered} on $R$ if $R \subset A$. \end{enumerate}


algebra-definition-value-group

Let $A$ be a valuation ring. \begin{enumerate} \item The totally ordered abelian group $(\Gamma, \geq)$ of Lemma \ref{lemma-valuation-group} is called the {\it value group} of the valuation ring $A$. \item The map $v : A - \{0\} \to \Gamma$ and also $v : K^* \to \Gamma$ is called the {\it valuation} associated to $A$. \item The valuation ring $A$ is called a {\it discrete valuation ring} if $\Gamma \cong \mathbf{Z}$. \end{enumerate}


algebra-definition-length

Let $R$ be a ring. For any $R$-module $M$ we define the {\it length} of $M$ over $R$ by the formula $$ \text{length}_R(M) = \sup \{ n \mid \exists\ 0 = M_0 \subset M_1 \subset \ldots \subset M_n = M, \text{ }M_i \not = M_{i + 1} \}. $$


algebra-definition-simple-module

Let $R$ be a ring. Let $M$ be an $R$-module. We say $M$ is {\it simple} if $M \not = 0$ and every submodule of $M$ is either equal to $M$ or to $0$.


algebra-definition-artinian

A ring $R$ is {\it Artinian} if it satisfies the descending chain condition for ideals.


algebra-definition-essentially-finite-p-t

Let $R \to S$ be a ring map. \begin{enumerate} \item We say that $R \to S$ is {\it essentially of finite type} if $S$ is the localization of an $R$-algebra of finite type. \item We say that $R \to S$ is {\it essentially of finite presentation} if $S$ is the localization of an $R$-algebra of finite presentation. \end{enumerate}


algebra-definition-proj

Let $S$ be a graded ring. We define $\text{Proj}(S)$ to be the set of homogeneous prime ideals $\mathfrak p$ of $S$ such that $S_{+} \not \subset \mathfrak p$. The set $\text{Proj}(S)$ is a subset of $\Spec(S)$ and we endow it with the induced topology. The topological space $\text{Proj}(S)$ is called the {\it homogeneous spectrum} of the graded ring $S$.


algebra-definition-numerical-polynomial

Let $A$ be an abelian group. We say that a function $f : n \mapsto f(n) \in A$ defined for all sufficient large integers $n$ is a {\it numerical polynomial} if there exists $r \geq 0$, elements $a_0, \ldots, a_r\in A$ such that $$ f(n) = \sum\nolimits_{i = 0}^r \binom{n}{i} a_i $$ for all $n \gg 0$.


algebra-definition-ideal-definition

Let $(R, \mathfrak m)$ be a local Noetherian ring. An ideal $I \subset R$ such that $\sqrt{I} = \mathfrak m$ is called {\it an ideal of definition of $R$}.


algebra-definition-hilbert-polynomial

Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. The {\it Hilbert polynomial} of $M$ over $R$ is the element $P(t) \in \mathbf{Q}[t]$ such that $P(n) = \varphi_M(n)$ for $n \gg 0$.


algebra-definition-d

Let $R$ be a local Noetherian ring and $M$ a finite $R$-module. We denote {\it $d(M)$} the element of $\{-\infty, 0, 1, 2, \ldots \}$ defined as follows: \begin{enumerate} \item If $M = 0$ we set $d(M) = -\infty$, \item if $M \not = 0$ then $d(M)$ is the degree of the numerical polynomial $\chi_M$. \end{enumerate}


algebra-definition-Krull

The {\it Krull dimension} of the ring $R$ is the Krull dimension of the topological space $\Spec(R)$, see Topology, Definition \ref{topology-definition-Krull}. In other words it is the supremum of the integers $n\geq 0$ such that there exists a chain of prime ideals of length $n$: $$ \mathfrak p_0 \subset \mathfrak p_1 \subset \ldots \subset \mathfrak p_n, \quad \mathfrak p_i \not = \mathfrak p_{i + 1}. $$


algebra-definition-height

The {\it height} of a prime ideal $\mathfrak p$ of a ring $R$ is the dimension of the local ring $R_{\mathfrak p}$.


algebra-definition-regular-local

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d$. \begin{enumerate} \item A {\it system of parameters of $R$} is a sequence of elements $x_1, \ldots, x_d \in \mathfrak m$ which generates an ideal of definition of $R$, \item if there exist $x_1, \ldots, x_d \in \mathfrak m$ such that $\mathfrak m = (x_1, \ldots, x_d)$ then we call $R$ a {\it regular local ring} and $x_1, \ldots, x_d$ a {\it regular system of parameters}. \end{enumerate}


algebra-definition-associated

Let $R$ be a ring. Let $M$ be an $R$-module. A prime $\mathfrak p$ of $R$ is {\it associated} to $M$ if there exists an element $m \in M$ whose annihilator is $\mathfrak p$. The set of all such primes is denoted $\text{Ass}_R(M)$ or $\text{Ass}(M)$.


algebra-definition-symbolic-power

Let $R$ be a ring. Let $\mathfrak p$ be a prime ideal. For $n \geq 0$ the $n$th {\it symbolic power} of $\mathfrak p$ is the ideal $\mathfrak p^{(n)} = \Ker(R \to R_\mathfrak p/\mathfrak p^nR_\mathfrak p)$.


algebra-definition-relative-assassin

Let $R \to S$ be a ring map. Let $N$ be an $S$-module. The {\it relative assassin of $N$ over $S/R$} is the set $$ \text{Ass}_{S/R}(N) = \{ \mathfrak q \subset S \mid \mathfrak q \in \text{Ass}_S(N \otimes_R \kappa(\mathfrak p)) \text{ with }\mathfrak p = R \cap \mathfrak q\}. $$ This is the set named $A$ in Lemma \ref{lemma-compare-relative-assassins}.


algebra-definition-weakly-associated

Let $R$ be a ring. Let $M$ be an $R$-module. A prime $\mathfrak p$ of $R$ is {\it weakly associated} to $M$ if there exists an element $m \in M$ such that $\mathfrak p$ is minimal among the prime ideals containing the annihilator $\text{Ann}(m) = \{f \in R \mid fm = 0\}$. The set of all such primes is denoted $\text{WeakAss}_R(M)$ or $\text{WeakAss}(M)$.


algebra-definition-embedded-primes

Let $R$ be a ring. Let $M$ be an $R$-module. \begin{enumerate} \item  The associated primes of $M$ which are not minimal among the associated primes of $M$ are called the {\it embedded associated primes} of $M$. \item The {\it embedded primes of $R$} are the embedded associated primes of $R$ as an $R$-module. \end{enumerate}


algebra-definition-regular-sequence

Let $R$ be a ring. Let $M$ be an $R$-module. A sequence of elements $f_1, \ldots, f_r$ of $R$ is called an {\it $M$-regular sequence} if the following conditions hold: \begin{enumerate} \item $f_i$ is a nonzerodivisor on $M/(f_1, \ldots, f_{i - 1})M$ for each $i = 1, \ldots, r$, and \item the module $M/(f_1, \ldots, f_r)M$ is not zero. \end{enumerate} If $I$ is an ideal of $R$ and $f_1, \ldots, f_r \in I$ then we call $f_1, \ldots, f_r$ a {\it $M$-regular sequence in $I$}. If $M = R$, we call $f_1, \ldots, f_r$ simply a {\it regular sequence} (in $I$).


algebra-definition-quasi-regular-sequence

Let $R$ be a ring. Let $M$ be an $R$-module. A sequence of elements $f_1, \ldots, f_c$ of $R$ is called {\it $M$-quasi-regular} if (\ref{equation-quasi-regular}) is an isomorphism. If $M = R$, we call $f_1, \ldots, f_c$ simply a {\it quasi-regular sequence}.


algebra-definition-blow-up

Let $R$ be a ring. Let $I \subset R$ be an ideal. \begin{enumerate} \item The {\it blowup algebra}, or the {\it Rees algebra}, associated to the pair $(R, I)$ is the graded $R$-algebra $$ \text{Bl}_I(R) = \bigoplus\nolimits_{n \geq 0} I^n = R \oplus I \oplus I^2 \oplus \ldots $$ where the summand $I^n$ is placed in degree $n$. \item Let $a \in I$ be an element. Denote $a^{(1)}$ the element $a$ seen as an element of degree $1$ in the Rees algebra. Then the {\it affine blowup algebra} $R[\frac{I}{a}]$ is the algebra $(\text{Bl}_I(R))_{(a^{(1)})}$ constructed in Section \ref{section-proj}. \end{enumerate}


algebra-definition-finite-free-resolution

Let $R$ be a ring. Let $M$ be an $R$-module. \begin{enumerate} \item A (left) {\it resolution} $F_\bullet \to M$ of $M$ is an exact complex $$ \ldots \to F_2 \to F_1 \to F_0 \to M \to 0 $$ of $R$-modules. \item A {\it resolution of $M$ by free $R$-modules} is a resolution $F_\bullet \to M$ where each $F_i$ is a free $R$-module. \item A {\it resolution of $M$ by finite free $R$-modules} is a resolution $F_\bullet \to M$ where each $F_i$ is a finite free $R$-module. \end{enumerate}


algebra-definition-depth

Let $R$ be a ring, and $I \subset R$ an ideal. Let $M$ be a finite $R$-module. The {\it $I$-depth} of $M$, denoted $\text{depth}_I(M)$, is defined as follows: \begin{enumerate} \item if $IM \not = M$, then $\text{depth}_I(M)$ is the supremum in $\{0, 1, 2, \ldots, \infty\}$ of the lengths of $M$-regular sequences in $I$, \item if $IM = M$ we set $\text{depth}_I(M) = \infty$. \end{enumerate} If $(R, \mathfrak m)$ is local we call $\text{depth}_{\mathfrak m}(M)$ simply the {\it depth} of $M$.


algebra-definition-projective

Let $R$ be a ring. An $R$-module $P$ is {\it projective} if and only if the functor $\Hom_R(P, -) : \text{Mod}_R \to \text{Mod}_R$ is an exact functor.


algebra-definition-locally-free

Let $R$ be a ring and $M$ an $R$-module. \begin{enumerate} \item We say that $M$ is {\it locally free} if we can cover $\Spec(R)$ by standard opens $D(f_i)$, $i \in I$ such that $M_{f_i}$ is a free $R_{f_i}$-module for all $i \in I$. \item We say that $M$ is {\it finite locally free} if we can choose the covering such that each $M_{f_i}$ is finite free. \item We say that $M$ is {\it finite locally free of rank $r$} if we can choose the covering such that each $M_{f_i}$ is isomorphic to $R_{f_i}^{\oplus r}$. \end{enumerate}


algebra-definition-universally-injective

Let $f: M \to N$ be a map of $R$-modules.  Then $f$ is called {\it universally injective} if for every $R$-module $Q$, the map $f \otimes_R \text{id}_Q: M \otimes_R Q \to N \otimes_R Q$ is injective.  A sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ of $R$-modules is called {\it universally exact} if it is exact and $M_1 \to M_2$ is universally injective.


algebra-definition-devissage

Let $M$ be an $R$-module.  A {\it direct sum d\'evissage} of $M$ is a family of submodules $(M_{\alpha})_{\alpha \in S}$, indexed by an ordinal $S$ and increasing (with respect to inclusion), such that: \begin{enumerate} \item[(0)] $M_0 = 0$; \item[(1)] $M = \bigcup_{\alpha} M_{\alpha}$; \item[(2)] if $\alpha \in S$ is a limit ordinal, then $M_{\alpha} = \bigcup_{\beta < \alpha} M_{\beta}$; \item[(3)] if $\alpha + 1 \in S$, then $M_{\alpha}$ is a direct summand of $M_{\alpha + 1}$. \end{enumerate} If moreover \begin{enumerate} \item[(4)] $M_{\alpha + 1}/M_{\alpha}$ is countably generated for $\alpha + 1 \in S$, \end{enumerate} then $(M_{\alpha})_{\alpha \in S}$ is called a {\it Kaplansky d\'evissage} of $M$.


algebra-definition-ML-system

Let $(A_i, \varphi_{ji})$ be a directed inverse system of sets over $I$.  Then we say  $(A_i, \varphi_{ji})$ is {\it Mittag-Leffler} if for each $i \in I$, the family $\varphi_{ji}(A_j) \subset A_i$ for $j \geq i$ stabilizes.  Explicitly, this means that for each $i \in I$, there exists $j \geq i$ such that for $k \geq j$ we have $\varphi_{ki}(A_k) = \varphi_{ji}( A_j)$.  If $(A_i, \varphi_{ji})$ is a directed inverse system of modules over a ring $R$, we say that it is Mittag-Leffler if the underlying inverse system of sets is Mittag-Leffler.


algebra-definition-ML-inductive-system

Let $(M_i, f_{ij})$ be a directed system of $R$-modules.  We say that $(M_i, f_{ij})$ is a {\it Mittag-Leffler directed system of modules} if each $M_i$ is an $R$-module of finite presentation and if for every $R$-module $N$, the inverse system $$ (\Hom_R(M_i, N), \Hom_R(f_{ij}, N)) $$ is Mittag-Leffler.


algebra-definition-domination

Let $f: M \to N$ and $g: M \to M'$ be maps of $R$-modules. Then we say $g$ {\it dominates} $f$ if for any $R$-module $Q$, we have $\Ker(f \otimes_R \text{id}_Q) \subset \Ker(g \otimes_R \text{id}_Q)$.


algebra-definition-mittag-leffler-module

Let $M$ be an $R$-module.  We say that $M$ is {\it Mittag-Leffler} if the equivalent conditions of Proposition \ref{proposition-ML-characterization} hold.


algebra-definition-coherent

Let $R$ be a ring. Let $M$ be an $R$-module. \begin{enumerate} \item We say $M$ is a {\it coherent module} if it is finitely generated and every finitely generated submodule of $M$ is finitely presented over $R$. \item We say $R$ is a {\it coherent ring} if it is coherent as a module over itself. \end{enumerate}


algebra-definition-complete

Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $M$ be an $R$-module. We say $M$ is {\it $I$-adically complete} if the map $$ M \longrightarrow M^\wedge = \lim_n M/I^nM $$ is an isomorphism\footnote{This includes the condition that $\bigcap I^nM = (0)$.}. We say $R$ is {\it $I$-adically complete} if $R$ is $I$-adically complete as an $R$-module.


algebra-definition-rank

Let $R$ be a ring. Suppose that $\varphi : R^m \to R^n$ is a map of finite free modules. \begin{enumerate} \item The {\it rank} of $\varphi$ is the maximal $r$ such that $\wedge^r \varphi : \wedge^r R^m \to \wedge^r R^n$ is nonzero. \item We let $I(\varphi) \subset R$ be the ideal generated by the $r \times r$ minors of the matrix of $\varphi$, where $r$ is the rank as defined above. \end{enumerate}


algebra-definition-CM

Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. We say $M$ is {\it Cohen-Macaulay} if $\dim(\text{Supp}(M)) = \text{depth}(M)$.


algebra-definition-maximal-CM

Let $R$ be a Noetherian local ring. A finite module $M$ over $R$ is called a {\it maximal Cohen-Macaulay} module if $\text{depth}(M) = \dim(R)$.


algebra-definition-module-CM

Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. We say $M$ is {\it Cohen-Macaulay} if $M_\mathfrak p$ is a Cohen-Macaulay module over $R_\mathfrak p$ for all primes $\mathfrak p$ of $R$.


algebra-definition-local-ring-CM

A Noetherian local ring $R$ is called {\it Cohen-Macaulay} if it is Cohen-Macaulay as a module over itself.


algebra-definition-ring-CM

A Noetherian ring $R$ is called {\it Cohen-Macaulay} if all its local rings are Cohen-Macaulay.


algebra-definition-catenary

A ring $R$ is said to be {\it catenary} if for any pair of prime ideals $\mathfrak p \subset \mathfrak q$, all maximal chains of primes $\mathfrak p = \mathfrak p_0 \subset \mathfrak p_1 \subset \ldots \subset \mathfrak p_e = \mathfrak q$ have the same (finite) length.


algebra-definition-universally-catenary

A Noetherian ring $R$ is said to be {\it universally catenary} if every $R$ algebra of finite type is catenary.


algebra-definition-pure-ideal

Let $R$ be a ring. We say that $I \subset R$ is {\it pure} if the quotient ring $R/I$ is flat over $R$.


algebra-definition-finite-proj-dim

Let $R$ be a ring. Let $M$ be an $R$-module. We say $M$ has {\it finite projective dimension} if it has a finite length resolution by projective $R$-modules. The minimal length of such a resolution is called the {\it projective dimension} of $M$.


algebra-definition-finite-gl-dim

Let $R$ be a ring. The ring $R$ is said to have {\it finite global dimension} if there exists an integer $n$ such that every $R$-module has a resolution by projective $R$-modules of length at most $n$. The minimal such $n$ is then called the {\it global dimension} of $R$.


algebra-definition-regular

A Noetherian ring $R$ is said to be {\it regular} if all the localizations $R_{\mathfrak p}$ at primes are regular local rings.


algebra-definition-fibre

Suppose that $R \to S$ is a ring map. Let $\mathfrak q \subset S$ be a prime lying over the prime $\mathfrak p$ of $R$. The {\it local ring of the fibre at $\mathfrak q$} is the local ring $$ S_{\mathfrak q}/\mathfrak pS_{\mathfrak q} = (S/\mathfrak pS)_{\mathfrak q} = (S \otimes_R \kappa(\mathfrak p))_{\mathfrak q} $$


algebra-definition-uniformizer

Let $A$ be a discrete valuation ring. A {\it uniformizer} is an element $\pi \in A$ which generates the maximal ideal of $A$.


algebra-definition-irreducible-prime-element

Let $R$ be a domain. \begin{enumerate} \item Elements $x, y \in R$ are called {\it associates} if there exists a unit $u \in R^*$ such that $x = uy$. \item An element $x \in R$ is called {\it irreducible} if it is nonzero, not a unit and whenever $x = yz$, $y, z \in R$, then $y$ is either a unit or an associate of $x$. \item An element $x \in R$ is called {\it prime} if the ideal generated by $x$ is a prime ideal. \end{enumerate}


algebra-definition-UFD

A {\it unique factorization domain}, abbreviated {\it UFD}, is a domain $R$ such that if $x \in R$ is a nonzero, nonunit, then $x$ has a factorization into irreducibles, and if $$ x = a_1 \ldots a_m = b_1 \ldots b_n $$ are factorizations into irreducibles then $n = m$ and there exists a permutation $\sigma : \{1, \ldots, n\} \to \{1, \ldots, n\}$ such that $a_i$ and $b_{\sigma(i)}$ are associates.


algebra-definition-PID

A {\it principal ideal domain}, abbreviated {\it PID}, is a domain $R$ such that every ideal is a principal ideal.


algebra-definition-dedekind-domain

A {\it Dedekind domain} is a domain $R$ such that every nonzero ideal $I \subset R$ can be written as a product $$ I = \mathfrak p_1 \ldots \mathfrak p_r $$ of nonzero prime ideals uniquely up to permutation of the $\mathfrak p_i$.


algebra-definition-ord

Suppose that $K$ is a field, and $R \subset K$ is a local\footnote{We could also define this when $R$ is only semi-local but this is probably never really what you want!} Noetherian subring of dimension $1$ with fraction field $K$. In this case we define the {\it order of vanishing along $R$} $$ \text{ord}_R : K^* \longrightarrow \mathbf{Z} $$ by the rule $$ \text{ord}_R(x) = \text{length}_R(R/(x)) $$ if $x \in R$ and we set $\text{ord}_R(x/y) = \text{ord}_R(x) - \text{ord}_R(y)$ for $x, y \in R$ both nonzero.


algebra-definition-lattice

Let $R$ be a Noetherian local domain of dimension $1$ with fraction field $K$. Let $V$ be a finite dimensional $K$-vector space. A {\it lattice in $V$} is a finite $R$-submodule $M \subset V$ such that $V = K \otimes_R M$.


algebra-definition-distance

Let $R$ be a Noetherian local domain of dimension $1$ with fraction field $K$. Let $V$ be a finite dimensional $K$-vector space. Let $M$, $M'$ be two lattices in $V$. The {\it distance between $M$ and $M'$} is the integer $$ d(M, M') = \text{length}_R(M/M \cap M') - \text{length}_R(M'/M \cap M') $$ of Lemma \ref{lemma-compare-lattices} part (5).


algebra-definition-quasi-finite

Let $R \to S$ be a finite type ring map. Let $\mathfrak q \subset S$ be a prime. \begin{enumerate} \item If the equivalent conditions of Lemma \ref{lemma-isolated-point-fibre} are satisfied then we say $R \to S$ is {\it quasi-finite at $\mathfrak q$}. \item We say a ring map $A \to B$ is {\it quasi-finite} if it is of finite type and quasi-finite at all primes of $B$. \end{enumerate}


algebra-definition-strongly-transcendental

Given an inclusion of rings $R \subset S$ and an element $x \in S$ we say that $x$ is {\it strongly transcendental over $R$} if whenever $u(a_0 + a_1 x + \ldots + a_k x^k) = 0$ with $u \in S$ and $a_i \in R$, then we have $ua_i = 0$ for all $i$.


algebra-definition-relative-dimension

Suppose that $R \to S$ is of finite type, and let $\mathfrak q \subset S$ be a prime lying over a prime $\mathfrak p$ of $R$. We define the {\it relative dimension of $S/R$ at $\mathfrak q$}, denoted $\dim_{\mathfrak q}(S/R)$, to be the dimension of $\Spec(S \otimes_R \kappa(\mathfrak p))$ at the point corresponding to $\mathfrak q$. We let $\dim(S/R)$ be the supremum of $\dim_{\mathfrak q}(S/R)$ over all $\mathfrak q$. This is called the {\it relative dimension of} $S/R$.


algebra-definition-derivation

Let $\varphi : R \to S$ be a ring map and let $M$ be an $S$-module. A {\it derivation}, or more precisely an {\it $R$-derivation} into $M$ is a map $D : S \to M$ which is additive, annihilates elements of $\varphi(R)$, and satisfies the {\it Leibniz rule}: $D(ab) = aD(b) + bD(a)$.


algebra-definition-differentials

The pair $(\Omega_{S/R}, \text{d})$ is called the {\it module of K\"ahler differentials} or the {\it module of differentials} of $S$ over $R$.


algebra-definition-differential-operators

Let $R \to S$ be a ring map. Let $M$, $N$ be $S$-modules. Let $k \geq 0$ be an integer. We inductively define a {\it differential operator $D : M \to N$ of order $k$} to be an $R$-linear map such that for all $g \in S$ the map $m \mapsto D(gm) - gD(m)$ is a differential operator of order $k - 1$. For the base case $k = 0$ we define a differential operator of order $0$ to be an $S$-linear map.


algebra-definition-module-principal-parts

Let $R \to S$ be a ring map. Let $M$ be an $S$-module. The module $P^k_{S/R}(M)$ constructed in Lemma \ref{lemma-module-principal-parts} is called the {\it module of principal parts of order $k$} of $M$.


algebra-definition-naive-cotangent-complex

Let $R \to S$ be a ring map. The {\it naive cotangent complex} $\NL_{S/R}$ is the chain complex (\ref{equation-naive-cotangent-complex}) $$ \NL_{S/R} = \left(I/I^2 \longrightarrow \Omega_{R[S]/R} \otimes_{R[S]} S\right) $$ with $I/I^2$ placed in (homological) degree $1$ and $\Omega_{R[S]/R} \otimes_{R[S]} S$ placed in degree $0$. We will denote $H_1(L_{S/R}) = H_1(\NL_{S/R})$\footnote{This module is sometimes denoted $\Gamma_{S/R}$ in the literature.} the homology in degree $1$.


algebra-definition-lci-field

Let $k$ be a field. Let $S$ be a finite type $k$-algebra. \begin{enumerate} \item We say that $S$ is a {\it global complete intersection over $k$} if there exists a presentation $S = k[x_1, \ldots, x_n]/(f_1, \ldots, f_c)$ such that $\dim(S) = n - c$. \item We say that $S$ is a {\it local complete intersection over $k$} if there exists a covering $\Spec(S) = \bigcup D(g_i)$ such that each of the rings $S_{g_i}$ is a global complete intersection over $k$. \end{enumerate} We will also use the convention that the zero ring is a global complete intersection over $k$.


algebra-definition-lci-local-ring

Let $k$ be a field. Let $S$ be a local $k$-algebra essentially of finite type over $k$. We say $S$ is a {\it complete intersection (over $k$)} if there exists a local $k$-algebra $R$ and elements $f_1, \ldots, f_c \in \mathfrak m_R$ such that \begin{enumerate} \item $R$ is essentially of finite type over $k$, \item $R$ is a regular local ring, \item $f_1, \ldots, f_c$ form a regular sequence in $R$, and \item $S \cong R/(f_1, \ldots, f_c)$ as $k$-algebras. \end{enumerate}


algebra-definition-lci

A ring map $R \to S$ is called {\it syntomic}, or we say $S$ is a {\it flat local complete intersection over $R$} if it is flat, of finite presentation, and if all of its fibre rings $S \otimes_R \kappa(\mathfrak p)$ are local complete intersections, see Definition \ref{definition-lci-field}.


algebra-definition-relative-global-complete-intersection

Let $R \to S$ be a ring map. We say that $R \to S$ is a {\it relative global complete intersection} if there exists a presentation $S = R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)$ and every nonempty fibre of $\Spec(S) \to \Spec(R)$ has dimension $n - c$. We will say ``let $S = R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)$ be a relative global complete intersection'' to indicate this situation.


algebra-definition-smooth

A ring map $R \to S$ is {\it smooth} if it is of finite presentation and the naive cotangent complex $\NL_{S/R}$ is quasi-isomorphic to a finite projective $S$-module placed in degree $0$.


algebra-definition-standard-smooth

Let $R$ be a ring. Given integers $n \geq c \geq 0$ and $f_1, \ldots, f_c \in R[x_1, \ldots, x_n]$ we say $$ S = R[x_1, \ldots, x_n]/(f_1, \ldots, f_c) $$ is a {\it standard smooth algebra over $R$} if the polynomial $$ g = \det \left( \begin{matrix} \partial f_1/\partial x_1 & \partial f_2/\partial x_1 & \ldots & \partial f_c/\partial x_1 \\ \partial f_1/\partial x_2 & \partial f_2/\partial x_2 & \ldots & \partial f_c/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial f_1/\partial x_c & \partial f_2/\partial x_c & \ldots & \partial f_c/\partial x_c \end{matrix} \right) $$ maps to an invertible element in $S$.


algebra-definition-smooth-at-prime

Let $R \to S$ be a ring map. Let $\mathfrak q$ be a prime of $S$. We say $R \to S$ is {\it smooth at $\mathfrak q$} if there exists a $g \in S$, $g \not \in \mathfrak q$ such that $R \to S_g$ is smooth.


algebra-definition-formally-smooth

Let $R \to S$ be a ring map. We say $S$ is {\it formally smooth over $R$} if for every commutative solid diagram $$ \xymatrix{ S \ar[r] \ar@{-->}[rd] & A/I \\ R \ar[r] \ar[u] & A \ar[u] } $$ where $I \subset A$ is an ideal of square zero, a dotted arrow exists which makes the diagram commute.


algebra-definition-small-extension

Let $\varphi : B' \to B$ be a ring map. We say $\varphi$ is a {\it small extension} if $B'$ and $B$ are local Artinian rings, $\varphi$ is surjective and $I = \Ker(\varphi)$ has length $1$ as a $B'$-module.


algebra-definition-etale

Let $R \to S$ be a ring map. We say $R \to S$ is {\it \'etale} if it is of finite presentation and the naive cotangent complex $\NL_{S/R}$ is quasi-isomorphic to zero. Given a prime $\mathfrak q$ of $S$ we say that $R \to S$ is {\it \'etale at $\mathfrak q$} if there exists a $g \in S$, $g \not \in \mathfrak q$ such that $R \to S_g$ is \'etale.


algebra-definition-standard-etale

Let $R$ be a ring. Let $g , f  \in R[x]$. Assume that $f$ is monic and the derivative $f'$ is invertible in the localization $R[x]_g/(f)$. In this case the ring map $R \to R[x]_g/(f)$ is said to be {\it standard \'etale}.


algebra-definition-formally-unramified

Let $R \to S$ be a ring map. We say $S$ is {\it formally unramified over $R$} if for every commutative solid diagram $$ \xymatrix{ S \ar[r] \ar@{-->}[rd] & A/I \\ R \ar[r] \ar[u] & A \ar[u] } $$ where $I \subset A$ is an ideal of square zero, there exists at most one dotted arrow making the diagram commute.


algebra-definition-universal-thickening

Let $R \to S$ be a formally unramified ring map. \begin{enumerate} \item The {\it universal first order thickening} of $S$ over $R$ is the surjection of $R$-algebras $S' \to S$ of Lemma \ref{lemma-universal-thickening}. \item The {\it conormal module} of $R \to S$ is the kernel $I$ of the universal first order thickening $S' \to S$, seen as an $S$-module. \end{enumerate} We often denote the conormal module {\it $C_{S/R}$} in this situation.


algebra-definition-formally-etale

Let $R \to S$ be a ring map. We say $S$ is {\it formally \'etale over $R$} if for every commutative solid diagram $$ \xymatrix{ S \ar[r] \ar@{-->}[rd] & A/I \\ R \ar[r] \ar[u] & A \ar[u] } $$ where $I \subset A$ is an ideal of square zero, there exists a unique dotted arrow making the diagram commute.


algebra-definition-unramified

Let $R \to S$ be a ring map. \begin{enumerate} \item We say $R \to S$ is {\it unramified} if $R \to S$ is of finite type and $\Omega_{S/R} = 0$. \item We say $R \to S$ is {\it G-unramified} if $R \to S$ is of finite presentation and $\Omega_{S/R} = 0$. \item Given a prime $\mathfrak q$ of $S$ we say that $S$ is {\it unramified at $\mathfrak q$} if there exists a $g \in S$, $g \not \in \mathfrak q$ such that $R \to S_g$ is unramified. \item Given a prime $\mathfrak q$ of $S$ we say that $S$ is {\it G-unramified at $\mathfrak q$} if there exists a $g \in S$, $g \not \in \mathfrak q$ such that $R \to S_g$ is G-unramified. \end{enumerate}


algebra-definition-henselian

Let $(R, \mathfrak m, \kappa)$ be a local ring. \begin{enumerate} \item We say $R$ is {\it henselian} if for every monic $f \in R[T]$ and every root $a_0 \in \kappa$ of $\overline{f}$ such that $\overline{f'}(a_0) \not = 0$ there exists an $a \in R$ such that $f(a) = 0$ and $a_0 = \overline{a}$. \item We say $R$ is {\it strictly henselian} if $R$ is henselian and its residue field is separably algebraically closed. \end{enumerate}


algebra-definition-henselization

Let $(R, \mathfrak m, \kappa)$ be a local ring. \begin{enumerate} \item The local ring map $R \to R^h$ constructed in Lemma \ref{lemma-henselization} is called the {\it henselization} of $R$. \item Given a separable algebraic closure $\kappa \subset \kappa^{sep}$ the local ring map $R \to R^{sh}$ constructed in Lemma \ref{lemma-strict-henselization} is called the {\it strict henselization of $R$ with respect to $\kappa \subset \kappa^{sep}$}. \item A local ring map $R \to R^{sh}$ is called a {\it strict henselization} of $R$ if it is isomorphic to one of the local ring maps constructed in Lemma \ref{lemma-strict-henselization} \end{enumerate}


algebra-definition-conditions

Let $R$ be a Noetherian ring. Let $k \geq 0$ be an integer. \begin{enumerate} \item We say $R$ has property {\it $(R_k)$} if for every prime $\mathfrak p$ of height $\leq k$ the local ring $R_{\mathfrak p}$ is regular. We also say that $R$ is {\it regular in codimension $\leq k$}. \item We say $R$ has property {\it $(S_k)$} if for every prime $\mathfrak p$ the local ring $R_{\mathfrak p}$ has depth at least $\min\{k, \dim(R_{\mathfrak p})\}$. \item Let $M$ be a finite $R$-module. We say $M$ has property $(S_k)$ if for every prime $\mathfrak p$ the module $M_{\mathfrak p}$ has depth at least $\min\{k, \dim(\text{Supp}(M_{\mathfrak p}))\}$. \end{enumerate}


algebra-definition-complete-local-ring

Let $(R, \mathfrak m)$ be a local ring. We say $R$ is a {\it complete local ring} if the canonical map $$ R \longrightarrow \lim_n R/\mathfrak m^n $$ to the completion of $R$ with respect to $\mathfrak m$ is an isomorphism\footnote{This includes the condition that $\bigcap \mathfrak m^n = (0)$; in some texts this may be indicated by saying that $R$ is complete and separated. Warning: It can happen that the completion $\lim_n R/\mathfrak m^n$ of a local ring is non-complete, see Examples, Lemma \ref{examples-lemma-noncomplete-completion}. This does not happen when $\mathfrak m$ is finitely generated, see Lemma \ref{lemma-hathat-finitely-generated} in which case the completion is Noetherian, see Lemma \ref{lemma-completion-Noetherian}.}.


algebra-definition-coefficient-ring

Let $(R, \mathfrak m)$ be a complete local ring. A subring $\Lambda \subset R$ is called a {\it coefficient ring} if the following conditions hold: \begin{enumerate} \item $\Lambda$ is a complete local ring with maximal ideal $\Lambda \cap \mathfrak m$, \item the residue field of $\Lambda$ maps isomorphically to the residue field of $R$, and \item $\Lambda \cap \mathfrak m = p\Lambda$, where $p$ is the characteristic of the residue field of $R$. \end{enumerate}


algebra-definition-cohen-ring

A {\it Cohen ring} is a complete discrete valuation ring with uniformizer $p$ a prime number.


algebra-definition-N

\begin{reference} \cite[Chapter 0, Definition 23.1.1]{EGA} \end{reference} Let $R$ be a domain with field of fractions $K$. \begin{enumerate} \item We say $R$ is {\it N-1} if the integral closure of $R$ in $K$ is a finite $R$-module. \item We say $R$ is {\it N-2} or {\it Japanese} if for any finite extension $K \subset L$ of fields the integral closure of $R$ in $L$ is finite over $R$. \end{enumerate}


algebra-definition-nagata

Let $R$ be a ring. \begin{enumerate} \item We say $R$ is {\it universally Japanese} if for any finite type ring map $R \to S$ with $S$ a domain we have that $S$ is N-2 (i.e., Japanese). \item We say that $R$ is a {\it Nagata ring} if $R$ is Noetherian and for every prime ideal $\mathfrak p$ the ring $R/\mathfrak p$ is N-2. \end{enumerate}


algebra-definition-analytically-unramified

Let $(R, \mathfrak m)$ be a Noetherian local ring. We say $R$ is {\it analytically unramified} if its completion $R^\wedge = \lim_n R/\mathfrak m^n$ is reduced. A prime ideal $\mathfrak p \subset R$ is said to be {\it analytically unramified} if $R/\mathfrak p$ is analytically unramified.


algebra-definition-geometrically-normal

Let $k$ be a field. A $k$-algebra $R$ is called {\it geometrically normal} over $k$ if the equivalent conditions of Lemma \ref{lemma-geometrically-normal} hold.


algebra-definition-geometrically-regular

Let $k$ be a field. Let $R$ be a Noetherian $k$-algebra. The $k$-algebra $R$ is called {\it geometrically regular} over $k$ if the equivalent conditions of Lemma \ref{lemma-geometrically-regular} hold.


moduli-curves-definition-deligne-mumford-smooth

\begin{reference} \cite{DM} \end{reference} We denote $\mathcal{M}$ and we name it the {\it moduli stack of smooth proper curves} the algebraic stack $\Curvesstack^{smooth, h0}$ parametrizing families of curves introduced in Lemma \ref{lemma-smooth-curves-h0}. For $g \geq 0$ we denote $\mathcal{M}_g$ and we name it the {\it moduli stack of smooth proper curves of genus $g$} the algebraic stack introduced in Lemma \ref{lemma-smooth-one-piece-per-genus}.


moduli-curves-definition-relative-dualizing-sheaf

Let $f : X \to S$ be a family of curves with Cohen-Macaulay fibres equidimensional of dimension $1$ (Lemma \ref{lemma-CM-1-curves}). Then the $\mathcal{O}_X$-module $$ \omega_{X/S} = H^{-1}(\omega_{X/S}^\bullet) $$ studied in Lemma \ref{lemma-CM-dualizing} is called the {\it relative dualizing sheaf} of $f$.


moduli-curves-definition-prestable

Let $f : X \to S$ be a family of curves. We say $f$ is a {\it prestable family of curves} if \begin{enumerate} \item $f$ is at-worst-nodal of relative dimension $1$, and \item $f_*\mathcal{O}_X = \mathcal{O}_S$ and this holds after any base change\footnote{In fact, it suffices to require $f_*\mathcal{O}_X = \mathcal{O}_S$ because the Stein factorization of $f$ is \'etale in this case, see More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-stein-factorization-etale}. The condition may also be replaced by asking the geometric fibres to be connected, see Lemma \ref{lemma-geomredcon-in-h0-1}.}. \end{enumerate}


moduli-curves-definition-semistable

Let $f : X \to S$ be a family of curves. We say $f$ is a {\it semistable family of curves} if \begin{enumerate} \item $X \to S$ is a prestable family of curves, and \item $X_s$ has genus $\geq 1$ and does not have a rational tail for all $s \in S$. \end{enumerate}


moduli-curves-definition-stable

Let $f : X \to S$ be a family of curves. We say $f$ is a {\it stable family of curves} if \begin{enumerate} \item $X \to S$ is a prestable family of curves, and \item $X_s$ has genus $\geq 2$ and does not have a rational tails or bridges for all $s \in S$. \end{enumerate}


moduli-curves-definition-deligne-mumford

\begin{reference} \cite{DM} \end{reference} We denote $\overline{\mathcal{M}}$ and we name the {\it moduli stack of stable curves} the algebraic stack $\Curvesstack^{stable}$ parametrizing stable families of curves introduced in Lemma \ref{lemma-stable-curves}. For $g \geq 2$ we denote $\overline{\mathcal{M}}_g$ and we name the {\it moduli stack of stable curves of genus $g$} the algebraic stack introduced in Lemma \ref{lemma-stable-one-piece-per-genus}.


dpa-definition-divided-powers

Let $A$ be a ring. Let $I$ be an ideal of $A$. A collection of maps $\gamma_n : I \to I$, $n > 0$ is called a {\it divided power structure} on $I$ if for all $n \geq 0$, $m > 0$, $x, y \in I$, and $a \in A$ we have \begin{enumerate} \item $\gamma_1(x) = x$, we also set $\gamma_0(x) = 1$, \item $\gamma_n(x)\gamma_m(x) = \frac{(n + m)!}{n! m!} \gamma_{n + m}(x)$, \item $\gamma_n(ax) = a^n \gamma_n(x)$, \item $\gamma_n(x + y) = \sum_{i = 0, \ldots, n} \gamma_i(x)\gamma_{n - i}(y)$, \item $\gamma_n(\gamma_m(x)) = \frac{(nm)!}{n! (m!)^n} \gamma_{nm}(x)$. \end{enumerate}


dpa-definition-divided-power-ring

A {\it divided power ring} is a triple $(A, I, \gamma)$ where $A$ is a ring, $I \subset A$ is an ideal, and $\gamma = (\gamma_n)_{n \geq 1}$ is a divided power structure on $I$. A {\it homomorphism of divided power rings} $\varphi : (A, I, \gamma) \to (B, J, \delta)$ is a ring homomorphism $\varphi : A \to B$ such that $\varphi(I) \subset J$ and such that $\delta_n(\varphi(x)) = \varphi(\gamma_n(x))$ for all $x \in I$ and $n \geq 1$.


dpa-definition-extends

Given a divided power ring $(A, I, \gamma)$ and a ring map $A \to B$ we say $\gamma$ {\it extends} to $B$ if there exists a divided power structure $\bar \gamma$ on $IB$ such that $(A, I, \gamma) \to (B, IB, \bar\gamma)$ is a homomorphism of divided power rings.


dpa-definition-divided-powers-graded

Let $R$ be a ring. Let $A = \bigoplus_{d \geq 0} A_d$ be a graded $R$-algebra which is strictly graded commutative. A collection of maps $\gamma_n : A_{even, +} \to A_{even, +}$ defined for all $n > 0$ is called a {\it divided power structure} on $A$ if we have \begin{enumerate} \item $\gamma_n(x) \in A_{2nd}$ if $x \in A_{2d}$, \item $\gamma_1(x) = x$ for any $x$, we also set $\gamma_0(x) = 1$, \item $\gamma_n(x)\gamma_m(x) = \frac{(n + m)!}{n! m!} \gamma_{n + m}(x)$, \item $\gamma_n(xy) = x^n \gamma_n(y)$ for all $x \in A_{even}$ and $y \in A_{even, +}$, \item $\gamma_n(xy) = 0$ if $x, y \in A_{odd}$ homogeneous and $n > 1$ \item if $x, y \in A_{even, +}$ then $\gamma_n(x + y) = \sum_{i = 0, \ldots, n} \gamma_i(x)\gamma_{n - i}(y)$, \item $\gamma_n(\gamma_m(x)) = \frac{(nm)!}{n! (m!)^n} \gamma_{nm}(x)$ for $x \in A_{even, +}$. \end{enumerate}


dpa-definition-divided-powers-dga

Let $R$ be a ring. Let $A = \bigoplus_{d \geq 0} A_d$ be a differential graded $R$-algebra which is strictly graded commutative. A divided power structure $\gamma$ on $A$ is {\it compatible with the differential graded structure} if $\text{d}(\gamma_n(x)) = \text{d}(x) \gamma_{n - 1}(x)$ for all $x \in A_{even, +}$.


dpa-definition-lci

Let $A$ be a Noetherian ring. \begin{enumerate} \item If $A$ is local, then we say $A$ is a {\it complete intersection} if its completion is a complete intersection in the sense above. \item In general we say $A$ is a {\it local complete intersection} if all of its local rings are complete intersections. \end{enumerate}


derived-definition-triangle

Let $\mathcal{D}$ be an additive category. Let $[n] : \mathcal{D} \to \mathcal{D}$, $E \mapsto E[n]$ be a collection of additive functors indexed by $n \in \mathbf{Z}$ such that $[n] \circ [m] = [n + m]$  and $[0] = \text{id}$ (equality as functors). In this situation we define a {\it triangle} to be a sextuple $(X, Y, Z, f, g, h)$ where $X, Y, Z \in \Ob(\mathcal{D})$ and $f : X \to Y$, $g : Y \to Z$ and $h : Z \to X[1]$ are morphisms of $\mathcal{D}$. A {\it morphism of triangles} $(X, Y, Z, f, g, h) \to (X', Y', Z', f', g', h')$ is given by morphisms $a : X \to X'$, $b : Y \to Y'$ and $c : Z \to Z'$ of $\mathcal{D}$ such that $b \circ f = f' \circ a$, $c  \circ g = g' \circ b$ and $a[1] \circ h = h' \circ c$.


derived-definition-triangulated-category

A {\it triangulated category} consists of a triple $(\mathcal{D}, \{[n]\}_{n\in \mathbf{Z}}, \mathcal{T})$ where \begin{enumerate} \item $\mathcal{D}$ is an additive category, \item $[n] : \mathcal{D} \to \mathcal{D}$, $E \mapsto E[n]$ is a collection of additive functors indexed by $n \in \mathbf{Z}$ such that $[n] \circ [m] = [n + m]$ and $[0] = \text{id}$ (equality as functors), and \item $\mathcal{T}$ is a set of triangles called the {\it distinguished triangles} \end{enumerate} subject to the following conditions \begin{enumerate} \item[TR1] Any triangle isomorphic to a distinguished triangle is a distinguished triangle. Any triangle of the form $(X, X, 0, \text{id}, 0, 0)$ is distinguished. For any morphism $f : X \to Y$ of $\mathcal{D}$ there exists a distinguished triangle of the form $(X, Y, Z, f, g, h)$. \item[TR2] The triangle $(X, Y, Z, f, g, h)$ is distinguished if and only if the triangle $(Y, Z, X[1], g, h, -f[1])$ is. \item[TR3] Given a solid diagram $$ \xymatrix{ X \ar[r]^f \ar[d]^a & Y \ar[r]^g \ar[d]^b & Z \ar[r]^h \ar@{-->}[d] & X[1] \ar[d]^{a[1]} \\ X' \ar[r]^{f'} & Y' \ar[r]^{g'} & Z' \ar[r]^{h'} & X'[1] } $$ whose rows are distinguished triangles and which satisfies $b \circ f = f' \circ a$, there exists a morphism $c : Z \to Z'$ such that $(a, b, c)$ is a morphism of triangles. \item[TR4] Given objects $X$, $Y$, $Z$ of $\mathcal{D}$, and morphisms $f : X \to Y$, $g : Y \to Z$, and distinguished triangles $(X, Y, Q_1, f, p_1, d_1)$, $(X, Z, Q_2, g \circ f, p_2, d_2)$, and $(Y, Z, Q_3, g, p_3, d_3)$, there exist morphisms $a : Q_1 \to Q_2$ and $b : Q_2 \to Q_3$ such that \begin{enumerate} \item $(Q_1, Q_2, Q_3, a, b, p_1[1] \circ d_3)$ is a distinguished triangle, \item the triple $(\text{id}_X, g, a)$ is a morphism of triangles $(X, Y, Q_1, f, p_1, d_1) \to (X, Z, Q_2, g \circ f, p_2, d_2)$, and \item the triple $(f, \text{id}_Z, b)$ is a morphism of triangles $(X, Z, Q_2, g \circ f, p_2, d_2) \to (Y, Z, Q_3, g, p_3, d_3)$. \end{enumerate} \end{enumerate} We will call $(\mathcal{D}, [\ ], \mathcal{T})$ a {\it pre-triangulated category} if TR1, TR2 and TR3 hold.\footnote{We use $[\ ]$ as an abbreviation for the family $\{[n]\}_{n\in \mathbf{Z}}$.}


derived-definition-exact-functor-triangulated-categories

Let $\mathcal{D}$, $\mathcal{D}'$ be pre-triangulated categories. An {\it exact functor}, or a {\it triangulated functor} from $\mathcal{D}$ to $\mathcal{D}'$ is a functor $F : \mathcal{D} \to \mathcal{D}'$ together with given functorial isomorphisms $\xi_X : F(X[1]) \to F(X)[1]$ such that for every distinguished triangle $(X, Y, Z, f, g, h)$ of $\mathcal{D}$ the triangle $(F(X), F(Y), F(Z), F(f), F(g), \xi_X \circ F(h))$ is a distinguished triangle of $\mathcal{D}'$.


derived-definition-triangulated-subcategory

Let $(\mathcal{D}, [\ ], \mathcal{T})$ be a pre-triangulated category. A {\it pre-triangulated subcategory}\footnote{This definition may be nonstandard. If $\mathcal{D}'$ is a full subcategory then $\mathcal{T}'$ is the intersection of the set of triangles in $\mathcal{D}'$ with $\mathcal{T}$, see Lemma \ref{lemma-triangulated-subcategory}. In this case we drop $\mathcal{T}'$ from the notation.} is a pair $(\mathcal{D}', \mathcal{T}')$ such that \begin{enumerate} \item $\mathcal{D}'$ is an additive subcategory of $\mathcal{D}$ which is preserved under $[1]$ and $[-1]$, \item $\mathcal{T}' \subset \mathcal{T}$ is a subset such that for every $(X, Y, Z, f, g, h) \in \mathcal{T}'$ we have $X, Y, Z \in \Ob(\mathcal{D}')$ and $f, g, h \in \text{Arrows}(\mathcal{D}')$, and \item $(\mathcal{D}', [\ ], \mathcal{T}')$ is a pre-triangulated category. \end{enumerate} If $\mathcal{D}$ is a triangulated category, then we say $(\mathcal{D}', \mathcal{T}')$ is a {\it triangulated subcategory} if it is a pre-triangulated subcategory and $(\mathcal{D}', [\ ], \mathcal{T}')$ is a triangulated category.


derived-definition-homological

Let $\mathcal{D}$ be a pre-triangulated category. Let $\mathcal{A}$ be an abelian category. An additive functor $H : \mathcal{D} \to \mathcal{A}$ is called {\it homological} if for every distinguished triangle $(X, Y, Z, f, g, h)$ the sequence $$ H(X) \to H(Y) \to H(Z) $$ is exact in the abelian category $\mathcal{A}$. An additive functor $H : \mathcal{D}^{opp} \to \mathcal{A}$ is called {\it cohomological} if the corresponding functor $\mathcal{D} \to \mathcal{A}^{opp}$ is homological.


derived-definition-delta-functor

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{D}$ be a triangulated category. A {\it $\delta$-functor from $\mathcal{A}$ to $\mathcal{D}$} is given by a functor $G : \mathcal{A} \to \mathcal{D}$ and a rule which assigns to every short exact sequence $$ 0 \to A \xrightarrow{a} B \xrightarrow{b} C \to 0 $$ a morphism $\delta = \delta_{A \to B \to C} : G(C) \to G(A)[1]$ such that \begin{enumerate} \item the triangle $(G(A), G(B), G(C), G(a), G(b), \delta_{A \to B \to C})$ is a distinguished triangle of $\mathcal{D}$ for any short exact sequence as above, and \item for every morphism $(A \to B \to C) \to (A' \to B' \to C')$ of short exact sequences the diagram $$ \xymatrix{ G(C) \ar[d] \ar[rr]_{\delta_{A \to B \to C}} & & G(A)[1] \ar[d] \\ G(C') \ar[rr]^{\delta_{A' \to B' \to C'}} & & G(A')[1] } $$ is commutative. \end{enumerate} In this situation we call $(G(A), G(B), G(C), G(a), G(b), \delta_{A \to B \to C})$ the {\it image of the short exact sequence under the given $\delta$-functor}.


derived-definition-localization

Let $\mathcal{D}$ be a pre-triangulated category. We say a multiplicative system $S$ is {\it compatible with the triangulated structure} if the following two conditions hold: \begin{enumerate} \item[MS5] For $s \in S$ we have $s[n] \in S$ for all $n \in \mathbf{Z}$. \item[MS6] Given a solid commutative square $$ \xymatrix{ X \ar[r] \ar[d]^s & Y \ar[r] \ar[d]^{s'} & Z \ar[r] \ar@{-->}[d] & X[1] \ar[d]^{s[1]} \\ X' \ar[r] & Y' \ar[r] & Z' \ar[r] & X'[1] } $$ whose rows are distinguished triangles with $s, s' \in S$ there exists a morphism $s'' : Z \to Z'$ in $S$ such that $(s, s', s'')$ is a morphism of triangles. \end{enumerate}


derived-definition-saturated

Let $\mathcal{D}$ be a pre-triangulated category. We say a full pre-triangulated subcategory $\mathcal{D}'$ of $\mathcal{D}$ is {\it saturated} if whenever $X \oplus Y$ is isomorphic to an object of $\mathcal{D}'$ then both $X$ and $Y$ are isomorphic to objects of $\mathcal{D}'$.


derived-definition-kernel-category

Let $\mathcal{D}$ be a (pre-)triangulated category. \begin{enumerate} \item Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor. The {\it kernel of $F$} is the strictly full saturated (pre-)triangulated subcategory described in Lemma \ref{lemma-triangle-functor-kernel}. \item Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor. The {\it kernel of $H$} is the strictly full saturated (pre-)triangulated subcategory described in Lemma \ref{lemma-homological-functor-kernel}. \end{enumerate} These are sometimes denoted $\Ker(F)$ or $\Ker(H)$.


derived-definition-quotient-category

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{B}$ be a full triangulated subcategory. We define the {\it quotient category $\mathcal{D}/\mathcal{B}$} by the formula $\mathcal{D}/\mathcal{B} = S^{-1}\mathcal{D}$, where $S$ is the multiplicative system of $\mathcal{D}$ associated to $\mathcal{B}$ via Lemma \ref{lemma-construct-multiplicative-system}. The localization functor $Q : \mathcal{D} \to \mathcal{D}/\mathcal{B}$ is called the {\it quotient functor} in this case.


derived-definition-complexes-notation

Let $\mathcal{A}$ be an additive category. \begin{enumerate} \item We set $\text{Comp}(\mathcal{A}) = \text{CoCh}(\mathcal{A})$ be the {\it category of (cochain) complexes}. \item A complex $K^\bullet$ is said to be {\it bounded below} if $K^n = 0$ for all $n \ll 0$. \item A complex $K^\bullet$ is said to be {\it bounded above} if $K^n = 0$ for all $n \gg 0$. \item A complex $K^\bullet$ is said to be {\it bounded} if $K^n = 0$ for all $|n| \gg 0$. \item We let $\text{Comp}^{+}(\mathcal{A})$, $\text{Comp}^{-}(\mathcal{A})$, resp.\ $\text{Comp}^b(\mathcal{A})$ be the full subcategory of $\text{Comp}(\mathcal{A})$ whose objects are the complexes which are bounded below, bounded above, resp.\ bounded. \item We let $K(\mathcal{A})$ be the category with the same objects as $\text{Comp}(\mathcal{A})$ but as morphisms homotopy classes of maps of complexes (see Homology, Lemma \ref{homology-lemma-compose-homotopy-cochain}). \item We let $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, resp.\ $K^b(\mathcal{A})$ be the full subcategory of $K(\mathcal{A})$ whose objects are bounded below, bounded above, resp.\ bounded complexes of $\mathcal{A}$. \end{enumerate}


derived-definition-cone

Let $\mathcal{A}$ be an additive category. Let $f : K^\bullet \to L^\bullet$ be a morphism of complexes of $\mathcal{A}$. The {\it cone} of $f$ is the complex $C(f)^\bullet$ given by $C(f)^n = L^n \oplus K^{n + 1}$ and differential $$ d_{C(f)}^n = \left( \begin{matrix} d^n_L & f^{n + 1} \\ 0 & -d_K^{n + 1} \end{matrix} \right) $$ It comes equipped with canonical morphisms of complexes $i : L^\bullet \to C(f)^\bullet$ and $p : C(f)^\bullet \to K^\bullet[1]$ induced by the obvious maps $L^n \to C(f)^n \to K^{n + 1}$.


derived-definition-termwise-split-map

Let $\mathcal{A}$ be an additive category. A {\it termwise split injection $\alpha : A^\bullet \to B^\bullet$} is a morphism of complexes such that each $A^n \to B^n$ is isomorphic to the inclusion of a direct summand. A {\it termwise split surjection $\beta : B^\bullet \to C^\bullet$} is a morphism of complexes such that each $B^n \to C^n$ is isomorphic to the projection onto a direct summand.


derived-definition-split-ses

Let $\mathcal{A}$ be an additive category. A {\it termwise split exact sequence of complexes of $\mathcal{A}$} is a complex of complexes $$ 0 \to A^\bullet \xrightarrow{\alpha} B^\bullet \xrightarrow{\beta} C^\bullet \to 0 $$ together with given direct sum decompositions $B^n = A^n \oplus C^n$ compatible with $\alpha^n$ and $\beta^n$. We often write $s^n : C^n \to B^n$ and $\pi^n : B^n \to A^n$ for the maps induced by the direct sum decompositions. According to Homology, Lemma \ref{homology-lemma-ses-termwise-split-cochain} we get an associated morphism of complexes $$ \delta : C^\bullet \longrightarrow A^\bullet[1] $$ which in degree $n$ is the map $\pi^{n + 1} \circ d_B^n \circ s^n$. In other words $(A^\bullet, B^\bullet, C^\bullet, \alpha, \beta, \delta)$ forms a triangle $$ A^\bullet \to B^\bullet \to C^\bullet \to A^\bullet[1] $$ This will be the {\it triangle associated to the termwise split sequence of complexes}.


derived-definition-distinguished-triangle

Let $\mathcal{A}$ be an additive category. A triangle $(X, Y, Z, f, g, h)$ of $K(\mathcal{A})$ is called a {\it distinguished triangle of $K(\mathcal{A})$} if it is isomorphic to the triangle associated to a termwise split exact sequence of complexes, see Definition \ref{definition-split-ses}. Same definition for $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, and $K^b(\mathcal{A})$.


derived-definition-unbounded-derived-category

Let $\mathcal{A}$ be an abelian category. Let $\text{Ac}(\mathcal{A})$ and $\text{Qis}(\mathcal{A})$ be as in Lemma \ref{lemma-acyclic}. The {\it derived category of $\mathcal{A}$} is the triangulated category $$ D(\mathcal{A}) = K(\mathcal{A})/\text{Ac}(\mathcal{A}) = \text{Qis}(\mathcal{A})^{-1} K(\mathcal{A}). $$ We denote $H^0 : D(\mathcal{A}) \to \mathcal{A}$ the unique functor whose composition with the quotient functor gives back the functor $H^0$ defined above. Using Lemma \ref{lemma-homological-functor-bounded} we introduce the strictly full saturated triangulated subcategories $D^{+}(\mathcal{A}), D^{-}(\mathcal{A}), D^b(\mathcal{A})$ whose sets of objects are $$ \begin{matrix} \Ob(D^{+}(\mathcal{A})) = \{X \in \Ob(D(\mathcal{A})) \mid H^n(X) = 0\text{ for all }n \ll 0\} \\ \Ob(D^{-}(\mathcal{A})) = \{X \in \Ob(D(\mathcal{A})) \mid H^n(X) = 0\text{ for all }n \gg 0\} \\ \Ob(D^b(\mathcal{A})) = \{X \in \Ob(D(\mathcal{A})) \mid H^n(X) = 0\text{ for all }|n| \gg 0\} \end{matrix} $$ The category $D^b(\mathcal{A})$ is called the {\it bounded derived category} of $\mathcal{A}$.


derived-definition-finite-filtered

Let $\mathcal{A}$ be an abelian category. The {\it category of finite filtered objects of $\mathcal{A}$} is the category of filtered objects $(A, F)$ of $\mathcal{A}$ whose filtration $F$ is finite. We denote it $\text{Fil}^f(\mathcal{A})$.


derived-definition-filtered-acyclic

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item Let $\alpha : K^\bullet \to L^\bullet$ be a morphism of $K(\text{Fil}^f(\mathcal{A}))$. We say that $\alpha$ is a {\it filtered quasi-isomorphism} if the morphism $\text{gr}(\alpha)$ is a quasi-isomorphism. \item Let $K^\bullet$ be an object of $K(\text{Fil}^f(\mathcal{A}))$. We say that $K^\bullet$ is {\it filtered acyclic} if the complex $\text{gr}(K^\bullet)$ is acyclic. \end{enumerate}


derived-definition-filtered-derived

Let $\mathcal{A}$ be an abelian category. Let $\text{FAc}(\mathcal{A})$ and $\text{FQis}(\mathcal{A})$ be as in Lemma \ref{lemma-filtered-acyclic}. The {\it filtered derived category of $\mathcal{A}$} is the triangulated category $$ DF(\mathcal{A}) = K(\text{Fil}^f(\mathcal{A}))/\text{FAc}(\mathcal{A}) = \text{FQis}(\mathcal{A})^{-1} K(\text{Fil}^f(\mathcal{A})). $$


derived-definition-filtered-derived-bounded

Let $\mathcal{A}$ be an abelian category. The {\it bounded filtered derived category} $DF^b(\mathcal{A})$ is the full subcategory of $DF(\mathcal{A})$ with objects those $X$ such that $\text{gr}(X) \in D^b(\mathcal{A})$. Similarly for the bounded below filtered derived category $DF^{+}(\mathcal{A})$ and the bounded above filtered derived category $DF^{-}(\mathcal{A})$.


derived-definition-right-derived-functor-defined

Assumptions and notation as in Situation \ref{situation-derived-functor}. Let $X \in \Ob(\mathcal{D})$. \begin{enumerate} \item we say the {\it right derived functor $RF$ is defined at} $X$ if the ind-object $$ (X/S) \longrightarrow \mathcal{D}', \quad (s : X \to X') \longmapsto F(X') $$ is essentially constant\footnote{For a discussion of when an ind-object or pro-object of a category is essentially constant we refer to Categories, Section \ref{categories-section-essentially-constant}.}; in this case the value $Y$ in $\mathcal{D}'$ is called the {\it value of $RF$ at $X$}. \item we say the {\it left derived functor $LF$ is defined at} $X$ if the pro-object $$ (S/X) \longrightarrow \mathcal{D}', \quad (s: X' \to X) \longmapsto F(X') $$ is essentially constant; in this case the value $Y$ in $\mathcal{D}'$ is called the {\it value of $LF$ at $X$}. \end{enumerate} By abuse of notation we often denote the values simply $RF(X)$ or $LF(X)$.


derived-definition-everywhere-defined

In Situation \ref{situation-derived-functor}. We say $F$ is {\it right derivable}, or that $RF$ {\it everywhere defined} if $RF$ is defined at every object of $\mathcal{D}$. We say $F$ is {\it left derivable}, or that $LF$ {\it everywhere defined} if $LF$ is defined at every object of $\mathcal{D}$.


derived-definition-computes

In Situation \ref{situation-derived-functor}. \begin{enumerate} \item An object $X$ of $\mathcal{D}$ {\it computes} $RF$ if $RF$ is defined at $X$ and the canonical map $F(X) \to RF(X)$ is an isomorphism. \item An object $X$ of $\mathcal{D}$ {\it computes} $LF$ if $LF$ is defined at $X$ and the canonical map $LF(X) \to F(X)$ is an isomorphism. \end{enumerate}


derived-definition-derived-functor

In Situation \ref{situation-classical}. \begin{enumerate} \item The {\it right derived functors of $F$} are the partial functors $RF$ associated to cases (1) and (2) of Situation \ref{situation-classical}. \item The {\it left derived functors of $F$} are the partial functors $LF$ associated to cases (3) and (4) of Situation \ref{situation-classical}. \item An object $A$ of $\mathcal{A}$ is said to be {\it right acyclic for $F$}, or {\it acyclic for $RF$} if $A[0]$ computes $RF$. \item An object $A$ of $\mathcal{A}$ is said to be {\it left acyclic for $F$}, or {\it acyclic for $LF$} if $A[0]$ computes $LF$. \end{enumerate}


derived-definition-higher-derived-functors

Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories. Assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. Let $i \in \mathbf{Z}$. The {\it $i$th right derived functor $R^iF$ of $F$} is the functor $$ R^iF = H^i \circ RF : \mathcal{A} \longrightarrow \mathcal{B} $$