spaces-properties-lemma-codimension-0-points

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $x \in |X|$. Consider \'etale morphisms $a : U \to X$ where $U$ is a scheme. The following are equivalent \begin{enumerate} \item $x$ is a point of codimension $0$ on $X$, \item for some $U \to X$ as above and $u \in U$ with $a(u) = x$, the point $u$ is the generic point of an irreducible component of $U$, and \item for any $U \to X$ as above and any $u \in U$ mapping to $x$, the point $u$ is the generic point of an irreducible component of $U$. \end{enumerate} If $X$ is representable, this is equivalent to $x$ being a generic point of an irreducible component of $|X|$.


spaces-properties-lemma-codimension-0-points-dense

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. The set of codimension $0$ points of $X$ is dense in $|X|$.


spaces-properties-lemma-subspace-induced-topology

Let $S$ be a scheme. Let $Z \to X$ be an immersion of algebraic spaces. Then $|Z| \to |X|$ is a homeomorphism of $|Z|$ onto a locally closed subset of $|X|$.


spaces-properties-lemma-subspaces-presentation

Let $S$ be a scheme. Let $j : R \to U \times_S U$ be an \'etale equivalence relation. Let $X = U/R$ be the associated algebraic space (Spaces, Theorem \ref{spaces-theorem-presentation}). There is a canonical bijection $$ R\text{-invariant locally closed subschemes }Z'\text{ of }U \leftrightarrow \text{locally closed subspaces }Z\text{ of }X $$ Moreover, if $Z \to X$ is closed (resp.\ open) if and only if $Z' \to U$ is closed (resp.\ open).


spaces-properties-lemma-reduced-closed-subspace

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset. There exists a unique closed subspace $Z \subset X$ with the following properties: (a) we have $|Z| = T$, and (b) $Z$ is reduced.


spaces-properties-lemma-map-into-reduction

Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $Z \subset X$ be a closed subspace. Assume $Y$ is reduced. A morphism $f : Y \to X$ factors through $Z$ if and only if $f(|Y|) \subset |Z|$.


spaces-properties-lemma-subscheme

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a largest open subspace $X' \subset X$ which is a scheme.


spaces-properties-lemma-quasi-separated-finite-etale-cover-dense-open-scheme

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a finite, \'etale, surjective morphism $U \to X$ where $U$ is a quasi-separated scheme, then there exists a dense open subspace $X'$ of $X$ which is a scheme. More precisely, every point $x \in |X|$ of codimension $0$ in $X$ is contained in $X'$.


spaces-properties-lemma-quotient-scheme

Let $S$ be a scheme. Let $G \to S$ be a group scheme. Let $X \to S$ be a morphism of schemes. Let $a : G \times_S X \to X$ be an action. Assume that \begin{enumerate} \item $G \to S$ is finite locally free, \item the action $a$ is free, \item $X \to S$ is affine, or quasi-affine, or projective, or quasi-projective, or $X$ is isomorphic to an open subscheme of an affine scheme, or $X$ is isomorphic to an open subscheme of $\text{Proj}(A)$ for some graded ring $A$, or $G \to S$ is radicial. \end{enumerate} Then the fppf quotient sheaf $X/G$ is a scheme and $X \to X/G$ is an fppf $G$-torsor.


spaces-properties-lemma-quotient-separated

Notation and assumptions as in Proposition \ref{proposition-finite-flat-equivalence-global}. Then \begin{enumerate} \item if $U$ is quasi-separated over $S$, then $U/R$ is quasi-separated over $S$, \item if $U$ is quasi-separated, then $U/R$ is quasi-separated, \item if $U$ is separated over $S$, then $U/R$ is separated over $S$, \item if $U$ is separated, then $U/R$ is separated, and \item add more here. \end{enumerate} Similar results hold in the setting of Lemma \ref{lemma-quotient-scheme}.


spaces-properties-lemma-quasi-separated-sober

Let $S$ be a scheme. Let $X$ be a Zariski locally quasi-separated algebraic space over $S$. Then the topological space $|X|$ is sober (see Topology, Definition \ref{topology-definition-generic-point}).


spaces-properties-lemma-quasi-compact-quasi-separated-spectral

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The topological space $|X|$ is a spectral space.


spaces-properties-lemma-point-like-spaces

Let $S$ be a scheme. Let $k$ be a field. Let $X$ be an algebraic space over $S$ and assume that there exists a surjective \'etale morphism $\Spec(k) \to X$. If $X$ is quasi-separated, then $X \cong \Spec(k')$ where $k' \subset k$ is a finite separable extension.


spaces-properties-lemma-etale-over-space

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$, $U'$ be schemes over $S$. \begin{enumerate} \item If $U \to U'$ is an \'etale morphism of schemes, and if $U' \to X$ is an \'etale morphism from $U'$ to $X$, then the composition $U \to X$ is an \'etale morphism from $U$ to $X$. \item If $\varphi : U \to X$ and $\varphi' : U' \to X$ are \'etale morphisms towards $X$, and if $\chi : U \to U'$ is a morphism of schemes such that $\varphi = \varphi' \circ \chi$, then $\chi$ is an \'etale morphism of schemes. \item If $\chi : U \to U'$ is a surjective \'etale morphism of schemes and $\varphi' : U' \to X$ is a morphism such that $\varphi = \varphi' \circ \chi$ is \'etale, then $\varphi'$ is \'etale. \end{enumerate}


spaces-properties-lemma-etale-local

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is \'etale, \item there exists a surjective \'etale morphism $\varphi : U \to X$, where $U$ is a scheme, such that the composition $f \circ \varphi$ is \'etale (as a morphism of algebraic spaces), \item there exists a surjective \'etale morphism $\psi : V \to Y$, where $V$ is a scheme, such that the base change $V \times_X Y \to V$ is \'etale (as a morphism of algebraic spaces), \item there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \'etale, and the left vertical arrow is surjective such that the horizontal arrow is \'etale. \end{enumerate}


spaces-properties-lemma-composition-etale

The composition of two \'etale morphisms of algebraic spaces is \'etale.


spaces-properties-lemma-base-change-etale

The base change of an \'etale morphism of algebraic spaces by any morphism of algebraic spaces is \'etale.


spaces-properties-lemma-etale-permanence

Let $S$ be a scheme. Let $X, Y, Z$ be algebraic spaces. Let $g : X \to Z$, $h : Y \to Z$ be \'etale morphisms and let $f : X \to Y$ be a morphism such that $h \circ f = g$. Then $f$ is \'etale.


spaces-properties-lemma-etale-open

Let $S$ be a scheme. If $X \to Y$ is an \'etale morphism of algebraic spaces over $S$, then the associated map $|X| \to |Y|$ of topological spaces is open.


spaces-properties-lemma-etale-over-field-scheme

Let $S$ be a scheme. Let $X \to \Spec(k)$ be \'etale morphism over $S$, where $k$ is a field. Then $X$ is a scheme.


spaces-properties-lemma-compare-etale-sites

The functor $$ X_\etale \longrightarrow X_{spaces, \etale}, \quad U/X \longmapsto U/X $$ is a special cocontinuous functor (Sites, Definition \ref{sites-definition-special-cocontinuous-functor}) and hence induces an equivalence of topoi $\Sh(X_\etale) \to \Sh(X_{spaces, \etale})$.


spaces-properties-lemma-alternative

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X_{affine, \etale}$ denote the full subcategory of $X_\etale$ whose objects are those $U/X \in \Ob(X_\etale)$ with $U$ affine. A covering of $X_{affine, \etale}$ will be a standard \'etale covering, see Topologies, Definition \ref{topologies-definition-standard-etale}. Then restriction $$ \mathcal{F} \longmapsto \mathcal{F}|_{X_{affine, \etale}} $$ defines an equivalence of topoi $\Sh(X_\etale) \cong \Sh(X_{affine, \etale})$.


spaces-properties-lemma-functoriality-etale-site

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item The continuous functor $$ Y_{spaces, \etale} \longrightarrow X_{spaces, \etale}, \quad V \longmapsto X \times_Y V $$ induces a morphism of sites $$ f_{spaces, \etale} : X_{spaces, \etale} \to Y_{spaces, \etale}. $$ \item The rule $f \mapsto f_{spaces, \etale}$ is compatible with compositions, in other words $(f \circ g)_{spaces, \etale} = f_{spaces, \etale} \circ g_{spaces, \etale}$ (see Sites, Definition \ref{sites-definition-composition-morphisms-sites}). \item The morphism of topoi associated to $f_{spaces, \etale}$ induces, via Lemma \ref{lemma-compare-etale-sites}, a morphism of topoi $f_{small} : \Sh(X_\etale) \to \Sh(Y_\etale)$ whose construction is compatible with compositions. \item If $f$ is a representable morphism of algebraic spaces, then $f_{small}$ comes from a morphism of sites $X_\etale \to Y_\etale$, corresponding to the continuous functor $V \mapsto X \times_Y V$. \end{enumerate}


spaces-properties-lemma-f-map

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf of sets on $X_\etale$ and let $\mathcal{G}$ be a sheaf of sets on $Y_\etale$. There are canonical bijections between the following three sets: \begin{enumerate} \item The set of maps $\mathcal{G} \to f_{small, *}\mathcal{F}$. \item The set of maps $f_{small}^{-1}\mathcal{G} \to \mathcal{F}$. \item The set of $f$-maps $\varphi : \mathcal{G} \to \mathcal{F}$. \end{enumerate}


spaces-properties-lemma-etale-morphism-topoi

Let $S$ be a scheme, and let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is \'etale. In this case there is a functor $$ j : X_\etale \to Y_\etale, \quad (\varphi : U \to X) \mapsto (f \circ \varphi : U \to Y) $$ which is cocontinuous. The morphism of topoi $f_{small}$ is the morphism of topoi associated to $j$, see Sites, Lemma \ref{sites-lemma-cocontinuous-morphism-topoi}. Moreover, $j$ is continuous as well, hence Sites, Lemma \ref{sites-lemma-when-shriek} applies. In particular $f_{small}^{-1}\mathcal{G}(U) = \mathcal{G}(jU)$ for all sheaves $\mathcal{G}$ on $Y_\etale$.


spaces-properties-lemma-pushforward-etale-base-change

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[r] \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a cartesian square of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf on $X_\etale$. If $g$ is \'etale, then \begin{enumerate} \item $f'_{small, *}(\mathcal{F}|_{X'}) = (f_{small, *}\mathcal{F})|_{Y'}$ in $\Sh(Y'_\etale)$\footnote{Also $(f')_{small}^{-1}(\mathcal{G}|_{Y'}) = (f_{small}^{-1}\mathcal{G})|_{X'}$ because of commutativity of the diagram and (\ref{equation-restrict})}, and \item if $\mathcal{F}$ is an abelian sheaf, then $R^if'_{small, *}(\mathcal{F}|_{X'}) = (R^if_{small, *}\mathcal{F})|_{Y'}$. \end{enumerate}


spaces-properties-lemma-characterize-sheaf-small-etale

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A sheaf $\mathcal{F}$ on $X_\etale$ is given by the following data: \begin{enumerate} \item for every $U \in \Ob(X_\etale)$ a sheaf $\mathcal{F}_U$ on $U_\etale$, \item for every $f : U' \to U$ in $X_\etale$ an isomorphism $c_f : f_{small}^{-1}\mathcal{F}_U \to \mathcal{F}_{U'}$. \end{enumerate} These data are subject to the condition that given any $f : U' \to U$ and $g : U'' \to U'$ in $X_\etale$ the composition $g_{small}^{-1}c_f \circ c_g$ is equal to $c_{f \circ g}$.


spaces-properties-lemma-descent-sheaf

With $S$, $\varphi : U \to X$, and $(U, R, s, t, c, e, i)$ as above. For any sheaf $\mathcal{F}$ on $X_\etale$ the sheaf\footnote{In this lemma and its proof we write simply $\varphi^{-1}$ instead of $\varphi_{small}^{-1}$ and similarly for all the other pullbacks.} $\mathcal{G} = \varphi^{-1}\mathcal{F}$ comes equipped with a canonical isomorphism $$ \alpha : t^{-1}\mathcal{G} \longrightarrow s^{-1}\mathcal{G} $$ such that the diagram $$ \xymatrix{ & \text{pr}_1^{-1}t^{-1}\mathcal{G} \ar[r]_-{\text{pr}_1^{-1}\alpha} & \text{pr}_1^{-1}s^{-1}\mathcal{G} \ar@{=}[rd] & \\ \text{pr}_0^{-1}s^{-1}\mathcal{G} \ar@{=}[ru] & & & c^{-1}s^{-1}\mathcal{G} \\ & \text{pr}_0^{-1}t^{-1}\mathcal{G} \ar[lu]^{\text{pr}_0^{-1}\alpha} \ar@{=}[r] & c^{-1}t^{-1}\mathcal{G} \ar[ru]_{c^{-1}\alpha} } $$ is a commutative. The functor $\mathcal{F} \mapsto (\mathcal{G}, \alpha)$ defines an equivalence of categories between sheaves on $X_\etale$ and pairs $(\mathcal{G}, \alpha)$ as above.


spaces-properties-lemma-cofinal-etale

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. The category of \'etale neighborhoods is cofiltered. More precisely: \begin{enumerate} \item Let $(U_i, \overline{u}_i)_{i = 1, 2}$ be two \'etale neighborhoods of $\overline{x}$ in $X$. Then there exists a third \'etale neighborhood $(U, \overline{u})$ and morphisms $(U, \overline{u}) \to (U_i, \overline{u}_i)$, $i = 1, 2$. \item Let $h_1, h_2: (U, \overline{u}) \to (U', \overline{u}')$ be two morphisms between \'etale neighborhoods of $\overline{s}$. Then there exist an \'etale neighborhood $(U'', \overline{u}'')$ and a morphism $h : (U'', \overline{u}'') \to (U, \overline{u})$ which equalizes $h_1$ and $h_2$, i.e., such that $h_1 \circ h = h_2 \circ h$. \end{enumerate} Moreover, given any \'etale neighbourhood $(U, \overline{u}) \to (X, \overline{x})$ there exists a morphism of \'etale neighbourhoods $(U', \overline{u}') \to (U, \overline{u})$ where $U'$ is a scheme.


spaces-properties-lemma-geometric-lift-to-usual

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x} : \Spec(k) \to X$ be a geometric point of $X$ lying over $x \in |X|$. Let $\varphi : U \to X$ be an \'etale morphism of algebraic spaces and let $u \in |U|$ with $\varphi(u) = x$. Then there exists a geometric point $\overline{u} : \Spec(k) \to U$ lying over $u$ with $\overline{x} = \varphi \circ \overline{u}$.


spaces-properties-lemma-geometric-lift-to-cover

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Let $(U, \overline{u})$ an \'etale neighborhood of $\overline{x}$. Let $\{\varphi_i : U_i \to U\}_{i \in I}$ be an \'etale covering in $X_{spaces, \etale}$. Then there exist $i \in I$ and $\overline{u}_i : \overline{x} \to U_i$ such that $\varphi_i : (U_i, \overline{u}_i) \to (U, \overline{u})$ is a morphism of \'etale neighborhoods.


spaces-properties-lemma-stalk-gives-point

\begin{slogan} A geometric point of an algebraic space gives a point of its \'etale topos. \end{slogan} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Consider the functor $$ u : X_\etale \longrightarrow \textit{Sets}, \quad U \longmapsto |U_{\overline{x}}| $$ Then $u$ defines a point $p$ of the site $X_\etale$ (Sites, Definition \ref{sites-definition-point}) and its associated stalk functor $\mathcal{F} \mapsto \mathcal{F}_p$ (Sites, Equation \ref{sites-equation-stalk}) is the functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ defined above.


spaces-properties-lemma-stalk-exact

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. \begin{enumerate} \item The stalk functor $\textit{PAb}(X_\etale) \to \textit{Ab}$, $\mathcal{F}  \mapsto  \mathcal{F}_{\overline{x}}$ is exact. \item We have $(\mathcal{F}^\#)_{\overline{x}} = \mathcal{F}_{\overline{x}}$ for any presheaf of sets $\mathcal{F}$ on $X_\etale$. \item The functor $\textit{Ab}(X_\etale) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is exact. \item Similarly the functors $\textit{PSh}(X_\etale) \to \textit{Sets}$ and $\Sh(X_\etale) \to \textit{Sets}$ given by the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ are exact (see Categories, Definition \ref{categories-definition-exact}) and commute with arbitrary colimits. \end{enumerate}


spaces-properties-lemma-stalk-pullback

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item The functor $f_{small}^{-1} : \textit{Ab}(Y_\etale) \to \textit{Ab}(X_\etale)$ is exact. \item The functor $f_{small}^{-1} : \Sh(Y_\etale) \to \Sh(X_\etale)$ is exact, i.e., it commutes with finite limits and colimits, see Categories, Definition \ref{categories-definition-exact}. \item For any \'etale morphism $V \to Y$ of algebraic spaces we have $f_{small}^{-1}h_V = h_{X \times_Y V}$. \item Let $\overline{x} \to X$ be a geometric point. Let $\mathcal{G}$ be a sheaf on $Y_\etale$. Then there is a canonical identification $$ (f_{small}^{-1}\mathcal{G})_{\overline{x}} = \mathcal{G}_{\overline{y}}. $$ where $\overline{y} = f \circ \overline{x}$. \end{enumerate}


spaces-properties-lemma-points-small-etale-site

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $p : \Sh(pt) \to \Sh(X_\etale)$ be a point of the small \'etale topos of $X$. Then there exists a geometric point $\overline{x}$ of $X$ such that the stalk functor $\mathcal{F} \mapsto \mathcal{F}_p$ is isomorphic to the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$.


spaces-properties-lemma-support-subsheaf-final

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a subsheaf of the final object of the \'etale topos of $X$ (see Sites, Example \ref{sites-example-singleton-sheaf}). Then there exists a unique open $W \subset X$ such that $\mathcal{F} = h_W$.


spaces-properties-lemma-zero-over-image

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be an abelian sheaf on $X_{spaces, \etale}$. Let $\sigma \in \mathcal{F}(U)$ be a local section. There exists an open subspace $W \subset U$ such that \begin{enumerate} \item $W \subset U$ is the largest open subspace of $U$ such that $\sigma|_W = 0$, \item for every $\varphi : V \to U$ in $X_\etale$ we have $$ \sigma|_V = 0 \Leftrightarrow \varphi(V) \subset W, $$ \item for every geometric point $\overline{u}$ of $U$ we have $$ (U, \overline{u}, \sigma) = 0\text{ in }\mathcal{F}_{\overline{s}} \Leftrightarrow \overline{u} \in W $$ where $\overline{s} = (U \to S) \circ \overline{u}$. \end{enumerate}


spaces-properties-lemma-support-sheaf-rings-closed

The support of a sheaf of rings on the small \'etale site of an algebraic space is closed.


spaces-properties-lemma-sheaf-condition-holds

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The rule $U \mapsto \Gamma(U, \mathcal{O}_U)$ defines a sheaf of rings on $X_\etale$.


spaces-properties-lemma-morphism-ringed-topoi

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then there is a canonical map $f^\sharp : f_{small}^{-1}\mathcal{O}_Y \to \mathcal{O}_X$ such that $$ (f_{small}, f^\sharp) : (\Sh(X_\etale), \mathcal{O}_X) \longrightarrow (\Sh(Y_\etale), \mathcal{O}_Y) $$ is a morphism of ringed topoi. Furthermore, \begin{enumerate} \item The construction $f \mapsto (f_{small}, f^\sharp)$ is compatible with compositions. \item If $f$ is a morphism of schemes, then $f^\sharp$ is the map described in Descent, Remark \ref{descent-remark-change-topologies-ringed}. \end{enumerate}


spaces-properties-lemma-reduced-space

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \begin{enumerate} \item $X$ is reduced, \item for every $x \in |X|$ the local ring of $X$ at $x$ is reduced (Remark \ref{remark-list-properties-local-ring-local-etale-topology}). \end{enumerate} In this case $\Gamma(X, \mathcal{O}_X)$ is a reduced ring and if $f \in \Gamma(X, \mathcal{O}_X)$ has $X = V(f)$, then $f = 0$.


spaces-properties-lemma-describe-etale-local-ring

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Let $(U, \overline{u})$ be an \'etale neighbourhood of $\overline{x}$ where $U$ is a scheme. Then we have $$ \mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, \overline{u}} = \mathcal{O}_{U, u}^{sh} $$ where the left hand side is the stalk of the structure sheaf of $X$, and the right hand side is the strict henselization of the local ring of $U$ at the point $u$ at which $\overline{u}$ is centered.


spaces-properties-lemma-etale-site-locally-ringed

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The small \'etale site $X_\etale$ endowed with its structure sheaf $\mathcal{O}_X$ is a locally ringed site, see Modules on Sites, Definition \ref{sites-modules-definition-locally-ringed}.


spaces-properties-lemma-dimension-local-ring

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. Let $d \in \{0, 1, 2, \ldots, \infty\}$. The following are equivalent \begin{enumerate} \item the dimension of the local ring of $X$ at $x$ (Definition \ref{definition-dimension-local-ring}) is $d$, \item $\dim(\mathcal{O}_{X, \overline{x}}) = d$ for some geometric point $\overline{x}$ lying over $x$, and \item $\dim(\mathcal{O}_{X, \overline{x}}) = d$ for any geometric point $\overline{x}$ lying over $x$. \end{enumerate}


spaces-properties-lemma-dimension-decent-invariant-under-etale

Let $S$ be a scheme. Let $f : X \to Y$ be an \'etale morphism of algebraic spaces over $S$. Let $x \in X$. Then (1) $\dim_x(X) = \dim_{f(x)}(Y)$ and (2) the dimension of the local ring of $X$ at $x$ equals the dimension of the local ring of $Y$ at $f(x)$. If $f$ is surjective, then (3) $\dim(X) = \dim(Y)$.


spaces-properties-lemma-irreducible-local-ring

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. The following are equivalent \begin{enumerate} \item for any scheme $U$ and \'etale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ the local ring $\mathcal{O}_{U, u}$ has a unique minimal prime, \item for any scheme $U$ and \'etale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ there is a unique irreducible component of $U$ through $u$, \item for any scheme $U$ and \'etale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ the local ring $\mathcal{O}_{U, u}$ is unibranch, \item for any scheme $U$ and \'etale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ the local ring $\mathcal{O}_{U, u}$ is geometrically unibranch, \item $\mathcal{O}_{X, \overline{x}}$ has a unique minimal prime for any geometric point $\overline{x}$ lying over $x$. \end{enumerate}


spaces-properties-lemma-nr-branches-local-ring

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. Let $n \in \{1, 2, \ldots\}$ be an integer. The following are equivalent \begin{enumerate} \item for any scheme $U$ and \'etale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ the number of minimal primes of the local ring $\mathcal{O}_{U, u}$ is $\leq n$ and for at least one choice of $U, a, u$ it is $n$, \item for any scheme $U$ and \'etale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ the number irreducible components of $U$ passing through $u$ is $\leq n$ and for at least one choice of $U, a, u$ it is $n$, \item for any scheme $U$ and \'etale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ the number of branches of $U$ at $u$ is $\leq n$ and for at least one choice of $U, a, u$ it is $n$, \item for any scheme $U$ and \'etale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ the number of geometric branches of $U$ at $u$ is $n$, and \item the number of minimal prime ideals of $\mathcal{O}_{X, \overline{x}}$ is $n$. \end{enumerate}


spaces-properties-lemma-Noetherian-topology

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item If $X$ is locally Noetherian then $|X|$ is a locally Noetherian topological space. \item If $X$ is quasi-compact and locally Noetherian, then $|X|$ is a Noetherian topological space. \end{enumerate}


spaces-properties-lemma-Noetherian-sober

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Noetherian, then $|X|$ is a sober Noetherian topological space.


spaces-properties-lemma-Noetherian-local-ring-Noetherian

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Then $\mathcal{O}_{X, \overline{x}}$ is a Noetherian local ring.


spaces-properties-lemma-regular

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. The following are equivalent \begin{enumerate} \item $X$ is regular, and \item every \'etale local ring $\mathcal{O}_{X, \overline{x}}$ is regular. \end{enumerate}


spaces-properties-lemma-regular-normal

A regular algebraic space is normal.


spaces-properties-lemma-etale-exact-pullback

Let $S$ be a scheme. Let $f : X \to Y$ be an \'etale morphism of algebraic spaces over $S$. Then $f^{-1}\mathcal{O}_Y = \mathcal{O}_X$, and $f^*\mathcal{G} = f_{small}^{-1}\mathcal{G}$ for any sheaf of $\mathcal{O}_Y$-modules $\mathcal{G}$. In particular, $f^* : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_Y)$ is exact.


spaces-properties-lemma-pushforward-etale-base-change-modules

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[r] \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a cartesian square of algebraic spaces over $S$. Let $\mathcal{F} \in \textit{Mod}(\mathcal{O}_X)$. If $g$ is \'etale, then $f'_*(\mathcal{F}|_{X'}) = (f_*\mathcal{F})|_{Y'}$\footnote{Also $(f')^*(\mathcal{G}|_{Y'}) = (f^*\mathcal{G})|_{X'}$ by commutativity of the diagram and (\ref{equation-restrict-modules})} and $R^if'_*(\mathcal{F}|_{X'}) = (R^if_*\mathcal{F})|_{Y'}$ in $\textit{Mod}(\mathcal{O}_{Y'})$.


spaces-properties-lemma-characterize-module-small-etale

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules is given by the following data: \begin{enumerate} \item for every $U \in \Ob(X_\etale)$ a sheaf $\mathcal{F}_U$ of $\mathcal{O}_U$-modules on $U_\etale$, \item for every $f : U' \to U$ in $X_\etale$ an isomorphism $c_f : f_{small}^*\mathcal{F}_U \to \mathcal{F}_{U'}$. \end{enumerate} These data are subject to the condition that given any $f : U' \to U$ and $g : U'' \to U'$ in $X_\etale$ the composition $g_{small}^{-1}c_f \circ c_g$ is equal to $c_{f \circ g}$.


spaces-properties-lemma-relocalize-morphism

Let $S$ be a scheme. Let $$ \xymatrix{ U \ar[d]_p \ar[r]_g & V \ar[d]^q \\ X \ar[r]^f & Y } $$ be a commutative diagram of algebraic spaces over $S$ with $p$ and $q$ \'etale. Via the identifications (\ref{equation-localize-ringed}) for $U \to X$ and $V \to Y$ the morphism of ringed topoi $$ (g_{spaces, \etale}, g^\sharp) : (\Sh(U_{spaces, \etale}), \mathcal{O}_U) \longrightarrow (\Sh(V_{spaces, \etale}), \mathcal{O}_V) $$ is $2$-isomorphic to the morphism $(f_{spaces, \etale, c}, f_c^\sharp)$ constructed in Modules on Sites, Lemma \ref{sites-modules-lemma-relocalize-morphism-ringed-sites} starting with the morphism of ringed sites $(f_{spaces, \etale}, f^\sharp)$ and the map $c : U \to V \times_Y X$ corresponding to $g$.


spaces-properties-lemma-relocalize-morphism-at-schemes

Same notation and assumptions as in Lemma \ref{lemma-relocalize-morphism} except that we also assume $U$ and $V$ are schemes. Via the identifications (\ref{equation-localize-at-scheme-ringed}) for $U \to X$ and $V \to Y$ the morphism of ringed topoi $$ (g_{small}, g^\sharp) : (\Sh(U_\etale), \mathcal{O}_U) \longrightarrow (\Sh(V_\etale), \mathcal{O}_V) $$ is $2$-isomorphic to the morphism $(f_{small, s}, f_s^\sharp)$ constructed in Modules on Sites, Lemma \ref{sites-modules-lemma-relocalize-morphism-ringed-topoi} starting with $(f_{small}, f^\sharp)$ and the map $s : h_U \to f_{small}^{-1}h_V$ corresponding to $g$.


spaces-properties-lemma-sheaf-gives-space

Let $S$ be a scheme and let $Y$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf of sets on $Y_\etale$. Provided a set theoretic condition is satisfied (see proof) the functor $X$ associated to $\mathcal{F}$ above is an algebraic space and there is an \'etale morphism $f : X \to Y$ of algebraic spaces such that $\mathcal{F} = f_{small, *}*$ where $*$ is the final object of the category $\Sh(X_\etale)$ (constant sheaf with value a singleton).


spaces-properties-lemma-morphism-locally-ringed

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The morphism of ringed topoi $(f_{small}, f^\sharp)$ associated to $f$ is a morphism of locally ringed topoi, see Modules on Sites, Definition \ref{sites-modules-definition-morphism-locally-ringed-topoi}.


spaces-properties-lemma-faithful

Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Any two morphisms $a, b : X \to Y$ of algebraic spaces over $S$ for which there exists a $2$-isomorphism $(a_{small}, a^\sharp) \cong (b_{small}, b^\sharp)$ in the $2$-category of ringed topoi are equal.


spaces-properties-lemma-pullback-quasi-coherent

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The pullback functor $f^* : \textit{Mod}(\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$ preserves quasi-coherent sheaves.


spaces-properties-lemma-characterize-quasi-coherent-small-etale

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ is given by the following data: \begin{enumerate} \item for every $U \in \Ob(X_\etale)$ a quasi-coherent $\mathcal{O}_U$-module $\mathcal{F}_U$ on $U_\etale$, \item for every $f : U' \to U$ in $X_\etale$ an isomorphism $c_f : f_{small}^*\mathcal{F}_U \to \mathcal{F}_{U'}$. \end{enumerate} These data are subject to the condition that given any $f : U' \to U$ and $g : U'' \to U'$ in $X_\etale$ the composition $g_{small}^{-1}c_f \circ c_g$ is equal to $c_{f \circ g}$.


spaces-properties-lemma-stalk-quasi-coherent

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in |X|$ be a point and let $\overline{x}$ be a geometric point lying over $x$. Finally, let $\varphi : (U, \overline{u}) \to (X, \overline{x})$ be an \'etale neighbourhood where $U$ is a scheme. Then $$ (\varphi^*\mathcal{F})_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} = \mathcal{F}_{\overline{x}} $$ where $u \in U$ is the image of $\overline{u}$.


spaces-properties-lemma-stalk-pullback-quasi-coherent

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. Let $\overline{x}$ be a geometric point of $X$ and let $\overline{y} = f \circ \overline{x}$ be the image in $Y$. Then there is a canonical isomorphism $$ (f^*\mathcal{G})_{\overline{x}} = \mathcal{G}_{\overline{y}} \otimes_{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{X, \overline{x}} $$ of the stalk of the pullback with the tensor product of the stalk with the local ring of $X$ at $\overline{x}$.


spaces-properties-lemma-characterize-quasi-coherent

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module, \item there exists an \'etale morphism $f : Y \to X$ of algebraic spaces over $S$ with $|f| : |Y| \to |X|$ surjective such that $f^*\mathcal{F}$ is quasi-coherent on $Y$, \item there exists a scheme $U$ and a surjective \'etale morphism $\varphi : U \to X$ such that $\varphi^*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_U$-module, and \item for every affine scheme $U$ and \'etale morphism $\varphi : U \to X$ the restriction $\varphi^*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_U$-module. \end{enumerate}


spaces-properties-lemma-properties-quasi-coherent

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The category $\QCoh(\mathcal{O}_X)$ of quasi-coherent sheaves on $X$ has the following properties: \begin{enumerate} \item Any direct sum of quasi-coherent sheaves is quasi-coherent. \item Any colimit of quasi-coherent sheaves is quasi-coherent. \item The kernel and cokernel of a morphism of quasi-coherent sheaves is quasi-coherent. \item Given a short exact sequence of $\mathcal{O}_X$-modules $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ if two out of three are quasi-coherent so is the third. \item Given two quasi-coherent $\mathcal{O}_X$-modules the tensor product is quasi-coherent. \item Given two quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$ is of finite presentation (see Section \ref{section-properties-modules}), then the internal hom $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is quasi-coherent. \end{enumerate}


spaces-properties-lemma-locally-projective

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item for some scheme $U$ and surjective \'etale morphism $U \to X$ the restriction $\mathcal{F}|_U$ is locally projective on $U$, and \item for any scheme $U$ and any \'etale morphism $U \to X$ the restriction $\mathcal{F}|_U$ is locally projective on $U$. \end{enumerate}


spaces-properties-lemma-locally-projective-pullback

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. If $\mathcal{G}$ is locally projective on $Y$, then $f^*\mathcal{G}$ is locally projective on $X$.


spaces-properties-lemma-morphism-to-affine-scheme

Let $X$ be an algebraic space over $\mathbf{Z}$. Let $T$ be an affine scheme. The map $$ \Mor(X, T) \longrightarrow \Hom(\Gamma(T, \mathcal{O}_T), \Gamma(X, \mathcal{O}_X)) $$ which maps $f$ to $f^\sharp$ (on global sections) is bijective.


spaces-properties-lemma-quotient

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $G$ be an abstract group with a free action on $X$. Then the quotient sheaf $X/G$ is an algebraic space.


spaces-properties-proposition-locally-quasi-separated-open-dense-scheme

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Zariski locally quasi-separated (for example if $X$ is quasi-separated), then there exists a dense open subspace $X'$ of $X$ which is a scheme. More precisely, every point $x \in |X|$ of codimension $0$ on $X$ is contained in $X'$.


spaces-properties-proposition-finite-flat-equivalence-global

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume \begin{enumerate} \item $s, t : R \to U$ finite locally free, \item $j = (t, s)$ is an equivalence, and \item for a dense set of points $u \in U$ the $R$-equivalence class $t(s^{-1}(\{u\}))$ is contained in an affine open of $U$. \end{enumerate} Then there exists a finite locally free morphism $U \to M$ of schemes over $S$ such that $R = U \times_M U$ and such that $M$ represents the quotient sheaf $U/R$ in the fppf topology.


spaces-properties-proposition-sheaf-fpqc

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ satisfies the sheaf property for the fpqc topology.


spaces-properties-proposition-quasi-coherent

With $S$, $\varphi : U \to X$, and $(U, R, s, t, c)$ as above. For any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the sheaf $\varphi^*\mathcal{F}$ comes equipped with a canonical isomorphism $$ \alpha : t^*\varphi^*\mathcal{F} \longrightarrow s^*\varphi^*\mathcal{F} $$ which satisfies the conditions of Groupoids, Definition \ref{groupoids-definition-groupoid-module} and therefore defines a quasi-coherent sheaf on $(U, R, s, t, c)$. The functor $\mathcal{F} \mapsto (\varphi^*\mathcal{F}, \alpha)$ defines an equivalence of categories $$ \begin{matrix} \text{Quasi-coherent} \\ \mathcal{O}_X\text{-modules} \end{matrix} \longleftrightarrow \begin{matrix} \text{Quasi-coherent modules}\\ \text{on }(U, R, s, t, c) \end{matrix} $$


spaces-properties-proposition-coherator

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item The category $\QCoh(\mathcal{O}_X)$ is a Grothendieck abelian category. Consequently, $\QCoh(\mathcal{O}_X)$ has enough injectives and all limits. \item The inclusion functor $\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$ has a right adjoint\footnote{This functor is sometimes called the {\it coherator}.} $$ Q : \textit{Mod}(\mathcal{O}_X) \longrightarrow \QCoh(\mathcal{O}_X) $$ such that for every quasi-coherent sheaf $\mathcal{F}$ the adjunction mapping $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism. \end{enumerate}


intersection-theorem-multiplicity-with-koszul

\begin{reference} \cite[Theorem 1 in part B of Chapter IV]{Serre_algebre_locale} \end{reference} Let $A$ be a Noetherian local ring. Let $I = (f_1, \ldots, f_r) \subset A$ be an ideal of definition. Let $M$ be a finite $A$-module. Then $$ e_I(M, r) = \sum (-1)^i\text{length}_A H_i(K_\bullet(f_1, \ldots, f_r) \otimes_A M) $$


intersection-theorem-well-defined

Let $X$ be a nonsingular projective variety. Let $\alpha$, resp.\ $\beta$ be an $r$, resp.\ $s$ cycle on $X$. Assume that $\alpha$ and $\beta$ intersect properly so that $\alpha \cdot \beta$ is defined. Finally, assume that $\alpha \sim_{rat} 0$. Then $\alpha \cdot \beta \sim_{rat} 0$.


intersection-lemma-push-coherent

\begin{reference} See \cite[Chapter V]{Serre_algebre_locale}. \end{reference} Suppose that $f : X \to Y$ is a proper morphism of varieties. Let $\mathcal{F}$ be a coherent sheaf with $\dim(\text{Supp}(\mathcal{F})) \leq k$, then $f_*[\mathcal{F}]_k = [f_*\mathcal{F}]_k$. In particular, if $Z \subset X$ is a closed subscheme of dimension $\leq k$, then $f_*[Z]_k = [f_*\mathcal{O}_Z]_k$.


intersection-lemma-compose-pushforward

Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms of varieties. Then $g_* \circ f_* = (g \circ f)_*$ as maps $Z_k(X) \to Z_k(Z)$.


intersection-lemma-pullback

Let $f : X \to Y$ be a flat morphism of varieties. Set $r = \dim(X) - \dim(Y)$. Then $f^*[\mathcal{F}]_k = [f^*\mathcal{F}]_{k + r}$ if $\mathcal{F}$ is a coherent sheaf on $Y$ and the dimension of the support of $\mathcal{F}$ is at most $k$.


intersection-lemma-dimension-product-varieties

Let $X$ and $Y$ be varieties. Then $X \times Y$ is a variety and $\dim(X \times Y) = \dim(X) + \dim(Y)$.


intersection-lemma-pullback-by-regular-immersion

Let $f : X \to Y$ be a morphism of varieties. \begin{enumerate} \item If $Z \subset Y$ is a subvariety dimension $d$ and $f$ is a regular immersion of codimension $c$, then every irreducible component of $f^{-1}(Z)$ has dimension $\geq d - c$. \item If $Z \subset Y$ is a subvariety of dimension $d$ and $f$ is a local complete intersection morphism of relative dimension $r$, then every irreducible component of $f^{-1}(Z)$ has dimension $\geq d + r$. \end{enumerate}


intersection-lemma-diagonal-regular-immersion

Let $X$ be a nonsingular variety. Then the diagonal $\Delta : X \to X \times X$ is a regular immersion of codimension $\dim(X)$.


intersection-lemma-intersect-in-smooth

Let $X$ be a nonsingular variety and let $W,V \subset X$ be closed subvarieties with $\dim(W) = s$ and $\dim(V) = r$. Then every irreducible component $Z$ of $V \cap W$ has dimension $\geq r + s - \dim(X)$.


intersection-lemma-tensor-coherent

Let $X$ be a locally Noetherian scheme. \begin{enumerate} \item If $\mathcal{F}$ and $\mathcal{G}$ are coherent $\mathcal{O}_X$-modules, then $\text{Tor}_p^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is too. \item If $L$ and $K$ are in $D^-_{\textit{Coh}}(\mathcal{O}_X)$, then so is $L \otimes_{\mathcal{O}_X}^\mathbf{L} K$. \end{enumerate}


intersection-lemma-transversal

Let $X$ be a nonsingular variety. Let $V, W \subset X$ be closed subvarieties which intersect properly. Let $Z$ be an irreducible component of $V \cap W$ and assume that the multiplicity (in the sense of Section \ref{section-cycle-of-closed}) of $Z$ in the closed subscheme $V \cap W$ is $1$. Then $e(X, V \cdot W, Z) = 1$ and $V$ and $W$ are smooth in a general point of $Z$.


intersection-lemma-multiplicity-ses

Let $A$ be a Noetherian local ring. Let $I \subset A$ be an ideal of definition. Let $0 \to M' \to M \to M'' \to 0$ be a short exact sequence of finite $A$-modules. Let $d \geq \dim(\text{Supp}(M))$. Then $$ e_I(M, d) = e_I(M', d) + e_I(M'', d) $$


intersection-lemma-leading-coefficient

Let $P$ be a polynomial of degree $r$ with leading coefficient $a$. Then $$ r! a = \sum\nolimits_{i = 0, \ldots, r} (-1)^i{r \choose i} P(t - i) $$ for any $t$.


intersection-lemma-intersection-multiplicity-CM

Let $X$ be a nonsingular variety and $W, V \subset X$ closed subvarieties which intersect properly. Let $Z$ be an irreducible component of $V \cap W$ with generic point $\xi$. Assume that $\mathcal{O}_{W, \xi}$ and $\mathcal{O}_{V, \xi}$ are Cohen-Macaulay. Then $$ e(X, V \cdot W, Z) = \text{length}_{\mathcal{O}_{X, \xi}}(\mathcal{O}_{V \cap W, \xi}) $$ where $V \cap W$ is the scheme theoretic intersection. In particular, if both $V$ and $W$ are Cohen-Macaulay, then $V \cdot W = [V \cap W]_{\dim(V) + \dim(W) - \dim(X)}$.


intersection-lemma-one-ideal-ci

Let $A$ be a Noetherian local ring. Let $I = (f_1, \ldots, f_r)$ be an ideal generated by a regular sequence. Let $M$ be a finite $A$-module. Assume that $\dim(\text{Supp}(M/IM)) = 0$. Then $$ e_I(M, r) = \sum (-1)^i\text{length}_A(\text{Tor}_i^A(A/I, M)) $$ Here $e_I(M, r)$ is as in Remark \ref{remark-trivial-generalization}.


intersection-lemma-multiplicity-with-lci

Let $X$ be a nonsingular variety. Let $W,V \subset X$ be closed subvarieties which intersect properly. Let $Z$ be an irreducible component of $V \cap W$ with generic point $\xi$. Suppose the ideal of $V$ in $\mathcal{O}_{X, \xi}$ is cut out by a regular sequence $f_1, \ldots, f_c \in \mathcal{O}_{X, \xi}$. Then $e(X, V\cdot W, Z)$ is equal to $c!$ times the leading coefficient in the Hilbert polynomial $$ t \mapsto \text{length}_{\mathcal{O}_{X, \xi}} \mathcal{O}_{W, \xi}/(f_1, \ldots, f_c)^t,\quad t \gg 0. $$ In particular, this coefficient is $> 0$.


intersection-lemma-multiplicity-with-effective-Cartier-divisor

In Lemma \ref{lemma-multiplicity-with-lci} assume that  $c = 1$, i.e., $V$ is an effective Cartier divisor. Then $$ e(X, V \cdot W, Z) = \text{length}_{\mathcal{O}_{X, \xi}} (\mathcal{O}_{W, \xi}/f_1\mathcal{O}_{W, \xi}). $$


intersection-lemma-multiplicity-lci-CM

In Lemma \ref{lemma-multiplicity-with-lci} assume that the local ring $\mathcal{O}_{W, \xi}$ is Cohen-Macaulay. Then we have $$ e(X, V \cdot W, Z) = \text{length}_{\mathcal{O}_{X, \xi}} (\mathcal{O}_{W, \xi}/ f_1\mathcal{O}_{W, \xi} + \ldots + f_c\mathcal{O}_{W, \xi}). $$


intersection-lemma-rational-equivalence-and-intersection

Let $X$ be a nonsingular variety. Let $a, b \in \mathbf{P}^1$ be distinct closed points. Let $k \geq 0$. \begin{enumerate} \item If $W \subset X \times \mathbf{P}^1$ is a closed subvariety of dimension $k + 1$ which intersects $X \times a$ properly, then \begin{enumerate} \item $[W_a]_k = W \cdot X \times a$ as cycles on $X \times \mathbf{P}^1$, and \item $[W_a]_k = \text{pr}_{X, *}(W \cdot X \times a)$ as cycles on $X$. \end{enumerate} \item Let $\alpha$ be a $(k + 1)$-cycle on $X \times \mathbf{P}^1$ which intersects $X \times a$ and $X \times b$ properly. Then $pr_{X,*}( \alpha \cdot X \times a - \alpha \cdot X \times b)$ is rationally equivalent to zero. \item Conversely, any $k$-cycle which is rationally equivalent to $0$ is of this form. \end{enumerate}


intersection-lemma-transversal-subschemes

Let $X$ be a nonsingular variety. Let $r, s \geq 0$ and let $Y, Z \subset X$ be closed subschemes with $\dim(Y) \leq r$ and $\dim(Z) \leq s$. Assume $[Y]_r = \sum n_i[Y_i]$ and $[Z]_s = \sum m_j[Z_j]$ intersect properly. Let $T$ be an irreducible component of $Y_{i_0} \cap Z_{j_0}$ for some $i_0$ and $j_0$ and assume that the multiplicity (in the sense of Section \ref{section-cycle-of-closed}) of $T$ in the closed subscheme $Y \cap Z$ is $1$. Then \begin{enumerate} \item the coefficient of $T$ in $[Y]_r \cdot [Z]_s$ is $1$, \item $Y$ and $Z$ are nonsingular at the generic point of $Z$, \item $n_{i_0} = 1$, $m_{j_0} = 1$, and \item $T$ is not contained in $Y_i$ or $Z_j$ for $i \not = i_0$ and $j \not = j_0$. \end{enumerate}


intersection-lemma-tor-and-diagonal

Let $X$ be a nonsingular variety. \begin{enumerate} \item If $\mathcal{F}$ and $\mathcal{G}$ are coherent $\mathcal{O}_X$-modules, then there are canonical isomorphisms $$ \text{Tor}_i^{\mathcal{O}_{X \times X}}(\mathcal{O}_\Delta, \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times X}} \text{pr}_2^*\mathcal{G}) = \Delta_*\text{Tor}_i^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) $$ \item If $K$ and $M$ are in $D_\QCoh(\mathcal{O}_X)$, then there is a canonical isomorphism $$ L\Delta^* \left( L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} L\text{pr}_2^*M \right) = K \otimes_{\mathcal{O}_X}^\mathbf{L} M $$ in $D_\QCoh(\mathcal{O}_X)$ and a canonical isomorphism $$ \mathcal{O}_\Delta \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} L\text{pr}_2^*M = \Delta_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} M) $$ in $D_\QCoh(X \times X)$. \end{enumerate}


intersection-lemma-reduction-diagonal

Let $X$ be a nonsingular variety. Let $\alpha$, resp.\ $\beta$ be an $r$-cycle, resp.\ $s$-cycle on $X$. Assume $\alpha$ and $\beta$ intersect properly. Then \begin{enumerate} \item $\alpha \times \beta$ and $[\Delta]$ intersect properly \item we have $\Delta_*(\alpha \cdot \beta) = [\Delta] \cdot \alpha\times\beta$ as cycles on $X \times X$, \item if $X$ is proper, then $\text{pr}_{1, *}([\Delta] \cdot \alpha\times\beta) = \alpha\cdot\beta$, where $pr_1 : X\times X \to X$ is the projection. \end{enumerate}


intersection-lemma-tor-sheaf

\begin{reference} \cite[Chapter V]{Serre_algebre_locale} \end{reference} Let $X$ be a nonsingular variety. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $X$ with $\dim(\text{Supp}(\mathcal{F})) \leq r$, $\dim(\text{Supp}(\mathcal{G})) \leq s$, and $\dim(\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G}) ) \leq r + s - \dim X$. In this case $[\mathcal{F}]_r$ and $[\mathcal{G}]_s$ intersect properly and $$ [\mathcal{F}]_r \cdot [\mathcal{G}]_s = \sum (-1)^p [\text{Tor}_p^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})]_{r + s - \dim(X)}. $$


intersection-lemma-associative

Let $X$ be a nonsingular variety. Let $U, V, W$ be closed subvarieties. Assume that $U, V, W$ intersect properly pairwise and that $\dim(U \cap V \cap W) \leq \dim(U) + \dim(V) + \dim(W) - 2\dim(X)$. Then $$ U \cdot (V \cdot W) = (U \cdot V) \cdot W $$ as cycles on $X$.


intersection-lemma-flat-pull-back-and-intersections-sheaves

Let $f : X \to Y$ be a flat morphism of nonsingular varieties. Set $e = \dim(X) - \dim(Y)$. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $Y$ with $\dim(\text{Supp}(\mathcal{F})) \leq r$, $\dim(\text{Supp}(\mathcal{G})) \leq s$, and $\dim(\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G}) ) \leq r + s - \dim(Y)$. In this case the cycles $[f^*\mathcal{F}]_{r + e}$ and $[f^*\mathcal{G}]_{s + e}$ intersect properly and $$ f^*([\mathcal{F}]_r \cdot [\mathcal{G}]_s) = [f^*\mathcal{F}]_{r + e} \cdot [f^*\mathcal{G}]_{s + e} $$


intersection-lemma-projection-formula-flat

\begin{reference} See \cite[Chapter V, C), Section 7, formula (10)]{Serre_algebre_locale} for a more general formula. \end{reference} Let $f : X \to Y$ be a flat proper morphism of nonsingular varieties. Set $e = \dim(X) - \dim(Y)$. Let $\alpha$ be an $r$-cycle on $X$ and let $\beta$ be a $s$-cycle on $Y$. Assume that $\alpha$ and $f^*(\beta)$ intersect properly. Then $f_*(\alpha)$ and $\beta$ intersect properly and $$ f_*(\alpha) \cdot \beta = f_*( \alpha \cdot f^*\beta) $$


intersection-lemma-transfer

Let $X \to P$ be a closed immersion of nonsingular varieties. Let $C' \subset P \times \mathbf{P}^1$ be a closed subvariety of dimension $r + 1$. Assume \begin{enumerate} \item the fibre $C = C'_0$ has dimension $r$, i.e., $C' \to \mathbf{P}^1$ is dominant, \item $C'$ intersects $X \times \mathbf{P}^1$ properly, \item $[C]_r$ intersects $X$ properly. \end{enumerate} Then setting $\alpha = [C]_r \cdot X$ viewed as cycle on $X$ and $\beta = C' \cdot X \times \mathbf{P}^1$ viewed as cycle on $X \times \mathbf{P}^1$, we have $$ \alpha = \text{pr}_{X, *}(\beta \cdot X \times 0) $$ as cycles on $X$ where $\text{pr}_X : X \times \mathbf{P}^1 \to X$ is the projection.


intersection-lemma-projection-generically-finite

Let $V$ be a vector space of dimension $n + 1$. Let $X \subset \mathbf{P}(V)$ be a closed subscheme. If $X \not = \mathbf{P}(V)$, then there is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ the restriction of the projection $r_p$ defines a finite morphism $r_p|_X : X \to \mathbf{P}(W_p)$.


intersection-lemma-projection-generically-immersion

Let $V$ be a vector space of dimension $n + 1$. Let $X \subset \mathbf{P}(V)$ be a closed subvariety. Let $x \in X$ be a nonsingular point. \begin{enumerate} \item If $\dim(X) < n - 1$, then there is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ the morphism $r_p|_X : X \to r_p(X)$ is an isomorphism over an open neighbourhood of $r_p(x)$. \item If $\dim(X) = n - 1$, then there is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ the morphism $r_p|_X : X \to \mathbf{P}(W_p)$ is \'etale at $x$. \end{enumerate}


intersection-lemma-projection-injective

Let $V$ be a vector space of dimension $n + 1$. Let $Y, Z \subset \mathbf{P}(V)$ be closed subvarieties. There is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ we have $$ Y \cap r_p^{-1}(r_p(Z)) = (Y \cap Z) \cup E $$ with $E \subset Y$ closed and $\dim(E) \leq \dim(Y) + \dim(Z) + 1 - n$.


intersection-lemma-find-lines

Let $V$ be a vector space. Let $B \subset \mathbf{P}(V)$ be a closed subvariety of codimension $\geq 2$. Let $p \in \mathbf{P}(V)$ be a closed point, $p \not \in B$. Then there exists a line $\ell \subset \mathbf{P}(V)$ with $\ell \cap B = \emptyset$. Moreover, these lines sweep out an open subset of $\mathbf{P}(V)$.


intersection-lemma-doubly-transitive

Let $V$ be a vector space. Let $G = \text{PGL}(V)$. Then $G \times \mathbf{P}(V) \to \mathbf{P}(V)$ is doubly transitive.


intersection-lemma-determinant

Let $k$ be a field. Let $n \geq 1$ be an integer and let $x_{ij}, 1 \leq i, j \leq n$ be variables. Then $$ \det \left( \begin{matrix} x_{11} & x_{12} & \ldots & x_{1n} \\ x_{21} & \ldots & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots \\ x_{n1} & \ldots & \ldots & x_{nn} \end{matrix} \right) $$ is an irreducible element of the polynomial ring $k[x_{ij}]$.


intersection-lemma-make-family

With notation as above. Let $X, Y$ be closed subvarieties of $\mathbf{P}(V)$ which intersect properly such that $X \not = \mathbf{P}(V)$ and $X \cap Y \not = \emptyset$. For a general line $\ell \subset \mathbf{P}$ with $[\text{id}_V] \in \ell$ we have \begin{enumerate} \item $X \subset U_g$ for all $[g] \in \ell$, \item $g(X)$ intersects $Y$ properly for all $[g] \in \ell$. \end{enumerate}


intersection-lemma-moving

\begin{reference} See \cite{Roberts}. \end{reference} Let $X \subset \mathbf{P}^N$ be a nonsingular closed subvariety. Let $n = \dim(X)$ and $0 \leq d, d' < n$. Let $Z \subset X$ be a closed subvariety of dimension $d$ and $T_i \subset X$, $i \in I$ be a finite collection of closed subvarieties of dimension $d'$. Then there exists a subvariety $C \subset \mathbf{P}^N$ such that $C$ intersects $X$ properly and such that $$ C \cdot X = Z + \sum\nolimits_{j \in J} m_j Z_j $$ where $Z_j \subset X$ are irreducible of dimension $d$, distinct from $Z$, and $$ \dim(Z_j \cap T_i) \leq \dim(Z \cap T_i) $$ with strict inequality if $Z$ does not intersect $T_i$ properly in $X$.


intersection-lemma-move

Let $C \subset \mathbf{P}^N$ be a closed subvariety. Let $X \subset \mathbf{P}^N$ be subvariety and let $T_i \subset X$ be a finite collection of closed subvarieties. Assume that $C$ and $X$ intersect properly. Then there exists a closed subvariety $C' \subset \mathbf{P}^N \times \mathbf{P}^1$ such that \begin{enumerate} \item $C' \to \mathbf{P}^1$ is dominant, \item $C'_0 = C$ scheme theoretically, \item $C'$ and $X \times \mathbf{P}^1$ intersect properly, \item $C'_\infty$ properly intersects each of the given $T_i$. \end{enumerate}


intersection-lemma-moving-move

Let $X$ be a nonsingular projective variety. Let $\alpha$ be an $r$-cycle and $\beta$ be an $s$-cycle on $X$. Then there exists an $r$-cycle $\alpha'$ such that $\alpha' \sim_{rat} \alpha$ and such that $\alpha'$ and $\beta$ intersect properly.


intersection-lemma-well-defined-special-case

Let $X$ be a nonsingular variety. Let $W \subset X \times \mathbf{P}^1$ be an $(s + 1)$-dimensional subvariety dominating $\mathbf{P}^1$. Let $W_a$, resp.\ $W_b$ be the fibre of $W \to \mathbf{P}^1$ over $a$, resp.\ $b$. Let $V$ be a $r$-dimensional subvariety of $X$ such that $V$ intersects both $W_a$ and $W_b$ properly. Then $[V] \cdot [W_a]_r \sim_{rat} [V] \cdot [W_b]_r$.


intersection-proposition-positivity

\begin{reference} This is one of the main results of \cite{Serre_algebre_locale}. \end{reference} Let $X$ be a nonsingular variety. Let $V \subset X$ and $W \subset Y$ be closed subvarieties which intersect properly. Let $Z \subset V \cap W$ be an irreducible component. Then $e(X, V \cdot W, Z) > 0$.


homology-lemma-preadditive-zero

Let $\mathcal{A}$ be a preadditive category. Let $x$ be an object of $\mathcal{A}$. The following are equivalent \begin{enumerate} \item $x$ is an initial object, \item $x$ is a final object, and \item $\text{id}_x = 0$ in $\Mor_\mathcal{A}(x, x)$. \end{enumerate} Furthermore, if such an object $0$ exists, then a morphism $\alpha : x \to y$ factors through $0$ if and only if $\alpha = 0$.


homology-lemma-preadditive-direct-sum

Let $\mathcal{A}$ be a preadditive category. Let $x, y \in \Ob(\mathcal{A})$. If the product $x \times y$ exists, then so does the coproduct $x \amalg y$. If the coproduct $x \amalg y$ exists, then so does the product $x \times y$. In this case also $x \amalg y \cong x \times y$.


homology-lemma-additive-additive

Let $\mathcal{A}$, $\mathcal{B}$ be preadditive categories. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor. Then $F$ transforms direct sums to direct sums and zero to zero.


homology-lemma-additive-cat-biproduct-kernel

Let $\mathcal{C}$ be a preadditive category. Let $x \oplus y$ with morphisms $i, j, p, q$ as in Lemma \ref{lemma-preadditive-direct-sum} be a direct sum in $\mathcal{C}$. Then $i : x \to x \oplus y$ is a kernel of $q : x \oplus y \rightarrow y$. Dually, $p$ is a cokernel for $j$.


homology-lemma-kernel-mono

Let $\mathcal{C}$ be a preadditive category. Let $f : x \to y$ be a morphism in $\mathcal{C}$. \begin{enumerate} \item If a kernel of $f$ exists, then this kernel is a monomorphism. \item If a cokernel of $f$ exists, then this cokernel is an epimorphism. \item If a kernel and coimage of $f$ exist, then the coimage is an epimorphism. \item If a cokernel and image of $f$ exist, then the image is a monomorphism. \end{enumerate}


homology-lemma-coim-im-map

Let $f : x \to y$ be a morphism in a preadditive category such that the kernel, cokernel, image and coimage all exist. Then $f$ can be factored uniquely as $x \to \Coim(f) \to \Im(f) \to y$.


homology-lemma-karoubian

Let $\mathcal{C}$ be a preadditive category. The following are equivalent \begin{enumerate} \item $\mathcal{C}$ is Karoubian, \item every idempotent endomorphism of an object of $\mathcal{C}$ has a cokernel, and \item given an idempotent endomorphism $p : z \to z$ of $\mathcal{C}$ there exists a direct sum decomposition $z = x \oplus y$ such that $p$ corresponds to the projection onto $y$. \end{enumerate}


homology-lemma-projectors-have-images

Let $\mathcal{D}$ be a preadditive category. \begin{enumerate} \item If $\mathcal{D}$ has countable products and kernels of maps which have a right inverse, then $\mathcal{D}$ is Karoubian. \item If $\mathcal{D}$ has countable coproducts and cokernels of maps which have a left inverse, then $\mathcal{D}$ is Karoubian. \end{enumerate}


homology-lemma-abelian-opposite

Let $\mathcal{A}$ be a preadditive category. The additions on sets of morphisms make $\mathcal{A}^{opp}$ into a preadditive category. Furthermore, $\mathcal{A}$ is additive if and only if $\mathcal{A}^{opp}$ is additive, and $\mathcal{A}$ is abelian if and only if $\mathcal{A}^{opp}$ is abelian.


homology-lemma-characterize-injective

Let $f : x \to y$ be a morphism in an abelian category $\mathcal{A}$. Then \begin{enumerate} \item $f$ is injective if and only if $f$ is a monomorphism, and \item $f$ is surjective if and only if $f$ is an epimorphism. \end{enumerate}


homology-lemma-colimit-abelian-category

Let $\mathcal{A}$ be an abelian category. All finite limits and finite colimits exist in $\mathcal{A}$.


homology-lemma-check-exactness

Let $\mathcal{A}$ be an abelian category. Let $0 \to M_1 \to M_2 \to M_3 \to 0$ be a complex of $\mathcal{A}$. \begin{enumerate} \item $M_1 \to M_2 \to M_3 \to 0$ is exact if and only if $$ 0 \to \Hom_\mathcal{A}(M_3, N) \to \Hom_\mathcal{A}(M_2, N) \to \Hom_\mathcal{A}(M_1, N) $$ is an exact sequence of abelian groups for all objects $N$ of $\mathcal{A}$, and \item $0 \to M_1 \to M_2 \to M_3$ is exact if and only if $$ 0 \to \Hom_\mathcal{A}(N, M_1) \to \Hom_\mathcal{A}(N, M_2) \to \Hom_\mathcal{A}(N, M_1) $$ is an exact sequence of abelian groups for all objects $N$ of $\mathcal{A}$. \end{enumerate}


homology-lemma-ses-split

Let $\mathcal{A}$ be an abelian category. Let $0 \to A \to B \to C \to 0$ be a short exact sequence. \begin{enumerate} \item Given a morphism $s : C \to B$ left inverse to $B \to C$, there exists a unique $\pi : B \to A$ such that $(s, \pi)$ splits the short exact sequence as in Definition \ref{definition-ses-split}. \item Given a morphism $\pi : B \to A$ right inverse to $A \to B$, there exists a unique $s : C \to B$ such that $(s, \pi)$ splits the short exact sequence as in Definition \ref{definition-ses-split}. \end{enumerate}


homology-lemma-characterize-cartesian

Let $\mathcal{A}$ be an abelian category. Let $$ \xymatrix{ w\ar[r]^f\ar[d]_g & y\ar[d]^h\\ x\ar[r]^k & z } $$ be a commutative diagram.  \begin{enumerate} \item The diagram is cartesian if and only if  $$ 0 \to w \xrightarrow{(g, f)} x \oplus y \xrightarrow{(k, -h)} z $$ is exact. \item The diagram is cocartesian if and only if  $$ w \xrightarrow{(g, -f)} x \oplus y \xrightarrow{(k, h)} z \to 0 $$ is exact. \end{enumerate}


homology-lemma-cartesian-kernel

Let $\mathcal{A}$ be an abelian category. Let $$ \xymatrix{ w\ar[r]^f\ar[d]_g & y\ar[d]^h\\ x\ar[r]^k & z } $$ be a commutative diagram. \begin{enumerate} \item If the diagram is cartesian, then the morphism  $\Ker(f)\to\Ker(k)$ induced by $g$ is an isomorphism. \item If the diagram is cocartesian, then the morphism  $\Coker(f)\to\Coker(k)$ induced by $h$ is an isomorphism. \end{enumerate}


homology-lemma-cartesian-cocartesian

Let $\mathcal{A}$ be an abelian category. Let $$ \xymatrix{ w\ar[r]^f\ar[d]_g & y\ar[d]^h\\ x\ar[r]^k & z } $$ be a commutative diagram. \begin{enumerate} \item If the diagram is cartesian and $k$ is an epimorphism,  then the diagram is cocartesian and $f$ is an epimorphism. \item If the diagram is cocartesian and $g$ is a monomorphism,  then the diagram is cartesian and $h$ is a monomorphism. \end{enumerate}


homology-lemma-epimorphism-universal-abelian-category

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item If $x \to y$ is surjective, then for every $z \to y$ the projection $x \times_y z \to z$ is surjective. \item If $x \to y$ is injective, then for every $x \to z$ the morphism $z \to z \amalg_x y$ is injective. \end{enumerate}


homology-lemma-check-exactness-fibre-product

Let $\mathcal{A}$ be an abelian category. Let $f:x\to y$ and $g:y\to z$  be morphisms with $g\circ f=0$. Then, the following statements are equivalent: \begin{enumerate} \item The sequence $x\overset{f}\to y\overset{g}\to z$ is exact. \item For every $h:w\to y$ with $g\circ h=0$ there exist an object $v$,  an epimorphism $k:v\to w$ and a morphism $l:v\to x$ with $h\circ k=f\circ l$. \end{enumerate}


homology-lemma-exact-kernel-sequence

Let $\mathcal{A}$ be an abelian category. Let $$ \xymatrix{ x \ar[r]^f \ar[d]^\alpha & y \ar[r]^g \ar[d]^\beta & z \ar[d]^\gamma\\ u \ar[r]^k & v \ar[r]^l & w } $$ be a commutative diagram. \begin{enumerate} \item If the first row is exact and $k$ is a monomorphism, then the induced  sequence $\Ker(\alpha) \to \Ker(\beta) \to \Ker(\gamma)$  is exact. \item If the second row is exact and $g$ is an epimorphism, then the induced  sequence $\Coker(\alpha) \to \Coker(\beta) \to \Coker(\gamma)$  is exact. \end{enumerate}


homology-lemma-snake

Let $\mathcal{A}$ be an abelian category. Let  $$ \xymatrix{ & x \ar[r]^f \ar[d]^\alpha & y \ar[r]^g \ar[d]^\beta & z \ar[r] \ar[d]^\gamma & 0 \\ 0 \ar[r] & u \ar[r]^k & v \ar[r]^l & w } $$ be a commutative diagram with exact rows. \begin{enumerate} \item There exists a unique morphism  $\delta : \Ker(\gamma) \rightarrow \Coker(\alpha)$  such that the diagram $$ \xymatrix{ y \ar[d]_\beta & y \times_z \Ker(\gamma) \ar[l]_{\pi'} \ar[r]^{\pi} & \Ker(\gamma) \ar[d]^\delta \\ v \ar[r]^{\iota'} & \Coker(\alpha) \amalg_u v & \Coker(\alpha) \ar[l]_\iota } $$ commutes, where $\pi$ and $\pi'$ are the canonical projections  and $\iota$ and $\iota'$ are the canonical coprojections. \item The induced sequence  $$ \Ker(\alpha)\overset{f'} \to \Ker(\beta)  \overset{g'}\to \Ker(\gamma)\overset{\delta}\to \Coker(\alpha) \overset{k'}\to \Coker(\beta)  \overset{l'}\to \Coker(\gamma) $$ is exact. If $f$ is injective then so is $f'$, and if $l$ is  surjective then so is $l'$. \end{enumerate}


homology-lemma-snake-natural

Let $\mathcal{A}$ be an abelian category. Let  $$ \xymatrix{ & & & x\ar[ld]\ar[rr]\ar[dd]^(.4)\alpha & & y\ar[ld]\ar[rr]\ar[dd]^(.4)\beta & & z\ar[ld]\ar[rr]\ar[dd]^(.4)\gamma & & 0\\ & & x'\ar[rr]\ar[dd]^(.4){\alpha'} & & y'\ar[rr]\ar[dd]^(.4){\beta'} & & z'\ar[rr]\ar[dd]^(.4){\gamma'} & & 0 & \\ & 0\ar[rr] & & u\ar[ld]\ar[rr] & & v\ar[ld]\ar[rr] & & w\ar[ld] & & \\ 0\ar[rr] & & u'\ar[rr] & & v'\ar[rr] & & w' & & & } $$ be a commutative diagram with exact rows. Then, the induced diagram $$ \xymatrix@C=15pt{ \Ker(\alpha) \ar[r] \ar[d] & \Ker(\beta) \ar[r] \ar[d] & \Ker(\gamma) \ar[r]^(.45){\delta} \ar[d] & \Coker(\alpha) \ar[r] \ar[d] & \Coker(\beta) \ar[r] \ar[d] & \Coker(\gamma) \ar[d] \\ \Ker(\alpha') \ar[r] & \Ker(\beta') \ar[r] & \Ker(\gamma') \ar[r]^(.45){\delta'} & \Coker(\alpha') \ar[r] & \Coker(\beta') \ar[r] & \Coker(\gamma') } $$ commutes.


homology-lemma-four-lemma

Let $\mathcal{A}$ be an abelian category. Let $$ \xymatrix{ w \ar[r] \ar[d]^\alpha & x \ar[r] \ar[d]^\beta & y \ar[r] \ar[d]^\gamma & z \ar[d]^\delta \\ w' \ar[r] & x' \ar[r] & y' \ar[r] & z' } $$ be a commutative diagram with exact rows. \begin{enumerate} \item If $\alpha, \gamma$ are surjective and $\delta$ is injective, then $\beta$ is surjective. \item If $\beta, \delta$ are injective and $\alpha$ is surjective, then $\gamma$ is injective. \end{enumerate}


homology-lemma-five-lemma

\begin{reference} \cite[Lemma 4.5 page 16]{Eilenberg-Steenrod} \end{reference} Let $\mathcal{A}$ be an abelian category. Let $$ \xymatrix{ v \ar[r] \ar[d]^\alpha & w \ar[r] \ar[d]^\beta & x \ar[r] \ar[d]^\gamma & y \ar[r] \ar[d]^\delta & z \ar[d]^\epsilon \\ v' \ar[r] & w' \ar[r] & x' \ar[r] & y' \ar[r] & z' } $$ be a commutative diagram with exact rows. If $\beta, \delta$ are isomorphisms, $\epsilon$ is injective, and $\alpha$ is surjective then $\gamma$ is an isomorphism.


homology-lemma-baer-sum

The construction $(E_1, E_2) \mapsto E_1 + E_2$ above defines a commutative group law on $\Ext_\mathcal{A}(B, A)$ which is functorial in both variables.


homology-lemma-six-term-sequence-ext

Let $\mathcal{A}$ be an abelian category. Let $0 \to M_1 \to M_2 \to M_3 \to 0$ be a short exact sequence in $\mathcal{A}$. \begin{enumerate} \item There is a canonical six term exact sequence of abelian groups $$ \xymatrix{ 0 \ar[r] & \Hom_\mathcal{A}(M_3, N) \ar[r] & \Hom_\mathcal{A}(M_2, N) \ar[r] & \Hom_\mathcal{A}(M_1, N) \ar[lld] \\ & \Ext_\mathcal{A}(M_3, N) \ar[r] & \Ext_\mathcal{A}(M_2, N) \ar[r] & \Ext_\mathcal{A}(M_1, N) } $$ for all objects $N$ of $\mathcal{A}$, and \item there is a canonical six term exact sequence of abelian groups $$ \xymatrix{ 0 \ar[r] & \Hom_\mathcal{A}(N, M_1) \ar[r] & \Hom_\mathcal{A}(N, M_2) \ar[r] & \Hom_\mathcal{A}(N, M_3) \ar[lld] \\ & \Ext_\mathcal{A}(N, M_1) \ar[r] & \Ext_\mathcal{A}(N, M_2) \ar[r] & \Ext_\mathcal{A}(N, M_3) } $$ for all objects $N$ of $\mathcal{A}$. \end{enumerate}


homology-lemma-additive-functor

Let $\mathcal{A}$ and $\mathcal{B}$ be additive categories. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. The following are equivalent \begin{enumerate} \item $F$ is additive, \item $F(A) \oplus F(B) \to F(A \oplus B)$ is an isomorphism for all $A, B \in \mathcal{A}$, and \item $F(A \oplus B) \to F(A) \oplus F(B)$  is an isomorphism for all $A, B \in \mathcal{A}$. \end{enumerate}


homology-lemma-exact-functor

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. \begin{enumerate} \item If $F$ is either left or right exact, then it is additive. \item $F$ is left exact if and only if for every short exact sequence $0 \to A \to B \to C \to 0$ the sequence $0 \to F(A) \to F(B) \to F(C)$ is exact. \item $F$ is right exact if and only if for every short exact sequence $0 \to A \to B \to C \to 0$ the sequence $F(A) \to F(B) \to F(C) \to 0$ is exact. \item $F$ is exact if and only if for every short exact sequence $0 \to A \to B \to C \to 0$ the sequence $0 \to F(A) \to F(B) \to F(C) \to 0$ is exact. \end{enumerate}


homology-lemma-exact-functor-ext

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ be an exact functor. For every pair of objects $A, B$ of $\mathcal{A}$ the functor $F$ induces an abelian group homomorphism $$ \Ext_\mathcal{A}(B, A) \longrightarrow \Ext_\mathcal{B}(F(B), F(A)) $$ which maps the extension $E$ to $F(E)$.


homology-lemma-adjoint-get-abelian

Let $a : \mathcal{A} \to \mathcal{B}$ and $b : \mathcal{B} \to \mathcal{A}$ be functors. Assume that \begin{enumerate} \item $\mathcal{A}$, $\mathcal{B}$ are additive categories, $a$, $b$ are additive functors, and $a$ is right adjoint to $b$, \item $\mathcal{B}$ is abelian and $b$ is left exact, and \item $ba \cong \text{id}_\mathcal{A}$. \end{enumerate} Then $\mathcal{A}$ is abelian.


homology-lemma-localization-preadditive

Let $\mathcal{C}$ be a preadditive category. Let $S$ be a left or right multiplicative system. There exists a canonical preadditive structure on $S^{-1}\mathcal{C}$ such that the localization functor $Q : \mathcal{C} \to S^{-1}\mathcal{C}$ is additive.


homology-lemma-localization-additive

Let $\mathcal{C}$ be an additive category. Let $S$ be a left or right multiplicative system. Then $S^{-1}\mathcal{C}$ is an additive category and the localization functor $Q : \mathcal{C} \to S^{-1}\mathcal{C}$ is additive.


homology-lemma-kernel-localization

Let $\mathcal{C}$ be an additive category. Let $S$ be a multiplicative system. Let $X$ be an object of $\mathcal{C}$. The following are equivalent \begin{enumerate} \item $Q(X) = 0$ in $S^{-1}\mathcal{C}$, \item there exists $Y \in \Ob(\mathcal{C})$ such that $0 : X \to Y$ is an element of $S$, and \item there exists $Z \in \Ob(\mathcal{C})$ such that $0 : Z \to X$ is an element of $S$. \end{enumerate}


homology-lemma-localization-abelian

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item If $S$ is a left multiplicative system, then the category $S^{-1}\mathcal{A}$ has cokernels and the functor $Q : \mathcal{A} \to S^{-1}\mathcal{A}$ commutes with them. \item If $S$ is a right multiplicative system, then the category $S^{-1}\mathcal{A}$ has kernels and the functor $Q : \mathcal{A} \to S^{-1}\mathcal{A}$ commutes with them. \item If $S$ is a multiplicative system, then the category $S^{-1}\mathcal{A}$ is abelian and the functor $Q : \mathcal{A} \to S^{-1}\mathcal{A}$ is exact. \end{enumerate}


homology-lemma-ses-artinian

Let $\mathcal{A}$ be an abelian category. Let $0 \to A_1 \to A_2 \to A_3 \to 0$ be a short exact sequence of $\mathcal{A}$. Then $A_2$ is Artinian if and only if $A_1$ and $A_3$ are Artinian.


homology-lemma-ses-noetherian

Let $\mathcal{A}$ be an abelian category. Let $0 \to A_1 \to A_2 \to A_3 \to 0$ be a short exact sequence of $\mathcal{A}$. Then $A_2$ is Noetherian if and only if $A_1$ and $A_3$ are Noetherian.


homology-lemma-finite-length

Let $\mathcal{A}$ be an abelian category. Let $A$ be an object of $\mathcal{A}$. The following are equivalent \begin{enumerate} \item $A$ is Artinian and Noetherian, and \item there exists a filtration $0 \subset A_1 \subset A_2 \subset \ldots \subset A_n = A$ by subobjects such that $A_i/A_{i - 1}$ is simple for $i = 1, \ldots, n$. \end{enumerate}


homology-lemma-jordan-holder

Let $\mathcal{A}$ be an abelian category. Let $A$ be an object of $\mathcal{A}$ satisfying the equivalent conditions of Lemma \ref{lemma-finite-length}. Given two filtrations $$ 0 \subset A_1 \subset A_2 \subset \ldots \subset A_n = A \quad\text{and}\quad 0 \subset B_1 \subset B_2 \subset \ldots \subset B_m = A $$ with $S_i = A_i/A_{i - 1}$ and $T_j = B_j/B_{j - 1}$ simple objects we have $n = m$ and there exists a permutation $\sigma$ of $\{1, \ldots, n\}$ such that $S_i \cong T_{\sigma(i)}$ for all $i \in \{1, \ldots, n\}$.


homology-lemma-characterize-serre-subcategory

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C}$ be a subcategory of $\mathcal{A}$. Then $\mathcal{C}$ is a Serre subcategory if and only if the following conditions are satisfied: \begin{enumerate} \item $0 \in \Ob(\mathcal{C})$, \item $\mathcal{C}$ is a strictly full subcategory of $\mathcal{A}$, \item any subobject or quotient of an object of $\mathcal{C}$ is an object of $\mathcal{C}$, \item if $A \in \Ob(\mathcal{A})$ is an extension of objects of $\mathcal{C}$ then also $A \in \Ob(\mathcal{C})$. \end{enumerate} Moreover, a Serre subcategory is an abelian category and the inclusion functor is exact.


homology-lemma-characterize-weak-serre-subcategory

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C}$ be a subcategory of $\mathcal{A}$. Then $\mathcal{C}$ is a weak Serre subcategory if and only if the following conditions are satisfied: \begin{enumerate} \item $0 \in \Ob(\mathcal{C})$, \item $\mathcal{C}$ is a strictly full subcategory of $\mathcal{A}$, \item kernels and cokernels in $\mathcal{A}$ of morphisms between objects of $\mathcal{C}$ are in $\mathcal{C}$, \item if $A \in \Ob(\mathcal{A})$ is an extension of objects of $\mathcal{C}$ then also $A \in \Ob(\mathcal{C})$. \end{enumerate} Moreover, a weak Serre subcategory is an abelian category and the inclusion functor is exact.


homology-lemma-kernel-exact-functor

Let $\mathcal{A}$, $\mathcal{B}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ be an exact functor. Then the full subcategory of objects $C$ of $\mathcal{A}$ such that $F(C) = 0$ forms a Serre subcategory of $\mathcal{A}$.


homology-lemma-serre-subcategory-is-kernel

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C} \subset \mathcal{A}$ be a Serre subcategory. There exists an abelian category $\mathcal{A}/\mathcal{C}$ and an exact functor $$ F : \mathcal{A} \longrightarrow \mathcal{A}/\mathcal{C} $$ which is essentially surjective and whose kernel is $\mathcal{C}$. The category $\mathcal{A}/\mathcal{C}$ and the functor $F$ are characterized by the following universal property: For any exact functor $G : \mathcal{A} \to \mathcal{B}$ such that $\mathcal{C} \subset \Ker(G)$ there exists a factorization $G = H \circ F$ for a unique exact functor $H : \mathcal{A}/\mathcal{C} \to \mathcal{B}$.


homology-lemma-quotient-by-kernel-exact-functor

Let $\mathcal{A}$, $\mathcal{B}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ be an exact functor. Let $\mathcal{C} \subset \mathcal{A}$ be a Serre subcategory contained in the kernel of $F$. Then $\mathcal{C} = \Ker(F)$ if and only if the induced functor $\overline{F} : \mathcal{A}/\mathcal{C} \to \mathcal{B}$ (Lemma \ref{lemma-serre-subcategory-is-kernel}) is faithful.


homology-lemma-exact-functor-K-groups

Let $F : \mathcal{A} \to \mathcal{B}$ be an exact functor between abelian categories. Then $F$ induces a homomorphism of $K$-groups $K_0(F) : K_0(\mathcal{A}) \to K_0(\mathcal{B})$ by simply setting $K_0(F)([A]) = [F(A)]$.


homology-lemma-serre-subcategory-K-groups

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C} \subset \mathcal{A}$ be a Serre subcategory and set $\mathcal{B} = \mathcal{A}/\mathcal{C}$. \begin{enumerate} \item The exact functors $\mathcal{C} \to \mathcal{A}$ and $\mathcal{A} \to \mathcal{B}$ induce an exact sequence $$ K_0(\mathcal{C}) \to K_0(\mathcal{A}) \to K_0(\mathcal{B}) \to 0 $$ of $K$-groups, and \item the kernel of $K_0(\mathcal{C}) \to K_0(\mathcal{A})$ is equal to the collection of elements of the form $$ [H^0(M, \varphi, \psi)] - [H^1(M, \varphi, \psi)] $$ where $(M, \varphi, \psi)$ is a complex as in (\ref{equation-cyclic-complex}) with the property that it becomes exact in $\mathcal{B}$; in other words that $H^0(M, \varphi, \psi)$ and $H^1(M, \varphi, \psi)$ are objects of $\mathcal{C}$. \end{enumerate}


homology-lemma-efface-implies-universal

Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $F = (F^n, \delta_F)$ be a $\delta$-functor from $\mathcal{A}$ to $\mathcal{B}$. Suppose that for every $n > 0$ and any $A \in \Ob(\mathcal{A})$ there exists an injective morphism $u : A \to B$ (depending on $A$ and $n$) such that $F^n(u) : F^n(A) \to F^n(B)$ is zero. Then $F$ is a universal $\delta$-functor.


homology-lemma-uniqueness-universal-delta-functor

Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. If there exists a universal $\delta$-functor $(F^n, \delta_F)$ from $\mathcal{A}$ to $\mathcal{B}$ with $F^0 = F$, then it is determined up to unique isomorphism of $\delta$-functors.


homology-lemma-compose-homotopy

Let $\mathcal{A}$ be an additive category. Let $f, g : B_\bullet \to C_\bullet$ be morphisms of chain complexes. Suppose given morphisms of chain complexes $a : A_\bullet \to B_\bullet$, and $c : C_\bullet \to D_\bullet$. If $\{h_i : B_i \to C_{i + 1}\}$ defines a homotopy between $f$ and $g$, then $\{c_{i + 1} \circ h_i \circ a_i\}$ defines a homotopy between $c \circ f \circ a$ and $c \circ g \circ a$.


homology-lemma-cat-chain-abelian

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item The category of chain complexes in $\mathcal{A}$ is abelian. \item A morphism of complexes $f : A_\bullet \to B_\bullet$ is injective if and only if each $f_n : A_n \to B_n$ is injective. \item A morphism of complexes $f : A_\bullet \to B_\bullet$ is surjective if and only if each $f_n : A_n \to B_n$ is surjective. \item A sequence of chain complexes $$ A_\bullet \xrightarrow{f} B_\bullet \xrightarrow{g} C_\bullet $$ is exact at $B_\bullet$ if and only if each sequence $$ A_i \xrightarrow{f_i} B_i \xrightarrow{g_i} C_i $$ is exact at $B_i$. \end{enumerate}


homology-lemma-map-homology-homotopy

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item If the maps $f, g : A_\bullet \to B_\bullet$ are homotopic, then the induced maps $H_i(f)$ and $H_i(g)$ are equal. \item If the map $f : A_\bullet \to B_\bullet$ is a homotopy equivalence, then $f$ is a quasi-isomorphism. \end{enumerate}


homology-lemma-long-exact-sequence-chain

Let $\mathcal{A}$ be an abelian category. Suppose that $$ 0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 $$ is a short exact sequence of chain complexes of $\mathcal{A}$. Then there is a canonical long exact homology sequence $$ \xymatrix{ \ldots & \ldots & \ldots \ar[lld] \\ H_i(A_\bullet) \ar[r] & H_i(B_\bullet) \ar[r] & H_i(C_\bullet) \ar[lld] \\ H_{i - 1}(A_\bullet) \ar[r] & H_{i - 1}(B_\bullet) \ar[r] & H_{i - 1}(C_\bullet) \ar[lld] \\ \ldots & \ldots & \ldots \\ } $$


homology-lemma-compose-homotopy-cochain

Let $\mathcal{A}$ be an additive category. Let $f, g : B^\bullet \to C^\bullet$ be morphisms of cochain complexes. Suppose given morphisms of cochain complexes $a : A^\bullet \to B^\bullet$, and $c : C^\bullet \to D^\bullet$. If $\{h^i : B^i \to C^{i - 1}\}$ defines a homotopy between $f$ and $g$, then $\{c^{i - 1} \circ h^i \circ a^i\}$ defines a homotopy between $c \circ f \circ a$ and $c \circ g \circ a$.


homology-lemma-cat-cochain-abelian

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item The category of cochain complexes in $\mathcal{A}$ is abelian. \item A morphism of cochain complexes $f : A^\bullet \to B^\bullet$ is injective if and only if each $f^n : A^n \to B^n$ is injective. \item A morphism of cochain complexes $f : A^\bullet \to B^\bullet$ is surjective if and only if each $f^n : A^n \to B^n$ is surjective. \item A sequence of cochain complexes $$ A^\bullet \xrightarrow{f} B^\bullet \xrightarrow{g} C^\bullet $$ is exact at $B^\bullet$ if and only if each sequence $$ A^i \xrightarrow{f^i} B^i \xrightarrow{g^i} C^i $$ is exact at $B^i$. \end{enumerate}


homology-lemma-map-cohomology-homotopy-cochain

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item If the maps $f, g : A^\bullet \to B^\bullet$ are homotopic, then the induced maps $H^i(f)$ and $H^i(g)$ are equal. \item If $f : A^\bullet \to B^\bullet$ is a homotopy equivalence, then $f$ is a quasi-isomorphism. \end{enumerate}


homology-lemma-long-exact-sequence-cochain

\begin{slogan} Short exact sequences of complexes give rise to long exact sequences of (co)homology. \end{slogan} Let $\mathcal{A}$ be an abelian category. Suppose that $$ 0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0 $$ is a short exact sequence of chain complexes of $\mathcal{A}$. Then there is a canonical long exact cohomology sequence $$ \xymatrix{ \ldots & \ldots & \ldots \ar[lld] \\ H^i(A^\bullet) \ar[r] & H^i(B^\bullet) \ar[r] & H^i(C^\bullet) \ar[lld] \\ H^{i + 1}(A^\bullet) \ar[r] & H^{i + 1}(B^\bullet) \ar[r] & H^{i + 1}(C^\bullet) \ar[lld] \\ \ldots & \ldots & \ldots \\ } $$


homology-lemma-homotopy-shift

Let $\mathcal{A}$ be an additive category. Suppose that $A_\bullet$ and $B_\bullet$ are chain complexes. Given any morphism of chain complexes $a : A_\bullet \to B_\bullet$ there is a bijection between the set of homotopies from $a$ to $a$ and $\Mor_{\text{Ch}(\mathcal{A})}(A_\bullet, B[1]_\bullet)$. More generally, the set of homotopies between $a$ and $b$ is either empty or a principal homogeneous space under the group $\Mor_{\text{Ch}(\mathcal{A})}(A_\bullet, B[1]_\bullet)$.


homology-lemma-ses-termwise-split

Let $\mathcal{A}$ be an abelian category. Let $$ 0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 $$ be a short exact sequence of complexes. Suppose that $\{s_n : C_n \to B_n\}$ is a family of morphisms which split the short exact sequences $0 \to A_n \to B_n \to C_n \to 0$. Let $\pi_n : B_n \to A_n$ be the associated projections, see Lemma \ref{lemma-ses-split}. Then the family of morphisms $$ \pi_{n - 1} \circ d_{B, n} \circ s_n : C_n \to A_{n - 1} $$ define a morphism of complexes $\delta(s) : C_\bullet \to A[-1]_\bullet$.


homology-lemma-ses-termwise-split-long

Notation and assumptions as in Lemma \ref{lemma-ses-termwise-split} above. The morphism of complexes $\delta(s) : C_\bullet \to A[-1]_\bullet$ induces the maps $$ H_i(\delta(s)) : H_i(C_\bullet) \longrightarrow H_i(A[-1]_\bullet) = H_{i - 1}(A_\bullet) $$ which occur in the long exact homology sequence associated to the short exact sequence of chain complexes by Lemma \ref{lemma-long-exact-sequence-chain}.


homology-lemma-ses-termwise-split-homotopy

Notation and assumptions as in Lemma \ref{lemma-ses-termwise-split} above. Suppose $\{s'_n : C_n \to B_n\}$ is a second choice of splittings. Write $s'_n = s_n + i_n \circ h_n$ for some unique morphisms $h_n : C_n \to A_n$. The family of maps $\{h_n : C_n \to A[-1]_{n + 1}\}$ is a homotopy between the associated morphisms $\delta(s), \delta(s') : C_\bullet \to A[-1]_\bullet$.


homology-lemma-homotopy-shift-cochain

Let $\mathcal{A}$ be an additive category. Suppose that $A^\bullet$ and $B^\bullet$ are cochain complexes. Given any morphism of cochain complexes $a : A^\bullet \to B^\bullet$ there is a bijection between the set of homotopies from $a$ to $a$ and $\Mor_{\text{CoCh}(\mathcal{A})}(A^\bullet, B[-1]^\bullet)$. More generally, the set of homotopies between $a$ and $b$ is either empty or a principal homogeneous space under the group $\Mor_{\text{CoCh}(\mathcal{A})}(A^\bullet, B[-1]^\bullet)$.


homology-lemma-ses-termwise-split-cochain

Let $\mathcal{A}$ be an additive category. Let $$ 0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0 $$ be a complex (!) of complexes. Suppose that we are given splittings $B^n = A^n \oplus C^n$ compatible with the maps in the displayed sequence. Let $s^n : C^n \to B^n$ and $\pi^n : B^n \to A^n$ be the corresponding maps. Then the family of morphisms $$ \pi^{n + 1} \circ d_B^n \circ s^n : C^n \to A^{n + 1} $$ define a morphism of complexes $\delta : C^\bullet \to A[1]^\bullet$.


homology-lemma-ses-termwise-split-long-cochain

Notation and assumptions as in Lemma \ref{lemma-ses-termwise-split-cochain} above. Assume in addition that $\mathcal{A}$ is abelian. The morphism of complexes $\delta : C^\bullet \to A[1]^\bullet$ induces the maps $$ H^i(\delta) : H^i(C^\bullet) \longrightarrow H^i(A[1]^\bullet) = H^{i + 1}(A^\bullet) $$ which occur in the long exact homology sequence associated to the short exact sequence of cochain complexes by Lemma \ref{lemma-long-exact-sequence-cochain}.


homology-lemma-ses-termwise-split-homotopy-cochain

Notation and assumptions as in Lemma \ref{lemma-ses-termwise-split-cochain}. Let $\alpha : A^\bullet \to B^\bullet$, $\beta : B^\bullet \to C^\bullet$ be the given morphisms of complexes. Suppose $(s')^n : C^n \to B^n$ and $(\pi')^n : B^n \to A^n$ is a second choice of splittings. Write $(s')^n = s^n + \alpha^n \circ h^n$ and $(\pi')^n = \pi^n + g^n \circ \beta^n$ for some unique morphisms $h^n : C^n \to A^n$ and $g^n : C^n \to A^n$. Then \begin{enumerate} \item $g^n = - h^n$, and \item the family of maps $\{g^n : C^n \to A[1]^{n - 1}\}$ is a homotopy between $\delta, \delta' : C^\bullet \to A[1]^\bullet$, more precisely $(\delta')^n = \delta^n + g^{n + 1} \circ d_C^n + d_{A[1]}^{n - 1} \circ g^n$. \end{enumerate}


homology-lemma-graded

Let $\mathcal{A}$ be an abelian category. The category of graded objects $\text{Gr}(\mathcal{A})$ is abelian.


homology-lemma-additive-dual

Let $\mathcal{A}$ be an additive monoidal category. If $Y_i$, $i = 1, 2$ are left duals of $X_i$, $i = 1, 2$, then $Y_1 \oplus Y_2$ is a left dual of $X_1 \oplus X_2$.


homology-lemma-Karoubian-dual

In a Karoubian additive monoidal category every summand of an object which has a left dual has a left dual.


homology-lemma-left-dual-graded-vector-spaces

Let $F$ be a field. Let $\mathcal{C}$ be the category of graded $F$-vector spaces viewed as a monoidal category as in Example \ref{example-graded-vector-spaces}. If $V$ in $\mathcal{C}$ has a left dual $W$, then $\sum_n \dim_F V^n < \infty$ and the map $\epsilon$ defines nondegenerate pairings $W^{-n} \times V^n \to F$.


homology-lemma-filtered

Let $\mathcal{A}$ be an abelian category. The category of filtered objects $\text{Fil}(\mathcal{A})$ has the following properties: \begin{enumerate} \item It is an additive category. \item It has a zero object. \item It has kernels and cokernels, images and coimages. \item In general it is not an abelian category. \end{enumerate}


homology-lemma-characterize-strict-general

Let $\mathcal{A}$ be an abelian category. Let $f : A \to B$ be a morphism of filtered objects of $\mathcal{A}$. The following are equivalent \begin{enumerate} \item $f$ is strict, \item the morphism $\Coim(f) \to \Im(f)$ of Lemma \ref{lemma-coim-im-map} is an isomorphism. \end{enumerate}


homology-lemma-add-summand-strict-monomorphism

Let $\mathcal{A}$ be an abelian category. Let $f : A \to B$ be a strict monomorphism of filtered objects. Let $g : A \to C$ be a morphism of filtered objects. Then $f \oplus g : A \to B \oplus C$ is a strict monomorphism.


homology-lemma-add-summand-strict-epimorphism

Let $\mathcal{A}$ be an abelian category. Let $f : B \to A$ be a strict epimorphism of filtered objects. Let $g : C \to A$ be a morphism of filtered objects. Then $f \oplus g : B \oplus C \to A$ is a strict epimorphism.


homology-lemma-induced-and-quotient-strict

Let $\mathcal{A}$ be an abelian category. Let $(A, F)$, $(B, F)$ be filtered objects. Let $u : A \to B$ be a morphism of filtered objects. If $u$ is injective then $u$ is strict if and only if the filtration on $A$ is the induced filtration. If $u$ is surjective then $u$ is strict if and only if the filtration on $B$ is the quotient filtration.


homology-lemma-filtration-subobject

Let $\mathcal{A}$ be an abelian category. Let $(A, F)$ be a filtered object of $\mathcal{A}$. Let $X \subset Y \subset A$ be subobjects of $A$. On the object $$ Y/X = \Ker(A/X \to A/Y) $$ the quotient filtration coming from the induced filtration on $Y$ and the induced filtration coming from the quotient filtration on $A/X$ agree. Any of the morphisms $X \to Y$, $X \to A$, $Y \to A$, $Y \to A/X$, $Y \to Y/X$, $Y/X \to A/X$ are strict (with induced/quotient filtrations).


homology-lemma-pushout-filtered

Let $\mathcal{A}$ be an abelian category. Let $A, B, C \in \text{Fil}(\mathcal{A})$. Let $f : A \to B$ and $g : A \to C$ be morphisms. Then there exists a pushout $$ \xymatrix{ A \ar[r]_f \ar[d]_g & B \ar[d]^{g'} \\ C \ar[r]^{f'} & C \amalg_A B } $$ in $\text{Fil}(\mathcal{A})$. If $f$ is strict, so is $f'$.


homology-lemma-ses-gr

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item Let $A$ be a filtered object and $X \subset A$. Then for each $p$ the sequence $$ 0 \to \text{gr}^p(X) \to \text{gr}^p(A) \to \text{gr}^p(A/X) \to 0 $$ is exact (with induced filtration on $X$ and quotient filtration on $A/X$). \item Let $f : A \to B$ be a morphism of filtered objects of $\mathcal{A}$. Then for each $p$ the sequences $$ 0 \to \text{gr}^p(\Ker(f)) \to \text{gr}^p(A) \to \text{gr}^p(\Coim(f)) \to 0 $$ and $$ 0 \to \text{gr}^p(\Im(f)) \to \text{gr}^p(B) \to \text{gr}^p(\Coker(f)) \to 0 $$ are exact. \end{enumerate}


homology-lemma-characterize-strict

Let $\mathcal{A}$ be an abelian category. Let $f : A \to B$ be a morphism of finite filtered objects of $\mathcal{A}$. The following are equivalent \begin{enumerate} \item $f$ is strict, \item the morphism $\Coim(f) \to \Im(f)$ is an isomorphism, \item $\text{gr}(\Coim(f)) \to \text{gr}(\Im(f))$ is an isomorphism, \item the sequence $\text{gr}(\Ker(f)) \to \text{gr}(A) \to \text{gr}(B)$ is exact, \item the sequence $\text{gr}(A) \to \text{gr}(B) \to \text{gr}(\Coker(f))$ is exact, and \item the sequence $$ 0 \to \text{gr}(\Ker(f)) \to \text{gr}(A) \to \text{gr}(B) \to \text{gr}(\Coker(f)) \to 0 $$ is exact. \end{enumerate}


homology-lemma-filtered-acyclic

Let $\mathcal{A}$ be an abelian category. Let $A \to B \to C$ be a complex of filtered objects of $\mathcal{A}$. Assume $A, B, C$ have finite filtrations and that $\text{gr}(A) \to \text{gr}(B) \to \text{gr}(C)$ is exact. Then \begin{enumerate} \item for each $p \in \mathbf{Z}$ the sequence $\text{gr}^p(A) \to \text{gr}^p(B) \to \text{gr}^p(C)$ is exact, \item for each $p \in \mathbf{Z}$ the sequence $F^p(A) \to F^p(B) \to F^p(C)$ is exact, \item for each $p \in \mathbf{Z}$ the sequence $A/F^p(A) \to B/F^p(B) \to C/F^p(C)$ is exact, \item the maps $A \to B$ and $B \to C$ are strict, and \item $A \to B \to C$ is exact (as a sequence in $\mathcal{A}$). \end{enumerate}


homology-lemma-derived-exact-couple

Let $(A, E, \alpha, f, g)$ be an exact couple in an abelian category $\mathcal{A}$. Set \begin{enumerate} \item $d = g \circ f : E \to E$ so that $d \circ d = 0$, \item $E' = \Ker(d)/\Im(d)$, \item $A' = \Im(\alpha)$, \item $\alpha' : A' \to A'$ induced by $\alpha$, \item $f' : E' \to A'$ induced by $f$, \item $g' : A' \to E'$ induced by ``$g \circ \alpha^{-1}$''. \end{enumerate} Then we have \begin{enumerate} \item $\Ker(d) = f^{-1}(\Ker(g)) = f^{-1}(\Im(\alpha))$, \item $\Im(d) = g(\Im(f)) = g(\Ker(\alpha))$, \item $(A', E', \alpha', f', g')$ is an exact couple. \end{enumerate}


homology-lemma-spectral-sequence-associated-exact-couple

Let $\mathcal{A}$ be an abelian category. Let $(A, E, \alpha, f, g)$ be an exact couple. Let $(E_r, d_r)_{r \geq 1}$ be the spectral sequence associated to the exact couple. In this case we have $$ 0 = B_1 \subset \ldots \subset B_{r + 1} = g(\Ker(\alpha^r)) \subset \ldots \subset Z_{r + 1} = f^{-1}(\Im(\alpha^r)) \subset \ldots \subset Z_1 = E $$ and the map $d_{r + 1} : E_{r + 1} \to E_{r + 1}$ is described by the following rule: For any (test) object $T$ of $\mathcal{A}$ and any elements $x : T \to Z_{r + 1}$ and $y : T \to A$ such that $f \circ x = \alpha^r \circ y$ we have $$ d_{r + 1} \circ \overline{x} = \overline{g \circ y} $$ where $\overline{x} : T \to E_{r + 1}$ is the induced morphism.


homology-lemma-differential-objects-abelian

\begin{slogan} The category of differential objects of an abelian category is itself an abelian category. \end{slogan} Let $\mathcal{A}$ be an abelian category. The category of differential objects of $\mathcal{A}$ is abelian.


homology-lemma-spectral-sequence-filtered-differential

Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. There is a spectral sequence $(E_r, d_r)_{r \geq 0}$ in $\text{Gr}(\mathcal{A})$ associated to $(K, F, d)$ such that $d_r : E_r \to E_r[r]$ for all $r$ and such that the graded pieces $E_r^p$ and maps $d_r^p : E_r^p \to E_r^{p + r}$ are as given above. Furthermore, $E_0^p = \text{gr}^p K$, $d_0^p = \text{gr}^p(d)$, and $E_1^p = H(\text{gr}^pK, d)$.


homology-lemma-compute-filtered-cohomology

Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. If $Z_\infty^p$ and $B_\infty^p$ exist (see proof), then \begin{enumerate} \item the limit $E_\infty$ exists and is graded having $E_\infty^p = Z_\infty^p/B_\infty^p$ in degree $p$, and \item the associated graded $\text{gr}(H(K))$ of the cohomology of $K$ is a graded subquotient of the graded limit object $E_\infty$. \end{enumerate}


homology-lemma-filtered-differential-ss-converges

Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. The associated spectral sequence \begin{enumerate} \item weakly converges to $H(K)$ if and only if for every $p \in \mathbf{Z}$ we have equality in equations (\ref{equation-at-bottom}) and (\ref{equation-on-top}), \item abuts to $H(K)$ if and only if it weakly converges to $H(K)$ and $\bigcap_p (\Ker(d) \cap F^pK + \Im(d)) = \Im(d)$ and $\bigcup_p (\Ker(d) \cap F^pK + \Im(d)) = \Ker(d)$. \end{enumerate}


homology-lemma-spectral-sequence-filtered-complex

Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet, F)$ be a filtered complex of $\mathcal{A}$. There is a spectral sequence $(E_r, d_r)_{r \geq 0}$ in the category of bigraded objects of $\mathcal{A}$ associated to $(K^\bullet, F)$ such that $d_r$ has bidegree $(r, - r + 1)$ and such that $E_r$ has bigraded pieces $E_r^{p, q}$ and maps $d_r^{p, q} : E_r^{p, q} \to E_r^{p + r, q - r + 1}$ as given above. Furthermore, we have $E_0^{p, q} = \text{gr}^p(K^{p + q})$, $d_0^{p, q} = \text{gr}^p(d^{p + q})$, and $E_1^{p, q} = H^{p + q}(\text{gr}^p(K^\bullet))$.


homology-lemma-spectral-sequence-filtered-complex-d1

Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet, F)$ be a filtered complex of $\mathcal{A}$. Assume $\mathcal{A}$ has countable direct sums. Let $(E_r, d_r)_{r \geq 0}$ be the spectral sequence associated to $(K^\bullet, F)$. \begin{enumerate} \item The map $$ d_1^{p, q} : E_1^{p, q} = H^{p + q}(\text{gr}^p(K^\bullet)) \longrightarrow E_1^{p + 1, q} = H^{p + q + 1}(\text{gr}^{p + 1}(K^\bullet)) $$ is equal to the boundary map in cohomology associated to the short exact sequence of complexes $$ 0 \to \text{gr}^{p + 1}(K^\bullet) \to F^pK^\bullet/F^{p + 2}K^\bullet \to \text{gr}^p(K^\bullet) \to 0. $$ \item Assume that $d(F^pK) \subset F^{p + 1}K$ for all $p \in \mathbf{Z}$. Then $d$ induces the zero differential on $\text{gr}^p(K^\bullet)$ and hence $E_1^{p, q} = \text{gr}^p(K^\bullet)^{p + q}$. Furthermore, in this case $$ d_1^{p, q} : E_1^{p, q} = \text{gr}^p(K^\bullet)^{p + q} \longrightarrow E_1^{p + 1, q} = \text{gr}^{p + 1}(K^\bullet)^{p + q + 1} $$ is the morphism induced by $d$. \end{enumerate}


homology-lemma-spectral-sequence-filtered-complex-functorial

Let $\mathcal{A}$ be an abelian category. Let $\alpha : (K^\bullet, F) \to (L^\bullet, F)$ be a morphism of filtered complexes of $\mathcal{A}$. Let $(E_r(K), d_r)_{r \geq 0}$, resp.\ $(E_r(L), d_r)_{r \geq 0}$ be the spectral sequence associated to $(K^\bullet, F)$, resp.\ $(L^\bullet, F)$. The morphism $\alpha$ induces a canonical morphism of spectral sequences $\{\alpha_r : E_r(K) \to E_r(L)\}_{r \geq 0}$ compatible with the bigradings.


homology-lemma-compute-cohomology-filtered-complex

Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet, F)$ be a filtered complex of $\mathcal{A}$. If $Z_\infty^{p, q}$ and $B_\infty^{p, q}$ exist (see proof), then \begin{enumerate} \item the limit $E_\infty$ exists and is a bigraded object having $E_\infty^{p, q} = Z_\infty^{p, q}/B_\infty^{p, q}$ in bidegree $(p, q)$, \item the $p$th graded part $\text{gr}^pH^n(K^\bullet)$ of the $n$th cohomology object of $K^\bullet$ is a subquotient of $E_\infty^{p, n - p}$. \end{enumerate}


homology-lemma-relate-boundedness

In the situation of Definition \ref{definition-bounded-ss}. Let $Z_r^{p, q}, B_r^{p, q} \subset E_{r_0}^{p, q}$ be the $(p, q)$-graded parts of $Z_r, B_r$ defined as in Section \ref{section-spectral-sequence}. \begin{enumerate} \item The spectral sequence is regular if and only if for all $p, q$ there exists an $r = r(p, q)$ such that $Z_r^{p, q} = Z_{r + 1}^{p, q} = \ldots$ \item The spectral sequence is coregular if and only if for all $p, q$ there exists an $r = r(p, q)$ such that $B_r^{p, q} = B_{r + 1}^{p, q} = \ldots$ \item The spectral sequence is bounded if and only if it is both bounded below and bounded above. \item If the spectral sequence is bounded below, then it is regular. \item If the spectral sequence is bounded above, then it is coregular. \end{enumerate}


homology-lemma-filtered-complex-ss-converges

Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet, F)$ be a filtered complex of $\mathcal{A}$. The associated spectral sequence \begin{enumerate} \item weakly converges to $H^*(K^\bullet)$ if and only if for every $p, q \in \mathbf{Z}$ we have equality in equations (\ref{equation-at-bottom-bigraded}) and (\ref{equation-on-top-bigraded}), \item abuts to $H^*(K)$ if and only if it weakly converges to $H^*(K^\bullet)$ and we have $\bigcap_p (\Ker(d) \cap F^pK^n + \Im(d) \cap K^n) = \Im(d) \cap K^n$ and $\bigcup_p (\Ker(d) \cap F^pK^n + \Im(d) \cap K^n) = \Ker(d) \cap K^n$. \end{enumerate}


homology-lemma-biregular-ss-converges

Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet, F)$ be a filtered complex of $\mathcal{A}$. Assume that the filtration on each $K^n$ is finite (see Definition \ref{definition-filtered}). Then \begin{enumerate} \item the spectral sequence associated to $(K^\bullet, F)$ is bounded, \item the filtration on each $H^n(K^\bullet)$ is finite, \item the spectral sequence associated to $(K^\bullet, F)$ converges to $H^*(K^\bullet)$, \item if $\mathcal{C} \subset \mathcal{A}$ is a weak Serre subcategory and for some $r$ we have $E_r^{p, q} \in \mathcal{C}$ for all $p, q \in \mathbf{Z}$, then $H^n(K^\bullet)$ is in $\mathcal{C}$. \end{enumerate}


homology-lemma-biregular-ss-relation-in-K0

Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet, F)$ be a filtered complex of $\mathcal{A}$. Assume that the filtration on each $K^n$ is finite (see Definition \ref{definition-filtered}) and that for some $r$ we have only a finite number of nonzero $E_r^{p, q}$. Then only a finite number of $H^n(K^\bullet)$ are nonzero and we have $$ \sum (-1)^n[H^n(K^\bullet)] = \sum (-1)^{p + q} [E_r^{p, q}] $$ in $K_0(\mathcal{A}')$ where $\mathcal{A}'$ is the smallest weak Serre subcategory of $\mathcal{A}$ containing the objects $E_r^{p, q}$.


homology-lemma-ss-converges-trivial

Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet, F)$ be a filtered complex of $\mathcal{A}$. Assume \begin{enumerate} \item for every $n$ there exist $p_0(n)$ such that $H^n(F^pK^\bullet) = 0$ for $p \geq p_0(n)$, \item for every $n$ there exist $p_1(n)$ such that $H^n(F^pK^\bullet) \to H^n(K^\bullet)$ is an isomorphism for $p \leq p_1(n)$. \end{enumerate} Then \begin{enumerate} \item the spectral sequence associated to $(K^\bullet, F)$ is bounded, \item the filtration on each $H^n(K^\bullet)$ is finite, \item the spectral sequence associated to $(K^\bullet, F)$ converges to $H^*(K^\bullet)$. \end{enumerate}


homology-lemma-ss-double-complex

Let $\mathcal{A}$ be an abelian category. Let $K^{\bullet, \bullet}$ be a double complex. The spectral sequences associated to $K^{\bullet, \bullet}$ have the following terms: \begin{enumerate} \item ${}'E_0^{p, q} = K^{p, q}$ with ${}'d_0^{p, q} = (-1)^p d_2^{p, q} : K^{p, q} \to K^{p, q + 1}$, \item ${}''E_0^{p, q} = K^{q, p}$ with ${}''d_0^{p, q} = d_1^{q, p} : K^{q, p} \to K^{q + 1, p}$, \item ${}'E_1^{p, q} = H^q(K^{p, \bullet})$ with ${}'d_1^{p, q} = H^q(d_1^{p, \bullet})$, \item ${}''E_1^{p, q} = H^q(K^{\bullet, p})$ with ${}''d_1^{p, q} = (-1)^q H^q(d_2^{\bullet, p})$, \item ${}'E_2^{p, q} = H^p_I(H^q_{II}(K^{\bullet, \bullet}))$, \item ${}''E_2^{p, q} = H^p_{II}(H^q_I(K^{\bullet, \bullet}))$. \end{enumerate}


homology-lemma-first-quadrant-ss

Let $\mathcal{A}$ be an abelian category. Let $K^{\bullet, \bullet}$ be a double complex. Assume that for every $n \in \mathbf{Z}$ there are only finitely many nonzero $K^{p, q}$ with $p + q = n$. Then \begin{enumerate} \item the two spectral sequences associated to $K^{\bullet, \bullet}$ are bounded, \item the filtrations $F_I$, $F_{II}$ on each $H^n(K^\bullet)$ are finite, \item the spectral sequences $({}'E_r, {}'d_r)_{r \geq 0}$ and $({}''E_r, {}''d_r)_{r \geq 0}$ converge to $H^*(\text{Tot}(K^{\bullet, \bullet}))$, \item if $\mathcal{C} \subset \mathcal{A}$ is a weak Serre subcategory and for some $r$ we have ${}'E_r^{p, q} \in \mathcal{C}$ for all $p, q \in \mathbf{Z}$, then $H^n(\text{Tot}(K^{\bullet, \bullet}))$ is in $\mathcal{C}$. Similarly for $({}''E_r, {}''d_r)_{r \geq 0}$. \end{enumerate}