formal-spaces-remark-set-theoretic

Let $S$ be a scheme and let $(\Sch/S)_{fppf}$ be a big fppf site as in Topologies, Definition \ref{topologies-definition-big-small-fppf}. As our set theoretic condition on $X$ in Definitions \ref{definition-affine-formal-algebraic-space} and \ref{definition-formal-algebraic-space} we take: there exist objects $U, R$ of $(\Sch/S)_{fppf}$, a morphism $U \to X$ which is a surjection of fppf sheaves, and a morphism $R \to U \times_X U$ which is a surjection of fppf sheaves. In other words, we require our sheaf to be a coequalizer of two maps between representable sheaves. Here are some observations which imply this notion behaves reasonably well: \begin{enumerate} \item Suppose $X = \colim_{\lambda \in \Lambda} X_\lambda$ and the system satisfies conditions (1) and (2) of Definition \ref{definition-affine-formal-algebraic-space}. Then $U = \coprod_{\lambda \in \Lambda} X_\lambda \to X$ is a surjection of fppf sheaves. Moreover, $U \times_X U$ is a closed subscheme of $U \times_S U$ by Lemma \ref{lemma-diagonal-affine-formal-algebraic-space}. Hence if $U$ is representable by an object of $(\Sch/S)_{fppf}$ then $U \times_S U$ is too (see Sets, Lemma \ref{sets-lemma-what-is-in-it}) and the set theoretic condition is satisfied. This is always the case if $\Lambda$ is countable, see Sets, Lemma \ref{sets-lemma-what-is-in-it}. \item Sanity check. Let $\{X_i \to X\}_{i \in I}$ be as in Definition \ref{definition-formal-algebraic-space} (with the set theoretic condition as formulated above) and assume that each $X_i$ is actually an affine scheme. Then $X$ is an algebraic space. Namely, if we choose a larger big fppf site $(\Sch'/S)_{fppf}$ such that $U' = \coprod X_i$ and $R' = \coprod X_i \times_X X_j$ are representable by objects in it, then $X' = U'/R'$ will be an object of the category of algebraic spaces for this choice. Then an application of Spaces, Lemma \ref{spaces-lemma-fully-faithful} shows that $X$ is an algebraic space for $(\Sch/S)_{fppf}$. \item Let $\{X_i \to X\}_{i \in I}$ be a family of maps of sheaves satisfying conditions (1), (2), (3) of Definition \ref{definition-formal-algebraic-space}. For each $i$ we can pick $U_i \in \Ob((\Sch/S)_{fppf})$ and $U_i \to X_i$ which is a surjection of sheaves. Thus if $I$ is not too large (for example countable) then $U = \coprod U_i \to X$ is a surjection of sheaves and $U$ is representable by an object of $(\Sch/S)_{fppf}$. To get $R \in \Ob((\Sch/S)_{fppf})$ surjecting onto $U \times_X U$ it suffices to assume the diagonal $\Delta : X \to X \times_S X$ is not too wild, for example this always works if the diagonal of $X$ is quasi-compact, i.e., $X$ is quasi-separated. \end{enumerate}


formal-spaces-remark-weak-ideals-of-definition

Let $\mathfrak X$ be a formal scheme in the sense of McQuillan, see Remark \ref{remark-mcquillan}. An {\it weak ideal of definition} for $\mathfrak X$ is an ideal sheaf $\mathcal{I} \subset \mathcal{O}_\mathfrak X$ such that for all $\mathfrak U \subset \mathfrak X$ affine formal open subscheme the ideal $\mathcal{I}(\mathfrak U) \subset \mathcal{O}_\mathfrak X(\mathfrak U)$ is a weak ideal of definition of the weakly admissible topological ring $\mathcal{O}_\mathfrak X(\mathfrak U)$. It suffices to check the condition on the members of an affine open covering. There is a one-to-one correspondence $$ \{\text{weak ideals of definition for }\mathfrak X\} \leftrightarrow \{\text{thickenings }i : Z \to h_\mathfrak X\text{ as above}\} $$ This correspondence associates to $\mathcal{I}$ the scheme $Z = (\mathfrak X, \mathcal{O}_\mathfrak X/\mathcal{I})$ together with the obvious morphism to $\mathfrak X$. A {\it fundamental system of weak ideals of definition} is a collection of weak ideals of definition $\mathcal{I}_\lambda$ such that on every affine open formal subscheme $\mathfrak U \subset \mathfrak X$ the ideals $$ I_\lambda = \mathcal{I}_\lambda(\mathfrak U) \subset A = \Gamma(\mathfrak U, \mathcal{O}_\mathfrak X) $$ form a fundamental system of weak ideals of definition of the weakly admissible topological ring $A$. It suffices to check on the members of an affine open covering. We conclude that the formal algebraic space $h_\mathfrak X$ associated to the McQuillan formal scheme $\mathfrak X$ is a colimit of schemes as in Lemma \ref{lemma-colimit-is-formal} if and only if there exists a fundamental system of weak ideals of definition for $\mathfrak X$.


formal-spaces-remark-ideals-of-definition

Let $\mathfrak X$ be a formal scheme \`a la EGA. An {\it ideal of definition} for $\mathfrak X$ is an ideal sheaf $\mathcal{I} \subset \mathcal{O}_\mathfrak X$ such that for all $\mathfrak U \subset \mathfrak X$ affine formal open subscheme the ideal $\mathcal{I}(\mathfrak U) \subset \mathcal{O}_\mathfrak X(\mathfrak U)$ is an ideal of definition of the admissible topological ring $\mathcal{O}_\mathfrak X(\mathfrak U)$. It suffices to check the condition on the members of an affine open covering. We do {\bf not} get the same correspondence between ideals of definition and thickenings $Z \to h_\mathfrak X$ as in Remark \ref{remark-weak-ideals-of-definition}; an example is given in Example \ref{example-david-hansen}. A {\it fundamental system of ideals of definition} is a collection of ideals of definition $\mathcal{I}_\lambda$ such that on every affine open formal subscheme $\mathfrak U \subset \mathfrak X$ the ideals $$ I_\lambda = \mathcal{I}_\lambda(\mathfrak U) \subset A = \Gamma(\mathfrak U, \mathcal{O}_\mathfrak X) $$ form a fundamental system of ideals of definition of the admissible topological ring $A$. It suffices to check on the members of an affine open covering. Suppose that $\mathfrak X$ is quasi-compact and that $\{\mathcal{I}_\lambda\}_{\lambda \in \Lambda}$ is a fundamental system of weak ideals of definition. If $A$ is an admissible topological ring then all sufficiently small open ideals are ideals of definition (namely any open ideal contained in an ideal of definition is an ideal of definition). Thus since we only need to check on the finitely many members of an affine open covering we see that $\mathcal{I}_\lambda$ is an ideal of definition for $\lambda$ sufficiently large. Using the discussion in Remark \ref{remark-weak-ideals-of-definition} we conclude that the formal algebraic space $h_\mathfrak X$ associated to the quasi-compact formal scheme $\mathfrak X$ \`a la EGA is a colimit of schemes as in Lemma \ref{lemma-colimit-is-formal} if and only if there exists a fundamental system of ideals of definition for $\mathfrak X$.


formal-spaces-remark-structure-quasi-compact-quasi-separated

In this remark we translate the statement and proof of Lemma \ref{lemma-structure-quasi-compact-quasi-separated} into the language of formal schemes \`a la EGA. Looking at Remark \ref{remark-ideals-of-definition} we see that the lemma can be translated as follows \begin{itemize} \item[(*)] Every quasi-compact and quasi-separated formal scheme has a fundamental system of ideals of definition. \end{itemize} To prove this we first use the induction principle (reformulated for quasi-compact and quasi-separated formal schemes) of Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle} to reduce to the following situation: $\mathfrak X = \mathfrak U \cup \mathfrak V$ with $\mathfrak U$, $\mathfrak V$ open formal subschemes, with $\mathfrak V$ affine, and the result is true for $\mathfrak U$, $\mathfrak V$, and $\mathfrak U \cap \mathfrak V$. Pick any ideals of definition $\mathcal{I} \subset \mathcal{O}_\mathfrak U$ and $\mathcal{J} \subset \mathcal{O}_\mathfrak V$. By our assumption that we have a fundamental system of ideals of definition on $\mathfrak U$ and $\mathfrak V$ and because $\mathfrak U \cap \mathfrak V$ is quasi-compact, we can find ideals of definition $\mathcal{I}' \subset \mathcal{I}$ and $\mathcal{J}' \subset \mathcal{J}$ such that $$ \mathcal{I}'|_{\mathfrak U \cap \mathfrak V} \subset \mathcal{J}|_{\mathfrak U \cap \mathfrak V} \quad\text{and}\quad \mathcal{J}'|_{\mathfrak U \cap \mathfrak V} \subset \mathcal{I}|_{\mathfrak U \cap \mathfrak V} $$ Let $U \to U' \to \mathfrak U$ and $V \to V' \to \mathfrak V$ be the closed immersions determined by the ideals of definition $\mathcal{I}' \subset \mathcal{I} \subset \mathcal{O}_\mathfrak U$ and $\mathcal{J}' \subset \mathcal{J} \subset \mathcal{O}_\mathfrak V$. Let $\mathfrak U \cap V$ denote the open subscheme of $V$ whose underlying topological space is that of $\mathfrak U \cap \mathfrak V$. By our choice of $\mathcal{I}'$ there is a factorization $\mathfrak U \cap V \to U'$. We define similarly $U \cap \mathfrak V$ which factors through $V'$. Then we consider $$ Z_U = \text{scheme theoretic image of } U \amalg (\mathfrak U \cap V) \longrightarrow U' $$ and $$ Z_V = \text{scheme theoretic image of } (U \cap \mathfrak V) \amalg V \longrightarrow V' $$ Since taking scheme theoretic images of quasi-compact morphisms commutes with restriction to opens (Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}) we see that $Z_U \cap \mathfrak V = \mathfrak U \cap Z_V$. Thus $Z_U$ and $Z_V$ glue to a scheme $Z$ which comes equipped with a morphism $Z \to \mathfrak X$. Analogous to the discussion in Remark \ref{remark-weak-ideals-of-definition} we see that $Z$ corresponds to a weak ideal of definition $\mathcal{I}_Z \subset \mathcal{O}_\mathfrak X$. Note that $Z_U \subset U'$ and that $Z_V \subset V'$. Thus the collection of all $\mathcal{I}_Z$ constructed in this manner forms a fundamental system of weak ideals of definition. Hence a subfamily gives a fundamental system of ideals of definition, see Remark \ref{remark-ideals-of-definition}.


formal-spaces-remark-warning

Lemma \ref{lemma-representable-affine} is sharp in the following two senses: \begin{enumerate} \item If $A$ and $B$ are weakly admissible rings and $\varphi : A \to B$ is a continuous map, then $\text{Spf}(\varphi) : \text{Spf}(B) \to \text{Spf}(A)$ is in general not representable. \item If $f : Y \to X$ is a representable morphism of affine formal algebraic spaces and $X = \text{Spf}(A)$ is McQuillan, then it does not follow that $Y$ is McQuillan. \end{enumerate} An example for (1) is to take $A = k$ a field (with discrete topology) and $B = k[[t]]$ with the $t$-adic topology. An example for (2) is given in Examples, Section \ref{examples-section-affine-formal-algebraic-space}.


formal-spaces-remark-warning-completion

Suppose $X = \Spec(A)$ and $T \subset X$ is the zero locus of a finitely generated ideal $I \subset A$. Let $J = \sqrt{I}$ be the radical of $I$. Then from the definitions we see that $X_{/T} = \text{Spf}(A^\wedge)$ where $A^\wedge = \lim A/I^n$ is the $I$-adic completion of $A$. On the other hand, the map $A^\wedge \to \lim A/J^n$ from the $I$-adic completion to the $J$-adic completion can fail to be a ring isomorphisms. As an example let $$ A = \bigcup\nolimits_{n \geq 1} \mathbf{C}[t^{1/n}] $$ and $I = (t)$. Then $J = \mathfrak m$ is the maximal ideal of the valuation ring $A$ and $J^2 = J$. Thus the $J$-adic completion of $A$ is $\mathbf{C}$ whereas the $I$-adic completion is the valuation ring described in Example \ref{example-david-hansen} (but in particular it is easy to see that $A \subset A^\wedge$).


formal-spaces-remark-variant-adic-star

Let $P$ be a property of morphisms of $\textit{WAdm}^{adic*}$. We say $P$ is a {\it local property} if axioms (\ref{item-axiom-1}), (\ref{item-axiom-2}), (\ref{item-axiom-3}) of Situation \ref{situation-local-property} hold for morphisms of $\textit{WAdm}^{adic*}$. In exactly the same way we obtain a variant of Lemma \ref{lemma-property-defines-property-morphisms} for morphisms between locally adic* formal algebraic spaces over $S$.


formal-spaces-remark-variant-Noetherian

Let $P$ be a property of morphisms of $\textit{WAdm}^{Noeth}$. We say $P$ is a {\it local property} if axioms (\ref{item-axiom-1}), (\ref{item-axiom-2}), (\ref{item-axiom-3}), of Situation \ref{situation-local-property} hold for morphisms of $\textit{WAdm}^{Noeth}$. In exactly the same way we obtain a variant of Lemma \ref{lemma-property-defines-property-morphisms} for morphisms between locally Noetherian formal algebraic spaces over $S$.


formal-spaces-remark-base-change-variant-adic-star

Let $P$ be a local property of morphisms of $\textit{WAdm}^{adic*}$, see Remark \ref{remark-variant-adic-star}. We say $P$ is {\it stable under base change} if given $B \to A$ and $B \to C$ in $\textit{WAdm}^{adic*}$ we have $P(B \to A) \Rightarrow P(C \to A \widehat{\otimes}_B C)$. This makes sense as $A \widehat{\otimes}_B C$ is an object of $\textit{WAdm}^{adic*}$ by Lemma \ref{lemma-completed-tensor-product}. In exactly the same way we obtain a variant of Lemma \ref{lemma-base-change-property-morphisms} for morphisms between locally adic* formal algebraic spaces over $S$.


formal-spaces-remark-base-change-variant-Noetherian

Let $P$ be a local property of morphisms of $\textit{WAdm}^{Noeth}$, see Remark \ref{remark-variant-Noetherian}. We say $P$ is {\it stable under base change} if given $B \to A$ and $B \to C$ in $\textit{WAdm}^{Noeth}$ the property $P(B \to A)$ implies both that $A \widehat{\otimes}_B C$ is adic Noetherian\footnote{See Lemma \ref{lemma-completed-tensor-product} for a criterion.} and that $P(C \to A \widehat{\otimes}_B C)$. In exactly the same way we obtain a variant of Lemma \ref{lemma-base-change-property-morphisms} for morphisms between locally Noetherian formal algebraic spaces over $S$.


formal-spaces-remark-base-change-variant-variant-Noetherian

Let $P$ and $Q$ be local properties of morphisms of $\textit{WAdm}^{Noeth}$, see Remark \ref{remark-variant-Noetherian}. We say {\it $P$ is stable under base change by $Q$} if given $B \to A$ and $B \to C$ in $\textit{WAdm}^{Noeth}$ satisfying $P(B \to A)$ and $Q(B \to C)$, then $A \widehat{\otimes}_B C$ is adic Noetherian and $P(C \to A \widehat{\otimes}_B C)$ holds. Arguing exactly as in the proof of Lemma \ref{lemma-base-change-property-morphisms} we obtain the following statement: given morphisms $f : X \to Y$ and $g : Y \to Z$ of locally Noetherian formal algebraic spaces over $S$ such that \begin{enumerate} \item the equivalent conditions of Lemma \ref{lemma-property-defines-property-morphisms} hold for $f$ and $P$, \item the equivalent conditions of Lemma \ref{lemma-property-defines-property-morphisms} hold for $g$ and $Q$, \end{enumerate} then the equivalent conditions of Lemma \ref{lemma-property-defines-property-morphisms} hold for $\text{pr}_2 : X \times_Y Z \to Z$ and $P$.


formal-spaces-remark-composition-variant-adic-star

Let $P$ be a local property of morphisms of $\textit{WAdm}^{adic*}$, see Remark \ref{remark-variant-adic-star}. We say $P$ is {\it stable under composition} if given $B \to A$ and $C \to B$ in $\textit{WAdm}^{adic*}$ we have $P(B \to A) \wedge P(C \to B) \Rightarrow P(C \to A)$. In exactly the same way we obtain a variant of Lemma \ref{lemma-composition-property-morphisms} for morphisms between locally adic* formal algebraic spaces over $S$.


formal-spaces-remark-composition-variant-Noetherian

Let $P$ be a local property of morphisms of $\textit{WAdm}^{Noeth}$, see Remark \ref{remark-variant-Noetherian}. We say $P$ is {\it stable under composition} if given $B \to A$ and $C \to B$ in $\textit{WAdm}^{Noeth}$ we have $P(B \to A) \wedge P(C \to B) \Rightarrow P(C \to A)$. In exactly the same way we obtain a variant of Lemma \ref{lemma-composition-property-morphisms} for morphisms between locally Noetherian formal algebraic spaces over $S$.


formal-spaces-remark-permanence-variant-adic-star

Let $P$ be a local property of morphisms of $\textit{WAdm}^{adic*}$, see Remark \ref{remark-variant-adic-star}. We say $P$ {\it has the cancellation property} if given $B \to A$ and $C \to B$ in $\textit{WAdm}^{adic*}$ we have $P(C \to A) \wedge P(C \to B) \Rightarrow P(B \to A)$. In exactly the same way we obtain a variant of Lemma \ref{lemma-composition-property-morphisms} for morphisms between locally adic* formal algebraic spaces over $S$.


formal-spaces-remark-permanence-variant-Noetherian

Let $P$ be a local property of morphisms of $\textit{WAdm}^{Noeth}$, see Remark \ref{remark-variant-Noetherian}. We say $P$ {\it has the cancellation property} if given $B \to A$ and $C \to B$ in $\textit{WAdm}^{Noeth}$ we have $P(C \to B) \wedge P(C \to A) \Rightarrow P(C \to B)$. In exactly the same way we obtain a variant of Lemma \ref{lemma-composition-property-morphisms} for morphisms between locally Noetherian formal algebraic spaces over $S$.


formal-spaces-remark-universal-property

\begin{reference} \cite[Chapter 0, 7.5.3]{EGA} \end{reference} Let $A \to C$ be a continuous map of complete linearly topologized rings. Then any $A$-algebra map $A[x_1, \ldots x_r] \to C$ extends uniquely to a continuous map $A\{x_1, \ldots, x_r\} \to C$ on restricted power series.


formal-spaces-remark-I-adic-completion-and-restricted-power-series

Let $A$ be a ring and let $I \subset A$ be an ideal. If $A$ is $I$-adically complete, then the $I$-adic completion $A[x_1, \ldots, x_r]^\wedge$ of $A[x_1, \ldots, x_r]$ is the restricted power series ring over $A$ as a ring. However, it is not clear that $A[x_1, \ldots, x_r]^\wedge$ is $I$-adically complete. We think of the topology on $A\{x_1, \ldots, x_r\}$ as the limit topology (which is always complete) whereas we often think of the topology on $A[x_1, \ldots, x_r]^\wedge$ as the $I$-adic topology (not always complete). If $I$ is finitely generated, then $A\{x_1, \ldots, x_r\} = A[x_1, \ldots, x_r]^\wedge$ as topological rings, see Algebra, Lemma \ref{algebra-lemma-hathat-finitely-generated}.


formal-spaces-remark-questions

Let $A$ be a weakly admissible topological ring and let $(I_\lambda)$ be a fundamental system of weak ideals of definition. Let $X = \text{Spf}(A)$, in other words, $X$ is a McQuillan affine formal algebraic space. Let $f : Y \to X$ be a morphism of affine formal algebraic spaces. In general it will not be true that $Y$ is McQuillan. More specifically, we can ask the following questions: \begin{enumerate} \item Assume that $f : Y \to X$ is a closed immersion. Then $Y$ is McQuillan and $f$ corresponds to a continuous map $\varphi : A \to B$ of weakly admissible topological rings which is taut, whose kernel $K \subset A$ is a closed ideal, and whose image $\varphi(A)$ is dense in $B$, see Lemma \ref{lemma-closed-immersion-into-McQuillan}. What conditions on $A$ guarantee that $B = (A/K)^\wedge$ as in Example \ref{example-closed-immersion-from-quotient}? \item What conditions on $A$ guarantee that closed immersions $f : Y \to X$ correspond to quotients $A/K$ of $A$ by closed ideals, in other words, the corresponding continuous map $\varphi$ is surjective and open? \item Suppose that $f : Y \to X$ is of finite type. Then we get $Y = \colim \Spec(B_\lambda)$ where $(B_\lambda)$ is an object of $\mathcal{C}$ by Lemma \ref{lemma-category-affine-over}. In this case it is true that there exists a fixed integer $r$ such that $B_\lambda$ is generated by $r$ elements over $A/I_\lambda$ for all $\lambda$ (the argument is essentially already given in the proof of (1) $\Rightarrow$ (2) in Lemma \ref{lemma-topologically-finite-type-finite-type}). However, it is not clear that the projections $\lim B_\lambda \to B_\lambda$ are surjective, i.e., it is not clear that $Y$ is McQuillan. Is there an example where $Y$ is not McQuillan? \item Suppose that $f : Y \to X$ is of finite type and $Y$ is McQuillan. Then $f$ corresponds to a continuous map $\varphi : A \to B$ of weakly admissible topological rings. In fact $\varphi$ is taut and $B$ is topologically of finite type over $A$, see Lemma \ref{lemma-topologically-finite-type-finite-type}. In other words, $f$ factors as $$ Y \longrightarrow \mathbf{A}^r_X \longrightarrow X $$ where the first arrow is a closed immersion of McQuillan affine formal algebraic spaces. However, then questions (1) and (2) are in force for $Y \to \mathbf{A}^r_X$. \end{enumerate} Below we will answer these questions when $X$ is countably indexed, i.e., when $A$ has a countable fundamental system of open ideals. If you have answers to these questions in greater generality, or if you have counter examples, please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.


formal-spaces-remark-not-enough-sections

The structure sheaf does not always have ``enough sections''. In Examples, Section \ref{examples-section-affine-formal-algebraic-space} we have seen that there exist affine formal algebraic spaces which aren't McQuillan and there are even examples whose points are not separated by regular functions.


formal-spaces-remark-bad-quasi-coherent

Even if the structure sheaf has good properties, this does not mean there is a good theory of quasi-coherent modules. For example, in Examples, Section \ref{examples-section-nonabelian-QCoh} we have seen that for almost any Noetherian affine formal algebraic spaces the most natural notion of a quasi-coherent module leads to a category of modules which is not abelian.


pione-remark-covering-surjective

Under the correspondence of Lemma \ref{lemma-sheaves-point}, the coverings in the small \'etale site $\Spec(K)_\etale$ of $K$ correspond to surjective families of maps in $G\textit{-Sets}$.


pione-remark-colimits-commute-forgetful

Let $X$ be a scheme. Consider the natural functors $F_1 : \textit{F\'Et}_X \to \Sch$ and $F_2 : \textit{F\'Et}_X \to \Sch/X$. Then \begin{enumerate} \item The functors $F_1$ and $F_2$ commute with finite colimits. \item The functor $F_2$ commutes with finite limits, \item The functor $F_1$ commutes with connected finite limits, i.e., with equalizers and fibre products. \end{enumerate} The results on limits are immediate from the discussion in the proof of Lemma \ref{lemma-finite-etale-covers-limits-colimits} and Categories, Lemma \ref{categories-lemma-connected-limit-over-X}. It is clear that $F_1$ and $F_2$ commute with finite coproducts. By the dual of Categories, Lemma \ref{categories-lemma-characterize-left-exact} we need to show that $F_1$ and $F_2$ commute with coequalizers. In the proof of Lemma \ref{lemma-finite-etale-covers-limits-colimits} we saw that coequalizers in $\textit{F\'Et}_X$ look \'etale locally like this $$ \xymatrix{ \coprod_{j \in J} U \ar@<1ex>[r]^a \ar@<-1ex>[r]_b & \coprod_{i \in I} U \ar[r] & \coprod_{t \in \text{Coeq}(a, b)} U } $$ which is certainly a coequalizer in the category of schemes. Hence the statement follows from the fact that being a coequalizer is fpqc local as formulate precisely in Descent, Lemma \ref{descent-lemma-coequalizer-fpqc-local}.


pione-remark-variance

In the situation of Lemma \ref{lemma-fundamental-group-Galois-group} let us give a more explicit construction of the isomorphism $\text{Gal}(K^{sep}/K) \to \pi_1(X, \overline{x}) = \text{Aut}(F_{\overline{x}})$. Observe that $\text{Gal}(K^{sep}/K) = \text{Aut}(\overline{K}/K)$ as $\overline{K}$ is the perfection of $K^{sep}$. Since $F_{\overline{x}}(Y) = \Mor_X(\Spec(\overline{K}), Y)$ we may consider the map $$ \text{Aut}(\overline{K}/K) \times F_{\overline{x}}(Y) \to F_{\overline{x}}(Y), \quad (\sigma, \overline{y}) \mapsto \sigma \cdot \overline{y} = \overline{y} \circ \Spec(\sigma) $$ This is an action because $$ \sigma\tau \cdot \overline{y} = \overline{y} \circ \Spec(\sigma\tau) = \overline{y} \circ \Spec(\tau) \circ \Spec(\sigma) = \sigma \cdot (\tau \cdot \overline{y}) $$ The action is functorial in $Y \in \textit{F\'Et}_X$ and we obtain the desired map.


pione-remark-combine

Let $(A, \mathfrak m)$ be a complete local Noetherian ring and $f \in \mathfrak m$ nonzero. Suppose that $A_f$ is $(S_2)$ and every irreducible component of $\Spec(A)$ has dimension $\geq 4$. Then Lemma \ref{lemma-essentially-surjective-general-better} tells us that the category $$ \colim\nolimits_{U' \subset U\text{ open, }U_0 \subset U} \text{ category of schemes finite \'etale over }U' $$ is equivalent to the category of schemes finite \'etale over $U_0$. For example this holds if $A$ is a normal domain of dimension $\geq 4$!


stacks-cohomology-remark-bousfield-colocalization

Let $\mathcal{X}$ be an algebraic stack. The results of Lemmas \ref{lemma-adjoint} and \ref{lemma-adjoint-kernel-parasitic} imply that $$ \QCoh(\mathcal{O}_\mathcal{X}) = \mathcal{M}_\mathcal{X} / \text{Parasitic} \cap \mathcal{M}_\mathcal{X} $$ in words: the category of quasi-coherent modules is the category of locally quasi-coherent modules with the flat base change property divided out by the Serre subcategory consisting of parasitic objects. See Homology, Lemma \ref{homology-lemma-serre-subcategory-is-kernel}. The existence of the inclusion functor $i : \QCoh(\mathcal{O}_\mathcal{X}) \to \mathcal{M}_\mathcal{X}$ which is left adjoint to the quotient functor means that $\mathcal{M}_\mathcal{X} \to \QCoh(\mathcal{O}_\mathcal{X})$ is a {\it Bousfield colocalization} or a {\it right Bousfield localization} (insert future reference here). Our next goal is to show a similar result holds on the level of derived categories.


sites-cohomology-remark-before-Leray

As a consequence of the results above we find that Derived Categories, Lemma \ref{derived-lemma-compose-derived-functors} applies to a number of situations. For example, given a morphism $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ of ringed topoi we have $$ R\Gamma(\mathcal{D}, Rf_*\mathcal{F}) = R\Gamma(\mathcal{C}, \mathcal{F}) $$ for any sheaf of $\mathcal{O}_\mathcal{C}$-modules $\mathcal{F}$. Namely, for an injective $\mathcal{O}_\mathcal{X}$-module $\mathcal{I}$ the $\mathcal{O}_\mathcal{D}$-module $f_*\mathcal{I}$ is totally acyclic by Lemma \ref{lemma-direct-image-injective-sheaf} and a totally acyclic sheaf is acyclic for $\Gamma(\mathcal{D}, -)$ by Lemma \ref{lemma-limp-acyclic}.


sites-cohomology-remark-base-change

The construction of unbounded derived functor $Lf^*$ and $Rf_*$ allows one to construct the base change map in full generality. Namely, suppose that $$ \xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{g'} \ar[d]_{f'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^f \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^g & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) } $$ is a commutative diagram of ringed topoi. Let $K$ be an object of $D(\mathcal{O}_\mathcal{C})$. Then there exists a canonical base change map $$ Lg^*Rf_*K \longrightarrow R(f')_*L(g')^*K $$ in $D(\mathcal{O}_{\mathcal{D}'})$. Namely, this map is adjoint to a map $L(f')^*Lg^*Rf_*K \to L(g')^*K$. Since $L(f')^* \circ Lg^* = L(g')^* \circ Lf^*$ we see this is the same as a map $L(g')^*Lf^*Rf_*K \to L(g')^*K$ which we can take to be $L(g')^*$ of the adjunction map $Lf^*Rf_*K \to K$.


sites-cohomology-remark-compose-base-change

Consider a commutative diagram $$ \xymatrix{ (\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'}) \ar[r]_k \ar[d]_{f'} & (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B}) \ar[d]^f \\ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]^l \ar[d]_{g'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^g \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^m & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \\ } $$ of ringed topoi. Then the base change maps of Remark \ref{remark-base-change} for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \begin{align*} Lm^* \circ R(g \circ f)_* & = Lm^* \circ Rg_* \circ Rf_* \\ & \to Rg'_* \circ Ll^* \circ Rf_* \\ & \to Rg'_* \circ Rf'_* \circ Lk^* \\ & = R(g' \circ f')_* \circ Lk^* \end{align*} is the base change map for the rectangle. We omit the verification.


sites-cohomology-remark-compose-base-change-horizontal

Consider a commutative diagram $$ \xymatrix{ (\Sh(\mathcal{C}''), \mathcal{O}_{\mathcal{C}''}) \ar[r]_{g'} \ar[d]_{f''} & (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_g \ar[d]_{f'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^f \\ (\Sh(\mathcal{D}''), \mathcal{O}_{\mathcal{D}''}) \ar[r]^{h'} & (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^h & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) } $$ of ringed topoi. Then the base change maps of Remark \ref{remark-base-change} for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \begin{align*} L(h \circ h')^* \circ Rf_* & = L(h')^* \circ Lh^* \circ Rf_* \\ & \to L(h')^* \circ Rf'_* \circ Lg^* \\ & \to Rf''_* \circ L(g')^* \circ Lg^* \\ & = Rf''_* \circ L(g \circ g')^* \end{align*} is the base change map for the rectangle. We omit the verification.


sites-cohomology-remark-cup-product

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. The adjointness of $Lf^*$ and $Rf_*$ allows us to construct a relative cup product $$ Rf_*K \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*L \longrightarrow Rf_*(K \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} L) $$ in $D(\mathcal{O}_\mathcal{D})$ for all $K, L$ in $D(\mathcal{O}_\mathcal{C})$. Namely, this map is adjoint to a map $Lf^*(Rf_*K \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*L) \to K \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} L$ for which we can take the composition of the isomorphism $Lf^*(Rf_*K \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*L) = Lf^*Rf_*K \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} Lf^*Rf_*L$ (Lemma \ref{lemma-pullback-tensor-product}) with the map $Lf^*Rf_*K \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} Lf^*Rf_*L \to K \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} L$ coming from the counit $Lf^* \circ Rf_* \to \text{id}$.


sites-cohomology-remark-discuss-derived-limit

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_n)$ be an inverse system in $D(\mathcal{O})$. Set $K = R\lim K_n$. For each $n$ and $m$ let $\mathcal{H}^m_n = H^m(K_n)$ be the $m$th cohomology sheaf of $K_n$ and similarly set $\mathcal{H}^m = H^m(K)$. Let us denote $\underline{\mathcal{H}}^m_n$ the presheaf $$ U \longmapsto \underline{\mathcal{H}}^m_n(U) = H^m(U, K_n) $$ Similarly we set $\underline{\mathcal{H}}^m(U) = H^m(U, K)$. By Lemma \ref{lemma-sheafification-cohomology} we see that $\mathcal{H}^m_n$ is the sheafification of $\underline{\mathcal{H}}^m_n$ and $\mathcal{H}^m$ is the sheafification of $\underline{\mathcal{H}}^m$. Here is a diagram $$ \xymatrix{ K \ar@{=}[d] & \underline{\mathcal{H}}^m \ar[d] \ar[r] &  \mathcal{H}^m \ar[d] \\ R\lim K_n & \lim \underline{\mathcal{H}}^m_n \ar[r] &  \lim \mathcal{H}^m_n } $$ In general it may not be the case that $\lim \mathcal{H}^m_n$ is the sheafification of $\lim \underline{\mathcal{H}}^m_n$. If $U \in \mathcal{C}$, then we have short exact sequences \begin{equation} \label{equation-ses-Rlim-over-U} 0 \to R^1\lim \underline{\mathcal{H}}^{m - 1}_n(U) \to \underline{\mathcal{H}}^m(U) \to \lim \underline{\mathcal{H}}^m_n(U) \to 0 \end{equation} by Lemma \ref{lemma-RGamma-commutes-with-Rlim}.


sites-cohomology-remark-set-theoretic-LC

The category $\textit{LC}$ is a ``big'' category as its objects form a proper class. Similarly, the coverings form a proper class. Let us define the {\it size} of a topological space $X$ to be the cardinality of the set of points of $X$. Choose a function $Bound$ on cardinals, for example as in Sets, Equation (\ref{sets-equation-bound}). Finally, let $S_0$ be an initial set of objects of $\textit{LC}$, for example $S_0 = \{(\mathbf{R}, \text{euclidean topology})\}$. Exactly as in Sets, Lemma \ref{sets-lemma-construct-category} we can choose a limit ordinal $\alpha$ such that $\textit{LC}_\alpha = \textit{LC} \cap V_\alpha$ contains $S_0$ and is preserved under all countable limits and colimits which exist in $\textit{LC}$. Moreover, if $X \in \textit{LC}_\alpha$ and if $Y \in \textit{LC}$ and $\text{size}(Y) \leq Bound(\text{size}(X))$, then $Y$ is isomorphic to an object of $\textit{LC}_\alpha$. Next, we apply Sets, Lemma \ref{sets-lemma-coverings-site} to choose set $\text{Cov}$ of qc covering on $\textit{LC}_\alpha$ such that every qc covering in $\textit{LC}_\alpha$ is combinatorially equivalent to a covering this set. In this way we obtain a site $(\textit{LC}_\alpha, \text{Cov})$ which we will denote $\textit{LC}_{qc}$.


sites-cohomology-remark-projection-formula-for-internal-hom

Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. Let $K, L$ be objects of $D(\mathcal{O}_\mathcal{C})$. We claim there is a canonical map $$ Rf_*R\SheafHom(L, K) \longrightarrow R\SheafHom(Rf_*L, Rf_*K) $$ Namely, by (\ref{equation-internal-hom}) this is the same thing as a map $Rf_*R\SheafHom(L, K) \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*L \to Rf_*K$. For this we can use the composition $$ Rf_*R\SheafHom(L, K) \otimes_{\mathcal{O}_\mathcal{D}}^\mathbf{L} Rf_*L \to Rf_*(R\SheafHom(L, K) \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} L) \to Rf_*K $$ where the first arrow is the relative cup product (Remark \ref{remark-cup-product}) and the second arrow is $Rf_*$ applied to the canonical map $R\SheafHom(L, K) \otimes_{\mathcal{O}_\mathcal{C}}^\mathbf{L} L \to K$ coming from Lemma \ref{lemma-internal-hom-composition} (with $\mathcal{O}_\mathcal{C}$ in one of the spots).


sites-cohomology-remark-prepare-fancy-base-change

Let $h : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Let $K, L$ be objects of $D(\mathcal{O}')$. We claim there is a canonical map $$ Lh^*R\SheafHom(K, L) \longrightarrow R\SheafHom(Lh^*K, Lh^*L) $$ in $D(\mathcal{O})$. Namely, by (\ref{equation-internal-hom}) proved in Lemma \ref{lemma-internal-hom} such a map is the same thing as a map $$ Lh^*R\SheafHom(K, L) \otimes^\mathbf{L} Lh^*K \longrightarrow Lh^*L $$ The source of this arrow is $Lh^*(\SheafHom(K, L) \otimes^\mathbf{L} K)$ by Lemma \ref{lemma-pullback-tensor-product} hence it suffices to construct a canonical map $$ R\SheafHom(K, L) \otimes^\mathbf{L} K \longrightarrow L. $$ For this we take the arrow corresponding to $$ \text{id} : R\SheafHom(K, L) \longrightarrow R\SheafHom(K, L) $$ via (\ref{equation-internal-hom}).


sites-cohomology-remark-fancy-base-change

Suppose that $$ \xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_h \ar[d]_{f'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^f \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^g & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) } $$ is a commutative diagram of ringed topoi. Let $K, L$ be objects of $D(\mathcal{O}_\mathcal{C})$. We claim there exists a canonical base change map $$ Lg^*Rf_*R\SheafHom(K, L) \longrightarrow R(f')_*R\SheafHom(Lh^*K, Lh^*L) $$ in $D(\mathcal{O}_{\mathcal{D}'})$. Namely, we take the map adjoint to the composition \begin{align*} L(f')^*Lg^*Rf_*R\SheafHom(K, L) & = Lh^*Lf^*Rf_*R\SheafHom(K, L) \\ & \to Lh^*R\SheafHom(K, L) \\ & \to R\SheafHom(Lh^*K, Lh^*L) \end{align*} where the first arrow uses the adjunction mapping $Lf^*Rf_* \to \text{id}$ and the second arrow is the canonical map constructed in Remark \ref{remark-prepare-fancy-base-change}.


sites-cohomology-remark-when-derived-shriek-equal

Warning! Let $u : \mathcal{C} \to \mathcal{D}$, $g$, $\mathcal{O}_\mathcal{D}$, and $\mathcal{O}_\mathcal{C}$ be as in Lemma \ref{lemma-existence-derived-lower-shriek}. In general it is {\bf not} the case that the diagram $$ \xymatrix{ D(\mathcal{O}_\mathcal{C}) \ar[r]_{Lg_!} \ar[d]_{forget} & D(\mathcal{O}_\mathcal{D}) \ar[d]^{forget} \\ D(\mathcal{C}) \ar[r]^{Lg^{Ab}_!} & D(\mathcal{D}) } $$ commutes where the functor $Lg_!^{Ab}$ is the one constructed in Lemma \ref{lemma-existence-derived-lower-shriek} but using the constant sheaf $\mathbf{Z}$ as the structure sheaf on both $\mathcal{C}$ and $\mathcal{D}$. In general it isn't even the case that $g_! = g_!^{Ab}$ (see Modules on Sites, Remark \ref{sites-modules-remark-when-shriek-equal}), but this phenomenon {\bf can occur even if $g_! = g_!^{Ab}$}! Namely, the construction of $Lg_!$ in the proof of Lemma \ref{lemma-existence-derived-lower-shriek} shows that $Lg_!$ agrees with $Lg_!^{\textit{Ab}}$ if and only if the canonical maps $$ Lg^{Ab}_!j_{U!}\mathcal{O}_U \longrightarrow j_{u(U)!}\mathcal{O}_{u(U)} $$ are isomorphisms in $D(\mathcal{D})$ for all objects $U$ in $\mathcal{C}$. In general all we can say is that there exists a natural transformation $$ Lg_!^{Ab} \circ forget \longrightarrow forget \circ Lg_! $$


sites-cohomology-remark-fibred-category

Assumptions and notation as in Situation \ref{situation-fibred-category}. Note that setting $\mathcal{C}' = \mathcal{D}$ and $u$ equal to the structure functor of $\mathcal{C}$ gives a situation as in Situation \ref{situation-morphism-fibred-categories}. Hence Lemma \ref{lemma-properties-lower-shriek-fibred-category} tells us we have functors  $\pi_!$, $\pi_!^{\textit{Ab}}$, $L\pi_!$, and $L\pi_!^{\textit{Ab}}$ such that $forget \circ \pi_! = \pi_!^{\textit{Ab}} \circ forget$ and $forget \circ L\pi_! = L\pi_!^{\textit{Ab}} \circ forget$.


sites-cohomology-remark-morphism-fibred-categories

Assumptions and notation as in Situation \ref{situation-morphism-fibred-categories}. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$, let $\mathcal{F}'$ be an abelian sheaf on $\mathcal{C}'$, and let $t : \mathcal{F}' \to g^{-1}\mathcal{F}$ be a map. Then we obtain a canonical map $$ L\pi'_!(\mathcal{F}') \longrightarrow L\pi_!(\mathcal{F}) $$ by using the adjoint $g_!\mathcal{F}' \to \mathcal{F}$ of $t$, the map $Lg_!(\mathcal{F}') \to g_!\mathcal{F}'$, and the equality $L\pi'_! = L\pi_! \circ Lg_!$.


sites-cohomology-remark-map-evaluation-to-derived

Notation and assumptions as in Example \ref{example-category-to-point}. Let $\mathcal{F}^\bullet$ be a bounded complex of abelian sheaves on $\mathcal{C}$. For any object $U$ of $\mathcal{C}$ there is a canonical map $$ \mathcal{F}^\bullet(U) \longrightarrow L\pi_!(\mathcal{F}^\bullet) $$ in $D(\textit{Ab})$. If $\mathcal{F}^\bullet$ is a complex of $\underline{B}$-modules then this map is in $D(B)$. To prove this, note that we compute $L\pi_!(\mathcal{F}^\bullet)$ by taking a quasi-isomorphism $\mathcal{P}^\bullet \to \mathcal{F}^\bullet$ where $\mathcal{P}^\bullet$ is a complex of projectives. However, since the topology is chaotic this means that $\mathcal{P}^\bullet(U) \to \mathcal{F}^\bullet(U)$ is a quasi-isomorphism hence can be inverted in $D(\textit{Ab})$, resp.\ $D(B)$. Composing with the canonical map $\mathcal{P}^\bullet(U) \to \pi_!(\mathcal{P}^\bullet)$ coming from the computation of $\pi_!$ as a colimit we obtain the desired arrow.


sites-cohomology-remark-simplicial-modules

Let $\mathcal{C} = \Delta$ and let $B$ be any ring. This is a special case of Example \ref{example-category-to-point} where the assumptions of Lemma \ref{lemma-compute-by-cosimplicial-resolution} hold. Namely, let $U_\bullet$ be the cosimplicial object of $\Delta$ given by the identity functor. To verify the condition we have to show that for $[m] \in \Ob(\Delta)$ the simplicial set $\Delta[m] : n \mapsto \Mor_\Delta([n], [m])$ is homotopy equivalent to a point. This is explained in Simplicial, Example \ref{simplicial-example-simplex-contractible}. \medskip\noindent In this situation the category $\textit{Mod}(\underline{B})$ is just the category of simplicial $B$-modules and the functor $L\pi_!$ sends a simplicial $B$-module $M_\bullet$ to its associated complex $s(M_\bullet)$ of $B$-modules. Thus the results above can be reinterpreted in terms of results on simplicial modules. For example a special case of Lemma \ref{lemma-eilenberg-zilber} is: if $M_\bullet$, $M'_\bullet$ are flat simplicial $B$-modules, then the complex $s(M_\bullet \otimes_B M'_\bullet)$ is quasi-isomorphic to the total complex associated to the double complex $s(M_\bullet) \otimes_B s(M'_\bullet)$. (Hint: use flatness to convert from derived tensor products to usual tensor products.) This is a special case of the Eilenberg-Zilber theorem which can be found in \cite{Eilenberg-Zilber}.


sites-cohomology-remark-O-homology-B-homology-general

Let $\mathcal{C}$ and $B$ be as in Example \ref{example-category-to-point}. Assume there exists a cosimplicial object as in Lemma \ref{lemma-compute-by-cosimplicial-resolution}. Let $\mathcal{O} \to \underline{B}$ be a map sheaf of rings on $\mathcal{C}$ which induces an isomorphism $L\pi_!\mathcal{O} \to L\pi_!\underline{B}$. In this case we obtain an exact functor of triangulated categories $$ L\pi_! : D(\mathcal{O}) \longrightarrow D(B) $$ Namely, for any object $K$ of $D(\mathcal{O})$ we have $L\pi^{\textit{Ab}}_!(K) = L\pi^{\textit{Ab}}_!(K \otimes_{\mathcal{O}}^\mathbf{L} \underline{B})$ by Lemma \ref{lemma-O-homology-qis}. Thus we can define the displayed functor as the composition of $- \otimes^\mathbf{L}_\mathcal{O} \underline{B}$ with the functor $L\pi_! : D(\underline{B}) \to D(B)$. In other words, we obtain a $B$-module structure on $L\pi_!(K)$ coming from the (canonical, functorial) identification of $L\pi_!(K)$ with $L\pi_!(K \otimes_\mathcal{O}^\mathbf{L} \underline{B})$ of the lemma.


sites-cohomology-remark-homology-augmentation

Let $\mathcal{C}$ be a site. Let $\epsilon : \mathcal{A}_\bullet \to \mathcal{O}$ be an augmentation (Simplicial, Definition \ref{simplicial-definition-augmentation}) in the category of sheaves of rings. Assume $\epsilon$ induces a quasi-isomorphism $s(\mathcal{A}_\bullet) \to \mathcal{O}$. In this case we obtain an exact functor of triangulated categories $$ L\pi_! : D(\mathcal{A}_\bullet) \longrightarrow D(\mathcal{O}) $$ Namely, for any object $K$ of $D(\mathcal{A}_\bullet)$ we have $L\pi_!(K) = L\pi_!(K \otimes_{\mathcal{A}_\bullet}^\mathbf{L} \mathcal{O})$ by Lemma \ref{lemma-base-change-by-qis}. Thus we can define the displayed functor as the composition of $- \otimes^\mathbf{L}_{\mathcal{A}_\bullet} \mathcal{O}$ with the functor $L\pi_! : D(\Delta \times \mathcal{C}, \pi^{-1}\mathcal{O}) \to D(\mathcal{O})$ of Remark \ref{remark-fibred-category}. In other words, we obtain a $\mathcal{O}$-module structure on $L\pi_!(K)$ coming from the (canonical, functorial) identification of $L\pi_!(K)$ with $L\pi_!(K \otimes_{\mathcal{A}_\bullet}^\mathbf{L} \mathcal{O})$ of the lemma.


sites-cohomology-remark-compatible-with-diagram

The map (\ref{equation-projection-formula-map}) is compatible with the base change map of Remark \ref{remark-base-change} in the following sense. Namely, suppose that $$ \xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{g'} \ar[d]_{f'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^f \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^g & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) } $$ is a commutative diagram of ringed topoi. Let $E \in D(\mathcal{O}_\mathcal{C})$ and $K \in D(\mathcal{O}_\mathcal{D})$. Then the diagram $$ \xymatrix{ Lg^*(Rf_*E \otimes^\mathbf{L}_{\mathcal{O}_\mathcal{D}} K) \ar[r]_p \ar[d]_t & Lg^*Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_\mathcal{C}} Lf^*K) \ar[d]_b \\ Lg^*Rf_*E \otimes^\mathbf{L}_{\mathcal{O}_{\mathcal{D}'}} Lg^*K \ar[d]_b & Rf'_*L(g')^*(E \otimes^\mathbf{L}_{\mathcal{O}_\mathcal{C}} Lf^*K) \ar[d]_t \\ Rf'_*L(g')^*E \otimes^\mathbf{L}_{\mathcal{O}_{\mathcal{D}'}} Lg^*K \ar[rd]_p & Rf'_*(L(g')^*E \otimes^\mathbf{L}_{\mathcal{O}_{\mathcal{D}'}} L(g')^*Lf^*K) \ar[d]_c \\ & Rf'_*(L(g')^*E \otimes^\mathbf{L}_{\mathcal{O}_{\mathcal{D}'}} L(f')^*Lg^*K) } $$ is commutative. Here arrows labeled $t$ are gotten by an application of Lemma \ref{lemma-pullback-tensor-product}, arrows labeled $b$ by an application of Remark \ref{remark-base-change}, arrows labeled $p$ by an application of (\ref{equation-projection-formula-map}), and $c$ comes from $L(g')^* \circ Lf^* = L(f')^* \circ Lg^*$. We omit the verification.


spaces-limits-remark-limit-preserving

Here is an important special case of Proposition \ref{proposition-characterize-locally-finite-presentation}. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is locally of finite presentation over $S$ if and only if $X$, as a functor $(\Sch/S)^{opp} \to \textit{Sets}$, is limit preserving. Compare with Limits, Remark \ref{limits-remark-limit-preserving}. In fact, we will see in Lemma \ref{lemma-surjection-is-enough} below that it suffices if the map $$ \colim X(T_i) \longrightarrow X(T) $$ is surjective whenever $T = \lim T_i$ is a directed limit of affine schemes over $S$.


spaces-limits-remark-cannot-embed-in-general

We have seen in Examples, Section \ref{examples-section-embedding-affines} that Lemma \ref{lemma-embedding-into-affine-over-ls-qs} does not hold if we drop the assumption that $X$ be locally separated. This raises the question: Does Lemma \ref{lemma-embedding-into-affine-over-ls-qs} hold if we drop the assumption that $X$ be quasi-separated? If you know the answer, please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.


spaces-limits-remark-finite-type-gives-well-defined-system

In Situation \ref{situation-limit-noetherian} Lemmas \ref{lemma-good-diagram}, \ref{lemma-limit-from-good-diagram}, and \ref{lemma-morphism-good-diagram} tell us that the category of algebraic spaces quasi-separated and of finite type over $B$ is equivalent to certain types of inverse systems of algebraic spaces over $(B_i)_{i \in I}$, namely the ones produced by applying Lemma \ref{lemma-limit-from-good-diagram} to a diagram of the form (\ref{equation-good-diagram}). For example, given $X \to B$ finite type and quasi-separated if we choose two different diagrams $X \to V_1 \to B_{i_1}$ and $X \to V_2 \to B_{i_2}$ as in (\ref{equation-good-diagram}), then applying Lemma \ref{lemma-morphism-good-diagram} to $\text{id}_X$ (in two directions) we see that the corresponding limit descriptions of $X$ are canonically isomorphic (up to shrinking the directed set $I$). And so on and so forth.


stacks-geometry-remark-upgrade

In Situation \ref{situation-versal} let $x_0 : \Spec(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Let $A$ be a versal ring to $\mathcal{X}$ at $x_0$. By Artin's Axioms, Lemma \ref{artin-lemma-effective} our versal formal object in fact comes from a morphism $$ \Spec(A) \longrightarrow \mathcal{X} $$ over $S$. Moreover, the results above each can be upgraded to be compatible with this morphism. Here is a list: \begin{enumerate} \item in Lemma \ref{lemma-versal-ring} the isomorphism $A \cong A'[[t_1, \ldots, t_r]]$ or $A' \cong A[[t_1, \ldots, t_r]]$ may be chosen compatible with these morphisms, \item in Lemma \ref{lemma-versal-ring-field-extension} the homomorphism $A \to A'$ may be chosen compatible with these morphisms, \item in Lemma \ref{lemma-compare-versal-ring-completion} the morphism $\Spec(\mathcal{O}_{U, u_0}^\wedge) \to \mathcal{X}$ is the composition of the canonical map $\Spec(\mathcal{O}_{U, u_0}^\wedge) \to U$ and the given map $U \to \mathcal{X}$, \item in Lemma \ref{lemma-Artin-approximation-by-smooth-morphism} the isomorphism $\mathcal{O}_{U, u_0}^\wedge \cong A$ may be chosen so $\Spec(A) \to \mathcal{X}$ corresponds to the canonical map in the item above. \end{enumerate} In each case the statement follows from the fact that our maps are compatible with versal formal elements; we note however that the implied diagrams are $2$-commutative only up to a (noncanonical) choice of a $2$-arrow. Still, this means that the implied map $A' \to A$ or $A \to A'$ in (1) is well defined up to formal homotopy, see Formal Deformation Theory, Lemma \ref{formal-defos-lemma-versal-unique-up-to-homotopy}.


stacks-geometry-remark-groupoid-defo

In Situation \ref{situation-versal} let $x_0 : \Spec(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. By Lemma \ref{lemma-deformation-category} and Formal Deformation Theory, Theorem \ref{formal-defos-theorem-presentation-deformation-groupoid} we know that $\mathcal{F}_{\mathcal{X}, k, x_0}$ has a presentation by a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$. Unwinding the definitions, this means we can choose \begin{enumerate} \item a Noetherian complete local $\Lambda$-algebra $A$ with residue field $k$ and a versal formal object $\xi$ of $\mathcal{F}_{\mathcal{X}, k, x_0}$ over $A$, \item a Noetherian complete local $\Lambda$-algebra $B$ with residue field $k$ and an isomorphism $$ \underline{B}|_{\mathcal{C}_\Lambda} \longrightarrow \underline{A}|_{\mathcal{C}_\Lambda} \times_{\underline{\xi}, \mathcal{F}_{\mathcal{X}, k, x_0}, \underline{\xi}} \underline{A}|_{\mathcal{C}_\Lambda} $$ \end{enumerate} The projections correspond to formally smooth maps $t : A \to B$ and $s : A \to B$ (because $\xi$ is versal). There is a map $c : B \to B \widehat{\otimes}_{s, A, t} B$ which turns $(A, B, s, t, c)$ into a cogroupoid in the category of Noetherian complete local $\Lambda$-algebras with residue field $k$ (on prorepresentable functors this map is constructed in Formal Deformation Theory, Lemma \ref{formal-defos-lemma-presentation-construction}). Finally, the cited theorem tells us that $\xi$ induces an equivalence $$ [\underline{A}|_{\mathcal{C}_\Lambda} / \underline{B}|_{\mathcal{C}_\Lambda}] \longrightarrow \mathcal{F}_{\mathcal{X}, k, x_0} $$ of groupoids cofibred over $\mathcal{C}_\Lambda$. In fact, we also get an equivalence $$ [\underline{A}/\underline{B}] \longrightarrow \widehat{\mathcal{F}}_{\mathcal{X}, k, x_0} $$ of groupoids cofibred over the completed category $\widehat{\mathcal{C}}_\Lambda$ (see discussion in Formal Deformation Theory, Section \ref{formal-defos-section-prorepresentable-groupoids-in-functors} as to why this works). Of course $A$ is a versal ring to $\mathcal{X}$ at $x_0$.


stacks-geometry-remark-dimension-algebraic-space

In general, the dimension of the algebraic space $X$ at a point $x$ may not coincide with the dimension of the underlying topological space $|X|$ at $x$.  E.g.\ if $k$ is a field of characteristic zero and $X =  \mathbf{A}^1_k / \mathbf{Z}$, then $X$ has dimension $1$ (the dimension of $\mathbf{A}^1_k$) at each of its points, while $|X|$ has the indiscrete topology, and hence is of Krull dimension zero. On the other hand, in Algebraic Spaces, Example \ref{spaces-example-infinite-product} there is given an example of an algebraic space which is of dimension $0$ at each of its points, while $|X|$ is irreducible of Krull dimension $1$, and admits a generic point (so that the dimension of $|X|$ at any of its points is $1$); see also the discussion of this example in Properties of Spaces, Section \ref{spaces-properties-section-dimension}. \medskip\noindent On the other hand, if $X$ is a {\it decent} algebraic space, in the sense of Decent Spaces, Definition \ref{decent-spaces-definition-very-reasonable} (in particular, if $X$ is quasi-separated; see Decent Spaces, Section \ref{decent-spaces-section-reasonable-decent}) then in fact the dimension of $X$ at $x$ does coincide with the dimension of $|X|$ at $x$; see Decent Spaces, Lemma \ref{decent-spaces-lemma-dimension-decent-space}.


stacks-geometry-remark-relative-dimension

(1) One easily verifies (for example, by using the invariance of the relative dimension of locally of finite type morphisms of schemes under base-change; see for example Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-after-base-change}) that $\dim_t(T_x)$ is well-defined, independently of the choices used to compute it. \medskip\noindent (2) In the case that $\mathcal{X}$ is also an algebraic space, it is straightforward to confirm that this definition agrees with the definition of relative dimension given in Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-dimension-fibre}.


stacks-geometry-remark-dimension-DM

For Deligne--Mumford stacks which are suitably decent (e.g.\ quasi-separated), it will again be the case that $\dim_x(\mathcal{X})$ coincides with the topologically defined quantity $\dim_x |\mathcal{X}|$.  However, for more general Artin stacks, this will typically not be the case.  For example, if $\mathcal{X} = [\mathbf{A}^1/\mathbf{G}_m]$ (over some field, with the quotient being taken with respect to the usual multiplication action of $\mathbf{G}_m$ on $\mathbf{A}^1$), then  $|\mathcal{X}|$ has two points, one the specialisation of the other (corresponding to the two orbits of $\mathbf{G}_m$ on $\mathbf{A}^1$), and hence is of dimension $1$ as a topological space; but $\dim_x (\mathcal{X}) = 0$ for both points $x \in |\mathcal{X}|$. (An even more extreme example is given by the classifying space $[\Spec k/\mathbf{G}_m]$, whose dimension at its unique point is equal to $-1$.)


stacks-geometry-remark-dimension-tangent-space-well-defined

Standard manipulations show that $\dim_t(\mathcal{T}_x)$ is well-defined, independently of the choices made to compute it.


stacks-geometry-remark-negative-dimension

We note that in the context of the preceding lemma, it need not be that $\dim \mathcal{T} \geq \dim \mathcal{Z}$; this does not contradict the inequality in the statement of the lemma, because the fibres of the morphism $f$ are again algebraic stacks, and so may have negative dimension.  This is illustrated by taking $k$ to be a field, and applying the lemma to the morphism $[\Spec k/\mathbf{G}_m] \to \Spec k$. \medskip\noindent If the morphism $f$ in the statement of the lemma is assumed to be quasi-DM (in the sense of Morphisms of Stacks, Definition \ref{stacks-morphisms-definition-separated}; e.g.\ morphisms that are representable by algebraic spaces are quasi-DM), then the fibres of the morphism over points of the target are quasi-DM algebraic stacks, and hence are of non-negative dimension.  In this case, the lemma implies that indeed $\dim \mathcal{T} \geq \dim \mathcal{Z}$.  In fact, we obtain the following more general result.


spaces-morphisms-remark-immersion

Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. Since $i$ is a monomorphism we may think of $|Z|$ as a subset of $|X|$; in the rest of this remark we do so. Let $\partial |Z|$ be the boundary of $|Z|$ in the topological space $|X|$. In a formula $$ \partial |Z| = \overline{|Z|} \setminus |Z|. $$ Let $\partial Z$ be the reduced closed subspace of $X$ with $|\partial Z| = \partial |Z|$ obtained by taking the reduced induced closed subspace structure, see Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}. By construction we see that $|Z|$ is closed in $|X| \setminus |\partial Z| = |X \setminus \partial Z|$. Hence it is true that any immersion of algebraic spaces can be factored as a closed immersion followed by an open immersion (but not the other way in general, see Morphisms, Example \ref{morphisms-example-thibaut}).


spaces-morphisms-remark-space-structure-locally-closed-subset

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a locally closed subset. Let $\partial T$ be the boundary of $T$ in the topological space $|X|$. In a formula $$ \partial T = \overline{T} \setminus T. $$ Let $U \subset X$ be the open subspace of $X$ with $|U| = |X| \setminus \partial T$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-open-subspaces}. Let $Z$ be the reduced closed subspace of $U$ with $|Z| = T$ obtained by taking the reduced induced closed subspace structure, see Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}. By construction $Z \to U$ is a closed immersion of algebraic spaces and $U \to X$ is an open immersion, hence $Z \to X$ is an immersion of algebraic spaces over $S$ (see Spaces, Lemma \ref{spaces-lemma-composition-immersions}). Note that $Z$ is a reduced algebraic space and that $|Z| = T$ as subsets of $|X|$. We sometimes say $Z$ is the {\it reduced induced subspace structure} on $T$.


spaces-morphisms-remark-universally-injective-not-separated

A universally injective morphism of schemes is separated, see Morphisms, Lemma \ref{morphisms-lemma-universally-injective-separated}. This is not the case for morphisms of algebraic spaces. Namely, the algebraic space $X = \mathbf{A}^1_k/\{x \sim -x \mid x \not = 0\}$ constructed in Spaces, Example \ref{spaces-example-affine-line-involution} comes equipped with a morphism $X \to \mathbf{A}^1_k$ which maps the point with coordinate $x$ to the point with coordinate $x^2$. This is an isomorphism away from $0$, and there is a unique point of $X$ lying above $0$. As $X$ isn't separated this is a universally injective morphism of algebraic spaces which is not separated.


spaces-morphisms-remark-factorization-quasi-compact-quasi-separated

Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Then $f$ has a canonical factorization $$ Y \longrightarrow \underline{\Spec}_X(f_*\mathcal{O}_Y) \longrightarrow X $$ This makes sense because $f_*\mathcal{O}_Y$ is quasi-coherent by Lemma \ref{lemma-pushforward}. The morphism $Y \to \underline{\Spec}_X(f_*\mathcal{O}_Y)$ comes from the canonical $\mathcal{O}_Y$-algebra map $f^*f_*\mathcal{O}_Y \to \mathcal{O}_Y$ which corresponds to a canonical morphism $Y \to Y \times_X \underline{\Spec}_X(f_*\mathcal{O}_Y)$ over $Y$ (see Lemma \ref{lemma-affine-equivalence-algebras}) whence a factorization of $f$ as above.


spaces-morphisms-remark-composition-P

Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of schemes which is \'etale local on the source-and-target. Suppose that moreover $\mathcal{P}$ is stable under compositions. Then the class of morphisms of algebraic spaces having property $\mathcal{P}$ is stable under composition.


spaces-morphisms-remark-base-change-P

Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of schemes which is \'etale local on the source-and-target. Suppose that moreover $\mathcal{P}$ is stable under base change. Then the class of morphisms of algebraic spaces having property $\mathcal{P}$ is stable under base change.


spaces-morphisms-remark-when-apply

We will apply Lemma \ref{lemma-local-source-target-global-implies-local} above to all cases listed in Descent, Remark \ref{descent-remark-list-local-source-target} except ``flat''. In each case we will do this by defining $f$ to have property $\mathcal{P}$ at $x$ if $f$ has $\mathcal{P}$ in a neighbourhood of $x$.


weil-remark-transpose

Let $X$ and $Y$ be smooth projective schemes over $k$. Assume $X$ is equidimensional of dimension $d$ and $Y$ is equidimensional of dimension $e$. Then the isomorphism $X \times Y \to Y \times X$ switching the factors determines an isomorphism $$ \text{Corr}^r(X, Y) \longrightarrow \text{Corr}^{d - e + r}(Y, X),\quad c \longmapsto c^t $$ called the {\it transpose}. It acts on cycles as well as cycle classes. An example which is sometimes useful, is the transpose $[\Gamma_f]^t = [\Gamma_f^t]$ of the graph of a morphism $f : Y \to X$.


weil-remark-lefschetz-tate

Let $X = \mathbf{P}^1_k$ and $c_2$ be as in Example \ref{example-decompose-P1}. In the literature the motive $(X, c_2, 0)$ is sometimes called the {\it Lefschetz motive} and depending on the reference the notation $L$, $\mathbf{L}$, $\mathbf{Q}(-1)$, or $h^2(\mathbf{P}^1_k)$ may be used to denote it. By Lemma \ref{lemma-inverse-h2} the Lefschetz motive is isomorphic to $\mathbf{1}(-1)$. Hence the Lefschetz motive is invertible (Categories, Definition \ref{categories-definition-invertible}) with inverse $\mathbf{1}(1)$. The motive $\mathbf{1}(1)$ is sometimes called the {\it Tate motive} and depending on the reference the notation $L^{-1}$, $\mathbf{L}^{-1}$, $\mathbf{T}$, or $\mathbf{Q}(1)$ may be used to denote it.


weil-remark-replace-cup-product-classical

Let $X$ be a smooth projective variety. We obtain maps $$ H^*(X) \otimes_F H^*(X) \longrightarrow H^*(X \times X) \xrightarrow{\Delta^*} H^*(X) $$ where the first arrow is as in axiom (B) and $\Delta^*$ is pullback along the diagonal morphism $\Delta : X \to X \times X$. The composition is the cup product as pullback is an algebra homomorphism and $\text{pr}_i \circ \Delta = \text{id}$. On the other hand, given cycles $\alpha, \beta$ on $X$  the intersection product is defined by the formula $$ \alpha \cdot \beta = \Delta^!(\alpha \times \beta) $$ In other words, $\alpha \cdot \beta$ is the pullback of the exterior product $\alpha \times \beta$ on $X \times X$ by the diagonal. Note also that $\alpha \times \beta = \text{pr}_1^*\alpha \cdot \text{pr}_2^*\beta$ in $\CH^*(X \times X)$ (we omit the proof). Hence, given axiom (C)(a), axiom (C)(c) is equivalent to the statement that $\gamma$ is compatible with exterior product in the sense that $\gamma(\alpha \times \beta)$ is equal to $\text{pr}_1^*\gamma(\alpha) \cup \text{pr}_2^*\gamma(\beta)$. This is how axiom (C)(c) is formulated in \cite{Kleiman-cycles}.


weil-remark-replace-cup-product

Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C)(a). Let $X$ be a smooth projective scheme over $k$. We obtain maps $$ H^*(X) \otimes_F H^*(X) \longrightarrow H^*(X \times X) \xrightarrow{\Delta^*} H^*(X) $$ where the first arrow is as in axiom (B) and $\Delta^*$ is pullback along the diagonal morphism $\Delta : X \to X \times X$. The composition is the cup product as pullback is an algebra homomorphism and $\text{pr}_i \circ \Delta = \text{id}$. On the other hand, given cycles $\alpha, \beta$ on $X$  the intersection product is defined by the formula $$ \alpha \cdot \beta = \Delta^!(\alpha \times \beta) $$ In other words, $\alpha \cdot \beta$ is the pullback of the exterior product $\alpha \times \beta$ on $X \times X$ by the diagonal. Note also that $\alpha \times \beta = \text{pr}_1^*\alpha \cdot \text{pr}_2^*\beta$ in $\CH^*(X \times X)$ (we omit the proof). Hence, given axiom (C)(a), axiom (C)(c) is equivalent to the statement that $\gamma$ is compatible with exterior product in the sense that $\gamma(\alpha \times \beta)$ is equal to $\text{pr}_1^*\gamma(\alpha) \cup \text{pr}_2^*\gamma(\beta)$.


weil-remark-betti-numbers-in-some-sense

Let $H^*$ be a Weil cohomology theory (Definition \ref{definition-weil-cohomology-theory}). Let $X$ be a geometrically irreducible smooth projective scheme of dimension $d$ over $k'$ with $k'/k$ a finite separable extension of fields. Suppose that $$ H^0(\Spec(k')) = F_1 \times \ldots \times F_r $$ for some fields $F_i$. Then we accordingly can write $$ H^*(X) = \prod\nolimits_{i = 1, \ldots, r} H^*(X) \otimes_{H^0(\Spec(k'))} F_i $$ Now, our final assumption in Definition \ref{definition-weil-cohomology-theory} tells us that $H^0(X)$ is free of rank $1$ over $\prod F_i$. In other words, each of the factors $H^0(X) \otimes_{H^0(\Spec(k'))} F_i$ has dimension $1$ over $F_i$. Poincar\'e duality then tells us that the same is true for cohomology in degree $2d$. What isn't clear however is that the same holds in other degrees. Namely, we don't know that given $0 < n < \dim(X)$ the integers $$ \dim_{F_i} H^n(X) \otimes_{H^0(\Spec(k'))} F_i $$ are independent of $i$! This question is closely related to the following open question: given an algebraically closed base field $\overline{k}$, a field of characteristic zero $F$, a classical Weil cohomology theory $H^*$ over $\overline{k}$ with coefficient field $F$, and a smooth projective variety $X$ over $\overline{k}$ is it true that the betti numbers of $X$ $$ \beta_i = \dim_F H^i(X) $$ are independent of $F$ and the Weil cohomology theory $H^*$?


weil-remark-trace

Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Let $X$ be a smooth projective scheme over $k$ which is nonempty and equidimensional of dimension $d$. Combining what was said in the proofs of Lemma \ref{lemma-poincare-duality} and Homology, Lemma \ref{homology-lemma-left-dual-graded-vector-spaces} we see that $$ \gamma([\Delta]) \in \bigoplus\nolimits_i H^i(X) \otimes H^{2d - i}(X)(d) $$ defines a perfect duality between $H^i(X)$ and $H^{2d - i}(X)(d)$ for all $i$. In particular, the linear map $\int_X = \lambda : H^{2d}(X)(d) \to F$ of axiom (A6) is unique! We will call the linear map $\int_X$ the trace map of $X$ from now on.


morphisms-remark-direct-argument

We can also argue directly that (2) implies (1) in Lemma \ref{lemma-characterize-affine} above as follows. Assume $S = \bigcup W_j$ is an affine open covering such that each $f^{-1}(W_j)$ is affine. First argue that $\mathcal{A} = f_*\mathcal{O}_X$ is quasi-coherent as in the proof above. Let $\Spec(R) = V \subset S$ be affine open. We have to show that $f^{-1}(V)$ is affine. Set $A = \mathcal{A}(V) = f_*\mathcal{O}_X(V) = \mathcal{O}_X(f^{-1}(V))$. By Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} there is a canonical morphism $\psi : f^{-1}(V) \to \Spec(A)$ over $\Spec(R) = V$. By Schemes, Lemma \ref{schemes-lemma-good-subcover} there exists an integer $n \geq 0$, a standard open covering $V = \bigcup_{i = 1, \ldots, n} D(h_i)$, $h_i \in R$, and a map $a : \{1, \ldots, n\} \to J$ such that each $D(h_i)$ is also a standard open of the affine scheme $W_{a(i)}$. The inverse image of a standard open under a morphism of affine schemes is standard open, see Algebra, Lemma \ref{algebra-lemma-spec-functorial}. Hence we see that $f^{-1}(D(h_i))$ is a standard open of $f^{-1}(W_{a(i)})$, in particular that $f^{-1}(D(h_i))$ is affine. Because $\mathcal{A}$ is quasi-coherent we have $A_{h_i} = \mathcal{A}(D(h_i)) = \mathcal{O}_X(f^{-1}(D(h_i)))$, so $f^{-1}(D(h_i))$ is the spectrum of $A_{h_i}$. It follows that the morphism $\psi$ induces an isomorphism of the open $f^{-1}(D(h_i))$ with the open $\Spec(A_{h_i})$ of $\Spec(A)$. Since $f^{-1}(V) = \bigcup f^{-1}(D(h_i))$ and $\Spec(A) = \bigcup \Spec(A_{h_i})$ we win.


morphisms-remark-affine-s-opens-cover-family

In Properties, Lemma \ref{properties-lemma-affine-s-opens-cover-quasi-separated} we see that a scheme which has an ample invertible module is separated. This is wrong for schemes having an ample family of invertible modules. Namely, let $X$ be as in Schemes, Example \ref{schemes-example-affine-space-zero-doubled} with $n = 1$, i.e., the affine line with zero doubled.  We use the notation of that example except that we write $x$ for $x_1$ and $y$ for $y_1$.  There is, for every integer $n$, an invertible sheaf $\mathcal{L}_n$ on $X$ which is trivial on $X_1$ and $X_2$ and whose transition function $U_{12} \to U_{21}$ is $f(x) \mapsto y^n f(y)$.  The global sections of $\mathcal{L}_n$ are pairs $(f(x), g(y)) \in k[x] \oplus k[y]$ such that $y^n f(y) = g(y)$.  The sections $s = (1, y)$ of $\mathcal{L}_1$ and $t = (x, 1)$ of $\mathcal{L}_{-1}$ determine an open affine cover because $X_s = X_1$ and $X_t = X_2$.  Therefore $X$ has an ample family of invertible modules but it is not separated.


morphisms-remark-flattening

The results above are a first step towards more refined flattening techniques for morphisms of schemes. The article \cite{GruRay} by Raynaud and Gruson contains many wonderful results in this direction.


morphisms-remark-differentials-glue

The lemma above gives a second way of constructing the module of differentials. Namely, let $f : X \to S$ be a morphism of schemes. Consider the collection of all affine opens $U \subset X$ which map into an affine open of $S$. These form a basis for the topology on $X$. Thus it suffices to define $\Gamma(U, \Omega_{X/S})$ for such $U$. We simply set $\Gamma(U, \Omega_{X/S}) = \Omega_{A/R}$ if $A$, $R$ are as in Lemma \ref{lemma-differentials-affine} above. This works, but it takes somewhat more algebraic preliminaries to construct the restriction mappings and to verify the sheaf condition with this ansatz.


morphisms-remark-differentials-diagonal

Let $X \to S$ be a morphism of schemes. According to Lemma \ref{lemma-differential-product} we have $$ \Omega_{X \times_S X/S} = \text{pr}_1^*\Omega_{X/S} \oplus \text{pr}_2^*\Omega_{X/S} $$ On the other hand, the diagonal morphism $\Delta : X \to X \times_S X$ is an immersion, which locally has a left inverse. Hence by Lemma \ref{lemma-differentials-relative-immersion-section} we obtain a canonical short exact sequence $$ 0 \to \mathcal{C}_{X/X \times_S X} \to \Omega_{X/S} \oplus \Omega_{X/S} \to \Omega_{X/S} \to 0 $$ Note that the right arrow is $(1, 1)$ which is indeed a split surjection. On the other hand, by Lemma \ref{lemma-differentials-diagonal} we have an identification $\Omega_{X/S} = \mathcal{C}_{X/X \times_S X}$. Because we chose $\text{d}_{X/S}(f) = s_2(f) - s_1(f)$ in this identification it turns out that the left arrow is the map $(-1, 1)$\footnote{Namely, the local section $\text{d}_{X/S}(f) = 1 \otimes f - f \otimes 1$ of the ideal sheaf of $\Delta$ maps via $\text{d}_{X \times_S X/X}$ to the local section $1 \otimes 1 \otimes 1 \otimes f - 1 \otimes f \otimes 1 \otimes 1 -1 \otimes 1 \otimes f \otimes 1 + f \otimes 1 \otimes 1 \otimes 1 = \text{pr}_2^*\text{d}_{X/S}(f) - \text{pr}_1^*\text{d}_{X/S}(f)$.}.


morphisms-remark-base-change-differential-operators

Let $a : X \to S$ and $b : Y \to S$ be morphisms of schemes. Denote $p : X \times_S Y \to X$ and $q : X \times_S Y \to Y$ the projections. In this remark, given an $\mathcal{O}_X$-module $\mathcal{F}$ and an $\mathcal{O}_Y$-module $\mathcal{G}$ let us set $$ \mathcal{F} \boxtimes \mathcal{G} = p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G} $$ Denote $\mathcal{A}_{X/S}$ the additive category whose objects are quasi-coherent $\mathcal{O}_X$-modules and whose morphisms are differential operators of finite order on $X/S$. Similarly for $\mathcal{A}_{Y/S}$ and $\mathcal{A}_{X \times_S Y/S}$. The construction of Lemma \ref{lemma-base-change-differential-operators} determines a functor $$ \boxtimes :  \mathcal{A}_{X/S} \times \mathcal{A}_{Y/S} \longrightarrow \mathcal{A}_{X \times_S Y/S}, \quad (\mathcal{F}, \mathcal{G}) \longmapsto \mathcal{F} \boxtimes \mathcal{G} $$ which is bilinear on morphisms. If $X = \Spec(A)$, $Y = \Spec(B)$, and $S = \Spec(R)$, then via the identification of quasi-coherent sheaves with modules this functor is given by $(M, N) \mapsto M \otimes_R N$ on objects and sends the morphism $(D, D') : (M, N) \to (M', N')$ to $D \otimes D' : M \otimes_R N \to M' \otimes_R N'$.


morphisms-remark-check-val-on-open

The assumption on uniqueness of the dotted arrows in Lemma \ref{lemma-refined-valuative-criterion-universally-closed} is necessary (details omitted). Of course, uniqueness is guaranteed if $f$ is separated (Schemes, Lemma \ref{schemes-lemma-separated-implies-valuative}).


morphisms-remark-generalize-category

Here is a generalization of the category of irreducible schemes and dominant rational maps. For a scheme $X$ denote $X^0$ the set of points $x \in X$ with $\dim(\mathcal{O}_{X, x}) = 0$, in other words, $X^0$ is the set of generic points of irreducible components of $X$. Then we can consider the category with \begin{enumerate} \item objects are schemes $X$ such that every quasi-compact open has finitely many irreducible components, and \item morphisms from $X$ to $Y$ are rational maps $f : U \to Y$ from $X$ to $Y$ such that $f(U^0) = Y^0$. \end{enumerate} If $U \subset X$ is a dense open of a scheme, then $U^0 \subset X^0$ need not be an equality, but if $X$ is an object of our category, then this is the case. Thus given two morphisms in our category, the composition is well defined and a morphism in our category.


morphisms-remark-pseudo-morphisms

There is a variant of Definition \ref{definition-rational-map} where we consider only those morphism $U \to Y$ defined on scheme theoretically dense open subschemes $U \subset X$. We use Lemma \ref{lemma-intersection-scheme-theoretically-dense} to see that we obtain an equivalence relation. An equivalence class of these is called a {\it pseudo-morphism from $X$ to $Y$}. If $X$ is reduced the two notions coincide.


morphisms-remark-quasi-finite-finite-over-dense-open

An alternative to Lemma \ref{lemma-generically-finite} is the statement that a quasi-finite morphism is finite over a dense open of the target. This will be shown in More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-finite-finite-over-dense-open}.


morphisms-remark-definition-generically-finite

Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. There are (at least) two properties that we could use to define {\it generically finite} morphisms. These correspond to whether you want the property to be local on the source or local on the target: \begin{enumerate} \item (Local on the target; suggested by Ravi Vakil.) Assume every quasi-compact open of $Y$ has finitely many irreducible components (for example if $Y$ is locally Noetherian). The requirement is that the inverse image of each generic point is finite, see Lemma \ref{lemma-generically-finite}. \item (Local on the source.) The requirement is that there exists a dense open $U \subset X$ such that $U \to Y$ is locally quasi-finite. \end{enumerate} In case (1) the requirement can be formulated without the auxiliary condition on $Y$, but probably doesn't give the right notion for general schemes. Property (2) as formulated doesn't imply that the fibres over generic points are finite; however, if $f$ is quasi-compact and $Y$ is as in (1) then it does.


chow-remark-gersten-complex-milnor

For a field $k$ let us denote $K^M_*(k)$ the quotient of the tensor algebra on $k^*$ divided by the two-sided ideal generated by the elements $x \otimes 1 - x$. Thus $K^M_0(k) = \mathbf{Z}$, $K_1^M(k) = k^*$, and $$ K^M_2(k) = k^* \otimes_\mathbf{Z} k^* / \langle x \otimes 1 - x \rangle $$ If $(A, \mathfrak m)$ is a $1$-dimensional Noetherian local domain with fraction field $Q(A)$ and residue field $\kappa$ there is a tame symbol $$ \partial_A : K_{i + 1}^M(Q(A)) \to K_i^M(\kappa(\mathfrak m)) $$ You can use the method of Section \ref{section-tame-symbol} to define these maps, provided you extend the norm map to $K_i^M$ for all $i$. Next, let $X$ be a Noetherian scheme with a dimension function $\delta$. Then we can use these tame symbols to get the arrows in the following: $$ \bigoplus\nolimits_{\delta(x) = j + 1} K^M_{i + 1}(\kappa(x)) \longrightarrow \bigoplus\nolimits_{\delta(x) = j} K^M_i(\kappa(x)) \longrightarrow \bigoplus\nolimits_{\delta(x) = j - 1} K^M_{i - 1}(\kappa(x)) $$ However, it is not clear, if you define the maps as suggested above, that the composition is zero. When $i = 1$ and $j$ arbitrary, this follows from Lemma \ref{lemma-milnor-gersten-low-degree}. For excellent $X$ this follows from \cite{Kato-Milnor-K} modulo the verification that Kato's maps are the same as ours.


chow-remark-infinite-sums-rational-equivalences

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Suppose we have infinite collections $\alpha_i, \beta_i \in Z_k(X)$, $i \in I$ of $k$-cycles on $X$. Suppose that the supports of $\alpha_i$ and $\beta_i$ form locally finite collections of closed subsets of $X$ so that $\sum \alpha_i$ and $\sum \beta_i$ are defined as cycles. Moreover, assume that $\alpha_i \sim_{rat} \beta_i$ for each $i$. Then it is not clear that $\sum \alpha_i \sim_{rat} \sum \beta_i$. Namely, the problem is that the rational equivalences may be given by locally finite families $\{W_{i, j}, f_{i, j} \in R(W_{i, j})^*\}_{j \in J_i}$ but the union $\{W_{i, j}\}_{i \in I, j\in J_i}$ may not be locally finite. \medskip\noindent In many cases in practice, one has a locally finite family of closed subsets $\{T_i\}_{i \in I}$ such that $\alpha_i, \beta_i$ are supported on $T_i$ and such that $\alpha_i = \beta_i$ in $\CH_k(T_i)$, in other words, the families $\{W_{i, j}, f_{i, j} \in R(W_{i, j})^*\}_{j \in J_i}$ consist of subschemes $W_{i, j} \subset T_i$. In this case it is true that $\sum \alpha_i \sim_{rat} \sum \beta_i$ on $X$, simply because the family $\{W_{i, j}\}_{i \in I, j\in J_i}$ is automatically locally finite in this case.


chow-remark-good-cases-K-A

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. We will see later (in Lemma \ref{lemma-cycles-rational-equivalence-K-group}) that the map $$ \CH_k(X) \longrightarrow K_0(\textit{Coh}_{k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)) $$ of Lemma \ref{lemma-from-chow-to-K} is injective. Composing with the canonical map $$ K_0(\textit{Coh}_{k + 1}(X)/\textit{Coh}_{\leq k - 1}(X)) \longrightarrow K_0(\textit{Coh}(X)/\textit{Coh}_{\leq k - 1}(X)) $$ we obtain a canonical map $$ \CH_k(X) \longrightarrow K_0(\textit{Coh}(X)/\textit{Coh}_{\leq k - 1}(X)). $$ We have not been able to find a statement or conjecture in the literature as to whether this map is should be injective or not. It seems reasonable to expect the kernel of this map to be torsion. We will return to this question (insert future reference).


chow-remark-generalize-to-virtual

Let $X$ be a scheme locally of finite type over $S$ as in Situation \ref{situation-setup}. Let $(D, \mathcal{N}, \sigma)$ be a triple consisting of a locally principal (Divisors, Definition \ref{divisors-definition-effective-Cartier-divisor}) closed subscheme $i : D \to X$, an invertible $\mathcal{O}_D$-module $\mathcal{N}$, and a surjection $\sigma : \mathcal{N}^{\otimes -1} \to i^*\mathcal{I}_D$ of $\mathcal{O}_D$-modules\footnote{This condition assures us that if $D$ is an effective Cartier divisor, then $\mathcal{N} = \mathcal{O}_X(D)|_D$.}. Here $\mathcal{N}$ should be thought of as a {\it virtual normal bundle of $D$ in $X$}. The construction of $i^* : Z_{k + 1}(X) \to \CH_k(D)$ in Definition \ref{definition-gysin-homomorphism} generalizes to such triples, see Section \ref{section-gysin-higher-codimension}.


chow-remark-generalize-to-pseudo-divisor

Let $X$ be a scheme locally of finite type over $S$ as in Situation \ref{situation-setup}. In \cite{F} a {\it pseudo-divisor} on $X$ is defined as a triple $D = (\mathcal{L}, Z, s)$ where $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, $Z \subset X$ is a closed subset, and $s \in \Gamma(X \setminus Z, \mathcal{L})$ is a nowhere vanishing section. Similarly to the above, one can define for every $\alpha$ in $\CH_{k + 1}(X)$ a product $D \cdot \alpha$ in $\CH_k(Z \cap |\alpha|)$ where $|\alpha|$ is the support of $\alpha$.


chow-remark-gysin-on-cycles

Let $X \to S$, $\mathcal{L}$, $s$, $i : D \to X$ be as in Definition \ref{definition-gysin-homomorphism} and assume that $\mathcal{L}|_D \cong \mathcal{O}_D$. In this case we can define a canonical map $i^* : Z_{k + 1}(X) \to Z_k(D)$ on cycles, by requiring that $i^*[W] = 0$ whenever $W \subset D$ is an integral closed subscheme. The possibility to do this will be useful later on.


chow-remark-pullback-pairs

Let $f : X' \to X$ be a morphism of schemes locally of finite type over $S$ as in Situation \ref{situation-setup}. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition \ref{definition-gysin-homomorphism}. Then we can set $\mathcal{L}' = f^*\mathcal{L}$, $s' = f^*s$, and $D' = X' \times_X D = Z(s')$. This gives a commutative diagram $$ \xymatrix{ D' \ar[d]_g \ar[r]_{i'} & X' \ar[d]^f \\ D \ar[r]^i & X } $$ and we can ask for various compatibilities between $i^*$ and $(i')^*$.


chow-remark-when-isomorphism

We will see later (Lemma \ref{lemma-vectorbundle}) that if $X$ is a vector bundle of rank $r$ over $Y$ then the pullback map $\CH_k(Y) \to \CH_{k + r}(X)$ is an isomorphism. This is true whenever $X \to Y$ satisfies the assumptions of Lemma \ref{lemma-pullback-affine-fibres-surjective}, see \cite[Lemma 2.2]{Totaro-group}.


chow-remark-restriction-bivariant

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X \to Y$ and $Y' \to Y$ be morphisms of schemes locally of finite type over $S$. Let $X' = Y' \times_Y X$. Then there is an obvious restriction map $$ A^p(X \to Y) \longrightarrow A^p(X' \to Y'),\quad c \longmapsto res(c) $$ obtained by viewing a scheme $Y''$ locally of finite type over $Y'$ as a scheme locally of finite type over $Y$ and settting $res(c) \cap \alpha'' = c \cap \alpha''$ for any $\alpha'' \in \CH_k(Y'')$. This restriction operation is compatible with compositions in an obvious manner.


chow-remark-bivariant-commute

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. For $i = 1, 2$ let $Z_i \to X$ be a morphism of schemes locally of finite type. Let $c_i \in A^{p_i}(Z_i \to X)$, $i = 1, 2$ be bivariant classes. For any $\alpha \in \CH_k(X)$ we can ask whether $$ c_1 \cap c_2 \cap \alpha = c_2 \cap c_1 \cap \alpha $$ in $\CH_{k - p_1 - p_2}(Z_1 \times_X Z_2)$. If this is true and if it holds after any base change by $X' \to X$ locally of finite type, then we say $c_1$ and $c_2$ {\it commute}. Of course this is the same thing as saying that $$ res(c_1) \circ c_2 = res(c_2) \circ c_1 $$ in $A^{p_1 + p_2}(Z_1 \times_X Z_2 \to X)$. Here $res(c_1) \in A^{p_1}(Z_1 \times_X Z_2 \to Z_2)$ is the restriction of $c_1$ as in Remark \ref{remark-restriction-bivariant}; similarly for $res(c_2)$.


chow-remark-more-general-bivariant

There is a more general type of bivariant class that doesn't seem to be considered in the literature. Namely, suppose we are given a diagram $$ X \longrightarrow Z \longleftarrow Y $$ of schemes locally of finite type over $(S, \delta)$ as in Situation \ref{situation-setup}. Let $p \in \mathbf{Z}$. Then we can consider a rule $c$ which assigns to every $Z' \to Z$ locally of finite type maps $$ c \cap - : \CH_k(Y') \longrightarrow \CH_{k - p}(X') $$ for all $k \in \mathbf{Z}$ where $X' = X \times_Z Z'$ and $Y' = Z' \times_Z Y$ compatible with \begin{enumerate} \item proper pushforward if given $Z'' \to Z'$ proper, \item flat pullback if given $Z'' \to Z'$ flat of fixed relative dimension, and \item gysin maps if given $D' \subset Z'$ as in Definition \ref{definition-gysin-homomorphism}. \end{enumerate} We omit the detailed formulations. Suppose we denote the collection of all such operations $A^p(X \to Z \leftarrow Y)$. A simple example of the utility of this concept is when we have a proper morphism $f : X_2 \to X_1$. Then $f_*$ isn't a bivariant operation in the sense of Definition \ref{definition-bivariant-class} but it is in the above generalized sense, namely, $f_* \in A^0(X_1 \to X_1 \leftarrow X_2)$.


chow-remark-pullback-cohomology

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : Y' \to Y$ be a morphism of schemes locally of finite type over $S$. As a special case of Remark \ref{remark-restriction-bivariant} there is a canonical $\mathbf{Z}$-algebra map $res : A^*(Y) \to A^*(Y')$. This map is often denoted $f^*$ in the literature.


chow-remark-ring-loc-classes

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $Z \to X$ be a closed immersion of schemes locally of finite type over $S$ and let $p \geq 0$. In this setting we define $$ A^{(p)}(Z \to X) = \prod\nolimits_{i \leq p - 1} A^i(X) \times \prod\nolimits_{i \geq p} A^i(Z \to X). $$ Then $A^{(p)}(Z \to X)$ canonically comes equipped with the structure of a graded algebra. In fact, more generally there is a multiplication $$ A^{(p)}(Z \to X) \times A^{(q)}(Z \to X) \longrightarrow A^{(\max(p, q))}(Z \to X) $$ In order to define these we define maps \begin{align*} A^i(Z \to X) \times A^j(X) & \to A^{i + j}(Z \to X) \\ A^i(X) \times A^j(Z \to X) & \to A^{i + j}(Z \to X) \\ A^i(Z \to X) \times A^j(Z \to X) & \to A^{i + j}(Z \to X) \end{align*} For the first we use composition of bivariant classes. For the second we use restriction $A^i(X) \to A^i(Z)$ (Remark \ref{remark-restriction-bivariant}) and composition $A^i(Z) \times A^j(Z \to X) \to A^{i + j}(Z \to X)$. For the third, we send $(c, c')$ to $res(c) \circ c'$ where $res : A^i(Z \to X) \to A^i(Z)$ is the restriction map (see Remark \ref{remark-restriction-bivariant}). We omit the verification that these multiplications are associative in a suitable sense.


chow-remark-res-push

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $Z \to X$ be a closed immersion of schemes locally of finite type over $S$. Denote $res : A^p(Z \to X) \to A^p(Z)$ the restriction map of Remark \ref{remark-restriction-bivariant}. For $c \in A^p(Z \to X)$ we have $res(c) \cap \alpha = c \cap i_*\alpha$ for $\alpha \in \CH_*(Z)$. Namely $res(c) \cap \alpha = c \cap \alpha$ and compatibility of $c$ with proper pushforward gives $(Z \to Z)_*(c \cap \alpha) = c \cap (Z \to X)_*\alpha$.


chow-remark-completion-bivariant

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $X = \coprod_{i \in I} X_i$ and $Y = \coprod_{j \in J} Y_j$ be the decomposition of $X$ and $Y$ into their connected components (the connected components are open as $X$ and $Y$ are locally Noetherian, see Topology, Lemma \ref{topology-lemma-locally-Noetherian-locally-connected} and Properties, Lemma \ref{properties-lemma-Noetherian-topology}). Let $a(i) \in J$ be the index such that $f(X_i) \subset Y_{a(i)}$. Then $A^p(X \to Y) = \prod A^p(X_i \to Y_{a(i)})$ by Lemma \ref{lemma-disjoint-decomposition-bivariant}. In this setting it is convenient to set $$ A^*(X \to Y)^\wedge = \prod\nolimits_i A^*(X_i \to Y_{a(i)}) $$ as a kind of natural completion of the graded $\mathbf{Z}$-module $A^*(X \to Y)$ of bivariant classes (we omit specifying the precise sense in which this is a completion). As a special case we set $$ A^*(X)^\wedge = \prod A^*(X_i) $$ If $Y \to Z$ is a second morphism, then the composition $A^*(X \to Y) \times A^*(Y \to Z) \to A^*(X \to Z)$ extends to a composition $A^*(X \to Y)^\wedge \times A^*(Y \to Z)^\wedge \to A^*(X \to Z)^\wedge$ of completions.


chow-remark-equation-signs

We could also rewrite equation \ref{equation-chern-classes} as \begin{equation} \label{equation-signs} \sum\nolimits_{i = 0}^r c_1(\mathcal{O}_P(-1))^i \cap \pi^*c_{r - i} = 0. \end{equation} but we find it easier to work with the tautological quotient sheaf $\mathcal{O}_P(1)$ instead of its dual.


chow-remark-extend-to-finite-locally-free

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module. If the rank of $\mathcal{E}$ is not constant then we can still define the Chern classes of $\mathcal{E}$. Namely, in this case we can write $$ X = X_0 \amalg X_1 \amalg X_2 \amalg \ldots $$ where $X_r \subset X$ is the open and closed subspace where the rank of $\mathcal{E}$ is $r$. By  Lemma \ref{lemma-disjoint-decomposition-bivariant} we have $A^p(X) = \prod A^p(X_r)$. Hence we can define $c_i(\mathcal{E})$ to be the product of the classes $c_i(\mathcal{E}|_{X_r})$ in $A^i(X_r)$. Explicitly, if $X' \to X$ is a morphism locally of finite type, then we obtain by pullback a corresponding decomposition of $X'$ and we find that $$ \CH_*(X') = \prod\nolimits_{r \geq 0} \CH_*(X'_r) $$ by our definitions. Then $c_i(\mathcal{E}) \in A^i(X)$ is the bivariant class which preserves these direct product decompositions and acts by the already defined operations $c_i(\mathcal{E}|_{X_r}) \cap -$ on the factors. Observe that in this setting it may happen that $c_i(\mathcal{E})$ is nonzero for infinitely many $i$. In this setting we moreover define the ``rank'' of $\mathcal{E}$ to be the element $r(\mathcal{E}) \in A^0(X)$ as the bivariant operation which sends $(\alpha_r) \in \prod \CH_*(X'_r)$ to $(r\alpha_r) \in \prod \CH_*(X'_r)$. Note that it is still true that $c_i(\mathcal{E})$ and $r(\mathcal{E})$ are in the center of $A^*(X)$.


chow-remark-top-chern-class

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module. In general we write $X = \coprod X_r$ as in Remark \ref{remark-extend-to-finite-locally-free}. If only a finite number of the $X_r$ are nonempty, then we can set $$ c_{top}(\mathcal{E}) = \sum\nolimits_r c_r(\mathcal{E}|_{X_r}) \in A^*(X) = \bigoplus A^*(X_r) $$ where the equality is Lemma \ref{lemma-disjoint-decomposition-bivariant}. If infinitely many $X_r$ are nonempty, we will use the same notation to denote $$ c_{top}(\mathcal{E}) = \prod c_r(\mathcal{E}|_{X_r}) \in \prod A^r(X_r) \subset A^*(X)^\wedge $$ see Remark \ref{remark-completion-bivariant} for notation.


chow-remark-fundamental-class

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$ satisfying the equivalent conditions of Lemma \ref{lemma-locally-equidimensional}. Let $X = \coprod X_n$ be the decomposition into open and closed subschemes such that every irreducible component of $X_n$ has $\delta$-dimension $n$. In this situation we sometimes set $$ [X] = \sum\nolimits_n [X_n]_n \in \CH^0(X) $$ This class is a kind of ``fundamental class'' of $X$ in Chow theory.


chow-remark-the-proof-shows-more

The proof of Lemma \ref{lemma-splitting-principle} shows that the morphism $\pi : P \to X$ has the following additional properties: \begin{enumerate} \item $\pi$ is a finite composition of projective space bundles associated to locally free modules of finite constant rank, and \item for every $\alpha \in \CH_k(X)$ we have $\alpha = \pi_*(\xi_1 \cap \ldots \cap \xi_d \cap \pi^*\alpha)$ where $\xi_i$ is the first Chern class of some invertible $\mathcal{O}_P$-module. \end{enumerate} The second observation follows from the first and Lemma \ref{lemma-cap-projective-bundle}. We will add more observations here as needed.


chow-remark-equalities-nonconstant-rank

The equalities proven above remain true even when we work with finite locally free $\mathcal{O}_X$-modules whose rank is allowed to be nonconstant. In fact, we can work with polynomials in the rank and the Chern classes as follows. Consider the  graded polynomial ring $\mathbf{Z}[r, c_1, c_2, c_3, \ldots]$ where $r$ has degree $0$ and $c_i$ has degree $i$. Let $$ P \in \mathbf{Z}[r, c_1, c_2, c_3, \ldots] $$ be a homogeneous polynomial of degree $p$. Then for any finite locally free $\mathcal{O}_X$-module $\mathcal{E}$ on $X$ we can consider $$ P(\mathcal{E}) = P(r(\mathcal{E}), c_1(\mathcal{E}), c_2(\mathcal{E}), c_3(\mathcal{E}), \ldots) \in A^p(X) $$ see Remark \ref{remark-extend-to-finite-locally-free} for notation and conventions. To prove relations among these polynomials (for multiple finite locally free modules) we can work locally on $X$ and use the splitting principle as above. For example, we claim that $$ c_2(\SheafHom_{\mathcal{O}_X}(\mathcal{E}, \mathcal{E})) = P(\mathcal{E}) $$ where $P = 2rc_2 - (r - 1)c_1^2$. Namely, since $\SheafHom_{\mathcal{O}_X}(\mathcal{E}, \mathcal{E}) = \mathcal{E} \otimes \mathcal{E}^\vee$ this follows easily from Lemmas \ref{lemma-chern-classes-dual} and \ref{lemma-chern-classes-tensor-product} above by decomposing $X$ into parts where the rank of $\mathcal{E}$ is constant as in Remark \ref{remark-extend-to-finite-locally-free}.


chow-remark-extend-chern-character-to-finite-locally-free

In the discussion above we have defined the Chern character $ch(\mathcal{E})$ of $\mathcal{E}$ even if the rank of $\mathcal{E}$ is not constant. See Remarks \ref{remark-extend-to-finite-locally-free} and \ref{remark-equalities-nonconstant-rank}.


chow-remark-splitting-principle-perfect

The Chern classes of a perfect complex, when defined, satisfy a kind of splitting principle. Namely, suppose that $(S, \delta), X, E$ are as in Definition \ref{definition-defined-on-perfect} such that the Chern classes of $E$ are defined. Say we want to prove a relation between the bivariant classes $c_p(E)$, $P_p(E)$, and $ch_p(E)$. To do this, we may choose a bounded complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_X$-modules representing $E$. Using the splitting principle (Lemma \ref{lemma-splitting-principle}) we may assume each $\mathcal{E}^i$ has a filtration whose successive quotients $\mathcal{L}_{i, j}$ are invertible modules. Settting $x_{i, j} = c_1(\mathcal{L}_{i, j})$ we see that $$ c(E) = \prod\nolimits_{i\text{ even}} (1 + x_{i, j}) \prod\nolimits_{i\text{ odd}} (1 + x_{i, j})^{-1} $$ and $$ P_p(E) =  \sum\nolimits_{i\text{ even}} (x_{i, j})^p - \sum\nolimits_{i\text{ odd}} (x_{i, j})^p $$ Formally taking the logarithm for the expression for $c(E)$ above we find that $$ \log(c(E)) = \sum (-1)^{p - 1}\frac{P_p(E)}{p} $$ Looking at the construction of the polynomials $P_p$ in Example \ref{example-power-sum} it follows that $P_p(E)$ is the exact same expression in the Chern classes of $E$ as in the case of vector bundles, in other words, we have \begin{align*} P_1(E) & = c_1(E), \\ P_2(E) & = c_1(E)^2 - 2c_2(E), \\ P_3(E) & = c_1(E)^3 - 3c_1(E)c_2(E) + 3c_3(E), \\ P_4(E) & = c_1(E)^4 - 4c_1(E)^2c_2(E) + 4c_1(E)c_3(E) + 2c_2(E)^2 - 4c_4(E), \end{align*} and so on. On the other hand, the bivariant class $P_0(E) = r(E) = ch_0(E)$ cannot be recovered from the Chern class $c(E)$ of $E$; the chern class doesn't know about the rank of the complex.


chow-remark-loc-chern-classes

In the situation of Definition \ref{definition-localized-chern} assume $E|_{X \setminus Z}$ is finite locally free of rank $< p$. In this setting it is convenient to define $$ c^{(p)}(Z \to X, E) = 1 + c_1(E) + \ldots + c_{p - 1}(E) + c_p(Z \to X, E) + c_{p + 1}(Z \to X, E) + \ldots $$ as an element of the algebra $A^{(p)}(Z \to X)$ considered in Remark \ref{remark-ring-loc-classes}.


chow-remark-gysin-for-immersion

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $i : Z \to X$ be an immersion of schemes. In this situation \begin{enumerate} \item the conormal sheaf $\mathcal{C}_{Z/X}$ of $Z$ in $X$ is defined (Morphisms, Definition \ref{morphisms-definition-conormal-sheaf}), \item we say a pair consisting of a finite locally free $\mathcal{O}_Z$-module $\mathcal{N}$ and a surjection $\sigma : \mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ is a virtual normal bundle for the immersion $Z \to X$, \item choose an open subscheme $U \subset X$ such that $Z \to X$ factors through a closed immersion $Z \to U$ and set $c(Z \to X, \mathcal{N}) = c(Z \to U, \mathcal{N}) \circ (U \to X)^*$. \end{enumerate} The bivariant class $c(Z \to X, \mathcal{N})$ does not depend on the choice of the open subscheme $U$. All of the lemmas have immediate counterparts for this slightly more general construction. We omit the details.


chow-remark-adams-derived

Let $X$ be a scheme such that $2$ is invertible on $X$. Then the Adams operator $\psi^2$ can be defined on the $K$-group $K_0(X) = K_0(D_{perf}(\mathcal{O}_X))$ (Derived Categories of Schemes, Definition \ref{perfect-definition-K-group}) in a straightforward manner. Namely, given a perfect complex $L$ on $X$ we get an action of the group $\{\pm 1\}$ on $L \otimes^\mathbf{L} L$ by switching the factors. Then we can set $$ \psi^2(L) = [(L \otimes^\mathbf{L} L)^+] - [(L \otimes^\mathbf{L} L)^-] $$ where $(-)^+$ denotes taking invariants and $(-)^-$ denotes taking anti-invariants (suitably defined). Using exactness of taking invariants and anti-invariants one can argue similarly to the proof of Lemma \ref{lemma-second-adams-operator} to show that this is well defined. When $2$ is not invertible on $X$ the situation is a good deal more complicated and another approach has to be used.


chow-remark-chern-classes-K

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. The Chern class map defines a canonical map $$ c : K_0(\textit{Vect}(X)) \longrightarrow \prod\nolimits_{i \geq 0} A^i(X) $$ by sending a generator $[\mathcal{E}]$ on the left hand side to $c(\mathcal{E}) = 1 + c_1(\mathcal{E}) + c_2(\mathcal{E}) + \ldots$ and extending multiplicatively. Thus $-[\mathcal{E}]$ is sent to the formal inverse $c(\mathcal{E})^{-1}$ which is why we have the infinite product on the right hand side. This is well defined by Lemma \ref{lemma-additivity-chern-classes}.


chow-remark-chern-character-K

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. The Chern character map defines a canonical ring map $$ ch : K_0(\textit{Vect}(X)) \longrightarrow \prod\nolimits_{i \geq 0} A^i(X) \otimes \mathbf{Q} $$ by sending a generator $[\mathcal{E}]$ on the left hand side to $ch(\mathcal{E})$ and extending additively. This is well defined by Lemma \ref{lemma-chern-character-additive} and a ring homomorphism by Lemma \ref{lemma-chern-character-multiplicative}.


chow-remark-perf-Z-cohomology-K

Let $X$ be a locally Noetherian scheme. Let $Z \subset X$ be a closed subscheme. Consider the strictly full, saturated, triangulated subcategory $$ D_{Z, perf}(\mathcal{O}_X) \subset D(\mathcal{O}_X) $$ consisting of perfect complexes of $\mathcal{O}_X$-modules whose cohomology sheaves are settheoretically supported on $Z$. Denote $\textit{Coh}_Z(X) \subset \textit{Coh}(X)$ the Serre subcategory of coherent $\mathcal{O}_X$-modules whose set theoretic support is contained in $Z$. Observe that given $E \in D_{Z, perf}(\mathcal{O}_X)$ Zariski locally on $X$ only a finite number of the cohomology sheaves $H^i(E)$ are nonzero (and they are all settheoretically supported on $Z$). Hence we can define $$ K_0(D_{Z, perf}(\mathcal{O}_X)) \longrightarrow K_0(\textit{Coh}_Z(X)) = K'_0(Z) $$ (equality by Lemma \ref{lemma-K-coherent-supported-on-closed}) by the rule $$ E \longmapsto [\bigoplus\nolimits_{i \in \mathbf{Z}} H^{2i}(E)] - [\bigoplus\nolimits_{i \in \mathbf{Z}} H^{2i + 1}(E)] $$ This works because given a distinguished triangle in $D_{Z, perf}(\mathcal{O}_X)$ we have a long exact sequence of cohomology sheaves.


chow-remark-perf-Z-regular

Let $X$, $Z$, $D_{Z, perf}(\mathcal{O}_X)$ be as in Remark \ref{remark-perf-Z-cohomology-K}. Assume $X$ is Noetherian regular of finite dimension. Then there is a canonical map $$ K_0(\textit{Coh}(Z)) \longrightarrow K_0(D_{Z, perf}(\mathcal{O}_X)) $$ defined as follows. For any coherent $\mathcal{O}_Z$-module $\mathcal{F}$ denote $\mathcal{F}[0]$ the object of $D(\mathcal{O}_X)$ which has $\mathcal{F}$ in degree $0$ and is zero in other degrees. Then $\mathcal{F}[0]$ is a perfect complex on $X$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-regular}. Hence $\mathcal{F}[0]$ is an object of $D_{Z, perf}(\mathcal{O}_X)$. On the other hand, given a short exact sequence $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ of coherent $\mathcal{O}_Z$-modules we obtain a distinguished triangle $\mathcal{F}[0] \to \mathcal{F}'[0] \to \mathcal{F}''[0] \to \mathcal{F}[1]$, see Derived Categories, Section \ref{derived-section-canonical-delta-functor}. This shows that we obtain a map $K_0(\textit{Coh}(Z)) \to K_0(D_{Z, perf}(\mathcal{O}_X))$ by sending $[\mathcal{F}]$ to $[\mathcal{F}[0]]$ with apologies for the horrendous notation.