chow-remark-localized-chern-classes-K

Let $X$, $Z$, $D_{Z, perf}(\mathcal{O}_X)$ be as in Remark \ref{remark-perf-Z-cohomology-K}. Assume $X$ is quasi-compact, has the resolution property, and is of finite type over $(S, \delta)$ as in Situation \ref{situation-setup}. The localized Chern classes define a canonical map $$ c(Z \to X, -) : K_0(D_{Z, perf}(\mathcal{O}_X)) \longrightarrow A^0(X) \times \prod\nolimits_{i \geq 1} A^i(Z \to X) $$ by sending a generator $[E]$ on the left hand side to $$ c(Z \to X, E) = 1 + c_1(Z \to X, E) + c_2(Z \to X, E) + \ldots $$ and extending multiplicatively (with product on the right hand side as in Remark \ref{remark-ring-loc-classes}). This makes sense because by Derived Categories of Schemes, Lemma \ref{perfect-lemma-resolution-property-perfect-complex} and Definition \ref{definition-localized-chern} $c_i(Z \to X, E) $ are defined for all $i \geq 1$. It is well defined by Derived Categories of Schemes, Lemma \ref{perfect-lemma-resolution-property-map-perfect-complex} (every map in $D_{Z, perf}(\mathcal{O}_X)$ can be represented by a map of bounded complexes of finite locally frees) and Lemma \ref{lemma-additivity-loc-chern-c}.


chow-remark-localized-chern-character-K

Let $X$, $Z$, $D_{Z, perf}(\mathcal{O}_X)$ be as in Remark \ref{remark-perf-Z-cohomology-K}. Assume $X$ is quasi-compact, has the resolution property, and is of finite type over $(S, \delta)$ as in Situation \ref{situation-setup}. The localized Chern character defines a canonical additive and multiplicative map $$ ch(Z \to X, -) : K_0(D_{Z, perf}(\mathcal{O}_X)) \longrightarrow \prod\nolimits_{i \geq 0} A^i(Z \to X) $$ by sending a generator $[E]$ on the left hand side to $ch(Z \to X, E)$ and extending additively. This makes sense because because $ch(Z \to X, E)$ is defined by Derived Categories of Schemes, Lemma \ref{perfect-lemma-resolution-property-perfect-complex} and Definition \ref{definition-localized-chern}. It is well defined by Derived Categories of Schemes, Lemma \ref{perfect-lemma-resolution-property-map-perfect-complex} (every map in $D_{Z, perf}(\mathcal{O}_X)$ can be represented by a map of bounded complexes of finite locally frees) and Lemma \ref{lemma-additivity-loc-chern-P}. The multiplication on $K_0(D_{Z, perf}(X))$ is defined using derived tensor product (Derived Categories of Schemes, Remark \ref{perfect-remark-perf-Z}) hence $ch(\alpha \beta) = ch(\alpha) ch(\beta)$ by Lemma \ref{lemma-loc-chern-tensor-product}.


chow-remark-chern-classes-agree

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$ and assume $X$ is quasi-compact and has the resolution property. With $Z = X$ and notation as in Remarks \ref{remark-localized-chern-classes-K} and \ref{remark-localized-chern-character-K} we have $D_{Z, perf}(\mathcal{O}_X) = D_{perf}(\mathcal{O}_X)$ and we see that $$ K_0(D_{Z, perf}(\mathcal{O}_X)) = K_0(D_{perf}(\mathcal{O}_X)) = K_0(X) $$ see  Derived Categories of Schemes, Definition \ref{perfect-definition-K-group}. Hence we get $c : K_0(X) \to \prod A^i(X)$ and $ch : K_0(X) \to \prod A^i(X)$ from Remarks \ref{remark-localized-chern-classes-K} and \ref{remark-localized-chern-character-K}. Via the equality $K_0(\textit{Vect}(X)) = K_0(X)$ of Derived Categories of Schemes, Lemma \ref{perfect-lemma-K-is-old-K} these maps agree with the maps constructed in Remarks \ref{remark-chern-classes-K} and \ref{remark-chern-character-K}.


chow-remark-gysin-chern-classes

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. Assume the gysin map exists for $f$. Then $f^! \circ c_i(\mathcal{E}) = c_i(f^*\mathcal{E}) \circ f^!$ and similarly for the Chern character, see Lemma \ref{lemma-lci-gysin-commutes}. If $X$ and $Y$ satisfy the equivalent conditions of Lemma \ref{lemma-locally-equidimensional} and $Y$ is Cohen-Macaulay (for example), then $f^![Y] = [X]$ by Lemma \ref{lemma-lci-gysin-easy}. In this case we also get $f^!(c_i(\mathcal{E}) \cap [Y]) = c_i(f^*\mathcal{E}) \cap [X]$ and similarly for the Chern character.


chow-remark-commuting-exterior

The upshot of Lemmas \ref{lemma-chow-cohomology-towards-point} and \ref{lemma-chow-cohomology-towards-point-commutes} is the following. Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. Let $\alpha \in \CH_*(X)$. Let $Y \to Z$ be a morphism of schemes locally of finite type over $k$. Let $c' \in A^q(Y \to Z)$. Then $$ \alpha \times (c' \cap \beta) = c' \cap (\alpha \times \beta) $$ in $\CH_*(X \times_k Y)$ for any $\beta \in \CH_*(Z)$. Namely, this follows by taking $c = c_\alpha \in A^*(X \to \Spec(k))$ the bivariant class corresponding to $\alpha$, see proof of Lemma \ref{lemma-chow-cohomology-towards-point}.


chow-remark-commuting-exterior-dim-1

The upshot of Lemmas \ref{lemma-chow-cohomology-towards-base-dim-1} and \ref{lemma-chow-cohomology-towards-base-dim-1-commutes} is the following. Let $(S, \delta)$ be as above. Let $X$ be a scheme locally of finite type over $S$. Let $\alpha \in \CH_*(X)$. Let $Y \to Z$ be a morphism of schemes locally of finite type over $S$. Let $c' \in A^q(Y \to Z)$. Then $$ \alpha \times (c' \cap \beta) = c' \cap (\alpha \times \beta) $$ in $\CH_*(X \times_S Y)$ for any $\beta \in \CH_*(Z)$. Namely, this follows by taking $c = c_\alpha \in A^*(X \to S)$ the bivariant class corresponding to $\alpha$, see proof of Lemma \ref{lemma-chow-cohomology-towards-base-dim-1}.


chow-remark-explain-determinant

Let $(R, \mathfrak m, \kappa)$ be a local ring and assume either the characteristic of $\kappa$ is zero or it is $p$ and $p R = 0$. Let $M_1, \ldots, M_n$ be finite length $R$-modules. We will show below that there exists an ideal $I \subset \mathfrak m$ annihilating $M_i$ for $i = 1, \ldots, n$ and a section $\sigma : \kappa \to R/I$ of the canonical surjection $R/I \to \kappa$. The restriction $M_{i, \kappa}$ of $M_i$ via $\sigma$ is a $\kappa$-vector space of dimension $l_i = \text{length}_R(M_i)$ and using Lemma \ref{lemma-determinant-quotient-ring} we see that $$ \det\nolimits_\kappa(M_i) = \wedge_\kappa^{l_i}(M_{i, \kappa}) $$ These isomorphisms are compatible with the isomorphisms $\gamma_{K \to M \to L}$ of Lemma \ref{lemma-det-exact-sequences} for short exact sequences of finite length $R$-modules annihilated by $I$. The conclusion is that verifying a property of $\det_\kappa$ often reduces to verifying corresponding properties of the usual determinant on the category finite dimensional vector spaces. \medskip\noindent For $I$ we can take the annihilator (Algebra, Definition \ref{algebra-definition-annihilator}) of the module $M = \bigoplus M_i$. In this case we see that $R/I \subset \text{End}_R(M)$ hence has finite length. Thus $R/I$ is an Artinian local ring with residue field $\kappa$. Since an Artinian local ring is complete we see that $R/I$ has a coefficient ring by the Cohen structure theorem (Algebra, Theorem \ref{algebra-theorem-cohen-structure-theorem}) which is a field by our assumption on $R$.


chow-remark-more-elementary

Here is a more down to earth description of the determinant introduced above. Let $R$ be a local ring with residue field $\kappa$. Let $(M, \varphi, \psi)$ be a $(2, 1)$-periodic complex over $R$. Assume that $M$ has finite length and that $(M, \varphi, \psi)$ is exact. Let us abbreviate $I_\varphi = \Im(\varphi)$, $I_\psi = \Im(\psi)$ as above. Assume that $\text{length}_R(I_\varphi) = a$ and $\text{length}_R(I_\psi) = b$, so that $a + b = \text{length}_R(M)$ by exactness. Choose admissible sequences $x_1, \ldots, x_a \in I_\varphi$ and $y_1, \ldots, y_b \in I_\psi$ such that the symbol $[x_1, \ldots, x_a]$ generates $\det_\kappa(I_\varphi)$ and the symbol $[x_1, \ldots, x_b]$ generates $\det_\kappa(I_\psi)$. Choose $\tilde x_i \in M$ such that $\varphi(\tilde x_i) = x_i$. Choose $\tilde y_j \in M$ such that $\psi(\tilde y_j) = y_j$. Then $\det_\kappa(M, \varphi, \psi)$ is characterized by the equality $$ [x_1, \ldots, x_a, \tilde y_1, \ldots, \tilde y_b] = (-1)^{ab} \det\nolimits_\kappa(M, \varphi, \psi) [y_1, \ldots, y_b, \tilde x_1, \ldots, \tilde x_a] $$ in $\det_\kappa(M)$. This also explains the sign.


flat-remark-finite-presentation

Note that the $R$-algebras $B_i$ for all $i$ and $A_i$ for $i \geq 2$ are of finite presentation over $R$. If $S$ is of finite presentation over $R$, then it is also the case that $A_1$ is of finite presentation over $R$. In this case all the ring maps in the complete d\'evissage are of finite presentation. See Algebra, Lemma \ref{algebra-lemma-compose-finite-type}. Still assuming $S$ of finite presentation over $R$ the following are equivalent \begin{enumerate} \item $M$ is of finite presentation over $S$, \item $M_1$ is of finite presentation over $A_1$, \item $M_1$ is of finite presentation over $B_1$, \item each $M_i$ is of finite presentation both as an $A_i$-module and as a $B_i$-module. \end{enumerate} The equivalences (1) $\Leftrightarrow$ (2) and (2) $\Leftrightarrow$ (3) follow from Algebra, Lemma \ref{algebra-lemma-finite-finitely-presented-extension}. If $M_1$ is finitely presented, so is $\Coker(\alpha_1)$ (see Algebra, Lemma \ref{algebra-lemma-extension}) and hence $M_2$, etc.


flat-remark-same-notion

Let $A \to B$ be a finite type ring map and let $N$ be a finite $B$-module. Let $\mathfrak q$ be a prime of $B$ lying over the prime $\mathfrak r$ of $A$. Set $X = \Spec(B)$, $S = \Spec(A)$ and $\mathcal{F} = \widetilde{N}$ on $X$. Let $x$ be the point corresponding to $\mathfrak q$ and let $s \in S$ be the point corresponding to $\mathfrak p$. Then \begin{enumerate} \item if there exists a complete d\'evissage of $\mathcal{F}/X/S$ over $s$ then there exists a complete d\'evissage of $N/B/A$ over $\mathfrak p$, and \item there exists a complete d\'evissage of $\mathcal{F}/X/S$ at $x$ if and only if there exists a complete d\'evissage of $N/B/A$ at $\mathfrak q$. \end{enumerate} There is just a small twist in that we omitted the condition on the relative dimension in the formulation of ``a complete d\'evissage of $N/B/A$ over $\mathfrak p$'' which is why the implication in (1) only goes in one direction. The notion of a complete d\'evissage at $\mathfrak q$ does have this condition built in. In any case we will only use that existence for $\mathcal{F}/X/S$ implies the existence for $N/B/A$.


flat-remark-how-in-RG

Lemma \ref{lemma-fibres-irreducible-flat-projective-nonnoetherian} is a key step in the development of results in this chapter. The analogue of this lemma in \cite{GruRay} is \cite[I Proposition 3.3.1]{GruRay}: If $R \to S$ is smooth with geometrically integral fibres, then $S$ is projective as an $R$-module. This is a special case of Lemma \ref{lemma-fibres-irreducible-flat-projective-nonnoetherian}, but as we will later improve on this lemma anyway, we do not gain much from having a stronger result at this point. We briefly sketch the proof of this as it is given in \cite{GruRay}. \begin{enumerate} \item First reduce to the case where $R$ is Noetherian as above. \item Since projectivity descends through faithfully flat ring maps, see Algebra, Theorem \ref{algebra-theorem-ffdescent-projectivity} we may work locally in the fppf topology on $R$, hence we may assume that $R \to S$ has a section $\sigma : S \to R$. (Just by the usual trick of base changing to $S$.) Set $I = \Ker(S \to R)$. \item Localizing a bit more on $R$ we may assume that $I/I^2$ is a free $R$-module and that the completion $S^\wedge$ of $S$ with respect to $I$ is isomorphic to $R[[t_1, \ldots, t_n]]$, see Morphisms, Lemma \ref{morphisms-lemma-section-smooth-morphism}. Here we are using that $R \to S$ is smooth. \item To prove that $S$ is projective as an $R$-module, it suffices to prove that $S$ is flat, countably generated and Mittag-Leffler as an $R$-module, see Algebra, Lemma \ref{algebra-lemma-countgen-projective}. The first two properties are evident. Thus it suffices to prove that $S$ is Mittag-Leffler as an $R$-module. By Algebra, Lemma \ref{algebra-lemma-power-series-ML} the module $R[[t_1, \ldots, t_n]]$ is Mittag-Leffler over $R$. Hence Algebra, Lemma \ref{algebra-lemma-pure-submodule-ML} shows that it suffices to show that the $S \to S^\wedge$ is universally injective as a map of $R$-modules. \item Apply Lemma \ref{lemma-base-change-universally-flat} to see that $S \to S^\wedge$ is $R$-universally injective. Namely, as $R \to S$ has geometrically integral fibres, any associated point of any fibre ring is just the generic point of the fibre ring which is in the image of $\Spec(S^\wedge) \to \Spec(S)$. \end{enumerate} There is an analogy between the proof as sketched just now, and the development of the arguments leading to the proof of Lemma \ref{lemma-fibres-irreducible-flat-projective-nonnoetherian}. In both a completion plays an essential role, and both times the assumption of having geometrically integral fibres assures one that the map from $S$ to the completion of $S$ is $R$-universally injective.


flat-remark-complete-devissage-flat-finitely-presented-module

There is a variant of Lemma \ref{lemma-complete-devissage-flat-finitely-presented-module} where we weaken the flatness condition by assuming only that $N$ is flat at some given prime $\mathfrak q$ lying over $\mathfrak r$ but where we strengthen the d\'evissage condition by assuming the existence of a complete d\'evissage {\it at $\mathfrak q$}. Compare with Lemma \ref{lemma-complete-devissage-flat-finite-type-module}.


flat-remark-finite-type-flat

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $X \to S$ is locally of finite type, \item $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite type, and \item the set of weakly associated points of $S$ is locally finite in $S$. \end{enumerate} Then $U = \{x \in X \mid \mathcal{F}\text{ flat at }x\text{ over }S\}$ is open in $X$ and $\mathcal{F}|_U$ is flat over $S$ and locally finitely presented relative to $S$ (see More on Morphisms, Definition \ref{more-morphisms-definition-relatively-finitely-presented-sheaf}). If we ever need this result in the Stacks project we will convert this remark into a lemma with a proof.


flat-remark-finite-type-flat-algebra

Let $R \to S$ be a ring map of finite type. Let $M$ be a finite $S$-module. Assume $\text{WeakAss}_R(R)$ is finite. Then $$ U = \{\mathfrak q \subset S \mid M_{\mathfrak q}\text{ flat over }R\} $$ is open in $\Spec(S)$ and for every $g \in S$ such that $D(g) \subset U$ the localization $M_g$ is flat over $R$ and an $S_g$-module finitely presented relative to $R$ (see More on Algebra, Definition \ref{more-algebra-definition-relatively-finitely-presented}). If we ever need this result in the Stacks project we will convert this remark into a lemma with a proof.


flat-remark-discuss-finite-type

Let $f : X \to S$ be a morphism which is locally of finite type and $\mathcal{F}$ a quasi-coherent finite type $\mathcal{O}_X$-module. In this case it is still true that (1) and (2) above are equivalent because the proof of Lemma \ref{lemma-quasi-finite-impurity-elementary} does not use that $f$ is quasi-compact. It is also clear that (3) and (4) are equivalent. However, we don't know if (1) and (3) are equivalent. In this case it may sometimes be more convenient to define purity using the equivalent conditions (3) and (4) as is done in \cite{GruRay}. On the other hand, for many applications it seems that the correct notion is really that of being universally pure.


flat-remark-flattening-local-scheme-theoretic

Here is a scheme theoretic reformulation of Theorem \ref{theorem-flattening-local}. Let $(X, x) \to (S, s)$ be a morphism of pointed schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Assume $S$ henselian local with closed point $s$. There exists a closed subscheme $Z \subset S$ with the following property: for any morphism of pointed schemes $(T, t) \to (S, s)$ the following are equivalent \begin{enumerate} \item $\mathcal{F}_T$ is flat over $T$ at all points of the fibre $X_t$ which map to $x \in X_s$, and \item $\Spec(\mathcal{O}_{T, t}) \to S$ factors through $Z$. \end{enumerate} Moreover, if $X \to S$ is of finite presentation at $x$ and $\mathcal{F}_x$ of finite presentation over $\mathcal{O}_{X, x}$, then $Z \to S$ is of finite presentation.


flat-remark-flattening-complete-noetherian

Tracing the proof of Lemma \ref{lemma-freebie} to its origins we find a long and winding road. But if we assume that \begin{enumerate} \item $f$ is of finite type, \item $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module, \item $E = X_s$, and \item $S$ is the spectrum of a Noetherian complete local ring. \end{enumerate} then there is a proof relying completely on more elementary algebra as follows: first we reduce to the case where $X$ is affine by taking a finite affine open cover. In this case $Z$ exists by More on Algebra, Lemma \ref{more-algebra-lemma-flattening-complete-local-universal-property}. The key step in this proof is constructing the closed subscheme $Z$ step by step inside the truncations $\Spec(\mathcal{O}_{S, s}/\mathfrak m_s^n)$. This relies on the fact that flattening stratifications always exist when the base is Artinian, and the fact that $\mathcal{O}_{S, s} = \lim \mathcal{O}_{S, s}/\mathfrak m_s^n$.


flat-remark-correct-generality

The result in this section can be generalized. It is probably correct if we only assume $X \to \Spec(A)$ to be separated, of finite presentation, and $K_n$ pseudo-coherent relative to $A_n$ supported on a closed subset of $X_n$ proper over $A_n$. The outcome will be a $K$ which is pseudo-coherent relative to $A$ supported on a closed subset proper over $A$. If we ever need this, we will formulate a precise statement and prove it here.


flat-remark-successive-blowups

Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \subset S$ be a quasi-compact open subscheme. Given a $U$-admissible blowup $S' \to S$ we denote $X'$ the strict transform of $X$ and $\mathcal{F}'$ the strict transform of $\mathcal{F}$ which we think of as a quasi-coherent module on $X'$ (via Divisors, Lemma \ref{divisors-lemma-strict-transform}). Let $P$ be a property of $\mathcal{F}/X/S$ which is stable under strict transform (as above) for $U$-admissible blowups. The general problem in this section is: Show (under auxiliary conditions on $\mathcal{F}/X/S$) there exists a $U$-admissible blowup $S' \to S$ such that the strict transform $\mathcal{F}'/X'/S'$ has $P$. \medskip\noindent The general strategy will be to use that a composition of $U$-admissible blowups is a $U$-admissible blowup, see Divisors, Lemma \ref{divisors-lemma-composition-admissible-blowups}. In fact, we will make use of the more precise Divisors, Lemma \ref{divisors-lemma-composition-finite-type-blowups} and combine it with Divisors, Lemma \ref{divisors-lemma-strict-transform-composition-blowups}. The result is that it suffices to find a sequence of $U$-admissible blowups $$ S = S_0 \leftarrow S_1 \leftarrow \ldots \leftarrow S_n $$ such that, setting $\mathcal{F}_0 = \mathcal{F}$ and $X_0 = X$ and setting $\mathcal{F}_i/X_i$ equal to the strict transform of $\mathcal{F}_{i - 1}/X_{i - 1}$, we arrive at $\mathcal{F}_n/X_n/S_n$ with property $P$. \medskip\noindent In particular, choose a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_S$ such that $V(\mathcal{I}) = S \setminus U$, see Properties, Lemma \ref{properties-lemma-quasi-coherent-finite-type-ideals}. Let $S' \to S$ be the blowup in $\mathcal{I}$ and let $E \subset S'$ be the exceptional divisor (Divisors, Lemma \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}). Then we see that we've reduced the problem to the case where there exists an effective Cartier divisor $D \subset S$ whose support is $X \setminus U$. In particular we may assume $U$ is scheme theoretically dense in $S$ (Divisors, Lemma \ref{divisors-lemma-complement-effective-Cartier-divisor}). \medskip\noindent Suppose that $P$ is local on $S$: If $S = \bigcup S_i$ is a finite open covering by quasi-compact opens and $P$ holds for $\mathcal{F}_{S_i}/X_{S_i}/S_i$ then $P$ holds for $\mathcal{F}/X/S$. In this case the general problem above is local on $S$ as well, i.e., if given $s \in S$ we can find a quasi-compact open neighbourhood $W$ of $s$ such that the problem for $\mathcal{F}_W/X_W/W$ is solvable, then the problem is solvable for $\mathcal{F}/X/S$. This follows from Divisors, Lemmas \ref{divisors-lemma-extend-admissible-blowups} and \ref{divisors-lemma-dominate-admissible-blowups}.


flat-remark-when-you-have-a-complex

Let $X$ be a scheme. Let $E \in D(\mathcal{O}_X)$ be a perfect object such that $H^i(E)$ is a perfect $\mathcal{O}_X$-module of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$. This property sometimes allows one to reduce questions about $E$ to questions about $H^i(E)$. For example, suppose $$ \mathcal{E}^a \xrightarrow{d^a} \ldots \xrightarrow{d^{b - 2}} \mathcal{E}^{b - 1} \xrightarrow{d^{b - 1}} \mathcal{E}^b $$ is a bounded complex of finite locally free $\mathcal{O}_X$-modules representing $E$. Then $\Im(d^i)$ and $\Ker(d^i)$ are finite locally free $\mathcal{O}_X$-modules for all $i$. Namely, suppose by induction we know this for all indices bigger than $i$. Then we can first use the short exact sequence $$ 0 \to \Im(d^i) \to \Ker(d^{i + 1}) \to H^{i + 1}(E) \to 0 $$ and the assumption that $H^{i + 1}(E)$ is perfect of tor dimension $\leq 1$ to conclude that $\Im(d^i)$ is finite locally free. The same argument used again for the short exact sequence $$ 0 \to \Ker(d^i) \to \mathcal{E}^i \to \Im(d^i) \to 0 $$ then gives that $\Ker(d^i)$ is finite locally free. It follows that the distinguished triangles $$ \tau_{\leq k - 1}E \to \tau_{\leq k}E \to H^k(E)[-k] \to (\tau_{\leq k - 1}E)[1] $$ are represented by the following short exact sequences of bounded complexes of finite locally free modules $$ \begin{matrix} & & & & & & 0 \\ & & & & & & \downarrow \\ \mathcal{E}^a & \to & \ldots & \to & \mathcal{E}^{k - 2} & \to & \Ker(d^{k - 1}) \\ \downarrow & & & & \downarrow & & \downarrow \\ \mathcal{E}^a & \to & \ldots & \to & \mathcal{E}^{k - 2} & \to & \mathcal{E}^{k - 1} & \to & \Ker(d^k) \\ & & & & & & \downarrow & & \downarrow \\ & & & & & & \Im(d^{k - 1}) & \to & \Ker(d^k) \\ & & & & & & \downarrow \\ & & & & & & 0 \end{matrix} $$ Here the complexes are the rows and the ``obvious'' zeros are omitted from the display.


flat-remark-Leta

Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. Let $\mathcal{G}^\bullet$ be a complex of $\mathcal{O}_X$-modules. By Cohomology, Lemma \ref{cohomology-lemma-K-flat-resolution} there exists a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{G}^\bullet$ such that $\mathcal{F}^\bullet$ is a K-flat complex whose terms are flat $\mathcal{O}_X$-modules. (Even if $\mathcal{G}^\bullet$ is a complex of quasi-coherent $\mathcal{O}_X$-modules, in general $\mathcal{F}^\bullet$ will not be so.) It follows that $\mathcal{F}^i$ is $\mathcal{I}$-torsion free for all $i$. In this situation we define $$ L\eta_\mathcal{I} \mathcal{G}^\bullet = \eta_\mathcal{I} \mathcal{F}^\bullet $$ This is independent of the choice of the K-flat resolution by Lemma \ref{lemma-eta-qis}. We obtain a functor $L\eta_\mathcal{I} : D(\mathcal{O}_X) \to D(\mathcal{O}_X)$. Beware that this functor isn't exact, i.e., does not tranform distinguished triangles into distinguished triangles.


flat-remark-complex-and-divisor-ideal

In Situation \ref{situation-complex-and-divisor} for any $i \in \mathbf{Z}$ there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J}_i \subset \mathcal{O}_X$ with the following property: for any $U \subset X$ open such that $\mathcal{I}|_U$, $\mathcal{E}^i|_U$, and $\mathcal{E}^{i + 1}|_U$ are free of ranks $1$, $r_i$, and $r_{i + 1}$, the ideal $\mathcal{J}_i$ is generated by the $r_i \times r_i$ minors of the map $$ 1, d^i : \mathcal{I}\mathcal{E}^i \longrightarrow \mathcal{E}^i \oplus \mathcal{I}\mathcal{E}^{i + 1} $$ with notation as in Section \ref{section-eta}. By convention we set $\mathcal{J}_i|_U = \mathcal{O}_U$ if $r_i = 0$. Observe that $\mathcal{I}^{r_i}|_U \subset \mathcal{J}_i|_U$ in other words, the closed subscheme $V(\mathcal{J})$ is set theoretically contained in $D$. Formation of the ideal $\mathcal{J}_i$ commutes with base change by any morphism $f : Y \to X$ such that the pullback of $D$ by $f$ is defined (Divisors, Definition \ref{divisors-definition-pullback-effective-Cartier-divisor}).


curves-remark-classical-linear-series

Let $X$ be a smooth projective curve over an algebraically closed field $k$. We say two effective Cartier divisors $D, D' \subset X$ are {\it linearly equivalent} if and only if $\mathcal{O}_X(D) \cong \mathcal{O}_X(D')$ as $\mathcal{O}_X$-modules. Since $\Pic(X) = \text{Cl}(X)$ (Divisors, Lemma \ref{divisors-lemma-local-rings-UFD-c1-bijective}) we see that $D$ and $D'$ are linearly equivalent if and only if the Weil divisors associated to $D$ and $D'$ define the same element of $\text{Cl}(X)$. Given an effective Cartier divisor $D \subset X$ of degree $d$ the {\it complete linear system} or {\it complete linear series} $|D|$ of $D$ is the set of effective Cartier divisors $E \subset X$ which are linearly equivalent to $D$. Another way to say it is that $|D|$ is the set of closed points of the fibre of the morphism $$ \gamma_d : \underline{\Hilbfunctor}^d_{X/k} \longrightarrow \underline{\Picardfunctor}^d_{X/k} $$ (Picard Schemes of Curves, Lemma \ref{pic-lemma-picard-pieces}) over the closed point corresponding to $\mathcal{O}_X(D)$. This gives $|D|$ a natural scheme structure and it turns out that $|D| \cong \mathbf{P}^m_k$ with $m + 1 = h^0(\mathcal{O}_X(D))$. In fact, more canonically we have $$ |D| = \mathbf{P}(H^0(X, \mathcal{O}_X(D))^\vee) $$ where $(-)^\vee$ indicates $k$-linear dual and $\mathbf{P}$ is as in Constructions, Example \ref{constructions-example-projective-space}. In this language a {\it linear system} or a {\it linear series} on $X$ is a closed subvariety $L \subset |D|$ which can be cut out by linear equations. If $L$ has dimension $r$, then $L = \mathbf{P}(V^\vee)$ where $V \subset H^0(X, \mathcal{O}_X(D))$ is a linear subspace of dimension $r + 1$. Thus the classical linear series $L \subset |D|$ corresponds to the linear series $(\mathcal{O}_X(D), V)$ as defined above.


curves-remark-rework-duality-locally-free

Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. Let $\omega_X^\bullet$ and $\omega_X$ be as in Lemma \ref{lemma-duality-dim-1}. If $\mathcal{E}$ is a finite locally free $\mathcal{O}_X$-module with dual $\mathcal{E}^\vee$ then we have canonical isomorphisms $$ \Hom_k(H^{-i}(X, \mathcal{E}), k) = H^i(X, \mathcal{E}^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_X^\bullet) $$ This follows from the lemma and Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}. If $X$ is Cohen-Macaulay and equidimensional of dimension $1$, then we have canonical isomorphisms $$ \Hom_k(H^{-i}(X, \mathcal{E}), k) = H^{1 - i}(X, \mathcal{E}^\vee \otimes_{\mathcal{O}_X} \omega_X) $$ by Lemma \ref{lemma-duality-dim-1-CM}. In particular if $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then we have $$ \dim_k H^0(X, \mathcal{L}) = \dim_k H^1(X, \mathcal{L}^{\otimes -1} \otimes_{\mathcal{O}_X} \omega_X) $$ and $$ \dim_k H^1(X, \mathcal{L}) = \dim_k H^0(X, \mathcal{L}^{\otimes -1} \otimes_{\mathcal{O}_X} \omega_X) $$


curves-remark-genus-higher-dimension

Suppose that $X$ is a $d$-dimensional proper smooth variety over an algebraically closed field $k$. Then the {\it arithmetic genus} is often defined as $p_a(X) = (-1)^d(\chi(X, \mathcal{O}_X) - 1)$ and the {\it geometric genus} as $p_g(X) = \dim_k H^0(X, \Omega^d_{X/k})$. In this situation the arithmetic genus and the geometric genus no longer agree even though it is still true that $\omega_X \cong \Omega_{X/k}^d$. For example, if $d = 2$, then we have \begin{align*} p_a(X) - p_g(X) & = h^0(X, \mathcal{O}_X) - h^1(X, \mathcal{O}_X) + h^2(X, \mathcal{O}_X) - 1 - h^0(X, \Omega^2_{X/k}) \\ & = - h^1(X, \mathcal{O}_X) + h^2(X, \mathcal{O}_X) - h^0(X, \omega_X) \\ & = - h^1(X, \mathcal{O}_X) \end{align*} where $h^i(X, \mathcal{F}) = \dim_k H^i(X, \mathcal{F})$ and where the last equality follows from duality. Hence for a surface the difference $p_g(X) - p_a(X)$ is always nonnegative; it is sometimes called the irregularity of the surface. If $X = C_1 \times C_2$ is a product of smooth projective curves of genus $g_1$ and $g_2$, then the irregularity is $g_1 + g_2$.


curves-remark-quadratic-extension

Let $k$ be a field. Let $(A, \mathfrak m, \kappa)$ be a Noetherian local $k$-algebra. Assume that either $(A, \mathfrak m, \kappa)$ is as in Lemma \ref{lemma-nodal-algebraic}, or $A$ is Nagata as in Lemma \ref{lemma-2-branches-delta-1}, or $A$ is complete and as in Lemma \ref{lemma-fitting-ideal}. Then $A$ defines canonically a degree $2$ separable $\kappa$-algebra $\kappa'$ as follows \begin{enumerate} \item let $q = ax^2 + bxy + cy^2$ be a nondegenerate quadric as in Lemma \ref{lemma-nodal-algebraic} with coordinates $x, y$ chosen such that $a \not = 0$ and set $\kappa' = \kappa[x]/(ax^2 + bx + c)$, \item let $A' \supset A$ be the integral closure of $A$ in its total ring of fractions and set $\kappa' = A'/\mathfrak m A'$, or \item let $\kappa'$ be the $\kappa$-algebra such that $\text{Proj}(\bigoplus_{n \geq 0} \mathfrak m^n/\mathfrak m^{n + 1}) = \Spec(\kappa')$. \end{enumerate} The equivalence of (1) and (2) was shown in the proof of Lemma \ref{lemma-2-branches-delta-1}. We omit the equivalence of this with (3). If $X$ is a locally Noetherian $k$-scheme and $x \in X$ is a point such that $\mathcal{O}_{X, x} = A$, then (3) shows that $\Spec(\kappa') = X^\nu \times_X \Spec(\kappa)$ where $\nu : X^\nu \to X$ is the normalization morphism.


curves-remark-trivial-quadratic-extension

Let $k$ be a field. Let $(A, \mathfrak m, \kappa)$ be as in Remark \ref{remark-quadratic-extension} and let $\kappa'/\kappa$ be the associated separable algebra of degree $2$. Then the following are equivalent \begin{enumerate} \item $\kappa' \cong \kappa \times \kappa$ as $\kappa$-algebra, \item the form $q$ of Lemma \ref{lemma-nodal-algebraic} can be chosen to be $xy$, \item $A$ has two branches, \item the extension $A'/A$ of Lemma \ref{lemma-2-branches-delta-1} has two maximal ideals, and \item $A^\wedge \cong \kappa[[x, y]]/(xy)$ as a $k$-algebra. \end{enumerate} The equivalence between these conditions has been shown in the proof of Lemma \ref{lemma-2-branches-delta-1}. If $X$ is a locally Noetherian $k$-scheme and $x \in X$ is a point such that $\mathcal{O}_{X, x} = A$, then this means exactly that there are two points $x_1, x_2$ of the normalization $X^\nu$ lying over $x$ and that $\kappa(x) = \kappa(x_1) = \kappa(x_2)$.


etale-cohomology-remark-i-is-j

In the last statement, it is essential not to forget the case where $i = j$ which is in general a highly nontrivial condition (unlike in the Zariski topology). In fact, frequently important coverings have only one element.


etale-cohomology-remark-empty-covering

For the empty covering (where $I = \emptyset$), this implies that $\mathcal{F}(\emptyset)$ is an empty product, which is a final object in the corresponding category (a singleton, for both $\textit{Sets}$ and $\textit{Ab}$).


etale-cohomology-remark-fpqc

The first condition corresponds to fp, which stands for {\it fid\`element plat}, faithfully flat in french, and the second to qc, {\it quasi-compact}. The second part of the first condition is unnecessary when the second condition holds.


etale-cohomology-remark-fpqc-finest

The fpqc is finer than the Zariski, \'etale, smooth, syntomic, and fppf topologies. Hence any presheaf satisfying the sheaf condition for the fpqc topology will be a sheaf on the Zariski, \'etale, smooth, syntomic, and fppf sites. In particular representable presheaves will be sheaves on the \'etale site of a scheme for example.


etale-cohomology-remark-final-object

In the case where $\mathcal{C}$ has a final object, e.g.\ $S$, it suffices to check the condition of the definition for $U = S$ in the above statement. See Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}.


etale-cohomology-remark-presheaves-no-topology

Observe that all of the preceding statements are about presheaves so we haven't made use of the topology yet.


etale-cohomology-remark-grothendieck-ss

This is a Grothendieck spectral sequence for the composition of functors $$ \textit{Ab}(\mathcal{C}) \longrightarrow \textit{PAb}(\mathcal{C}) \xrightarrow{\check H^0} \textit{Ab}. $$


etale-cohomology-remark-refinement

In the statement of Lemma \ref{lemma-cech-complex} the covering $\mathcal{U}$ is a refinement of $\mathcal{V}$ but not the other way around. Coverings of the form $\{V \to U\}$ do not form an initial subcategory of the category of all coverings of $U$. Yet it is still true that we can compute {\v C}ech cohomology $\check H^n(U, \mathcal{F})$ (which is defined as the colimit over the opposite of the category of coverings $\mathcal{U}$ of $U$ of the {\v C}ech cohomology groups of $\mathcal{F}$ with respect to $\mathcal{U}$) in terms of the coverings $\{V \to U\}$. We will formulate a precise lemma (it only works for sheaves) and add it here if we ever need it.


etale-cohomology-remark-right-derived-global-sections

Comment on Theorem \ref{theorem-zariski-fpqc-quasi-coherent}. Since $S$ is a final object in the category $\mathcal{C}$, the cohomology groups on the right-hand side are merely the right derived functors of the global sections functor. In fact the proof shows that $H^p(U, f^*\mathcal{F}) = H^p_\tau(U, \mathcal{F}^a)$ for any object $f : U \to S$ of the site $\mathcal{C}$.


etale-cohomology-remark-constant-locally-constant-maps

Let $G$ be an abstract group. On any of the sites $(\Sch/S)_\tau$ or $S_\tau$ of Section \ref{section-big-small} the sheafification $\underline{G}$ of the constant presheaf associated to $G$ in the {\it Zariski topology} of the site already gives $$ \Gamma(U, \underline{G}) = \{\text{Zariski locally constant maps }U \to G\} $$ This Zariski sheaf is representable by the group scheme $G_S$ according to Groupoids, Example \ref{groupoids-example-constant-group}. By Lemma \ref{lemma-representable-sheaf-fpqc} any representable presheaf satisfies the sheaf condition for the $\tau$-topology as well, and hence we conclude that the Zariski sheafification $\underline{G}$ above is also the $\tau$-sheafification.


etale-cohomology-remark-special-case-fpqc-cohomology-quasi-coherent

In the terminology introduced above a special case of Theorem \ref{theorem-zariski-fpqc-quasi-coherent} is $$ H_{fppf}^p(X, \mathbf{G}_a) = H_\etale^p(X, \mathbf{G}_a) = H_{Zar}^p(X, \mathbf{G}_a) = H^p(X, \mathcal{O}_X) $$ for all $p \geq 0$. Moreover, we could use the notation $H^p_{fppf}(X, \mathcal{O}_X)$ to indicate the cohomology of the structure sheaf on the big fppf site of $X$.


etale-cohomology-remark-no-kummer-sequence-zariski

Lemma \ref{lemma-kummer-sequence} is false when ``\'etale'' is replaced with ``Zariski''. Since the \'etale topology is coarser than the smooth topology, see Topologies, Lemma \ref{topologies-lemma-zariski-etale-smooth} it follows that the sequence is also exact in the smooth topology.


etale-cohomology-remark-no-kummer-sequence-smooth-etale-zariski

Lemma \ref{lemma-kummer-sequence-syntomic} is false for the smooth, \'etale, or Zariski topology.


etale-cohomology-remark-etale-between-etale

Since $U$ and $U'$ are \'etale over $S$, any $S$-morphism between them is also \'etale, see Proposition \ref{proposition-etale-morphisms}. In particular all morphisms of \'etale neighborhoods are \'etale.


etale-cohomology-remark-etale-neighbourhoods

Let $S$ be a scheme and $s \in S$ a point. In More on Morphisms, Definition \ref{more-morphisms-definition-etale-neighbourhood} we defined the notion of an \'etale neighbourhood $(U, u) \to (S, s)$ of $(S, s)$. If $\overline{s}$ is a geometric point of $S$ lying over $s$, then any \'etale neighbourhood $(U, \overline{u}) \to (S, \overline{s})$ gives rise to an \'etale neighbourhood $(U, u)$ of $(S, s)$ by taking $u \in U$ to be the unique point of $U$ such that $\overline{u}$ lies over $u$. Conversely, given an \'etale neighbourhood $(U, u)$ of $(S, s)$ the residue field extension $\kappa(s) \subset \kappa(u)$ is finite separable (see Proposition \ref{proposition-etale-morphisms}) and hence we can find an embedding $\kappa(u) \subset \kappa(\overline{s})$ over $\kappa(s)$. In other words, we can find a geometric point $\overline{u}$ of $U$ lying over $u$ such that $(U, \overline{u})$ is an \'etale neighbourhood of $(S, \overline{s})$. We will use these observations to go between the two types of \'etale neighbourhoods.


etale-cohomology-remark-map-stalks

Let $S$ be a scheme and let $\overline{s} : \Spec(k) \to S$ and $\overline{s}' : \Spec(k') \to S$ be two geometric points of $S$. A {\it morphism $a : \overline{s} \to \overline{s}'$ of geometric points} is simply a morphism $a : \Spec(k) \to \Spec(k')$ such that $a \circ \overline{s}' = \overline{s}$. Given such a morphism we obtain a functor from the category of \'etale neighbourhoods of $\overline{s}'$ to the category of \'etale neighbourhoods of $\overline{s}$ by the rule $(U, \overline{u}') \mapsto (U, \overline{u}' \circ a)$. Hence we obtain a canonical map $$ \mathcal{F}_{\overline{s}'} = \colim_{(U, \overline{u}')} \mathcal{F}(U) \longrightarrow \colim_{(U, \overline{u})} \mathcal{F}(U) = \mathcal{F}_{\overline{s}} $$ from Categories, Lemma \ref{categories-lemma-functorial-colimit}. Using the description of elements of stalks as triples this maps the element of $\mathcal{F}_{\overline{s}'}$ represented by the triple $(U, \overline{u}', \sigma)$ to the element of $\mathcal{F}_{\overline{s}}$ represented by the triple $(U, \overline{u}' \circ a, \sigma)$. Since the functor above is clearly an equivalence we conclude that this canonical map is an isomorphism of stalk functors. \medskip\noindent Let us make sure we have the map of stalks corresponding to $a$ pointing in the correct direction. Note that the above means, according to Sites, Definition \ref{sites-definition-morphism-points}, that $a$ defines a morphism $a : p \to p'$ between the points $p, p'$ of the site $S_\etale$ associated to $\overline{s}, \overline{s}'$ by Lemma \ref{lemma-stalk-gives-point}. There are more general morphisms of points (corresponding to specializations of points of $S$) which we will describe later, and which will not be isomorphisms (insert future reference here).


etale-cohomology-remark-points-fppf-site

\begin{reference} This is discussed in \cite{Schroeer}. \end{reference} Let $S = \Spec(A)$ be an affine scheme. Let $(p, u)$ be a point of the site $(\textit{Aff}/S)_{fppf}$, see Sites, Sections \ref{sites-section-points} and \ref{sites-section-construct-points}. Let $B = \mathcal{O}_p$ be the stalk of the structure sheaf at the point $p$. Recall that $$ B = \colim_{(U, x)} \mathcal{O}(U) = \colim_{(\Spec(C), x_C)} C $$ where $x_C \in u(\Spec(C))$. It can happen that $\Spec(B)$ is an object of $(\textit{Aff}/S)_{fppf}$ and that there is an element $x_B \in u(\Spec(B))$ mapping to the compatible system $x_C$. In this case the system of neighbourhoods has an initial object and it follows that $\mathcal{F}_p = \mathcal{F}(\Spec(B))$ for any sheaf $\mathcal{F}$ on $(\textit{Aff}/S)_{fppf}$. It is straightforward to see that if $\mathcal{F} \mapsto \mathcal{F}(\Spec(B))$ defines a point of $\Sh((\textit{Aff}/S)_{fppf})$, then $B$ has to be a local $A$-algebra such that for every faithfully flat, finitely presented ring map $B \to B'$ there is a section $B' \to B$. Conversely, for any such $A$-algebra $B$ the functor $\mathcal{F} \mapsto \mathcal{F}(\Spec(B))$ is the stalk functor of a point. Details omitted. It is not clear what a general point of the site $(\textit{Aff}/S)_{fppf}$ looks like.


etale-cohomology-remark-henselization-Noetherian

Let $S$ be a scheme. Let $s \in S$. If $S$ is locally Noetherian then $\mathcal{O}_{S, s}^h$ is also Noetherian and it has the same completion: $$ \widehat{\mathcal{O}_{S, s}} \cong \widehat{\mathcal{O}_{S, s}^h}. $$ In particular, $\mathcal{O}_{S, s} \subset \mathcal{O}_{S, s}^h \subset \widehat{\mathcal{O}_{S, s}}$. The henselization of $\mathcal{O}_{S, s}$ is in general much smaller than its completion and inherits many of its properties. For example, if $\mathcal{O}_{S, s}$ is reduced, then so is $\mathcal{O}_{S, s}^h$, but this is not true for the completion in general. Insert future references here.


etale-cohomology-remark-direct-image-sheaf

We claim that the direct image of a sheaf is a sheaf. Namely, if $\{V_j \to V\}$ is an \'etale covering in $Y_\etale$ then $\{X \times_Y V_j \to X \times_Y V\}$ is an \'etale covering in $X_\etale$. Hence the sheaf condition for $\mathcal{F}$ with respect to $\{X \times_Y V_i \to X \times_Y V\}$ is equivalent to the sheaf condition for $f_*\mathcal{F}$ with respect to $\{V_i \to V\}$. Thus if $\mathcal{F}$ is a sheaf, so is $f_*\mathcal{F}$.


etale-cohomology-remark-functoriality-general

More generally, let $\mathcal{C}_1, \mathcal{C}_2$ be sites, and assume they have final objects and fibre products. Let $u: \mathcal{C}_2 \to \mathcal{C}_1$ be a functor satisfying: \begin{enumerate} \item if $\{V_i \to V\}$ is a covering of $\mathcal{C}_2$, then $\{u(V_i) \to u(V)\}$ is a covering of $\mathcal{C}_1$ (we say that $u$ is {\it continuous}), and \item $u$ commutes with finite limits (i.e., $u$ is left exact, i.e., $u$ preserves fibre products and final objects). \end{enumerate} Then one can define $f_*: \Sh(\mathcal{C}_1) \to \Sh(\mathcal{C}_2)$ by $ f_* \mathcal{F}(V) = \mathcal{F}(u(V))$. Moreover, there exists an exact functor $f^{-1}$ which is left adjoint to $f_*$, see Sites, Definition \ref{sites-definition-morphism-sites} and Proposition \ref{sites-proposition-get-morphism}. Warning: It is not enough to require simply that $u$ is continuous and commutes with fibre products in order to get a morphism of topoi.


etale-cohomology-remark-property-C-strong

Property (C) holds if $f : X \to Y$ is an open immersion. Namely, if $U \in \Ob(X_\etale)$, then we can view $U$ also as an object of $Y_\etale$ and $U \times_Y X = U$. Hence property (C) does not imply that $f_{small, *}$ is exact as this is not the case for open immersions (in general).


etale-cohomology-remark-affine-inside-equivalence

In the situation of Theorem \ref{theorem-topological-invariance} it is also true that $V \mapsto V_X$ induces an equivalence between those \'etale morphisms $V \to Y$ with $V$ affine and those \'etale morphisms $U \to X$ with $U$ affine. This follows for example from Limits, Proposition \ref{limits-proposition-affine}.


etale-cohomology-remark-push-pull-shriek

In the situation of Lemma \ref{lemma-monomorphism-big-push-pull} it is true that the canonical map $\mathcal{F} \to f_{big}^{-1}f_{big!}\mathcal{F}$ is an isomorphism for any sheaf of sets $\mathcal{F}$ on $(\Sch/X)_\tau$. The proof is the same. This also holds for sheaves of abelian groups. However, note that the functor $f_{big!}$ for sheaves of abelian groups is defined in Modules on Sites, Section \ref{sites-modules-section-exactness-lower-shriek} and is in general different from $f_{big!}$ on sheaves of sets. The result for sheaves of abelian groups follows from Modules on Sites, Lemma \ref{sites-modules-lemma-back-and-forth}.


etale-cohomology-remark-fppf-closed-immersion-not-closed

In Lemma \ref{lemma-closed-immersion-pushforward-exact} the case $\tau = fppf$ is missing. The reason is that given a ring $A$, an ideal $I$ and a faithfully flat, finitely presented ring map $A/I \to \overline{B}$, there is no reason to think that one can find {\it any} flat finitely presented ring map $A \to B$ with $B/IB \not = 0$ such that $A/I \to B/IB$ factors through $\overline{B}$. Hence the proof of Lemma \ref{lemma-closed-immersion-almost-cocontinuous} does not work for the fppf topology. In fact it is likely false that $f_{big, *} : \textit{Ab}((\Sch/X)_{fppf}) \to \textit{Ab}((\Sch/Y)_{fppf})$ is exact when $f$ is a closed immersion. If you know an example, please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.


etale-cohomology-remark-colimit-variant-complexes

Many of the results above have variants for bounded below complexes, but one has to be careful that the bounds have to be uniform. We explain this in the simplest case. Let $X$ be a quasi-compact and quasi-separated scheme. Let $I$ be a directed set. Let $\mathcal{F}_i^\bullet$ be a system over $I$ of complexes of sheaves on $X_\etale$. Assume there is an integer $a$ such that $\mathcal{F}_i^n = 0$ for $n < a$ and all $i \in I$. Then we have $$ H^p_\etale(X, \colim \mathcal{F}_i^\bullet) = \colim H^p_\etale(X, \mathcal{F}^\bullet_i) $$ If we ever need this we will state a precise lemma with full proof here.


etale-cohomology-remark-cohomological-descent-finite

In the situation of Lemma \ref{lemma-cohomological-descent-finite} if $\mathcal{G}$ is a sheaf of sets on $Y_\etale$, then we have $$ \Gamma(Y, \mathcal{G}) = \text{Equalizer}( \xymatrix{ \Gamma(X_0, f_0^{-1}\mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] & \Gamma(X_1, f_1^{-1}\mathcal{G}) } ) $$ This is proved in exactly the same way, by showing that the sheaf $\mathcal{G}$ is the equalizer of the two maps $f_{0, *}f_0^{-1}\mathcal{G} \to f_{1, *}f_1^{-1}\mathcal{G}$.


etale-cohomology-remark-every-sheaf-representable

Another way to state the conclusion of Theorem \ref{theorem-equivalence-sheaves-point} and Fundamental Groups, Lemma \ref{pione-lemma-sheaves-point} is to say that every sheaf on $\Spec(K)_\etale$ is representable by a scheme $X$ \'etale over $\Spec(K)$. This does not mean that every sheaf is representable in the sense of Sites, Definition \ref{sites-definition-representable-sheaf}. The reason is that in our construction of $\Spec(K)_\etale$ we chose a sufficiently large set of schemes \'etale over $\Spec(K)$, whereas sheaves on $\Spec(K)_\etale$ form a proper class.


etale-cohomology-remark-stalk-pullback

Let $S$ be a scheme and let $\overline{s} : \Spec(k) \to S$ be a geometric point of $S$. By definition this means that $k$ is algebraically closed. In particular the absolute Galois group of $k$ is trivial. Hence by Theorem \ref{theorem-equivalence-sheaves-point} the category of sheaves on $\Spec(k)_\etale$ is equivalent to the category of sets. The equivalence is given by taking sections over $\Spec(k)$. This finally provides us with an alternative definition of the stalk functor. Namely, the functor $$ \Sh(S_\etale) \longrightarrow \textit{Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{s}} $$ is isomorphic to the functor $$ \Sh(S_\etale) \longrightarrow \Sh(\Spec(k)_\etale) = \textit{Sets}, \quad \mathcal{F} \longmapsto \overline{s}^*\mathcal{F} $$ To prove this rigorously one can use Lemma \ref{lemma-stalk-pullback} part (3) with $f = \overline{s}$. Moreover, having said this the general case of Lemma \ref{lemma-stalk-pullback} part (3) follows from functoriality of pullbacks.


etale-cohomology-remark-functorial-locally-constant-on-connected

The equivalences of Lemma \ref{lemma-locally-constant-on-connected} are compatible with pullbacks. More precisely, suppose $f : Y \to X$ is a morphism of connected schemes. Let $\overline{y}$ be geometric point of $Y$ and set $\overline{x} = f(\overline{y})$. Then the diagram $$ \xymatrix{ \text{finite locally constant sheaves of sets on }Y_\etale \ar[r] & \text{finite }\pi_1(Y, \overline{y})\text{-sets} \\ \text{finite locally constant sheaves of sets on }X_\etale \ar[r] \ar[u]_{f^{-1}} & \text{finite }\pi_1(X, \overline{x})\text{-sets} \ar[u] } $$ is commutative, where the vertical arrow on the right comes from the continuous homomorphism $\pi_1(Y, \overline{y}) \to \pi_1(X, \overline{x})$ induced by $f$. This follows immediately from the commutative diagram in Fundamental Groups, Theorem \ref{pione-theorem-fundamental-group}.


etale-cohomology-remark-natural-proof

The ``natural'' way to prove the previous corollary is to excise $X$ from $\bar X$. This is possible, we just haven't developed that theory.


etale-cohomology-remark-normalize-H1-Gm

Let $k$ be an algebraically closed field. Let $n$ be an integer prime to the characteristic of $k$. Recall that $$ \mathbf{G}_{m, k} = \mathbf{A}^1_k \setminus \{0\} = \mathbf{P}^1_k \setminus \{0, \infty\} $$ We claim there is a canonical isomorphism $$ H^1_\etale(\mathbf{G}_{m, k}, \mu_n) = \mathbf{Z}/n\mathbf{Z} $$ What does this mean? This means there is an element $1_k$ in $H^1_\etale(\mathbf{G}_{m, k}, \mu_n)$ such that for every morphism $\Spec(k') \to \Spec(k)$ the pullback map on \'etale cohomology for the map $\mathbf{G}_{m, k'} \to \mathbf{G}_{m, k}$ maps $1_k$ to $1_{k'}$. (In particular this element is fixed under all automorphisms of $k$.) To see this, consider the $\mu_{n, \mathbf{Z}}$-torsor $\mathbf{G}_{m, \mathbf{Z}} \to \mathbf{G}_{m, \mathbf{Z}}$, $x \mapsto x^n$. By the identification of torsors with first cohomology, this pulls back to give our canonical elements $1_k$. Twisting back we see that there are canonical identifications $$ H^1_\etale(\mathbf{G}_{m, k}, \mathbf{Z}/n\mathbf{Z}) = \Hom(\mu_n(k), \mathbf{Z}/n\mathbf{Z}), $$ i.e., these isomorphisms are compatible with respect to maps of algebraically closed fields, in particular with respect to automorphisms of $k$.


etale-cohomology-remark-different

Objects in the derived category $D_{ctf}(X_\etale, \Lambda)$ in some sense have better global properties than the perfect objects in  $D(\mathcal{O}_X)$. Namely, it can happen that a complex of $\mathcal{O}_X$-modules is locally quasi-isomorphic to a finite complex of finite locally free $\mathcal{O}_X$-modules, without being globally quasi-isomorphic to a bounded complex of locally free $\mathcal{O}_X$-modules. The following lemma shows this does not happen for $D_{ctf}$ on a Noetherian scheme.


etale-cohomology-remark-projective-each-degree

Let $\Lambda$ be a Noetherian ring. Let $X$ be a scheme. For a bounded complex $K^\bullet$ of constructible flat $\Lambda$-modules on $X_\etale$ each stalk $K^p_{\overline{x}}$ is a finite projective $\Lambda$-module. Hence the stalks of the complex are perfect complexes of $\Lambda$-modules.


etale-cohomology-remark-invariance

Let $k$ be an algebraically closed field of characteristic $p > 0$. In Section \ref{section-artin-schreier} we have seen that there is an exact sequence $$ k[x] \to k[x] \to H^1_\etale(\mathbf{A}^1_k, \mathbf{Z}/p\mathbf{Z}) \to 0 $$ where the first arrow maps $f(x)$ to $f^p - f$. A set of representatives for the cokernel is formed by the polynomials $$ \sum\nolimits_{p \not | n} \lambda_n x^n $$ with $\lambda_n \in k$. (If $k$ is not algebraically closed you have to add some constants to this as well.) In particular when $k' \supset k$ is an algebraically closed overfield, then the map $$ H^1_\etale(\mathbf{A}^1_k, \mathbf{Z}/p\mathbf{Z}) \to H^1_\etale(\mathbf{A}^1_{k'}, \mathbf{Z}/p\mathbf{Z}) $$ is not an isomorphism in general. In particular, the map $\pi_1(\mathbf{A}^1_{k'}) \to \pi_1(\mathbf{A}^1_k)$ between \'etale fundamental groups (insert future reference here) is not an isomorphism either. Thus the \'etale homotopy type of the affine line depends on the algebraically closed ground field. From Lemma \ref{lemma-constant-smooth-statements} above we see that this is a phenomenon which only happens in characteristic $p$ with $p$-power torsion coefficients.


etale-cohomology-remark-base-change-holds

Let $f : X \to S$ be a morphism of schemes. Let $n$ be an integer. We will say $BC(f, n, q_0)$ is true if for every commutative diagram $$ \xymatrix{ X \ar[d]_f & X' \ar[l] \ar[d]_{f'} & Y \ar[l]^h \ar[d]^e \\ S & S' \ar[l] & T \ar[l]_g } $$ with $X' = X \times_S S'$ and $Y = X' \times_{S'} T$ and $g$ quasi-compact and quasi-separated, and every abelian sheaf $\mathcal{F}$ on $T_\etale$ annihilated by $n$ the base change map $$ (f')^{-1}R^qg_*\mathcal{F} \longrightarrow R^qh_*e^{-1}\mathcal{F} $$ is an isomorphism for $q \leq q_0$.


etale-cohomology-remark-define-kunneth-map

Consider a cartesian diagram in the category of schemes: $$ \xymatrix{ X \times_S Y \ar[d]_p \ar[r]_q \ar[rd]_c & Y \ar[d]^g \\ X \ar[r]^f & S } $$ Let $\Lambda$ be a ring and let $E \in D(X_\etale, \Lambda)$ and $K \in D(Y_\etale, \Lambda)$. Then there is a canonical map $$ Rf_*E \otimes_\Lambda^\mathbf{L} Rg_*K \longrightarrow Rc_*(p^{-1}E \otimes_\Lambda^\mathbf{L} q^{-1}K) $$ For example we can define this using the canonical maps $Rf_*E \to Rc_*p^{-1}E$ and $Rg_*K \to Rc_*q^{-1}K$ and the relative cup product defined in Cohomology on Sites, Remark \ref{sites-cohomology-remark-cup-product}. Or you can use the adjoint to the map $$ c^{-1}(Rf_*E \otimes_\Lambda^\mathbf{L} Rg_*K) = p^{-1}f^{-1}Rf_*E \otimes_\Lambda^\mathbf{L} q^{-1} g^{-1}Rg_*K \to p^{-1}E \otimes_\Lambda^\mathbf{L} q^{-1}K $$ which uses the adjunction maps $f^{-1}Rf_*E \to E$ and $g^{-1}Rg_*K \to K$.


etale-cohomology-remarks-theorem-modules-exactness

The results on descent of modules have several applications: \begin{enumerate} \item The exactness of the {\v C}ech complex in positive degrees for the covering $\{\Spec(B) \to \Spec(A)\}$ where $A \to B$ is faithfully flat. This will give some vanishing of cohomology. \item If $(N, \varphi)$ is a descent datum with respect to a faithfully flat map $A \to B$, then the corresponding $A$-module is given by $$ M = \Ker \left( \begin{matrix} N & \longrightarrow & B \otimes_A N \\ n & \longmapsto & 1 \otimes n - \varphi(n \otimes 1) \end{matrix} \right). $$ See Descent, Proposition \ref{descent-proposition-descent-module}. \end{enumerate}


etale-cohomology-remarks-enough-points

On points of the geometric sites. \begin{enumerate} \item Theorem \ref{theorem-exactness-stalks} says that the family of points of $S_\etale$ given by the geometric points of $S$ (Lemma \ref{lemma-stalk-gives-point}) is conservative, see Sites, Definition \ref{sites-definition-enough-points}. In particular $S_\etale$ has enough points. \item Suppose $\mathcal{F}$ is a sheaf on the big \'etale site \label{item-stalks-big} of $S$. Let $T \to S$ be an object of the big \'etale site of $S$, and let $\overline{t}$ be a geometric point of $T$. Then we define $\mathcal{F}_{\overline{t}}$ as the stalk of the restriction $\mathcal{F}|_{T_\etale}$ of $\mathcal{F}$ to the small \'etale site of $T$. In other words, we can define the stalk of $\mathcal{F}$ at any geometric point of any scheme $T/S \in \Ob((\Sch/S)_\etale)$. \item The big \'etale site of $S$ also has enough points, by considering all geometric points of all objects of this site, see (\ref{item-stalks-big}). \end{enumerate}


equiv-remark-affine-morphism

Below we will use that for an affine morphism $h : T \to S$ we have $h_*\mathcal{G} \otimes \mathcal{H} = h_*(\mathcal{G} \otimes h^*\mathcal{H})$ for $\mathcal{G} \in \QCoh(\mathcal{O}_T)$ and $\mathcal{H} \in \QCoh(\mathcal{O}_S)$. This follows immediately on translating into algebra.


equiv-remark-difficult

If $F, F' : D^b_{\textit{Coh}}(\mathcal{O}_X) \to \mathcal{D}$ are siblings, $F$ is fully faithful, and $X$ is reduced and projective over $k$ then $F \cong F'$; this follows from Proposition \ref{proposition-siblings-isomorphic} via the argument given in the proof of Theorem \ref{theorem-fully-faithful}. However, in general we do not know whether siblings are isomorphic. Even in the situation of Lemma \ref{lemma-exact-functor-preserving-Coh} it seems difficult to prove that the siblings $F$ and $F'$ are isomorphic functors. If $X$ is smooth and proper over $k$ and $F$ is fully faithful, then $F \cong F'$ as is shown in \cite{Noah}. If you have a proof or a counter example in more general situations, please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.


stacks-more-morphisms-remark-gerbe-of-lifts

Consider a diagram $$ \xymatrix{ W \ar[d]_x \\ \mathcal{X} \ar[r] & \mathcal{X}' } $$ where $\mathcal{X} \subset \mathcal{X}'$ is a thickening of algebraic stacks, $W$ is an algebraic space, and $W \to \mathcal{X}$ is smooth. We will construct a category $\mathcal{C}$ and a functor $$ p : \mathcal{C} \longrightarrow W_{spaces, \etale} $$ (see Properties of Spaces, Definition \ref{spaces-properties-definition-spaces-etale-site} for notation) as follows. An object of $\mathcal{C}$ will be a system $(U, U', a, i, x', \alpha)$ which forms a commutative diagram \begin{equation} \label{equation-object} \vcenter{ \xymatrix{ U \ar[d]_a \ar[r]_i & U' \ar[dd]^{x'} \\ W \ar[d]_x & \\ \mathcal{X} \ar[r] & \mathcal{X}' } } \end{equation} with commutativity witnessed by the $2$-morphism $\alpha : x \circ a \to x' \circ i$ such that $U$ and $U'$ are algebraic spaces, $a : U \to W$ is \'etale, $x' : U' \to \mathcal{X}'$ is smooth, and such that $U = \mathcal{X} \times_{\mathcal{X}'} U'$. In particular $U \subset U'$ is a thickening. A morphism $$ (U, U', a, i, x', \alpha) \to (V, V', b, j, y', \beta) $$ is given by $(f, f', \gamma)$ where $f : U \to V$ is a morphism over $W$, $f' : U' \to V'$ is a morphism whose restriction to $U$ gives $f$, and $\gamma : x' \circ f' \to y'$ is a $2$-morphism witnessing the commutativity in right triangle of the diagram below \begin{equation} \label{equation-morphism} \vcenter{ \xymatrix{ & V \ar[ld]_f \ar[ldd]^b \ar[rr]_j & & V' \ar[ld]_{f'} \ar[lddd]^{y'} \\ U \ar[d]_a \ar[rr]_i & & U' \ar[dd]_{x'} \\ W \ar[d]_x & \\ \mathcal{X} \ar[rr] & & \mathcal{X}' } } \end{equation} Finally, we require that $\gamma$ is compatible with $\alpha$ and $\beta$: in the calculus of $2$-categories of Categories, Sections \ref{categories-section-formal-cat-cat} and \ref{categories-section-2-categories} this reads $$ \beta = (\gamma \star \text{id}_j) \circ (\alpha \star \text{id}_f) $$ (more succinctly: $\beta = j^*\gamma \circ f^*\alpha$). Another formulation is that objects are commutative diagrams (\ref{equation-object}) with some additional properties and morphisms are commutative diagrams (\ref{equation-morphism}) in the category $\textit{Spaces}/\mathcal{X}'$ introduced in Properties of Stacks, Remark \ref{stacks-properties-remark-representable-over}. This makes it clear that $\mathcal{C}$ is a category and that the rule $p : \mathcal{C} \to W_{spaces, \etale}$ sending $(U, U', a, i, x', \alpha)$ to $a : U \to W$ is a functor.


perfect-remark-support-c-equations

Let $X$ be a scheme. Let $f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)$. Denote $Z \subset X$ the closed subscheme cut out by $f_1, \ldots, f_c$. For $0 \leq p < c$ and $1 \leq i_0 < \ldots < i_p \leq c$ we denote $U_{i_0 \ldots i_p} \subset X$ the open subscheme where $f_{i_0} \ldots f_{i_p}$ is invertible. For any $\mathcal{O}_X$-module $\mathcal{F}$ we set $$ \mathcal{F}_{i_0 \ldots i_p} = (U_{i_0 \ldots i_p} \to X)_*(\mathcal{F}|_{U_{i_0 \ldots i_p}}) $$ In this situation the {\it extended alternating {\v C}ech complex} is the complex of $\mathcal{O}_X$-modules \begin{equation} \label{equation-extended-alternating} 0 \to \mathcal{F} \to \bigoplus\nolimits_{i_0} \mathcal{F}_{i_0} \to \ldots \to \bigoplus\nolimits_{i_0 < \ldots < i_p} \mathcal{F}_{i_0 \ldots i_p} \to \ldots \to \mathcal{F}_{1 \ldots c} \to 0 \end{equation} where $\mathcal{F}$ is put in degree $0$. The maps are constructed as follows. Given $1 \leq i_0 < \ldots < i_{p + 1} \leq c$ and $0 \leq j \leq p + 1$ we have the canonical map $$ \mathcal{F}_{i_0 \ldots \hat i_j \ldots i_{p + 1}} \to \mathcal{F}_{i_0 \ldots i_p} $$ coming from the inclusion $U_{i_0 \ldots i_p} \subset U_{i_0 \ldots \hat i_j \ldots i_{p + 1}}$. The differentials in the extended alternating complex use these canonical maps with sign $(-1)^j$.


perfect-remark-extended-alternating-map-to-support

Let $X$, $f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)$, and $\mathcal{F}$ be as in Remark \ref{remark-support-c-equations}. Denote $\mathcal{F}^\bullet$ the complex (\ref{equation-extended-alternating}). By Lemma \ref{lemma-extended-alternating-zero} the cohomology sheaves of $\mathcal{F}^\bullet$ are supported on $Z$ hence $\mathcal{F}^\bullet$ is an object of $D_Z(\mathcal{O}_X)$. On the other hand, the equality $\mathcal{F}^0 = \mathcal{F}$ determines a canonical map $\mathcal{F}^\bullet \to \mathcal{F}$ in $D(\mathcal{O}_X)$. As $i_* \circ R\mathcal{H}_Z$ is a right adjoint to the inclusion functor $D_Z(\mathcal{O}_X) \to D(\mathcal{O}_X)$, see Cohomology, Lemma \ref{cohomology-lemma-complexes-with-support-on-closed}, we obtain a canonical commutative diagram $$ \xymatrix{ \mathcal{F}^\bullet \ar[rd] \ar[rr] & & \mathcal{F} \\ & i_*R\mathcal{H}_Z(\mathcal{F}) \ar[ru] } $$ in $D(\mathcal{O}_X)$ functorial in the $\mathcal{O}_X$-module $\mathcal{F}$.


perfect-remark-supported-map-c-equations

With $X$, $f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)$, and $\mathcal{F}$ as in Remark \ref{remark-support-c-equations}. There is a canonical $\mathcal{O}_X|_Z$-linear map $$ c_{f_1, \ldots, f_c} : i^*\mathcal{F} \longrightarrow \mathcal{H}^c_Z(\mathcal{F}) $$ functorial in $\mathcal{F}$. Namely, denoting $\mathcal{F}^\bullet$ the extended alternating {\v C}ech complex (\ref{equation-extended-alternating}) we have the canonical map $\mathcal{F}^\bullet \to i_*R\mathcal{H}_Z(\mathcal{F})$ of Remark \ref{remark-extended-alternating-map-to-support}. This determines a canonical map $$ \Coker\left(\bigoplus \mathcal{F}_{1 \ldots \hat i \ldots c} \to \mathcal{F}_{1 \ldots c}\right) \longrightarrow i_*\mathcal{H}^c_Z(\mathcal{F}) $$ on cohomology sheaves in degree $c$. Given a local section $s$ of $\mathcal{F}$ we can consider the local section $$ \frac{s}{f_1 \ldots f_c} $$ of $\mathcal{F}_{1 \ldots c}$. The class of this section in the cokernel displayed above depends only on $s$ modulo the image of $(f_1, \ldots, f_c) : \mathcal{F}^{\oplus c} \to \mathcal{F}$. Since $i_*i^*\mathcal{F}$ is equal to the cokernel of $(f_1, \ldots, f_c) : \mathcal{F}^{\oplus c} \to \mathcal{F}$ we see that we get an $\mathcal{O}_X$-module map $i_*i^*\mathcal{F} \to i_*\mathcal{H}_Z^c(\mathcal{F})$. As $i_*$ is fully faithful we get the map $c_{f_1, \ldots, f_c}$.


perfect-remark-supported-functorial

Let $g : X' \to X$ be a morphism of schemes. Let $f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)$. Set $f'_i = g^\sharp(f_i) \in \Gamma(X', \mathcal{O}_{X'})$. Denote $Z \subset X$, resp.\ $Z' \subset X'$ the closed subscheme cut out by $f_1, \ldots, f_c$, resp.\ $f'_1, \ldots, f'_c$. Then $Z' = Z \times_X X'$. Denote $h : Z' \to Z$ the induced morphism of schemes. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Set $\mathcal{F}' = g^*\mathcal{F}$. In this setting, if $\mathcal{F}$ is quasi-coherent, then the diagram $$ \xymatrix{ (i')^{-1}\mathcal{O}_{X'} \otimes_{h^{-1}i^{-1}\mathcal{O}_X} h^{-1}\mathcal{H}^c_Z(\mathcal{F}) \ar[r] & \mathcal{H}_{Z'}^c(\mathcal{F}') \\ h^*i^*\mathcal{F} \ar[r] \ar[u]_-{c_{f_1, \ldots, f_c}} & (i')^*\mathcal{F}' \ar[u]^-{c_{f'_1, \ldots, f'_c}} } $$ is commutative where the top horizonal arrow is the map of Cohomology, Remark \ref{cohomology-remark-support-functorial} on cohomology sheaves in degree $c$. Namely, denote $\mathcal{F}^\bullet$, resp.\ $(\mathcal{F}')^\bullet$ the extended alternating {\v C}ech complex constructed in Remark \ref{remark-support-c-equations} using $\mathcal{F}, f_1, \ldots, f_c$, resp.\ $\mathcal{F}', f'_1, \ldots, f'_c$. Note that $(\mathcal{F}')^\bullet = g^*\mathcal{F}^\bullet$. Then, without assuming $\mathcal{F}$ is quasi-coherent, the diagram $$ \xymatrix{ i'_* L(g|_{Z'})^* R\mathcal{H}_Z(\mathcal{F}) \ar[r] \ar@{=}[d] & i'_*R\mathcal{H}_{Z'}(Lg^*\mathcal{F}) \ar[d] \\ Lg^*i_*R\mathcal{H}_Z(\mathcal{F}) & i'_*R\mathcal{H}_{Z'}(\mathcal{F}') \\ Lg^*(\mathcal{F}^\bullet) \ar[u] \ar[r] & (\mathcal{F}')^\bullet \ar[u] } $$ is commutative where $g|_{Z'} : (Z', (i')^{-1}\mathcal{O}_{X'}) \to (Z, i^{-1}\mathcal{O}_X)$ is the induced morphism of ringed spaces. Here the top horizontal arrow is given in Cohomology, Remark \ref{cohomology-remark-support-functorial} as is the explanation for the equal sign. The arrows pointing up are from Remark \ref{remark-extended-alternating-map-to-support}. The lower horizonal arrow is the map $Lg^*\mathcal{F}^\bullet \to g^*\mathcal{F}^\bullet = (\mathcal{F}')^\bullet$ and the arrow pointing down is induced by $Lg^*\mathcal{F} \to g^*\mathcal{F} = \mathcal{F}'$. The diagram commutes because going around the diagram both ways we obtain two arrows $Lg^*\mathcal{F}^\bullet \to i'_*R\mathcal{H}_{Z'}(\mathcal{F}')$ whose composition with $i'_*R\mathcal{H}_{Z'}(\mathcal{F}') \to \mathcal{F}'$ is the canonical map $Lg^*\mathcal{F}^\bullet \to \mathcal{F}'$. Some details omitted. Now the commutativity of the first diagram follows by looking at this diagram on cohomology sheaves in degree $c$ and using that the construction of the map $i^*\mathcal{F} \to \Coker(\bigoplus \mathcal{F}_{1 \ldots \hat i \ldots c} \to \mathcal{F}_{1 \ldots c})$ used in Remark \ref{remark-supported-map-c-equations} is compatible with pullbacks.


perfect-remark-warning-coherator

Let $X$ be a quasi-compact scheme with affine diagonal. Even though we know that $D(\QCoh(\mathcal{O}_X)) = D_\QCoh(\mathcal{O}_X)$ by Proposition \ref{proposition-quasi-compact-affine-diagonal} strange things can happen and it is easy to make mistakes with this material. One pitfall is to carelessly assume that this equality means derived functors are the same. For example, suppose we have a quasi-compact open $U \subset X$. Then we can consider the higher right derived functors $$ R^i(\QCoh)\Gamma(U, -) : \QCoh(\mathcal{O}_X) \to \textit{Ab} $$ of the left exact functor $\Gamma(U, -)$. Since this is a universal $\delta$-functor, and since the functors $H^i(U, -)$ (defined for all abelian sheaves on $X$) restricted to $\QCoh(\mathcal{O}_X)$ form a $\delta$-functor, we obtain canonical tranformations $$ t^i : R^i(\QCoh)\Gamma(U, -) \to H^i(U, -). $$ These transformations aren't in general isomorphisms even if $X = \Spec(A)$ is affine! Namely, we have $R^1(\QCoh)\Gamma(U, \widetilde{I}) = 0$ if $I$ an injective $A$-module by construction of right derived functors and the equivalence of $\QCoh(\mathcal{O}_X)$ and $\text{Mod}_A$. But Examples, Lemma \ref{examples-lemma-nonvanishing} shows there exists $A$, $I$, and $U$ such that $H^1(U, \widetilde{I}) \not = 0$.


perfect-remark-addendum

The proof of Lemma \ref{lemma-lift-map-from-perfect-complex-with-support} shows that $$ R|_U = P \oplus P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m] $$ for some $m \geq 0$ and $n_j \geq 0$. Thus the highest degree cohomology sheaf of $R|_U$ equals that of $P$. By repeating the construction for the map $P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m] \to R|_U$, taking cones, and using induction we can achieve equality of cohomology sheaves of $R|_U$ and $P$ above any given degree.


perfect-remark-pullback-generator

Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $E \in D_\QCoh(\mathcal{O}_Y)$ be a generator (see Theorem \ref{theorem-bondal-van-den-Bergh}). Then the following are equivalent \begin{enumerate} \item for $K \in D_\QCoh(\mathcal{O}_X)$ we have $Rf_*K = 0$ if and only if $K = 0$, \item $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ reflects isomorphisms, and \item $Lf^*E$ is a generator for $D_\QCoh(\mathcal{O}_X)$. \end{enumerate} The equivalence between (1) and (2) is a formal consequence of the fact that $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ is an exact functor of triangulated categories. Similarly, the equivalence between (1) and (3) follows formally from the fact that $Lf^*$ is the left adjoint to $Rf_*$. These conditions hold if $f$ is affine (Lemma \ref{lemma-affine-morphism}) or if $f$ is an open immersion, or if $f$ is a composition of such. We conclude that \begin{enumerate} \item if $X$ is a quasi-affine scheme then $\mathcal{O}_X$ is a generator for $D_\QCoh(\mathcal{O}_X)$, \item if $X \subset \mathbf{P}^n_A$ is a quasi-compact locally closed subscheme, then $\mathcal{O}_X \oplus \mathcal{O}_X(-1) \oplus \ldots \oplus \mathcal{O}_X(-n)$ is a generator for $D_\QCoh(\mathcal{O}_X)$ by Lemma \ref{lemma-generator-P1}. \end{enumerate}


perfect-remark-classical-generator

Let $X$ be a quasi-compact and quasi-separated scheme. Let $G$ be a perfect object of $D(\mathcal{O}_X)$ which is a generator for $D_\QCoh(\mathcal{O}_X)$. By Theorem \ref{theorem-bondal-van-den-Bergh} there is at least one of these. Combining Lemma \ref{lemma-quasi-coherence-direct-sums} with Proposition \ref{proposition-compact-is-perfect} and with Derived Categories, Proposition \ref{derived-proposition-generator-versus-classical-generator} we see that $G$ is a classical generator for $D_{perf}(\mathcal{O}_X)$.


perfect-remark-classical-generator-with-support

Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. Let $G$ be a perfect object of $D_{\QCoh, T}(\mathcal{O}_X)$ which is a generator for $D_{\QCoh, T}(\mathcal{O}_X)$. By Lemma \ref{lemma-generator-with-support} there is at least one of these. Combining the fact that $D_{\QCoh, T}(\mathcal{O}_X)$ has direct sums with Lemma \ref{lemma-compact-is-perfect-with-support} and with Derived Categories, Proposition \ref{derived-proposition-generator-versus-classical-generator} we see that $G$ is a classical generator for $D_{perf, T}(\mathcal{O}_X)$.


perfect-remark-DQCoh-is-Ddga-with-support

Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. The analogue of Theorem \ref{theorem-DQCoh-is-Ddga} holds for $D_{\QCoh, T}(\mathcal{O}_X)$. This follows from the exact same argument as in the proof of the theorem, using Lemmas \ref{lemma-generator-with-support} and \ref{lemma-compact-is-perfect-with-support} and a variant of Lemma \ref{lemma-tensor-with-QCoh-complex} with supports. If we ever need this, we will precisely state the result here and give a detailed proof.


perfect-remark-independence-choice

Let $X$ be a quasi-compact and quasi-separated scheme over a ring $R$. By the construction of the proof of Theorem \ref{theorem-DQCoh-is-Ddga} there exists a differential graded algebra $(A, \text{d})$ over $R$ such that $D_\QCoh(X)$ is $R$-linearly equivalent to $D(A, \text{d})$ as a triangulated category. One may ask: how unique is $(A, \text{d})$? The answer is (only) slightly better than just saying that $(A, \text{d})$ is well defined up to derived equivalence. Namely, suppose that $(B, \text{d})$ is a second such pair. Then we have $$ (A, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet) $$ and $$ (B, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(L^\bullet, L^\bullet) $$ for some K-injective complexes $K^\bullet$ and $L^\bullet$ of $\mathcal{O}_X$-modules corresponding to perfect generators of $D_\QCoh(\mathcal{O}_X)$. Set $$ \Omega = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, L^\bullet) \quad \Omega' = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(L^\bullet, K^\bullet) $$ Then $\Omega$ is a differential graded $B^{opp} \otimes_R A$-module and $\Omega'$ is a differential graded $A^{opp} \otimes_R B$-module. Moreover, the equivalence $$ D(A, \text{d}) \to D_\QCoh(\mathcal{O}_X) \to D(B, \text{d}) $$ is given by the functor $- \otimes_A^\mathbf{L} \Omega'$ and similarly for the quasi-inverse. Thus we are in the situation of Differential Graded Algebra, Remark \ref{dga-remark-hochschild-cohomology}. If we ever need this remark we will provide a precise statement with a detailed proof here.


perfect-remark-explain-consequence

Let $X$ be a quasi-compact and quasi-separated scheme. Let $X = U \cup V$ with $U$ and $V$ quasi-compact open. By Lemma \ref{lemma-better-coherator} the functors $DQ_X$, $DQ_U$, $DQ_V$, $DQ_{U \cap V}$ exist. Moreover, there is a canonical distinguished triangle $$ DQ_X(K) \to Rj_{U, *}DQ_U(K|_U) \oplus Rj_{V, *}DQ_V(K|_V) \to Rj_{U \cap V, *}DQ_{U \cap V}(K|_{U \cap V}) \to $$ for any $K \in D(\mathcal{O}_X)$. This follows by applying the exact functor $DQ_X$ to the distinguished triangle of Cohomology, Lemma \ref{cohomology-lemma-exact-sequence-j-star} and using Lemma \ref{lemma-pushforward-better-coherator} three times.


perfect-remark-multiplication-map

With notation as in Lemma \ref{lemma-affine-morphism-and-hom-out-of-perfect}. The diagram $$ \xymatrix{ R\Hom_X(M, Rg'_*L) \otimes_R^\mathbf{L} R' \ar[r] \ar[d]_\mu & R\Hom_{X'}(L(g')^*M, L(g')^*Rg'_*L) \ar[d]^a \\ R\Hom_X(M, R(g')_*L) \ar@{=}[r] & R\Hom_{X'}(L(g')^*M, L) } $$ is commutative where the top horizontal arrow is the map from the lemma, $\mu$ is the multiplication map, and $a$ comes from the adjunction map $L(g')^*Rg'_*L \to L$. The multiplication map is the adjunction map $K' \otimes_R^\mathbf{L} R' \to K'$ for any $K' \in D(R')$.


perfect-remark-annoying-compatibility

Let $S = \Spec(A)$ be an affine scheme. Let $a : X \to S$ and $b : Y \to S$ be morphisms of schemes. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_X$-modules and let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_Y$-module. Let $\xi \in H^i(X, \mathcal{G})$ with pullback $p^*\xi \in H^i(X \times_S Y, p^*\mathcal{G})$. Then the following diagram is commutative $$ \xymatrix{ R\Gamma(X, \mathcal{F})[-i] \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{E}) \ar[d] \ar[rr]_-{\xi \otimes \text{id}} & & R\Gamma(X, \mathcal{G} \otimes_{\mathcal{O}_X} \mathcal{F}) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{E}) \ar[d] \\ R\Gamma(X \times_S Y, p^*\mathcal{F} \otimes q^*\mathcal{E})[-i] \ar[rr]^-{p^*\xi} & & R\Gamma(X \times_S Y, p^*(\mathcal{G} \otimes_{\mathcal{O}_X} \mathcal{F}) \otimes q^*\mathcal{E}) } $$ where the unadorned tensor products are over $\mathcal{O}_{X \times_S Y}$. The horizontal arrows are from Cohomology, Remark \ref{cohomology-remark-cup-with-element-map-total-cohomology} and the vertical arrows are (\ref{equation-kunneth-global}) hence given by pulling back followed by cup product on $X \times_S Y$. The diagram commutes because the global cup product (on $X \times_S Y$ with the sheaves $p^*\mathcal{G}$, $p^*\mathcal{F}$, and $q^*\mathcal{E}$) is associative, see Cohomology, Lemma \ref{cohomology-lemma-cup-product-associative}.


perfect-remark-base-change-of-L

The pseudo-coherent complex $L$ of part (B) of Lemma \ref{lemma-compute-ext} is canonically associated to the situation. For example, formation of $L$ as in (B) is compatible with base change. In other words, given a cartesian diagram $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ of schemes we have canonical functorial isomorphisms $$ \Ext^i_{\mathcal{O}_{S'}}(Lg^*L, \mathcal{F}') \longrightarrow \Ext^i_{\mathcal{O}_X}(L(g')^*E, (g')^*\mathcal{G}^\bullet \otimes_{\mathcal{O}_{X'}} (f')^*\mathcal{F}') $$ for $\mathcal{F}'$ quasi-coherent on $S'$. Obsere that we do {\bf not} use derived pullback on $\mathcal{G}^\bullet$ on the right hand side. If we ever need this, we will formulate a precise result here and give a detailed proof.


perfect-remark-explain-perfect-direct-image

Let $R$ be a ring. Let $X$ be a scheme of finite presentation over $R$. Let $\mathcal{G}$ be a finitely presented $\mathcal{O}_X$-module flat over $R$ with support proper over $R$. By Lemma \ref{lemma-base-change-tensor-perfect} there exists a finite complex of finite projective $R$-modules $M^\bullet$ such that we have $$ R\Gamma(X_{R'}, \mathcal{G}_{R'}) = M^\bullet \otimes_R R' $$ functorially in the $R$-algebra $R'$.


perfect-remark-compare-L

The reader may have noticed the similarity between Lemma \ref{lemma-compute-ext-rel-perfect} and Lemma \ref{lemma-compute-ext}. Indeed, the pseudo-coherent complex $L$ of Lemma \ref{lemma-compute-ext-rel-perfect} may be characterized as the unique pseudo-coherent complex on $S$ such that there are functorial isomorphisms $$ \Ext^i_{\mathcal{O}_S}(L, \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(K, E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F}) $$ compatible with boundary maps for $\mathcal{F}$ ranging over $\QCoh(\mathcal{O}_S)$. If we ever need this we will formulate a precise result here and give a detailed proof.


perfect-remark-discuss-rel-perfect

Our Definition \ref{definition-relatively-perfect} of a relatively perfect complex is equivalent to the one given in \cite{lieblich-complexes} whenever our definition applies\footnote{To see this, use Lemma \ref{lemma-affine-locally-rel-perfect} and More on Algebra, Lemma \ref{more-algebra-lemma-structure-relatively-perfect}.}. Next, suppose that $f : X \to S$ is only assumed to be locally of finite type (not necessarily flat, nor locally of finite presentation). The definition in the paper cited above is that $E \in D(\mathcal{O}_X)$ is relatively perfect if \begin{enumerate} \item[(A)] locally on $X$ the object $E$ should be quasi-isomorphic to a finite complex of $S$-flat, finitely presented $\mathcal{O}_X$-modules. \end{enumerate} On the other hand, the natural generalization of our Definition \ref{definition-relatively-perfect} is \begin{enumerate} \item[(B)] $E$ is pseudo-coherent relative to $S$ (More on Morphisms, Definition \ref{more-morphisms-definition-relative-pseudo-coherence}) and $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_S)$ (Cohomology, Definition \ref{cohomology-definition-tor-amplitude}). \end{enumerate} The advantage of condition (B) is that it clearly defines a triangulated subcategory of $D(\mathcal{O}_X)$, whereas we suspect this is not the case for condition (A). The advantage of condition (A) is that it is easier to work with in particular in regards to limits.


perfect-remark-K-ring

Let $X$ be a scheme. The K-group $K_0(X)$ is canonically a commutative ring. Namely, using the derived tensor product $$ \otimes = \otimes^\mathbf{L}_{\mathcal{O}_X} : D_{perf}(\mathcal{O}_X) \times D_{perf}(\mathcal{O}_X) \longrightarrow D_{perf}(\mathcal{O}_X) $$ and Derived Categories, Lemma \ref{derived-lemma-bilinear-map-K} we obtain a bilinear multiplication. Since $K \otimes L \cong L \otimes K$ we see that this product is commutative. Since $(K \otimes L) \otimes M = K \otimes (L \otimes M)$ we see that this product is associative. Finally, the unit of $K_0(X)$ is the element $1 = [\mathcal{O}_X]$. \medskip\noindent If $\textit{Vect}(X)$ and $K_0(\textit{Vect}(X))$ are as above, then it is clearly the case that $K_0(\textit{Vect}(X))$ also has a ring structure: if $\mathcal{E}$ and $\mathcal{F}$ are finite locally free $\mathcal{O}_X$-modules, then we set $$ [\mathcal{E}] \cdot [\mathcal{F}] = [\mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{F}] $$ The reader easily verifies that this indeed defines a bilinear commutative, associative product. Details omitted. The map $$ K_0(\textit{Vect}(X)) \longrightarrow K_0(X) $$ constructed above is a ring map with these definitions. \medskip\noindent Now assume $X$ is Noetherian. The derived tensor product also produces a map $$ \otimes = \otimes^\mathbf{L}_{\mathcal{O}_X} : D_{perf}(\mathcal{O}_X) \times D^b_{\textit{Coh}}(\mathcal{O}_X) \longrightarrow D^b_{\textit{Coh}}(\mathcal{O}_X) $$ Again using Derived Categories, Lemma \ref{derived-lemma-bilinear-map-K} we obtain a bilinear multiplication $K_0(X) \times K'_0(X) \to K'_0(X)$ since $K'_0(X) = K_0(D^b_{\textit{Coh}}(\mathcal{O}_X))$ by Lemma \ref{lemma-Noetherian-Kprime}. The reader easily shows that this gives $K'_0(X)$ the structure of a module over the ring $K_0(X)$.


perfect-remark-pushforward-K

Let $f : X \to Y$ be a proper morphism of locally Noetherian schemes. There is a map $$ f_* : K'_0(X) \longrightarrow K'_0(Y) $$ which sends $[\mathcal{F}]$ to $$ [\bigoplus\nolimits_{i \geq 0} R^{2i}f_*\mathcal{F}] - [\bigoplus\nolimits_{i \geq 0} R^{2i + 1}f_*\mathcal{F}] $$ This is well defined because the sheaves $R^if_*\mathcal{F}$ are coherent (Cohomology of Schemes, Lemma \ref{coherent-lemma-locally-projective-pushforward}), because locally only a finite number are nonzero, and because a short exact sequence of coherent sheaves on $X$ produces a long exact sequence of $R^if_*$ on $Y$. If $Y$ is quasi-compact (the only case most often used in practice), then we can rewrite the above as $$ f_*[\mathcal{F}] = \sum (-1)^i[R^if_*\mathcal{F}] = [Rf_*\mathcal{F}] $$ where we have used the equality $K'_0(Y) = K_0(D^b_{\textit{Coh}}(Y))$ from Lemma \ref{lemma-Noetherian-Kprime}.


perfect-remark-perf-Z

Let $X$ be a 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 $Z$. The zeroth $K$-group $K_0(D_{Z, perf}(\mathcal{O}_X))$ of this triangulated category is sometimes denoted $K_Z(X)$ or $K_{0, Z}(X)$. Using derived tensor product exactly as in Remark \ref{remark-K-ring} we see that $K_0(D_{Z, perf}(\mathcal{O}_X))$ has a multiplication which is associative and commutative, but in general $K_0(D_{Z, perf}(\mathcal{O}_X))$ doesn't have a unit.


perfect-remark-functorial-det

The construction of Lemma \ref{lemma-determinant-two-term-complexes} is compatible with pullbacks. More precisely, given a morphism $f : X \to Y$ of schemes and a perfect object $K$ of $D(\mathcal{O}_Y)$ of tor-amplitude in $[-1, 0]$ then $Lf^*K$ is a perfect object $K$ of $D(\mathcal{O}_X)$ of tor-amplitude in $[-1, 0]$ and we have a canonical identification $$ f^*\det(K) \longrightarrow \det(Lf^*K) $$ Moreover, if $K$ has rank $0$, then $\delta(K)$ pulls back to $\delta(Lf^*K)$ via this map. This is clear from the affine local construction of the determinant.


spaces-flat-remark-correct-generality

The result in this section can be generalized. It is probably correct if we only assume $X \to \Spec(A)$ to be separated, of finite presentation, and $K_n$ pseudo-coherent relative to $A_n$ supported on a closed subset of $X_n$ proper over $A_n$. The outcome will be a $K$ which is pseudo-coherent relative to $A$ supported on a closed subset proper over $A$. If we ever need this, we will formulate a precise statement and prove it here.


spaces-chow-remark-sober

In Situation \ref{situation-setup} if $X/B$ is good, then $|X|$ is a sober topological space. See Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-separated-sober} or Decent Spaces, Proposition \ref{decent-spaces-proposition-reasonable-sober}. We will use this without further mention to choose generic points of irreducible closed subsets of $|X|$.


spaces-chow-remark-integral

In Situation \ref{situation-setup} if $X/B$ is good, then $X$ is integral (Spaces over Fields, Definition \ref{spaces-over-fields-definition-integral-algebraic-space}) if and only if $X$ is reduced and $|X|$ is irreducible. Moreover, for any point $\xi \in |X|$ there is a unique integral closed subspace $Z \subset X$ such that $\xi$ is the generic point of the closed subset $|Z| \subset |X|$, see Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-decent-irreducible-closed}.


spaces-chow-remark-irreducible-component

In Situation \ref{situation-setup} let $X/B$ be good. Let $Y \subset X$ be a closed subspace. By Remarks \ref{remark-sober} and \ref{remark-integral} there are $1$-to-$1$ correspondences between \begin{enumerate} \item irreducible components $T$ of $|Y|$, \item generic points of irreducible components of $|Y|$, and \item integral closed subspaces $Z \subset Y$ with the property that $|Z|$ is an irreducible component of $|Y|$. \end{enumerate} In this chapter we will call $Z$ as in (3) an {\it irreducible component of $Y$} and we will call $\xi \in |Z|$ its {\it generic point}.


spaces-chow-remark-residue-field

In Situation \ref{situation-setup} let $X/B$ be good. Every $x \in |X|$ can be represented by a (unique) monomorphism $\Spec(k) \to X$ where $k$ is a field, see Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-points-monomorphism}. Then $k$ is the {\it residue field of} $x$ and is denoted $\kappa(x)$. Recall that $X$ has a dense open subscheme $U \subset X$ (Properties of Spaces, Proposition \ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}). If $x \in U$, then $\kappa(x)$ agrees with the residue field of $x$ on $U$ as a scheme. See Decent Spaces, Section \ref{decent-spaces-section-residue-fields-henselian-local-rings}.


spaces-chow-remark-function-field

In Situation \ref{situation-setup} let $X/B$ be good. Assume $X$ is integral. In this case the {\it function field} $R(X)$ of $X$ is defined and is equal to the residue field of $X$ at its generic point. See Spaces over Fields, Definition \ref{spaces-over-fields-definition-function-field}. Combining this with Remark \ref{remark-integral} we find that for any $x \in X$ the residue field $\kappa(x)$ is the function field of the unique integral closed subspace $Z \subset X$ whose generic point is $x$.


spaces-chow-remark-infinite-sums-rational-equivalences

In Situation \ref{situation-setup} let $X/B$ be good. 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 $\{T_i\}_{i \in I}$ of $|X|$ such that $\alpha_i, \beta_i$ are supported on $T_i$ and such that $\alpha_i \sim_{rat} \beta_i$ ``on'' $T_i$. More precisely, the families $\{W_{i, j}, f_{i, j} \in R(W_{i, j})^*\}_{j \in J_i}$ consist of integral closed subspaces $W_{i, j}$ with $|W_{i, j}| \subset T_i$. 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.


spaces-chow-remark-on-cycles

Let $S$, $B$, $X$, $\mathcal{L}$, $s$, $i : D \to X$ 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 $W \subset D$. The possibility to do this will be useful later on.


spaces-chow-remark-pullback-pairs

Let $f : X' \to X$ be a morphism of good algebraic spaces over $B$ 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 $(i')^*$.


spaces-chow-remark-pullback-cohomology

In Situation \ref{situation-setup} let $f : X \to Y$ be a morphism of good algebraic spaces over $B$. Then there is a canonical $\mathbf{Z}$-algebra map $A^*(Y) \to A^*(X)$. Namely, given $c \in A^p(Y)$ and $X' \to X$, then we can let $f^*c$ be defined by the map $c \cap - : \CH_k(X') \to \CH_{k - p}(X')$ which is given by thinking of $X'$ as an algebraic space over $Y$.


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

In Situation \ref{situation-setup} let $X/B$ be good. 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 $r$. If $X' \to X$ is a morphism of good algebraic spaces over $B$, 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 we simply define $c_i(\mathcal{E})$ to be 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 $i$.


sdga-remark-functoriality-ga

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. We have \begin{enumerate} \item Let $\mathcal{A}$ be a graded $\mathcal{O}_\mathcal{C}$-algebra. The multiplication maps of $\mathcal{A}$ induce multiplication maps $f_*\mathcal{A}^n \times f_*\mathcal{A}^m \to f_*\mathcal{A}^{n + m}$ and via $f^\sharp$ we may view these as $\mathcal{O}_\mathcal{D}$-bilinear maps. We will denote $f_*\mathcal{A}$ the graded $\mathcal{O}_\mathcal{D}$-algebra we so obtain. \item Let $\mathcal{B}$ be a graded $\mathcal{O}_\mathcal{D}$-algebra.  The multiplication maps of $\mathcal{B}$ induce multiplication maps $f^*\mathcal{B}^n \times f^*\mathcal{B}^m \to f^*\mathcal{B}^{n + m}$ and using $f^\sharp$ we may view these as $\mathcal{O}_\mathcal{C}$-bilinear maps. We will denote $f^*\mathcal{B}$ the graded $\mathcal{O}_\mathcal{C}$-algebra we so obtain. \item The set of homomorphisms $f^*\mathcal{B} \to \mathcal{A}$ of graded $\mathcal{O}_\mathcal{C}$-algebras is in $1$-to-$1$ correspondence with the set of homomorphisms $\mathcal{B} \to f_*\mathcal{A}$ of graded $\mathcal{O}_\mathcal{C}$-algebras. \end{enumerate} Part (3) follows immediately from the usual adjunction between $f^*$ and $f_*$ on sheaves of modules.


sdga-remark-functoriality-dga

Let $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi. \begin{enumerate} \item Let $(\mathcal{A}, \text{d})$ be a differential graded $\mathcal{O}_\mathcal{C}$-algebra. The pushforward will be the differential graded $\mathcal{O}_\mathcal{D}$-algebra $(f_*\mathcal{A}, \text{d})$ where $f_*\mathcal{A}$ is as in Remark \ref{remark-functoriality-ga} and $\text{d} = f_*\text{d}$ as maps $f_*\mathcal{A}^n \to f_*\mathcal{A}^{n + 1}$. We omit the verification that the Leibniz rule is satisfied. \item Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal{D}$-algebra. The pullback will be the differential graded $\mathcal{O}_\mathcal{C}$-algebra $(f^*\mathcal{B}, \text{d})$ where $f^*\mathcal{B}$ is as in Remark \ref{remark-functoriality-ga} and $\text{d} = f^*\text{d}$ as maps $f^*\mathcal{B}^n \to f^*\mathcal{B}^{n + 1}$. We omit the verification that the Leibniz rule is satisfied. \item The set of homomorphisms $f^*\mathcal{B} \to \mathcal{A}$ of differential graded $\mathcal{O}_\mathcal{C}$-algebras is in $1$-to-$1$ correspondence with the set of homomorphisms $\mathcal{B} \to f_*\mathcal{A}$ of differential graded $\mathcal{O}_\mathcal{D}$-algebras. \end{enumerate} Part (3) follows immediately from the usual adjunction between $f^*$ and $f_*$ on sheaves of modules.


sdga-remark-cone-identity

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $C = C(\text{id}_\mathcal{A})$ be the cone on the identity map $\mathcal{A} \to \mathcal{A}$ viewed as a map of differential graded $\mathcal{A}$-modules. Then $$ \Hom_{\text{Mod}_{(\mathcal{A}, \text{d})}}(C, \mathcal{M}) = \{(x, y) \in \Gamma(\mathcal{C}, \mathcal{M}^0) \times \Gamma(\mathcal{C}, \mathcal{M}^{-1}) \mid x = \text{d}(y)\} $$ where the map from left to right sends $f$ to the pair $(x, y)$ where $x$ is the image of the global section $(0, 1)$ of $C^{-1} = \mathcal{A}^{-1} \oplus \mathcal{A}^0$ and where $y$ is the image of the global section $(1, 0)$ of $C^0 = \mathcal{A}^0 \oplus \mathcal{A}^1$.


sdga-remark-sheaf-graded-sets

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A {\it sheaf of graded sets} on $\mathcal{C}$ is a sheaf of sets $\mathcal{S}$ endowed with a map $\deg : \mathcal{S} \to \underline{\mathbf{Z}}$ of sheaves of sets. Let us denote $\mathcal{O}[\mathcal{S}]$ the graded $\mathcal{O}$-module which is the free $\mathcal{O}$-module on the graded sheaf of sets $\mathcal{S}$. More precisely, the $n$th graded part of $\mathcal{O}[\mathcal{S}]$ is the sheafification of the rule $$ U \longmapsto \bigoplus\nolimits_{s \in \mathcal{S}(U),\ \deg(s) = n} s \cdot \mathcal{O}(U) $$ With zero differential we also may consider this as a differential graded $\mathcal{O}$-module. Let $\mathcal{A}$ be a sheaf of graded $\mathcal{O}$-algebras Then we similarly define $\mathcal{A}[\mathcal{S}]$ to be the graded $\mathcal{A}$-module whose $n$th graded part is the sheafification of the rule $$ U \longmapsto \bigoplus\nolimits_{s \in \mathcal{S}(U)} s \cdot \mathcal{A}^{n - \deg(s)}(U) $$ If $\mathcal{A}$ is a differential graded $\mathcal{O}$-algebra, the we turn this into a differential graded $\mathcal{O}$-module by setting $\text{d}(s) = 0$ for all $s \in \mathcal{S}(U)$ and sheafifying.


sdga-remark-why-graded-injective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{I}$ be a graded injective diffential graded $\mathcal{A}$-module. Let $$ 0 \to \mathcal{M}_1 \to \mathcal{M}_2 \to \mathcal{M}_3 \to 0 $$ be a short exact sequence of differential graded $\mathcal{A}$-modules. Since $\mathcal{I}$ is graded injective we obtain a short exact sequence of complexes $$ 0 \to \Hom_{\text{Mod}^{dg}_{(\mathcal{A}, \text{d})}}(\mathcal{M}_3, \mathcal{I}) \to \Hom_{\text{Mod}^{dg}_{(\mathcal{A}, \text{d})}}(\mathcal{M}_2, \mathcal{I}) \to \Hom_{\text{Mod}^{dg}_{(\mathcal{A}, \text{d})}}(\mathcal{M}_1, \mathcal{I}) \to 0 $$ of $\Gamma(\mathcal{C}, \mathcal{O})$-modules. Taking cohomology we obtain a long exact sequence $$ \xymatrix{ \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{M}_3, \mathcal{I}) \ar[d] & \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{M}_3, \mathcal{I})[1] \ar[d] \\ \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{M}_2, \mathcal{I}) \ar[d] & \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{M}_2, \mathcal{I})[1] \ar[d] \\ \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{M}_1, \mathcal{I}) \ar[ruu] & \Hom_{K(\text{Mod}_{(\mathcal{A}, \text{d})})}(\mathcal{M}_1, \mathcal{I})[1] } $$ of groups of homomorphisms in the homotopy category. The point is that we get this even though we didn't assume that our short exact sequence is admissible (so the short exact sequence in general does not define a distinguished triangle in the homotopy category).


stacks-morphisms-remark-inertia-is-group-in-spaces

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. In Properties of Stacks, Remark \ref{stacks-properties-remark-representable-over} we have seen that the $2$-category of morphisms $\mathcal{Z} \to \mathcal{X}$ representable by algebraic spaces with target $\mathcal{X}$ forms a category. In this category the inertia stack of $\mathcal{X}/\mathcal{Y}$ is a {\it group object}. Recall that an object of $\mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is just a pair $(x, \alpha)$ where $x$ is an object of $\mathcal{X}$ and $\alpha$ is an automorphism of $x$ in the fibre category of $\mathcal{X}$ that $x$ lives in with $f(\alpha) = \text{id}$. The composition $$ c : \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \times_\mathcal{X} \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \longrightarrow \mathcal{I}_{\mathcal{X}/\mathcal{Y}} $$ is given by the rule on objects $$ ((x, \alpha), (x', \alpha'), \beta) \mapsto (x, \alpha \circ \beta^{-1} \circ \alpha' \circ \beta) $$ which makes sense as $\beta : x \to x'$ is an isomorphism in the fibre category by our definition of fibre products. The neutral element $e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is given by the functor $x \mapsto (x, \text{id}_x)$. We omit the proof that the axioms of a group object hold.


stacks-morphisms-remark-composition

Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target and stable under composition. Then the property of morphisms of algebraic stacks defined in Definition \ref{definition-P} is stable under composition. Namely, let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks having property $\mathcal{P}$. Choose an algebraic space $W$ and a surjective smooth morphism $W \to \mathcal{Z}$. Choose an algebraic space $V$ and a surjective smooth morphism $V \to \mathcal{Y} \times_\mathcal{Z} W$. Finally, choose an algebraic space $U$ and a surjective and smooth morphism $U \to \mathcal{X} \times_\mathcal{Y} V$. Then the morphisms $V \to W$ and $U \to V$ have property $\mathcal{P}$ by definition. Whence $U \to W$ has property $\mathcal{P}$ as we assumed that $\mathcal{P}$ is stable under composition. Thus, by definition again, we see that $g \circ f : \mathcal{X} \to \mathcal{Z}$ has property $\mathcal{P}$.


stacks-morphisms-remark-base-change

Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target and stable under base change. Then the property of morphisms of algebraic stacks defined in Definition \ref{definition-P} is stable under base change. Namely, let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y}' \to \mathcal{Y}$ be morphisms of algebraic stacks and assume $f$ has property $\mathcal{P}$. Choose an algebraic space $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Choose an algebraic space $U$ and a surjective smooth morphism $U \to \mathcal{X} \times_\mathcal{Y} V$. Finally, choose an algebraic space $V'$ and a surjective and smooth morphism $V' \to \mathcal{Y}' \times_\mathcal{Y} V$. Then the morphism $U \to V$ has property $\mathcal{P}$ by definition. Whence $V' \times_V U \to V'$ has property $\mathcal{P}$ as we assumed that $\mathcal{P}$ is stable under base change. Considering the diagram $$ \xymatrix{ V' \times_V U \ar[r] \ar[d] & \mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ V' \ar[r] & \mathcal{Y}' \ar[r] & \mathcal{Y} } $$ we see that the left top horizontal arrow is smooth and surjective, whence by definition we see that the projection $\mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \to \mathcal{Y}'$ has property $\mathcal{P}$.