spaces-over-fields-remark-when-does-the-argument-work

Let $k$ be finite field. Let $K \supset k$ be a geometrically irreducible field extension. Then $K$ is the limit of geometrically irreducible finite type $k$-algebras $A$. Given $A$ the estimates of Lang and Weil \cite{LW}, show that for $n \gg 0$ there exists an $k$-algebra homomorphism $A \to k'$ with $k'/k$ of degree $n$. Analyzing the argument given in the proof of Lemma \ref{lemma-scheme-after-purely-transcendental-base-change} we see that if $X$ is a quasi-separated algebraic space over $k$ and $X_K$ is a scheme, then $X$ is a scheme. If we ever need this result we will precisely formulate it and prove it here.


dga-remark-evaluation-map-left

Let $R$ be a ring. Let $A$ be a differential graded $R$-algebra. Let $M$ be a left differential graded $A$-module. Let $N^\bullet$ be a complex of $R$-modules. The constructions above produce a right differential graded $A$-module $\Hom(M, N^\bullet)$ and then a leftt differential graded $A$-module $\Hom(\Hom(M, N^\bullet), N^\bullet)$. We claim there is an evaluation map $$ ev : M \longrightarrow \Hom(\Hom(M, N^\bullet), N^\bullet) $$ in the category of left differential graded $A$-modules. To define it, by Lemma \ref{lemma-characterize-hom} it suffices to construct an $A$-bilinear pairing $$ \Hom(M, N^\bullet) \times M \longrightarrow N^\bullet $$ compatible with grading and differentials. For this we take $$ (f, x) \longmapsto f(x) $$ We leave it to the reader to verify this is compatible with grading, differentials, and $A$-bilinear. The map $ev$ on underlying complexes of $R$-modules is More on Algebra, Item (\ref{more-algebra-item-evaluation}).


dga-remark-evaluation-map-right

Let $R$ be a ring. Let $A$ be a differential graded $R$-algebra. Let $M$ be a right differential graded $A$-module. Let $N^\bullet$ be a complex of $R$-modules. The constructions above produce a left differential graded $A$-module $\Hom(M, N^\bullet)$ and then a right differential graded $A$-module $\Hom(\Hom(M, N^\bullet), N^\bullet)$. We claim there is an evaluation map $$ ev : M \longrightarrow \Hom(\Hom(M, N^\bullet), N^\bullet) $$ in the category of right differential graded $A$-modules. To define it, by Lemma \ref{lemma-characterize-hom} it suffices to construct an $A$-bilinear pairing $$ M \times \Hom(M, N^\bullet) \longrightarrow N^\bullet $$ compatible with grading and differentials. For this we take $$ (x, f) \longmapsto (-1)^{\deg(x)\deg(f)}f(x) $$ We leave it to the reader to verify this is compatible with grading, differentials, and $A$-bilinear. The map $ev$ on underlying complexes of $R$-modules is More on Algebra, Item (\ref{more-algebra-item-evaluation}).


dga-remark-shift-dual

Let $R$ be a ring. Let $A$ be a differential graded $R$-algebra. Let $M^\bullet$ and $N^\bullet$ be complexes of $R$-modules. Let $k \in \mathbf{Z}$ and consider the isomorphism $$ \Hom^\bullet(M^\bullet, N^\bullet)[-k] \longrightarrow \Hom^\bullet(M^\bullet[k], N^\bullet) $$ of complexes of $R$-modules defined in More on Algebra, Item (\ref{more-algebra-item-shift-hom}). If $M^\bullet$ has the structure of a left, resp.\ right differential graded $A$-module, then this is a map of right, resp.\ left differential graded $A$-modules (with the module structures as defined in this section). We omit the verification; we warn the reader that the $A$-module structure on the shift of a left graded $A$-module is defined using a sign, see Definition \ref{definition-shift-graded-module}.


dga-remark-P-resolution

Let $R$ be a ring. Let $(A, \text{d})$ be a differential graded $R$-algebra. Using P-resolutions we can sometimes reduce statements about general objects of $D(A, \text{d})$ to statements about $A[k]$. Namely, let $T$ be a property of objects of $D(A, \text{d})$ and assume that \begin{enumerate} \item if $K_i$, $i \in I$ is a family of objects of $D(A, \text{d})$ and $T(K_i)$ holds for all $i \in I$, then $T(\bigoplus K_i)$, \item if $K \to L \to M \to K[1]$ is a distinguished triangle of $D(A, \text{d})$ and $T$ holds for two, then $T$ holds for the third object, and \item $T(A[k])$ holds for all $k \in \mathbf{Z}$. \end{enumerate} Then $T$ holds for all objects of $D(A, \text{d})$. This is clear from Lemmas \ref{lemma-property-P-sequence} and \ref{lemma-resolve}.


dga-remark-graded-shift-functors

Let $R$ be a ring. Let $\mathcal{D}$ be an $R$-linear category endowed with a collection of $R$-linear functors $[n] : \mathcal{D} \to \mathcal{D}$, $x \mapsto x[n]$ indexed by $n \in \mathbf{Z}$ such that $[n] \circ [m] = [n + m]$ and $[0] = \text{id}_\mathcal{D}$ (equality as functors). This allows us to construct a graded category $\mathcal{D}^{gr}$ over $R$ with the same objects of $\mathcal{D}$ setting $$ \Hom_{\mathcal{D}^{gr}}(x, y) = \bigoplus\nolimits_{n \in \mathbf{Z}} \Hom_\mathcal{D}(x, y[n]) $$ for $x, y$ in $\mathcal{D}$. Observe that $(\mathcal{D}^{gr})^0 = \mathcal{D}$ (see Definition \ref{definition-H0-of-graded-category}). Moreover, the graded category $\mathcal{D}^{gr}$ inherits $R$-linear graded functors $[n]$ satisfying $[n] \circ [m] = [n + m]$ and $[0] = \text{id}_{\mathcal{D}^{gr}}$ with the property that $$ \Hom_{\mathcal{D}^{gr}}(x, y[n]) = \Hom_{\mathcal{D}^{gr}}(x, y)[n] $$ as graded $R$-modules compatible with composition of morphisms. \medskip\noindent Conversely, suppose given a graded category $\mathcal{A}$ over $R$ endowed with a collection of $R$-linear graded functors $[n]$ satisfying $[n] \circ [m] = [n + m]$ and $[0] = \text{id}_\mathcal{A}$ which are moreover equipped with isomorphisms $$ \Hom_\mathcal{A}(x, y[n]) = \Hom_\mathcal{A}(x, y)[n] $$ as graded $R$-modules compatible with composition of morphisms. Then the reader easily shows that $\mathcal{A} = (\mathcal{A}^0)^{gr}$. \medskip\noindent Here are two examples of the relationship $\mathcal{D} \leftrightarrow \mathcal{A}$ we established above: \begin{enumerate} \item Let $\mathcal{B}$ be an additive category. If $\mathcal{D} = \text{Gr}(\mathcal{B})$, then $\mathcal{A} = \text{Gr}^{gr}(\mathcal{B})$ as in Example \ref{example-graded-category-graded-objects}. \item If $A$ is a graded ring and $\mathcal{D} = \text{Mod}_A$ is the category of graded right $A$-modules, then $\mathcal{A} = \text{Mod}^{gr}_A$, see Example \ref{example-gm-gr-cat}. \end{enumerate}


dga-remark-shift-tensor-no-sign

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Let $M$ be a right differential graded $A$-module. Then for every $k \in \mathbf{Z}$ there is an isomorphism $$ (M \otimes_A N)[k] \longrightarrow M[k] \otimes_A N $$ of right differential graded $B$-modules defined without the intervention of signs, see More on Algebra, Section \ref{more-algebra-section-sign-rules}.


dga-remark-shift-hom-no-sign

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Let $N'$ be a right differential graded $B$-module. Then for every $k \in \mathbf{Z}$ there is an isomorphism $$ \Hom_{\text{Mod}^{gr}_B}(N, N')[k] \longrightarrow \Hom_{\text{Mod}^{gr}_B}(N, N'[k]) $$ of right differential graded $A$-modules defined without the intervention of signs, see More on Algebra, Section \ref{more-algebra-section-sign-rules}.


dga-remark-source-graded-projective

Let $(A, \text{d})$ be a differential graded algebra. Is there a characterization of those differential graded $A$-modules $P$ for which we have $$ \Hom_{K(A, \text{d})}(P, M) =  \Hom_{D(A, \text{d})}(P, M) $$ for all differential graded $A$-modules $M$? Let $\mathcal{D} \subset K(A, \text{d})$ be the full subcategory whose objects are the objects $P$ satisfying the above. Then $\mathcal{D}$ is a strictly full saturated triangulated subcategory of $K(A, \text{d})$. If $P$ is projective as a graded $A$-module, then to see where $P$ is an object of $\mathcal{D}$ it is enough to check that $\Hom_{K(A, \text{d})}(P, M) = 0$ whenever $M$ is acyclic. However, in general it is not enough to assume that $P$ is projective as a graded $A$-module. Example: take $A = R = k[\epsilon]$ where $k$ is a field and $k[\epsilon] = k[x]/(x^2)$ is the ring of dual numbers. Let $P$ be the object with $P^n = R$ for all $n \in \mathbf{Z}$ and differential given by multiplication by $\epsilon$. Then $\text{id}_P \in \Hom_{K(A, \text{d})}(P, P)$ is a nonzero element but $P$ is acyclic.


dga-remark-graded-projective-is-compact

Let $(A, \text{d})$ be a differential graded algebra. Let us say a differential graded $A$-module $M$ is {\it finite} if $M$ is generated, as a right $A$-module, by finitely many elements. If $P$ is a differential graded $A$-module which is finite graded projective, then we can ask: Does $P$ give a compact object of $D(A, \text{d})$? Presumably, this is not true in general, but we do not know a counter example. However, if $P$ is also an object of the category $\mathcal{D}$ of Remark \ref{remark-source-graded-projective}, then this is the case (this follows from the fact that direct sums in $D(A, \text{d})$ are given by direct sums of modules; details omitted).


dga-remark-tilting-equivalence

In Lemma \ref{lemma-tilting-equivalence} we can replace condition (2) by the condition that $N$ is a classical generator for $D_{compact}(B, d)$, see Derived Categories, Proposition \ref{derived-proposition-generator-versus-classical-generator}. Moreover, if we knew that $R\Hom(N, B)$ is a compact object of $D(A, \text{d})$, then it suffices to check that $N$ is a weak generator for $D_{compact}(B, \text{d})$. We omit the proof; we will add it here if we ever need it in the Stacks project.


dga-remark-lift-equivalence-to-dga

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Suppose given an $R$-linear equivalence $$ F : D(A, \text{d}) \longrightarrow D(B, \text{d}) $$ of triangulated categories. Set $N = F(A)$. Then $N$ is a differential graded $B$-module. Since $F$ is an equivalence and $A$ is a compact object of $D(A, \text{d})$, we conclude that $N$ is a compact object of $D(B, \text{d})$. Since $A$ generates $D(A, \text{d})$ and $F$ is an equivalence, we see that $N$ generates $D(B, \text{d})$. Finally, $H^k(A) = \Hom_{D(A, \text{d})}(A, A[k])$ and as $F$ an equivalence we see that $F$ induces an isomorphism $H^k(A) = \Hom_{D(B, \text{d})}(N, N[k])$ for all $k$. In order to conclude that there is an equivalence $D(A, \text{d}) \longrightarrow D(B, \text{d})$ which arises from the construction in Lemma \ref{lemma-tilting-equivalence} all we need is a left $A$-module structure on $N$ compatible with derivation and commuting with the given right $B$-module structure. In fact, it suffices to do this after replacing $N$ by a quasi-isomorphic differential graded $B$-module. The module structure can be constructed in certain cases. For example, if we assume that $F$ can be lifted to a differential graded functor $$ F^{dg} : \text{Mod}^{dg}_{(A, \text{d})} \longrightarrow \text{Mod}^{dg}_{(B, \text{d})} $$ (for notation see Example \ref{example-dgm-dg-cat}) between the associated differential graded categories, then this holds. Another case is discussed in the proposition below.


dga-remark-rickard

Let $A, B, F, N$ be as in Proposition \ref{proposition-rickard}. It is not clear that $F$ and the functor $G(-) = - \otimes_A^\mathbf{L} N$ are isomorphic. By construction there is an isomorphism $N = G(A) \to F(A)$ in $D(B, \text{d})$. It is straightforward to extend this to a functorial isomorphism $G(M) \to F(M)$ for $M$ is a differential graded $A$-module which is graded projective (e.g., a sum of shifts of $A$). Then one can conclude that $G(M) \cong F(M)$ when $M$ is a cone of a map between such modules. We don't know whether more is true in general.


dga-remark-centers

Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. If $D(A)$ and $D(B)$ are equivalent as $R$-linear triangulated categories, then the centers of $A$ and $B$ are isomorphic as $R$-algebras. In particular, if $A$ and $B$ are commutative, then $A \cong B$. The rather tricky proof can be found in \cite[Proposition 9.2]{Rickard} or \cite[Proposition 6.3.2]{KZ}. Another approach might be to use Hochschild cohomology (see remark below).


dga-remark-hochschild-cohomology

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras which are derived equivalent, i.e., such that there exists an $R$-linear equivalence $D(A, \text{d}) \to D(B, \text{d})$ of triangulated categories. We would like to show that certain invariants of $(A, \text{d})$ and $(B, \text{d})$ coincide. In many situations one has more control of the situation. For example, it may happen that there is an equivalence of the form $$ - \otimes_A \Omega : D(A, \text{d}) \longrightarrow D(B, \text{d}) $$ for some differential graded $(A, B)$-bimodule $\Omega$ (this happens in the situation of Proposition \ref{proposition-rickard} and is often true if the equivalence comes from a geometric construction). If also the quasi-inverse of our functor is given as $$ - \otimes_B^\mathbf{L} \Omega' : D(B, \text{d}) \longrightarrow D(A, \text{d}) $$ for a differential graded $(B, A)$-bimodule $\Omega'$ (and as before such a module $\Omega'$ often exists in practice). In this case we can consider the functor $$ D(A^{opp} \otimes_R A, \text{d}) \longrightarrow D(B^{opp} \otimes_R B, \text{d}),\quad M \longmapsto \Omega' \otimes^\mathbf{L}_A M \otimes_A^\mathbf{L} \Omega $$ on derived categories of bimodules (use Lemma \ref{lemma-bimodule-over-tensor} to turn bimodules into right modules). Observe that this functor sends the $(A, A)$-bimodule $A$ to the $(B, B)$-bimodule $B$. Under suitable conditions (e.g., flatness of $A$, $B$, $\Omega$ over $R$, etc) this functor will be an equivalence as well. If this is the case, then it follows that we have isomorphisms of Hochschild cohomology groups $$ HH^i(A, \text{d}) = \Hom_{D(A^{opp} \otimes_R A, \text{d})}(A, A[i]) \longrightarrow \Hom_{D(B^{opp} \otimes_R B, \text{d})}(B, B[i]) = HH^i(B, \text{d}). $$ For example, if $A = H^0(A)$, then $HH^0(A, \text{d})$ is equal to the center of $A$, and this gives a conceptual proof of the result mentioned in Remark \ref{remark-centers}. If we ever need this remark we will provide a precise statement with a detailed proof here.


spaces-more-groupoids-remark-finite-monoid

Let $f : X \to Y$ be a separated morphism of algebraic spaces. The sheaf $(X/Y)_{fin}$ comes with a natural map $(X/Y)_{fin} \to Y$ by mapping the pair $(a, Z) \in (X/Y)_{fin}(T)$ to the element $a \in Y(T)$. We can use Lemma \ref{lemma-finite-separated} to define operations $$ \star_i : (X/Y)_{fin} \times_Y (X/Y)_{fin} \longrightarrow (X/Y)_{fin} $$ by the rules \begin{align*} \star_1 : ((a, Z_1), (a, Z_2)) & \longmapsto (a, Z_1 \cup Z_2) \\ \star_2 : ((a, Z_1), (a, Z_2)) & \longmapsto (a, Z_1 \cap Z_2) \\ \star_3 : ((a, Z_1), (a, Z_2)) & \longmapsto (a, Z_1 \setminus Z_2) \\ \star_4 : ((a, Z_1), (a, Z_2)) & \longmapsto (a, Z_2 \setminus Z_1). \end{align*} The reason this works is that $Z_1 \cap Z_2$ is both open and closed inside $Z_1$ and $Z_2$ (which also implies that $Z_1 \cup Z_2$ is the disjoint union of the other three pieces). Thus we can think of $(X/Y)_{fin}$ as an $\mathbf{F}_2$-algebra (without unit) over $Y$ with multiplication given by $ss' = \star_2(s, s')$, and addition given by $$ s + s' = \star_1(\star_3(s, s'), \star_4(s, s')) $$ which boils down to taking the symmetric difference. Note that in this sheaf of algebras $0 = (1_Y, \emptyset)$ and that indeed $s + s = 0$ for any local section $s$. If $f : X \to Y$ is finite, then this algebra has a unit namely $1 = (1_Y, X)$ and $\star_3(s, s') = s(1 + s')$, and $\star_4(s, s') = (1 + s)s'$.


spaces-more-groupoids-remark-finite-quasi-finite-separated-morphism-schemes

Let $f : X \to Y$ be a separated, locally quasi-finite morphism of schemes. In this case the sheaf $(X/Y)_{fin}$ is closely related to the sheaf $f_!\mathbf{F}_2$ (insert future reference here) on $Y_\etale$. Namely, if $V \to Y$ is \'etale, and $s \in \Gamma(V, f_!\mathbf{F}_2)$, then $s \in \Gamma(V \times_Y X, \mathbf{F}_2)$ is a section with proper support $Z = \text{Supp}(s)$ over $V$. Since $f$ is also locally quasi-finite we see that the projection $Z \to V$ is actually finite. Since the support of a section of a constant abelian sheaf is open we see that the pair $(V \to Y, \text{Supp}(s))$ satisfies \ref{equation-finite-conditions}. In fact, $f_!\mathbf{F}_2 \cong (X/Y)_{fin}|_{Y_\etale}$ in this case which also explains the $\mathbf{F}_2$-algebra structure introduced in Remark \ref{remark-finite-monoid}.


spaces-more-groupoids-remark-warning

The condition that $f$ be separated cannot be dropped from Proposition \ref{proposition-finite-algebraic-space}. An example is to take $X$ the affine line with zero doubled, see Schemes, Example \ref{schemes-example-affine-space-zero-doubled}, $Y = \mathbf{A}^1_k$ the affine line, and $X \to Y$ the obvious map. Recall that over $0 \in Y$ there are two points $0_1$ and $0_2$ in $X$. Thus $(X/Y)_{fin}$ has four points over $0$, namely $\emptyset, \{0_1\}, \{0_2\}, \{0_1, 0_2\}$. Of these four points only three can be lifted to an open subscheme of $U \times_Y X$ finite over $U$ for $U \to Y$ \'etale, namely $\emptyset, \{0_1\}, \{0_2\}$. This shows that $(X/Y)_{fin}$ if representable by an algebraic space is not \'etale over $Y$. Similar arguments show that $(X/Y)_{fin}$ is really not an algebraic space. Details omitted.


spaces-more-groupoids-remark-not-scheme

Let $Y = \mathbf{A}^1_{\mathbf{R}}$ be the affine line over the real numbers, and let $X = \Spec(\mathbf{C})$ mapping to the $\mathbf{R}$-rational point $0$ in $Y$. In this case the morphism $f : X \to Y$ is finite, but it is not the case that $(X/Y)_{fin}$ is a scheme. Namely, one can show that in this case the algebraic space $(X/Y)_{fin}$ is isomorphic to the algebraic space of Spaces, Example \ref{spaces-example-non-representable-descent} associated to the extension $\mathbf{R} \subset \mathbf{C}$. Thus it is really necessary to leave the category of schemes in order to represent the sheaf $(X/Y)_{fin}$, even when $f$ is a finite morphism.


modules-remark-sections-support-in-closed

Let $X$ be a topological space. Let $Z \subset X$ be a closed subset. Let $\mathcal{F}$ be an abelian sheaf on $X$. For $U \subset X$ open set $$ \mathcal{H}_Z(\mathcal{F})(U) = \{s \in \mathcal{F}(U) \mid \text{ the support of }s\text{ is contained in }Z \cap U\} $$ Then $\mathcal{H}_Z(\mathcal{F})$ is an abelian subsheaf of $\mathcal{F}$. It is the largest abelian subsheaf of $\mathcal{F}$ whose support is contained in $Z$. By Lemma \ref{lemma-i-star-exact} we may (and we do) view $\mathcal{H}_Z(\mathcal{F})$ as an abelian sheaf on $Z$. In this way we obtain a left exact functor $$ \textit{Ab}(X) \longrightarrow \textit{Ab}(Z),\quad \mathcal{F} \longmapsto \mathcal{H}_Z(\mathcal{F}) \text{ viewed as abelian sheaf on }Z $$ All of the statements made above follow directly from Lemma \ref{lemma-support-section-closed}.


modules-remark-i-star-right-adjoint

In Sheaves, Remark \ref{sheaves-remark-i-star-not-exact} we showed that $i_*$ as a functor on the categories of sheaves of sets does not have a right adjoint simply because it is not exact. However, it is very close to being true, in fact, the functor $i_*$ is exact on sheaves of pointed sets, sections with support in $Z$ can be defined for sheaves of pointed sets, and $\mathcal{H}_Z$ makes sense and is a right adjoint to $i_*$.


modules-remark-infinite-direct-sum-quasi-coherent-not

Warning: It is not true in general that an infinite direct sum of quasi-coherent $\mathcal{O}_X$-modules is quasi-coherent. For more esoteric behaviour of quasi-coherent modules see Example \ref{example-quasi-coherent}.


modules-remark-condition-necessary

In the lemma above some condition beyond the condition that $X$ is quasi-compact is necessary. See Sheaves, Example \ref{sheaves-example-conditions-needed-colimit}.


modules-remark-sections-support-in-closed-modules

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $Z \subset X$ be a closed subset. For an $\mathcal{O}_X$-module $\mathcal{F}$ we can consider the {\it submodule of sections with support in $Z$}, denoted $\mathcal{H}_Z(\mathcal{F})$, defined by the rule $$ \mathcal{H}_Z(\mathcal{F})(U) = \{s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset U \cap Z\} $$ Observe that $\mathcal{H}_Z(\mathcal{F})(U)$ is a module over $\mathcal{O}_X(U)$, i.e., $\mathcal{H}_Z(\mathcal{F})$ is an $\mathcal{O}_X$-module. By construction $\mathcal{H}_Z(\mathcal{F})$ is the largest $\mathcal{O}_X$-submodule of $\mathcal{F}$ whose support is contained in $Z$. Applying Lemma \ref{lemma-i-star-equivalence} to the morphism of ringed spaces $(Z, \mathcal{O}_X|_Z) \to (X, \mathcal{O}_X)$ we may (and we do) view $\mathcal{H}_Z(\mathcal{F})$ as an $\mathcal{O}_X|_Z$-module on $Z$. Thus we obtain a functor $$ \textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{Mod}(\mathcal{O}_X|_Z), \quad \mathcal{F} \longmapsto \mathcal{H}_Z(\mathcal{F}) \text{ viewed as an }\mathcal{O}_X|_Z\text{-module on }Z $$ This functor is left exact, but in general not exact. All of the statements made above follow directly from Lemma \ref{lemma-support-section-closed}. Clearly the construction is compatible with the construction in Remark \ref{remark-sections-support-in-closed}.


modules-remark-functoriality-principal-parts

Let $X$ be a topological space. Suppose given a commutative diagram of sheaves of rings $$ \xymatrix{ \mathcal{B} \ar[r] & \mathcal{B}' \\ \mathcal{A} \ar[u] \ar[r] & \mathcal{A}' \ar[u] } $$ on $X$, a $\mathcal{B}$-module $\mathcal{F}$, a $\mathcal{B}'$-module $\mathcal{F}'$, and a $\mathcal{B}$-linear map $\mathcal{F} \to \mathcal{F}'$. Then we get a compatible system of module maps $$ \xymatrix{ \ldots \ar[r] & \mathcal{P}^2_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \ar[r] & \mathcal{P}^1_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \ar[r] & \mathcal{P}^0_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \\ \ldots \ar[r] & \mathcal{P}^2_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[r] \ar[u] & \mathcal{P}^1_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[r] \ar[u] & \mathcal{P}^0_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[u] } $$ These maps are compatible with further composition of maps of this type. The easiest way to see this is to use the description of the modules $\mathcal{P}^k_{\mathcal{B}/\mathcal{A}}(\mathcal{M})$ in terms of (local) generators and relations in the proof of Lemma \ref{lemma-module-principal-parts} but it can also be seen directly from the universal property of these modules. Moreover, these maps are compatible with the short exact sequences of Lemma \ref{lemma-sequence-of-principal-parts}.


defos-remark-trivial-thickening

Let $(X, \mathcal{O}_X)$ be a ringed space. A first order thickening $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ is said to be {\it trivial} if there exists a morphism of ringed spaces $\pi : (X', \mathcal{O}_{X'}) \to (X, \mathcal{O}_X)$ which is a left inverse to $i$. The choice of such a morphism $\pi$ is called a {\it trivialization} of the first order thickening. Given $\pi$ we obtain a splitting \begin{equation} \label{equation-splitting} \mathcal{O}_{X'} = \mathcal{O}_X \oplus \mathcal{I} \end{equation} as sheaves of algebras on $X$ by using $\pi^\sharp$ to split the surjection $\mathcal{O}_{X'} \to \mathcal{O}_X$. Conversely, such a splitting determines a morphism $\pi$. The category of trivialized first order thickenings of $(X, \mathcal{O}_X)$ is equivalent to the category of  $\mathcal{O}_X$-modules.


defos-remark-trivial-extension

Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ be a trivial first order thickening of ringed spaces and let $\pi : (X', \mathcal{O}_{X'}) \to (X, \mathcal{O}_X)$ be a trivialization. Then given any triple $(\mathcal{F}, \mathcal{K}, c)$ consisting of a pair of $\mathcal{O}_X$-modules and a map $c : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}$ we may set $$ \mathcal{F}'_{c, triv} = \mathcal{F} \oplus \mathcal{K} $$ and use the splitting (\ref{equation-splitting}) associated to $\pi$ and the map $c$ to define the $\mathcal{O}_{X'}$-module structure and obtain an extension (\ref{equation-extension}). We will call $\mathcal{F}'_{c, triv}$ the {\it trivial extension} of $\mathcal{F}$ by $\mathcal{K}$ corresponding to $c$ and the trivialization $\pi$. Given any extension $\mathcal{F}'$ as in (\ref{equation-extension}) we can use $\pi^\sharp : \mathcal{O}_X \to \mathcal{O}_{X'}$ to think of $\mathcal{F}'$ as an $\mathcal{O}_X$-module extension, hence a class $\xi_{\mathcal{F}'}$ in $\Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})$. Lemma \ref{lemma-inf-ext} assures that $\mathcal{F}' \mapsto \xi_{\mathcal{F}'}$ induces a bijection $$ \left\{ \begin{matrix} \text{isomorphism classes of extensions}\\ \mathcal{F}'\text{ as in (\ref{equation-extension}) with }c = c_{\mathcal{F}'} \end{matrix} \right\} \longrightarrow \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}) $$ Moreover, the trivial extension $\mathcal{F}'_{c, triv}$ maps to the zero class.


defos-remark-extension-functorial

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $(X, \mathcal{O}_X) \to (X'_i, \mathcal{O}_{X'_i})$, $i = 1, 2$ be first order thickenings with ideal sheaves $\mathcal{I}_i$. Let $h : (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2})$ be a morphism of first order thickenings of $(X, \mathcal{O}_X)$. Picture $$ \xymatrix{ & (X, \mathcal{O}_X) \ar[ld] \ar[rd] & \\ (X'_1, \mathcal{O}_{X'_1}) \ar[rr]^h & &  (X'_2, \mathcal{O}_{X'_2}) } $$ Observe that $h^\sharp : \mathcal{O}_{X'_2} \to \mathcal{O}_{X'_1}$ in particular induces an $\mathcal{O}_X$-module map $\mathcal{I}_2 \to \mathcal{I}_1$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Let $(\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an $\mathcal{O}_X$-module $\mathcal{K}_i$ and a map $c_i : \mathcal{I}_i \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}_i$. Assume furthermore given a map of $\mathcal{O}_X$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that $$ \xymatrix{ \mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } $$ is commutative. Then there is a canonical functoriality $$ \left\{ \begin{matrix} \mathcal{F}'_2\text{ as in (\ref{equation-extension}) with }\\ c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2 \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \mathcal{F}'_1\text{ as in (\ref{equation-extension}) with }\\ c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1 \end{matrix} \right\} $$ Namely, thinking of all sheaves $\mathcal{O}_X$, $\mathcal{O}_{X'_i}$, $\mathcal{F}$, $\mathcal{K}_i$, etc as sheaves on $X$, we set given $\mathcal{F}'_2$ the sheaf $\mathcal{F}'_1$ equal to the pushout, i.e., fitting into the following diagram of extensions $$ \xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d] & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{K}_1 \ar[r] & \mathcal{F}'_1 \ar[r] & \mathcal{F} \ar[r] & 0 } $$ We omit the construction of the $\mathcal{O}_{X'_1}$-module structure on the pushout (this uses the commutativity of the diagram involving $c_1$ and $c_2$).


defos-remark-trivial-extension-functorial

Let $(X, \mathcal{O}_X)$, $(X, \mathcal{O}_X) \to (X'_i, \mathcal{O}_{X'_i})$, $\mathcal{I}_i$, and $h : (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2})$ be as in Remark \ref{remark-extension-functorial}. Assume that we are given trivializations $\pi_i : X'_i \to X$ such that $\pi_1 = h \circ \pi_2$. In other words, assume $h$ is a morphism of trivialized first order thickening of $(X, \mathcal{O}_X)$. Let $(\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an $\mathcal{O}_X$-module $\mathcal{K}_i$ and a map $c_i : \mathcal{I}_i \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}_i$. Assume furthermore given a map of $\mathcal{O}_X$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that $$ \xymatrix{ \mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } $$ is commutative. In this situation the construction of Remark \ref{remark-trivial-extension} induces a commutative diagram $$ \xymatrix{ \{\mathcal{F}'_2\text{ as in (\ref{equation-extension}) with } c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2\} \ar[d] \ar[rr] & & \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_2) \ar[d] \\ \{\mathcal{F}'_1\text{ as in (\ref{equation-extension}) with } c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1\} \ar[rr] & & \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_1) } $$ where the vertical map on the right is given by functoriality of $\Ext$ and the map $\mathcal{K}_2 \to \mathcal{K}_1$ and the vertical map on the left is the one from Remark \ref{remark-extension-functorial}.


defos-remark-short-exact-sequence-thickenings

Let $(X, \mathcal{O}_X)$ be a ringed space. We define a sequence of morphisms of first order thickenings $$ (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2}) \to (X'_3, \mathcal{O}_{X'_3}) $$ of $(X, \mathcal{O}_X)$ to be a {\it complex} if the corresponding maps between the ideal sheaves $\mathcal{I}_i$ give a complex of $\mathcal{O}_X$-modules $\mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1$ (i.e., the composition is zero). In this case the composition $(X'_1, \mathcal{O}_{X'_1}) \to (X_3', \mathcal{O}_{X'_3})$ factors through $(X, \mathcal{O}_X) \to (X'_3, \mathcal{O}_{X'_3})$, i.e., the first order thickening $(X'_1, \mathcal{O}_{X'_1})$ of $(X, \mathcal{O}_X)$ is trivial and comes with a canonical trivialization $\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_X)$. \medskip\noindent We say a sequence of morphisms of first order thickenings $$ (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2}) \to (X'_3, \mathcal{O}_{X'_3}) $$ of $(X, \mathcal{O}_X)$ is {\it a short exact sequence} if the corresponding maps between ideal sheaves is a short exact sequence $$ 0 \to \mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1 \to 0 $$ of $\mathcal{O}_X$-modules.


defos-remark-complex-thickenings-and-ses-modules

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Let $$ (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2}) \to (X'_3, \mathcal{O}_{X'_3}) $$ be a complex first order thickenings of $(X, \mathcal{O}_X)$, see Remark \ref{remark-short-exact-sequence-thickenings}. Let $(\mathcal{K}_i, c_i)$, $i = 1, 2, 3$ be pairs consisting of an $\mathcal{O}_X$-module $\mathcal{K}_i$ and a map $c_i : \mathcal{I}_i \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}_i$. Assume given a short exact sequence of $\mathcal{O}_X$-modules $$ 0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0 $$ such that $$ \vcenter{ \xymatrix{ \mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } } \quad\text{and}\quad \vcenter{ \xymatrix{ \mathcal{I}_3 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_3} \ar[d] & \mathcal{K}_3 \ar[d] \\ \mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]^-{c_2} & \mathcal{K}_2 } } $$ are commutative. Finally, assume given an extension $$ 0 \to \mathcal{K}_2 \to \mathcal{F}'_2 \to \mathcal{F} \to 0 $$ as in (\ref{equation-extension}) with $\mathcal{K} = \mathcal{K}_2$ of $\mathcal{O}_{X'_2}$-modules with $c_{\mathcal{F}'_2} = c_2$. In this situation we can apply the functoriality of Remark \ref{remark-extension-functorial} to obtain an extension $\mathcal{F}'_1$ on $X'_1$ (we'll describe $\mathcal{F}'_1$ in this special case below). By Remark \ref{remark-trivial-extension} using the canonical splitting $\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_X)$ of Remark \ref{remark-short-exact-sequence-thickenings} we obtain $\xi_{\mathcal{F}'_1} \in \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_1)$. Finally, we have the obstruction $$ o(\mathcal{F}, \mathcal{K}_3, c_3) \in \Ext^2_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_3) $$ see Lemma \ref{lemma-inf-obs-ext}. In this situation we {\bf claim} that the canonical map $$ \partial : \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_1) \longrightarrow \Ext^2_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_3) $$ coming from the short exact sequence $0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0$ sends $\xi_{\mathcal{F}'_1}$ to the obstruction class $o(\mathcal{F}, \mathcal{K}_3, c_3)$. \medskip\noindent To prove this claim choose an embedding $j : \mathcal{K}_3 \to \mathcal{K}$ where $\mathcal{K}$ is an injective $\mathcal{O}_X$-module. We can lift $j$ to a map $j' : \mathcal{K}_2 \to \mathcal{K}$. Set $\mathcal{E}'_2 = j'_*\mathcal{F}'_2$ equal to the pushout of $\mathcal{F}'_2$ by $j'$ so that $c_{\mathcal{E}'_2} = j' \circ c_2$. Picture: $$ \xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d]_{j'} & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[r] & \mathcal{E}'_2 \ar[r] & \mathcal{F} \ar[r] & 0 } $$ Set $\mathcal{E}'_3 = \mathcal{E}'_2$ but viewed as an $\mathcal{O}_{X'_3}$-module via $\mathcal{O}_{X'_3} \to \mathcal{O}_{X'_2}$. Then $c_{\mathcal{E}'_3} = j \circ c_3$. The proof of Lemma \ref{lemma-inf-obs-ext} constructs $o(\mathcal{F}, \mathcal{K}_3, c_3)$ as the boundary of the class of the extension of $\mathcal{O}_X$-modules $$ 0 \to \mathcal{K}/\mathcal{K}_3 \to \mathcal{E}'_3/\mathcal{K}_3 \to \mathcal{F} \to 0 $$ On the other hand, note that $\mathcal{F}'_1 = \mathcal{F}'_2/\mathcal{K}_3$ hence the class $\xi_{\mathcal{F}'_1}$ is the class of the extension $$ 0 \to \mathcal{K}_2/\mathcal{K}_3 \to \mathcal{F}'_2/\mathcal{K}_3 \to \mathcal{F} \to 0 $$ seen as a sequence of $\mathcal{O}_X$-modules using $\pi^\sharp$ where $\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_X)$ is the canonical splitting. Thus finally, the claim follows from the fact that we have a commutative diagram $$ \xymatrix{ 0 \ar[r] & \mathcal{K}_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F}'_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}/\mathcal{K}_3 \ar[r] & \mathcal{E}'_3/\mathcal{K}_3 \ar[r] & \mathcal{F} \ar[r] & 0 } $$ which is $\mathcal{O}_X$-linear (with the $\mathcal{O}_X$-module structures given above).


defos-remark-trivial-thickening-ringed-topoi

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. A first order thickening $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ is said to be {\it trivial} if there exists a morphism of ringed topoi $\pi : (\Sh(\mathcal{D}), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})$ which is a left inverse to $i$. The choice of such a morphism $\pi$ is called a {\it trivialization} of the first order thickening. Given $\pi$ we obtain a splitting \begin{equation} \label{equation-splitting-ringed-topoi} \mathcal{O}' = \mathcal{O} \oplus \mathcal{I} \end{equation} as sheaves of algebras on $\mathcal{C}$ by using $\pi^\sharp$ to split the surjection $\mathcal{O}' \to \mathcal{O}$. Conversely, such a splitting determines a morphism $\pi$. The category of trivialized first order thickenings of $(\Sh(\mathcal{C}), \mathcal{O})$ is equivalent to the category of  $\mathcal{O}$-modules.


defos-remark-trivial-extension-ringed-topoi

Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$ be a trivial first order thickening of ringed topoi and let $\pi : (\Sh(\mathcal{D}), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})$ be a trivialization. Then given any triple $(\mathcal{F}, \mathcal{K}, c)$ consisting of a pair of $\mathcal{O}$-modules and a map $c : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}$ we may set $$ \mathcal{F}'_{c, triv} = \mathcal{F} \oplus \mathcal{K} $$ and use the splitting (\ref{equation-splitting-ringed-topoi}) associated to $\pi$ and the map $c$ to define the $\mathcal{O}'$-module structure and obtain an extension (\ref{equation-extension-ringed-topoi}). We will call $\mathcal{F}'_{c, triv}$ the {\it trivial extension} of $\mathcal{F}$ by $\mathcal{K}$ corresponding to $c$ and the trivialization $\pi$. Given any extension $\mathcal{F}'$ as in (\ref{equation-extension-ringed-topoi}) we can use $\pi^\sharp : \mathcal{O} \to \mathcal{O}'$ to think of $\mathcal{F}'$ as an $\mathcal{O}$-module extension, hence a class $\xi_{\mathcal{F}'}$ in $\Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K})$. Lemma \ref{lemma-inf-ext-ringed-topoi} assures that $\mathcal{F}' \mapsto \xi_{\mathcal{F}'}$ induces a bijection $$ \left\{ \begin{matrix} \text{isomorphism classes of extensions}\\ \mathcal{F}'\text{ as in (\ref{equation-extension-ringed-topoi}) with } c = c_{\mathcal{F}'} \end{matrix} \right\} \longrightarrow \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K}) $$ Moreover, the trivial extension $\mathcal{F}'_{c, triv}$ maps to the zero class.


defos-remark-extension-functorial-ringed-topoi

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $(\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}_i), \mathcal{O}'_i)$, $i = 1, 2$ be first order thickenings with ideal sheaves $\mathcal{I}_i$. Let $h : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2)$ be a morphism of first order thickenings of $(\Sh(\mathcal{C}), \mathcal{O})$. Picture $$ \xymatrix{ & (\Sh(\mathcal{C}), \mathcal{O}) \ar[ld] \ar[rd] & \\ (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \ar[rr]^h & &  (\Sh(\mathcal{D}_2), \mathcal{O}'_2) } $$ Observe that $h^\sharp : \mathcal{O}'_2 \to \mathcal{O}'_1$ in particular induces an $\mathcal{O}$-module map $\mathcal{I}_2 \to \mathcal{I}_1$. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Let $(\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an $\mathcal{O}$-module $\mathcal{K}_i$ and a map $c_i : \mathcal{I}_i \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}_i$. Assume furthermore given a map of $\mathcal{O}$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that $$ \xymatrix{ \mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_\mathcal{O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } $$ is commutative. Then there is a canonical functoriality $$ \left\{ \begin{matrix} \mathcal{F}'_2\text{ as in (\ref{equation-extension-ringed-topoi}) with }\\ c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2 \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \mathcal{F}'_1\text{ as in (\ref{equation-extension-ringed-topoi}) with }\\ c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1 \end{matrix} \right\} $$ Namely, thinking of all sheaves $\mathcal{O}$, $\mathcal{O}'_i$, $\mathcal{F}$, $\mathcal{K}_i$, etc as sheaves on $\mathcal{C}$, we set given $\mathcal{F}'_2$ the sheaf $\mathcal{F}'_1$ equal to the pushout, i.e., fitting into the following diagram of extensions $$ \xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d] & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{K}_1 \ar[r] & \mathcal{F}'_1 \ar[r] & \mathcal{F} \ar[r] & 0 } $$ We omit the construction of the $\mathcal{O}'_1$-module structure on the pushout (this uses the commutativity of the diagram involving $c_1$ and $c_2$).


defos-remark-trivial-extension-functorial-ringed-topoi

Let $(\Sh(\mathcal{C}), \mathcal{O})$, $(\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}_i), \mathcal{O}'_i)$, $\mathcal{I}_i$, and $h : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2)$ be as in Remark \ref{remark-extension-functorial-ringed-topoi}. Assume that we are given trivializations $\pi_i : (\Sh(\mathcal{D}_i), \mathcal{O}'_i) \to (\Sh(\mathcal{C}), \mathcal{O})$ such that $\pi_1 = h \circ \pi_2$. In other words, assume $h$ is a morphism of trivialized first order thickenings of $(\Sh(\mathcal{C}), \mathcal{O})$. Let $(\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an $\mathcal{O}$-module $\mathcal{K}_i$ and a map $c_i : \mathcal{I}_i \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}_i$. Assume furthermore given a map of $\mathcal{O}$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that $$ \xymatrix{ \mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_\mathcal{O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } $$ is commutative. In this situation the construction of Remark \ref{remark-trivial-extension-ringed-topoi} induces a commutative diagram $$ \xymatrix{ \{\mathcal{F}'_2\text{ as in (\ref{equation-extension-ringed-topoi}) with } c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2\} \ar[d] \ar[rr] & & \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_2) \ar[d] \\ \{\mathcal{F}'_1\text{ as in (\ref{equation-extension-ringed-topoi}) with } c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1\} \ar[rr] & & \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_1) } $$ where the vertical map on the right is given by functoriality of $\Ext$ and the map $\mathcal{K}_2 \to \mathcal{K}_1$ and the vertical map on the left is the one from Remark \ref{remark-extension-functorial-ringed-topoi}.


defos-remark-obstruction-extension-functorial-ringed-topoi

Let $(\Sh(\mathcal{C}), \mathcal{O})$, $(\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}_i), \mathcal{O}'_i)$, $\mathcal{I}_i$, and $h : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2)$ be as in Remark \ref{remark-extension-functorial-ringed-topoi}. Observe that $h^\sharp : \mathcal{O}'_2 \to \mathcal{O}'_1$ in particular induces an $\mathcal{O}$-module map $\mathcal{I}_2 \to \mathcal{I}_1$. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Let $(\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an $\mathcal{O}$-module $\mathcal{K}_i$ and a map $c_i : \mathcal{I}_i \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}_i$. Assume furthermore given a map of $\mathcal{O}$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that $$ \xymatrix{ \mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_\mathcal{O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } $$ is commutative. Then we {\bf claim} the map $$ \Ext^2_\mathcal{O}(\mathcal{F}, \mathcal{K}_2) \longrightarrow \Ext^2_\mathcal{O}(\mathcal{F}, \mathcal{K}_1) $$ sends $o(\mathcal{F}, \mathcal{K}_2, c_2)$ to $o(\mathcal{F}, \mathcal{K}_1, c_1)$. \medskip\noindent To prove this claim choose an embedding $j_2 : \mathcal{K}_2 \to \mathcal{K}_2'$ where $\mathcal{K}_2'$ is an injective $\mathcal{O}$-module. As in the proof of Lemma \ref{lemma-inf-obs-ext-ringed-topoi} we can choose an extension of $\mathcal{O}_2$-modules $$ 0 \to \mathcal{K}_2' \to \mathcal{E}_2 \to \mathcal{F} \to 0 $$ such that $c_{\mathcal{E}_2} = j_2 \circ c_2$. The proof of Lemma \ref{lemma-inf-obs-ext-ringed-topoi} constructs $o(\mathcal{F}, \mathcal{K}_2, c_2)$ as the Yoneda extension class (in the sense of Derived Categories, Section \ref{derived-section-ext}) of the exact sequence of $\mathcal{O}$-modules $$ 0 \to \mathcal{K}_2 \to \mathcal{K}_2' \to \mathcal{E}_2/\mathcal{K}_2 \to \mathcal{F} \to 0 $$ Let $\mathcal{K}_1'$ be the cokernel of $\mathcal{K}_2 \to \mathcal{K}_1 \oplus \mathcal{K}_2'$. There is an injection $j_1 : \mathcal{K}_1 \to \mathcal{K}_1'$ and a map $\mathcal{K}_2' \to \mathcal{K}_1'$ forming a commutative square. We form the pushout: $$ \xymatrix{ 0 \ar[r] & \mathcal{K}_2' \ar[r] \ar[d] & \mathcal{E}_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}_1' \ar[r] & \mathcal{E}_1 \ar[r] & \mathcal{F} \ar[r] & 0 } $$ There is a canonical $\mathcal{O}_1$-module structure on $\mathcal{E}_1$ and for this structure we have $c_{\mathcal{E}_1} = j_1 \circ c_1$ (this uses the commutativity of the diagram involving $c_1$ and $c_2$ above). The procedure of Lemma \ref{lemma-inf-obs-ext-ringed-topoi} tells us that $o(\mathcal{F}, \mathcal{K}_1, c_1)$ is the Yoneda extension class of the exact sequence of $\mathcal{O}$-modules $$ 0 \to \mathcal{K}_1 \to \mathcal{K}_1' \to \mathcal{E}_1/\mathcal{K}_1 \to \mathcal{F} \to 0 $$ Since we have maps of exact sequences $$ \xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[d] \ar[r] & \mathcal{K}_2' \ar[d] \ar[r] & \mathcal{E}_2/\mathcal{K}_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{K}_2 \ar[r] & \mathcal{K}_2' \ar[r] & \mathcal{E}_2/\mathcal{K}_2 \ar[r] & \mathcal{F} \ar[r] & 0 } $$ we conclude that the claim is true.


defos-remark-short-exact-sequence-thickenings-ringed-topoi

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. We define a sequence of morphisms of first order thickenings $$ (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2) \to (\Sh(\mathcal{D}_3), \mathcal{O}'_3) $$ of $(\Sh(\mathcal{C}), \mathcal{O})$ to be a {\it complex} if the corresponding maps between the ideal sheaves $\mathcal{I}_i$ give a complex of $\mathcal{O}$-modules $\mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1$ (i.e., the composition is zero). In this case the composition $(\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_3), \mathcal{O}'_3)$ factors through $(\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}_3), \mathcal{O}'_3)$, i.e., the first order thickening $(\Sh(\mathcal{D}_1), \mathcal{O}'_1)$ of $(\Sh(\mathcal{C}), \mathcal{O})$ is trivial and comes with a canonical trivialization $\pi : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{C}), \mathcal{O})$. \medskip\noindent We say a sequence of morphisms of first order thickenings $$ (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2) \to (\Sh(\mathcal{D}_3), \mathcal{O}'_3) $$ of $(\Sh(\mathcal{C}), \mathcal{O})$ is {\it a short exact sequence} if the corresponding maps between ideal sheaves is a short exact sequence $$ 0 \to \mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1 \to 0 $$ of $\mathcal{O}$-modules.


defos-remark-complex-thickenings-and-ses-modules-ringed-topoi

Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Let $$ (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2) \to (\Sh(\mathcal{D}_3), \mathcal{O}'_3) $$ be a complex first order thickenings of $(\Sh(\mathcal{C}), \mathcal{O})$, see Remark \ref{remark-short-exact-sequence-thickenings-ringed-topoi}. Let $(\mathcal{K}_i, c_i)$, $i = 1, 2, 3$ be pairs consisting of an $\mathcal{O}$-module $\mathcal{K}_i$ and a map $c_i : \mathcal{I}_i \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}_i$. Assume given a short exact sequence of $\mathcal{O}$-modules $$ 0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0 $$ such that $$ \vcenter{ \xymatrix{ \mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_\mathcal{O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } } \quad\text{and}\quad \vcenter{ \xymatrix{ \mathcal{I}_3 \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_3} \ar[d] & \mathcal{K}_3 \ar[d] \\ \mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F} \ar[r]^-{c_2} & \mathcal{K}_2 } } $$ are commutative. Finally, assume given an extension $$ 0 \to \mathcal{K}_2 \to \mathcal{F}'_2 \to \mathcal{F} \to 0 $$ as in (\ref{equation-extension-ringed-topoi}) with $\mathcal{K} = \mathcal{K}_2$ of $\mathcal{O}'_2$-modules with $c_{\mathcal{F}'_2} = c_2$. In this situation we can apply the functoriality of Remark \ref{remark-extension-functorial-ringed-topoi} to obtain an extension $\mathcal{F}'_1$ of $\mathcal{O}'_1$-modules (we'll describe $\mathcal{F}'_1$ in this special case below). By Remark \ref{remark-trivial-extension-ringed-topoi} using the canonical splitting $\pi : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{C}), \mathcal{O})$ of Remark \ref{remark-short-exact-sequence-thickenings-ringed-topoi} we obtain $\xi_{\mathcal{F}'_1} \in \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_1)$. Finally, we have the obstruction $$ o(\mathcal{F}, \mathcal{K}_3, c_3) \in \Ext^2_\mathcal{O}(\mathcal{F}, \mathcal{K}_3) $$ see Lemma \ref{lemma-inf-obs-ext-ringed-topoi}. In this situation we {\bf claim} that the canonical map $$ \partial : \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_1) \longrightarrow \Ext^2_\mathcal{O}(\mathcal{F}, \mathcal{K}_3) $$ coming from the short exact sequence $0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0$ sends $\xi_{\mathcal{F}'_1}$ to the obstruction class $o(\mathcal{F}, \mathcal{K}_3, c_3)$. \medskip\noindent To prove this claim choose an embedding $j : \mathcal{K}_3 \to \mathcal{K}$ where $\mathcal{K}$ is an injective $\mathcal{O}$-module. We can lift $j$ to a map $j' : \mathcal{K}_2 \to \mathcal{K}$. Set $\mathcal{E}'_2 = j'_*\mathcal{F}'_2$ equal to the pushout of $\mathcal{F}'_2$ by $j'$ so that $c_{\mathcal{E}'_2} = j' \circ c_2$. Picture: $$ \xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d]_{j'} & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[r] & \mathcal{E}'_2 \ar[r] & \mathcal{F} \ar[r] & 0 } $$ Set $\mathcal{E}'_3 = \mathcal{E}'_2$ but viewed as an $\mathcal{O}'_3$-module via $\mathcal{O}'_3 \to \mathcal{O}'_2$. Then $c_{\mathcal{E}'_3} = j \circ c_3$. The proof of Lemma \ref{lemma-inf-obs-ext-ringed-topoi} constructs $o(\mathcal{F}, \mathcal{K}_3, c_3)$ as the boundary of the class of the extension of $\mathcal{O}$-modules $$ 0 \to \mathcal{K}/\mathcal{K}_3 \to \mathcal{E}'_3/\mathcal{K}_3 \to \mathcal{F} \to 0 $$ On the other hand, note that $\mathcal{F}'_1 = \mathcal{F}'_2/\mathcal{K}_3$ hence the class $\xi_{\mathcal{F}'_1}$ is the class of the extension $$ 0 \to \mathcal{K}_2/\mathcal{K}_3 \to \mathcal{F}'_2/\mathcal{K}_3 \to \mathcal{F} \to 0 $$ seen as a sequence of $\mathcal{O}$-modules using $\pi^\sharp$ where $\pi : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{C}), \mathcal{O})$ is the canonical splitting. Thus finally, the claim follows from the fact that we have a commutative diagram $$ \xymatrix{ 0 \ar[r] & \mathcal{K}_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F}'_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}/\mathcal{K}_3 \ar[r] & \mathcal{E}'_3/\mathcal{K}_3 \ar[r] & \mathcal{F} \ar[r] & 0 } $$ which is $\mathcal{O}$-linear (with the $\mathcal{O}$-module structures given above).


duality-remark-iso-on-RSheafHom

In the situation of Lemma \ref{lemma-iso-on-RSheafHom} we have $$ DQ_Y(Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K))) = Rf_* DQ_X(R\SheafHom_{\mathcal{O}_X}(L, a(K))) $$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-pushforward-better-coherator}. Thus if $R\SheafHom_{\mathcal{O}_X}(L, a(K)) \in D_\QCoh(\mathcal{O}_X)$, then we can ``erase'' the $DQ_Y$ on the left hand side of the arrow. On the other hand, if we know that $R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K) \in D_\QCoh(\mathcal{O}_Y)$, then we can ``erase'' the $DQ_Y$ from the right hand side of the arrow. If both are true then we see that (\ref{equation-sheafy-trace}) is an isomorphism. Combining this with Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-internal-hom} we see that $Rf_*R\SheafHom_{\mathcal{O}_X}(L, a(K)) \to R\SheafHom_{\mathcal{O}_Y}(Rf_*L, K)$ is an isomorphism if \begin{enumerate} \item $L$ and $Rf_*L$ are perfect, or \item $K$ is bounded below and $L$ and $Rf_*L$ are pseudo-coherent. \end{enumerate} For (2) we use that $a(K)$ is bounded below if $K$ is bounded below, see Lemma \ref{lemma-twisted-inverse-image-bounded-below}.


duality-remark-going-around

Consider a commutative diagram $$ \xymatrix{ X'' \ar[r]_{k'} \ar[d]_{f''} & X' \ar[r]_k \ar[d]_{f'} & X \ar[d]^f \\ Y'' \ar[r]^{l'} \ar[d]_{g''} & Y' \ar[r]^l \ar[d]_{g'} & Y \ar[d]^g \\ Z'' \ar[r]^{m'} & Z' \ar[r]^m & Z } $$ of quasi-compact and quasi-separated schemes where all squares are cartesian and where $(f, l)$, $(g, m)$, $(f', l')$, $(g', m')$ are Tor independent pairs of maps. Let $a$, $a'$, $a''$, $b$, $b'$, $b''$ be the right adjoints of Lemma \ref{lemma-twisted-inverse-image} for $f$, $f'$, $f''$, $g$, $g'$, $g''$. Let us label the squares of the diagram $A$, $B$, $C$, $D$ as follows $$ \begin{matrix} A & B \\ C & D \end{matrix} $$ Then the maps (\ref{equation-base-change-map}) for the squares are (where we use $k^* = Lk^*$, etc) $$ \begin{matrix} \gamma_A : (k')^* \circ a' \to a'' \circ (l')^* & \gamma_B : k^* \circ a \to a' \circ l^* \\ \gamma_C : (l')^* \circ b' \to b'' \circ (m')^* & \gamma_D : l^* \circ b \to b' \circ m^* \end{matrix} $$ For the $2 \times 1$ and $1 \times 2$ rectangles we have four further base change maps $$ \begin{matrix} \gamma_{A + B} : (k \circ k')^* \circ a \to a'' \circ (l \circ l')^* \\ \gamma_{C + D} : (l \circ l')^* \circ b \to b'' \circ (m \circ m')^* \\ \gamma_{A + C} : (k')^* \circ (a' \circ b') \to (a'' \circ b'') \circ (m')^* \\ \gamma_{B + D} : k^* \circ (a \circ b) \to (a' \circ b') \circ m^* \end{matrix} $$ By Lemma \ref{lemma-compose-base-change-maps-horizontal} we have $$ \gamma_{A + B} = \gamma_A \circ \gamma_B, \quad \gamma_{C + D} = \gamma_C \circ \gamma_D $$ and by Lemma \ref{lemma-compose-base-change-maps} we have $$ \gamma_{A + C} = \gamma_C \circ \gamma_A, \quad \gamma_{B + D} = \gamma_D \circ \gamma_B $$ Here it would be more correct to write $\gamma_{A + B} = (\gamma_A \star \text{id}_{l^*}) \circ (\text{id}_{(k')^*} \star \gamma_B)$ with notation as in Categories, Section \ref{categories-section-formal-cat-cat} and similarly for the others. However, we continue the abuse of notation used in the proofs of Lemmas \ref{lemma-compose-base-change-maps} and \ref{lemma-compose-base-change-maps-horizontal} of dropping $\star$ products with identities as one can figure out which ones to add as long as the source and target of the transformation is known. Having said all of this we find (a priori) two transformations $$ (k')^* \circ k^* \circ a \circ b \longrightarrow a'' \circ b'' \circ (m')^* \circ m^* $$ namely $$ \gamma_C \circ \gamma_A \circ \gamma_D \circ \gamma_B = \gamma_{A + C} \circ \gamma_{B + D} $$ and $$ \gamma_C \circ \gamma_D \circ \gamma_A \circ \gamma_B = \gamma_{C + D} \circ \gamma_{A + B} $$ The point of this remark is to point out that these transformations are equal. Namely, to see this it suffices to show that $$ \xymatrix{ (k')^* \circ a' \circ l^* \circ b \ar[r]_{\gamma_D} \ar[d]_{\gamma_A} & (k')^* \circ a' \circ b' \circ m^* \ar[d]^{\gamma_A} \\ a'' \circ (l')^* \circ l^* \circ b \ar[r]^{\gamma_D} & a'' \circ (l')^* \circ b' \circ m^* } $$ commutes. This is true by Categories, Lemma \ref{categories-lemma-properties-2-cat-cats} or more simply the discussion preceding Categories, Definition \ref{categories-definition-horizontal-composition}.


duality-remark-check-over-affines

Consider a cartesian diagram $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ of quasi-compact and quasi-separated schemes with $(g, f)$ Tor independent. Let $V \subset Y$ and $V' \subset Y'$ be affine opens with $g(V') \subset V$. Form the cartesian diagrams $$ \vcenter{ \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & Y } } \quad\text{and}\quad \vcenter{ \xymatrix{ U' \ar[r] \ar[d] & X' \ar[d] \\ V' \ar[r] & Y' } } $$ Assume (\ref{equation-sheafy}) with respect to $K$ and the first diagram and (\ref{equation-sheafy}) with respect to $Lg^*K$ and the second diagram are isomorphisms. Then the restriction of the base change map (\ref{equation-base-change-map}) $$ L(g')^*a(K) \longrightarrow a'(Lg^*K) $$ to $U'$ is isomorphic to the base change map (\ref{equation-base-change-map}) for $K|_V$ and the cartesian diagram $$ \xymatrix{ U' \ar[r] \ar[d] & U \ar[d] \\ V' \ar[r] & V } $$ This follows from the fact that (\ref{equation-sheafy}) is a special case of the base change map (\ref{equation-base-change-map}) and that the base change maps compose correctly if we stack squares horizontally, see Lemma \ref{lemma-compose-base-change-maps-horizontal}. Thus in order to check the base change map restricted to $U'$ is an isomorphism it suffices to work with the last diagram.


duality-remark-trace-map-finite

If $f : Y \to X$ is a finite morphism of Noetherian schemes, then the diagram $$ \xymatrix{ Rf_*a(K) \ar[r]_-{\text{Tr}_{f, K}} \ar@{=}[d] & K \ar@{=}[d] \\ R\SheafHom_{\mathcal{O}_X}(f_*\mathcal{O}_Y, K) \ar[r] & K } $$ is commutative for $K \in D_\QCoh^+(\mathcal{O}_X)$. This follows from Lemma \ref{lemma-finite-twisted}. The lower horizontal arrow is induced by the map $\mathcal{O}_X \to f_*\mathcal{O}_Y$ and the upper horizontal arrow is the trace map discussed in Section \ref{section-trace}.


duality-remark-relative-dualizing-complex

Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a proper, flat morphism of finite presentation. Let $a$ be the adjoint of Lemma \ref{lemma-twisted-inverse-image} for $f$. In this situation, $\omega_{X/Y}^\bullet = a(\mathcal{O}_Y)$ is sometimes called the {\it relative dualizing complex}. By Lemma \ref{lemma-compare-with-pullback-flat-proper} there is a functorial isomorphism $a(K) = Lf^*K \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}^\bullet$ for $K \in D_\QCoh(\mathcal{O}_Y)$. Moreover, the trace map $$ \text{Tr}_{f, \mathcal{O}_Y} : Rf_*\omega_{X/Y}^\bullet \to \mathcal{O}_Y $$ of Section \ref{section-trace} induces the trace map for all $K$ in $D_\QCoh(\mathcal{O}_Y)$. More precisely the diagram $$ \xymatrix{ Rf_*a(K) \ar[rrr]_{\text{Tr}_{f, K}} \ar@{=}[d] & & & K \ar@{=}[d] \\ Rf_*(Lf^*K \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}^\bullet) \ar@{=}[r] & K \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\omega_{X/Y}^\bullet \ar[rr]^-{\text{id}_K \otimes \text{Tr}_{f, \mathcal{O}_Y}} & & K } $$ where the equality on the lower right is Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change}. If $g : Y' \to Y$ is a morphism of quasi-compact and quasi-separated schemes and $X' = Y' \times_Y X$, then by Lemma \ref{lemma-proper-flat-base-change} we have $\omega_{X'/Y'}^\bullet = L(g')^*\omega_{X/Y}^\bullet$ where $g' : X' \to X$ is the projection and by Lemma \ref{lemma-trace-map-and-base-change} the trace map $$ \text{Tr}_{f', \mathcal{O}_{Y'}} : Rf'_*\omega_{X'/Y'}^\bullet \to \mathcal{O}_{Y'} $$ for $f' : X' \to Y'$ is the base change of $\text{Tr}_{f, \mathcal{O}_Y}$ via the base change isomorphism.


duality-remark-relative-dualizing-complex-relative-cup-product

Let $f : X \to Y$, $\omega^\bullet_{X/Y}$, and $\text{Tr}_{f, \mathcal{O}_Y}$ be as in Remark \ref{remark-relative-dualizing-complex}. Let $K$ and $M$ be in $D_\QCoh(\mathcal{O}_X)$ with $M$ pseudo-coherent (for example perfect). Suppose given a map $K \otimes_{\mathcal{O}_X}^\mathbf{L} M \to \omega^\bullet_{X/Y}$ which corresponds to an isomorphism $K \to R\SheafHom_{\mathcal{O}_X}(M, \omega^\bullet_{X/Y})$ via Cohomology, Equation (\ref{cohomology-equation-internal-hom}). Then the relative cup product (Cohomology, Remark \ref{cohomology-remark-cup-product}) $$ Rf_*K \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*M \to Rf_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} M) \to Rf_*\omega^\bullet_{X/Y} \xrightarrow{\text{Tr}_{f, \mathcal{O}_Y}} \mathcal{O}_Y $$ determines an isomorphism $Rf_*K \to R\SheafHom_{\mathcal{O}_Y}(Rf_*M, \mathcal{O}_Y)$. Namely, since $\omega^\bullet_{X/Y} = a(\mathcal{O}_Y)$ the canonical map (\ref{equation-sheafy-trace}) $$ Rf_*R\SheafHom_{\mathcal{O}_X}(M, \omega^\bullet_{X/Y}) \to R\SheafHom_{\mathcal{O}_Y}(Rf_*M, \mathcal{O}_Y) $$ is an isomorphism by Lemma \ref{lemma-iso-on-RSheafHom} and Remark \ref{remark-iso-on-RSheafHom} and the fact that $M$ and $Rf_*M$ are pseudo-coherent, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-pseudo-coherent-direct-image-general}. To see that the relative cup product induces this isomorphism use the commutativity of the diagram in Cohomology, Remark \ref{cohomology-remark-relative-cup-and-composition}.


duality-remark-van-den-bergh

Lemma \ref{lemma-van-den-bergh} means our relative dualizing complex is {\it rigid} in a sense analogous to the notion introduced in \cite{vdB-rigid}. Namely, since the functor on the right of (\ref{equation-rigid}) is ``quadratic'' in $\omega_{X/Y}^\bullet$ and the functor on the left of (\ref{equation-rigid}) is ``linear'' this ``pins down'' the complex $\omega_{X/Y}^\bullet$ to some extent. There is an approach to duality theory using ``rigid'' (relative) dualizing complexes, see for example \cite{Neeman-rigid}, \cite{Yekutieli-rigid}, and \cite{Yekutieli-Zhang}. We will return to this in Section \ref{section-relative-dualizing-complexes}.


duality-remark-local-calculation-shriek

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Using the lemmas above we can compute $f^!$ locally as follows. Suppose that we are given affine opens $$ \xymatrix{ U \ar[r]_j \ar[d]_g & X \ar[d]^f \\ V \ar[r]^i & Y } $$ Since $j^! \circ f^! = g^! \circ i^!$ (Lemma \ref{lemma-upper-shriek-composition}) and since $j^!$ and $i^!$ are given by restriction (Lemma \ref{lemma-shriek-open-immersion}) we see that $$ (f^!E)|_U = g^!(E|_V) $$ for any $E \in D^+_\QCoh(\mathcal{O}_X)$. Write $U = \Spec(A)$ and $V = \Spec(R)$ and let $\varphi : R \to A$ be the finite type ring map corresponding to $g$. Choose a presentation $A = P/I$ where $P = R[x_1, \ldots, x_n]$ is a polynomial algebra in $n$ variables over $R$. Choose an object $K \in D^+(R)$ corresponding to $E|_V$ (Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-compare-bounded}). Then we claim that $f^!E|_U$ corresponds to $$ \varphi^!(K) = R\Hom(A, K \otimes_R^\mathbf{L} P)[n] $$ where $R\Hom(A, -) : D(P) \to D(A)$ is the functor of Dualizing Complexes, Section \ref{dualizing-section-trivial} and where $\varphi^! : D(R) \to D(A)$ is the functor of Dualizing Complexes, Section \ref{dualizing-section-relative-dualizing-complex-algebraic}. Namely, the choice of presentation gives a factorization $$ U \rightarrow \mathbf{A}^n_V \to \mathbf{A}^{n - 1}_V \to \ldots \to \mathbf{A}^1_V \to V $$ Applying Lemma \ref{lemma-shriek-affine-line} exactly $n$ times we see that $(\mathbf{A}^n_V \to V)^!(E|_V)$ corresponds to $K \otimes_R^\mathbf{L} P[n]$. By Lemmas \ref{lemma-sheaf-with-exact-support-quasi-coherent} and \ref{lemma-shriek-closed-immersion} the last step corresponds to applying $R\Hom(A, -)$.


duality-remark-independent-omega-S

Let $S$ be a Noetherian scheme which has a dualizing complex. Let $f : X \to Y$ be a morphism of schemes of finite type over $S$. Then the functor $$ f_{new}^! : D^+_{Coh}(\mathcal{O}_Y) \to D^+_{Coh}(\mathcal{O}_X) $$ is independent of the choice of the dualizing complex $\omega_S^\bullet$ up to canonical isomorphism. We sketch the proof. Any second dualizing complex is of the form $\omega_S^\bullet \otimes_{\mathcal{O}_S}^\mathbf{L} \mathcal{L}$ where $\mathcal{L}$ is an invertible object of $D(\mathcal{O}_S)$, see Lemma \ref{lemma-dualizing-unique-schemes}. For any separated morphism $p : U \to S$ of finite type we have $p^!(\omega_S^\bullet \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{L}) = p^!(\omega_S^\bullet) \otimes^\mathbf{L}_{\mathcal{O}_U} Lp^*\mathcal{L}$ by Lemma \ref{lemma-compare-with-pullback-perfect}. Hence, if $\omega_X^\bullet$ and $\omega_Y^\bullet$ are the dualizing complexes normalized relative to $\omega_S^\bullet$ we see that $\omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} La^*\mathcal{L}$ and $\omega_Y^\bullet \otimes_{\mathcal{O}_Y}^\mathbf{L} Lb^*\mathcal{L}$ are the dualizing complexes normalized relative to $\omega_S^\bullet \otimes_{\mathcal{O}_S}^\mathbf{L} \mathcal{L}$ (where $a : X \to S$ and $b : Y \to S$ are the structure morphisms). Then the result follows as \begin{align*} & R\SheafHom_{\mathcal{O}_X}(Lf^*R\SheafHom_{\mathcal{O}_Y}(K, \omega_Y^\bullet \otimes_{\mathcal{O}_Y}^\mathbf{L} Lb^*\mathcal{L}), \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} La^*\mathcal{L}) \\ & = R\SheafHom_{\mathcal{O}_X}(Lf^*R(\SheafHom_{\mathcal{O}_Y}(K, \omega_Y^\bullet) \otimes_{\mathcal{O}_Y}^\mathbf{L} Lb^*\mathcal{L}), \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} La^*\mathcal{L}) \\ & = R\SheafHom_{\mathcal{O}_X}(Lf^*R\SheafHom_{\mathcal{O}_Y}(K, \omega_Y^\bullet) \otimes_{\mathcal{O}_X}^\mathbf{L} La^*\mathcal{L}, \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} La^*\mathcal{L}) \\ & = R\SheafHom_{\mathcal{O}_X}(Lf^*R\SheafHom_{\mathcal{O}_Y}(K, \omega_Y^\bullet), \omega_X^\bullet) \end{align*} for $K \in D^+_{Coh}(\mathcal{O}_Y)$. The last equality because $La^*\mathcal{L}$ is invertible in $D(\mathcal{O}_X)$.


duality-remark-dualizing-finite

Let $S$ be a Noetherian scheme and let $\omega_S^\bullet$ be a dualizing complex. Let $f : X \to Y$ be a finite morphism between schemes of finite type over $S$. Let $\omega_X^\bullet$ and $\omega_Y^\bullet$ be dualizing complexes normalized relative to $\omega_S^\bullet$. Then we have $$ f_*\omega_X^\bullet = R\SheafHom(f_*\mathcal{O}_X, \omega_Y^\bullet) $$ in $D_\QCoh^+(f_*\mathcal{O}_X)$ by Lemmas \ref{lemma-finite-twisted} and \ref{lemma-proper-map-good-dualizing-complex} and the trace map of Example \ref{example-trace-proper} is the map $$ \text{Tr}_f : Rf_*\omega_X^\bullet = f_*\omega_X^\bullet = R\SheafHom(f_*\mathcal{O}_X, \omega_Y^\bullet) \longrightarrow \omega_Y^\bullet $$ which often goes under the name ``evaluation at $1$''.


duality-remark-relative-dualizing-complex-shriek

Let $f : X \to Y$ be a flat proper morphism of finite type schemes over a pair $(S, \omega_S^\bullet)$ as in Situation \ref{situation-dualizing}. The relative dualizing complex (Remark \ref{remark-relative-dualizing-complex}) is $\omega_{X/Y}^\bullet = a(\mathcal{O}_Y)$. By Lemma \ref{lemma-proper-map-good-dualizing-complex} we have the first canonical isomorphism in $$ \omega_X^\bullet = a(\omega_Y^\bullet) = Lf^*\omega_Y^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}^\bullet $$ in $D(\mathcal{O}_X)$. The second canonical isomorphism follows from the discussion in Remark \ref{remark-relative-dualizing-complex}.


duality-remark-the-same-is-true

Let $S$ be a Noetherian scheme endowed with a dualizing complex $\omega_S^\bullet$. In this case Lemmas \ref{lemma-shriek}, \ref{lemma-flat-shriek}, \ref{lemma-flat-quasi-finite-shriek}, and \ref{lemma-CM-shriek} are true for any morphism $f : X \to Y$ of finite type schemes over $S$ but with $f^!$ replaced by $f_{new}^!$. This is clear because in each case the proof reduces immediately to the affine case and then $f^! = f_{new}^!$ by Lemma \ref{lemma-duality-bootstrap}.


duality-remark-CM-morphism-compare-dualizing

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Assume $f$ is a Cohen-Macaulay morphism of relative dimension $d$. Let $\omega_{X/Y} = H^{-d}(f^!\mathcal{O}_Y)$ be the unique nonzero cohomology sheaf of $f^!\mathcal{O}_Y$, see Lemma \ref{lemma-CM-shriek}. Then there is a canonical isomorphism $$ f^!K = Lf^*K \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}[d] $$ for $K \in D^+_\QCoh(\mathcal{O}_Y)$, see Lemma \ref{lemma-perfect-comparison-shriek}. In particular, if $S$ has a dualizing complex $\omega_S^\bullet$, $\omega_Y^\bullet = (Y \to S)^!\omega_S^\bullet$, and $\omega_X^\bullet = (X \to S)^!\omega_S^\bullet$ then we have $$ \omega_X^\bullet = Lf^*\omega_Y^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} \omega_{X/Y}[d] $$ Thus if further $X$ and $Y$ are connected and Cohen-Macaulay and if $\omega_Y$ and $\omega_X$ denote the unique nonzero cohomology sheaves of $\omega_Y^\bullet$ and $\omega_X^\bullet$, then we have $$ \omega_X = f^*\omega_Y \otimes_{\mathcal{O}_X} \omega_{X/Y}. $$ Similar results hold for $X$ and $Y$ arbitrary finite type schemes over $S$ (i.e., not necessarily separated over $S$) with dualizing complexes normalized with respect to $\omega_S^\bullet$ as in Section \ref{section-glue}.


duality-remark-duality-proper-over-field

Let $k$, $X$, and $\omega_X^\bullet$ be as in Lemma \ref{lemma-duality-proper-over-field}. The identity on the complex $\omega_X^\bullet$ corresponds, via the functorial isomorphism in part (5), to a map $$ t : H^0(X, \omega_X^\bullet) \longrightarrow k $$ For an arbitrary $K$ in $D_\QCoh(\mathcal{O}_X)$ the identification $\Hom(K, \omega_X^\bullet)$ with $H^0(X, K)^\vee$ in part (5) corresponds to the pairing $$ \Hom_X(K, \omega_X^\bullet) \times H^0(X, K) \longrightarrow k,\quad (\alpha, \beta) \longmapsto t(\alpha(\beta)) $$ This follows from the functoriality of the isomorphisms in (5). Similarly for any $i \in \mathbf{Z}$ we get the pairing $$ \Ext^i_X(K, \omega_X^\bullet) \times H^{-i}(X, K) \longrightarrow k,\quad (\alpha, \beta) \longmapsto t(\alpha(\beta)) $$ Here we think of $\alpha$ as a morphism $K[-i] \to \omega_X^\bullet$ and $\beta$ as an element of $H^0(X, K[-i])$ in order to define $\alpha(\beta)$. Observe that if $K$ is general, then we only know that this pairing is nondegenerate on one side: the pairing induces an isomorphism of $\Hom_X(K, \omega_X^\bullet)$, resp.\ $\Ext^i_X(K, \omega_X^\bullet)$ with the $k$-linear dual of $H^0(X, K)$, resp.\ $H^{-i}(X, K)$ but in general not vice versa. If $K$ is in $D^b_{\textit{Coh}}(\mathcal{O}_X)$, then $\Hom_X(K, \omega_X^\bullet)$, $\Ext_X(K, \omega_X^\bullet)$, $H^0(X, K)$, and $H^i(X, K)$ are finite dimensional $k$-vector spaces (by Derived Categories of Schemes, Lemmas \ref{perfect-lemma-coherent-internal-hom} and \ref{perfect-lemma-direct-image-coherent-bdd-below}) and the pairings are perfect in the usual sense.


duality-remark-coherent-duality-proper-over-field

We continue the discussion in Remark \ref{remark-duality-proper-over-field} and we use the same notation $k$, $X$, $\omega_X^\bullet$, and $t$. If $\mathcal{F}$ is a coherent $\mathcal{O}_X$-module we obtain perfect pairings $$ \langle -, - \rangle : \Ext^i_X(\mathcal{F}, \omega_X^\bullet) \times H^{-i}(X,\mathcal{F}) \longrightarrow k,\quad (\alpha, \beta) \longmapsto t(\alpha(\beta)) $$ of finite dimensional $k$-vector spaces. These pairings satisfy the following (obvious) functoriality: if $\varphi : \mathcal{F} \to \mathcal{G}$ is a homomorphism of coherent $\mathcal{O}_X$-modules, then we have $$ \langle \alpha \circ \varphi, \beta \rangle = \langle \alpha, \varphi(\beta) \rangle $$ for $\alpha \in \Ext^i_X(\mathcal{G}, \omega_X^\bullet)$ and $\beta \in H^{-i}(X, \mathcal{F})$. In other words, the $k$-linear map $\Ext^i_X(\mathcal{G}, \omega_X^\bullet) \to \Ext^i_X(\mathcal{F}, \omega_X^\bullet)$ induced by $\varphi$ is, via the pairings, the $k$-linear dual of the $k$-linear map $H^{-i}(X, \mathcal{F}) \to H^{-i}(X, \mathcal{G})$ induced by $\varphi$. Formulated in this manner, this still works if $\varphi$ is a homomorphism of quasi-coherent $\mathcal{O}_X$-modules.


duality-remark-rework-duality-locally-free-CM

Let $X$ be a proper Cohen-Macaulay scheme over a field $k$ which is equidimensional of dimension $d$. Let $\omega_X^\bullet$ and $\omega_X$ be as in Lemma \ref{lemma-duality-proper-over-field}. By Lemma \ref{lemma-duality-proper-over-field-CM} we have $\omega_X^\bullet = \omega_X[d]$. Let $t : H^d(X, \omega_X) \to k$ be the map of Remark \ref{remark-duality-proper-over-field}. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module with dual $\mathcal{E}^\vee$. Then we have perfect pairings $$ H^i(X, \omega_X \otimes_{\mathcal{O}_X} \mathcal{E}^\vee) \times H^{d - i}(X, \mathcal{E}) \longrightarrow k,\quad (\xi, \eta) \longmapsto t(1 \otimes \epsilon)(\xi \cup \eta)) $$ where $\cup$ is the cup-product and $\epsilon : \mathcal{E}^\vee \otimes_{\mathcal{O}_X} \mathcal{E} \to \mathcal{O}_X$ is the evaluation map. This is a special case of Lemma \ref{lemma-duality-proper-over-field-perfect}.


duality-remark-relative-dualizing-complex-bis

Let $X \to S$ be a morphism of schemes which is flat, proper, and of finite presentation. By Lemma \ref{lemma-existence-relative-dualizing} there exists a relative dualizing complex $(\omega_{X/S}^\bullet, \xi)$ in the sense of Definition \ref{definition-relative-dualizing-complex}. Consider any morphism $g : S' \to S$ where $S'$ is quasi-compact and quasi-separated (for example an affine open of $S$). By Lemma \ref{lemma-base-change-relative-dualizing} we see that $(L(g')^*\omega_{X/S}^\bullet, L(g')^*\xi)$ is a relative dualizing complex for the base change $f' : X' \to S'$ in the sense of Definition \ref{definition-relative-dualizing-complex}. Let $\omega_{X'/S'}^\bullet$ be the relative dualizing complex for $X' \to S'$ in the sense of Remark \ref{remark-relative-dualizing-complex}. Combining Lemmas \ref{lemma-flat-proper-relative-dualizing} and \ref{lemma-uniqueness-relative-dualizing} we see that there is a unique isomorphism $$ \omega_{X'/S'}^\bullet \longrightarrow L(g')^*\omega_{X/S}^\bullet $$ compatible with (\ref{equation-pre-rigid}) and $L(g')^*\xi$. These isomorphisms are compatible with morphisms between quasi-compact and quasi-separated schemes over $S$ and the base change isomorphisms of Lemma \ref{lemma-proper-flat-base-change} (if we ever need this compatibility we will carefully state and prove it here).


duality-remark-extension-by-zero

Let $j : U \to X$ be an open immersion of Noetherian schemes. Sending $K \in D^b_{\textit{Coh}}(\mathcal{O}_U)$ to a Deligne system whose restriction to $U$ is $K$ determines a functor $$ Rj_! : D^b_{\textit{Coh}}(\mathcal{O}_U) \longrightarrow \text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_X) $$ which is ``exact'' by Lemma \ref{lemma-extension-by-zero-triangle} and which is ``left adjoint'' to the functor $j^* : D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_U)$ by Lemma \ref{lemma-lift-map}.


duality-remark-extension-by-zero-linear-pro-system

Let $(A_n)$ and $(B_n)$ be inverse systems of a category $\mathcal{C}$. Let us say a linear-pro-morphism from $(A_n)$ to $(B_n)$ is given by a compatible family of morphisms $\varphi_n : A_{cn + d} \to B_n$ for all $n \geq 1$ for some fixed integers $c, d \geq 1$. We'll say $(\varphi_n : A_{cn + d} \to B_n)$ and $(\psi_n : A_{c'n + d'} \to B_n)$ determine the same morphism if there exist $c'' \geq \max(c, c')$ and $d'' \geq \max(d, d')$ such that the two induced morphisms $A_{c'' n + d''} \to B_n$ are the same for all $n$. It seems likely that Deligne systems $(K_n)$ with given value on $U$ are well defined up to linear-pro-isomorphisms. If we ever need this we will carefully formulate and prove this here.


duality-remark-covariance-open-j-lower-shriek

Let $X \supset U \supset U'$ be open subschemes of a Noetherian scheme $X$. Denote $j : U \to X$ and $j' : U' \to X$ the inclusion morphisms. We claim there is a canonical map $$ Rj'_!(K|_{U'}) \longrightarrow Rj_!K $$ functorial for $K$ in $D^b_{\textit{Coh}}(\mathcal{O}_U)$. Namely, by Lemma \ref{lemma-lift-map} we have for any $L$ in $D^b_{\textit{Coh}}(\mathcal{O}_X)$ the map \begin{align*} \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_X)}(Rj_!K, L) & = \Hom_U(K, L|_U) \\ & \to \Hom_{U'}(K|_{U'}, L|_{U'}) \\ & = \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_X)}(Rj'_!(K|_{U'}), L) \end{align*} functorial in $L$ and $K'$. The functoriality in $L$ shows by Categories, Remark \ref{categories-remark-pro-category-copresheaves} that we obtain a canonical map $Rj'_!(K|_{U'}) \to Rj_!K$ which is functorial in $K$ by the functoriality of the arrow above in $K$. \medskip\noindent Here is an explicit construction of this arrow. Namely, suppose that $\mathcal{F}^\bullet$ is a bounded complex of coherent $\mathcal{O}_X$-modules whose restriction to $U$ represents $K$ in the derived category. We have seen in the proof of Lemma \ref{lemma-extension-by-zero} that such a complex always exists. Let $\mathcal{I}$, resp.\ $\mathcal{I}'$ be a quasi-coherent sheaf of ideals on $X$ with $V(\mathcal{I}) = X \setminus U$, resp.\ $V(\mathcal{I}') = X \setminus U'$. After replacing $\mathcal{I}$ by $\mathcal{I} + \mathcal{I}'$ we may assume $\mathcal{I}' \subset \mathcal{I}$. By construction $Rj_!K$, resp.\ $Rj'_!(K|_{U'})$ is represented by the inverse system $(K_n)$, resp.\ $(K'_n)$ of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ with $$ K_n = \mathcal{I}^n\mathcal{F}^\bullet \quad\text{resp.}\quad K'_n = (\mathcal{I}')^n\mathcal{F}^\bullet $$ Clearly the map constructed above is given by the maps $K'_n \to K_n$ coming from the inclusions $(\mathcal{I}')^n \subset \mathcal{I}^n$.


duality-remark-compose-inverse-systems

Let $\mathcal{C}$ be a category. Suppose given an inverse system $$ \ldots \xrightarrow{\alpha_4} (M_{3, n}) \xrightarrow{\alpha_3} (M_{2, n}) \xrightarrow{\alpha_2} (M_{1, n}) $$ of inverse systems in the category of pro-objects of $\mathcal{C}$. In other words, the arrows $\alpha_i$ are morphisms of pro-objects. By Categories, Example \ref{categories-example-pro-morphism-inverse-systems} we can represent each $\alpha_i$ by a pair $(m_i, a_i)$ where $m_i : \mathbf{N} \to \mathbf{N}$ is an increasing function and $a_{i, n} : M_{i, m_i(n)} \to M_{i - 1, n}$ is a morphism of $\mathcal{C}$ making the diagrams $$ \xymatrix{ \ldots \ar[r] & M_{i, m_i(3)} \ar[d]^{a_{i, 3}} \ar[r] & M_{i, m_i(2)} \ar[d]^{a_{i, 2}} \ar[r] & M_{i, m_i(1)} \ar[d]^{a_{i, 1}} \\ \ldots \ar[r] & M_{i - 1, 3} \ar[r] & M_{i - 1, 2} \ar[r] & M_{i - 1, 1} } $$ commute. By replacing $m_i(n)$ by $\max(n, m_i(n))$ and adjusting the morphisms $a_i(n)$ accordingly (as in the example referenced) we may assume that $m_i(n) \geq n$. In this situation consider the inverse system $$ \ldots \to M_{4, m_4(m_3(m_2(4)))} \to M_{3, m_3(m_2(3))} \to M_{2, m_2(2)} \to M_{1, 1} $$ with general term $$ M_k = M_{k, m_k(m_{k - 1}(\ldots (m_2(k))\ldots))} $$ For any object $N$ of $\mathcal{C}$ we have $$ \colim_i \colim_n \Mor_\mathcal{C}(M_{i, n}, N) = \colim_k \Mor_\mathcal{C}(M_k, N)  $$ We omit the details. In other words, we see that the inverse system $(M_k)$ has the property $$ \colim_i \Mor_{\text{Pro-}\mathcal{C}}((M_{i, n}), N) = \Mor_{\text{Pro-}\mathcal{C}}((M_k), N) $$ This property determines the inverse system $(M_k)$ up to pro-isomorphism by the discussion in Categories, Remark \ref{categories-remark-pro-category-copresheaves}. In this way we can turn certain inverse systems in $\text{Pro-}\mathcal{C}$ into pro-objects with countable index categories.


duality-remark-composition-lower-shriek

In Situation \ref{situation-shriek} let $f : X \to Y$ and $g : Y \to Z$ be composable morphisms of $\textit{FTS}_S$. Let us define the composition $$ Rg_! \circ Rf_! : D^b_{\textit{Coh}}(\mathcal{O}_X) \longrightarrow \text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Z) $$ Namely, by the very construction of $Rf_!$ for $K$ in $D^b_{\textit{Coh}}(\mathcal{O}_X)$ the output $Rf_!K$ is the pro-isomorphism class of an inverse system $(M_n)$ in $D^b_{\textit{Coh}}(\mathcal{O}_Y)$. Then, since $Rg_!$ is constructed similarly, we see that $$ \ldots \to Rg_!M_3 \to Rg_!M_2 \to Rg_!M_1 $$ is an inverse system of $\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)$. By the discussion in Remark \ref{remark-compose-inverse-systems} there is a unique pro-isomorphism class, which we will denote $Rg_! Rf_! K$, of inverse systems in $D^b_{\textit{Coh}}(\mathcal{O}_Z)$ such that $$ \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Z)}(Rg_!Rf_!K, L) = \colim_n \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Z)}(Rg_!M_n, L) $$ We omit the discussion necessary to see that this construction is functorial in $K$ as it will immediately follow from the next lemma.


duality-remark-covariance-open-lower-shriek

In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$ and let $U \subset X$ be an open. Set $g = f|_U : U \to Y$. Then there is a canonical morphism $$ Rg_!(K|_U) \longrightarrow Rf_!K $$ functorial in $K$ in $D^b_{\textit{Coh}}(\mathcal{O}_X)$ which can be defined in at least 3 ways. \begin{enumerate} \item Denote $i : U \to X$ the inclusion morphism. We have $Rg_! = Rf_! \circ Ri_!$ by Lemma \ref{lemma-composition-lower-shriek} and we can use $Rf_!$ applied to the map $Ri_!(K|_U) \to K$ which is a special case of Remark \ref{remark-covariance-open-j-lower-shriek}. \item Choose a compactification $j : X \to \overline{X}$ of $X$ over $Y$ with structure morphism $\overline{f} : \overline{X} \to Y$. Set $j' = j \circ i : U \to \overline{X}$. We can use that $Rf_! = R\overline{f}_* \circ Rj_!$ and $Rg_! = R\overline{f}_* \circ Rj'_!$ and we can use $R\overline{f}_*$ applied to the map $Rj'_!(K|_U) \to Rj_!K$ of Remark \ref{remark-covariance-open-j-lower-shriek}. \item We can use \begin{align*} \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rf_!K, L) & = \Hom_X(K, f^!L) \\ & \to \Hom_U(K|_U, f^!L|_U) \\ & = \Hom_U(K|_U, g^!L) \\ & = \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rg_!(K|_U), L) \end{align*} functorial in $L$ and $K$. Here we have used Proposition \ref{proposition-duality-compactly-supported} twice and the construction of upper shriek functors which shows that $g^! = i^* \circ f^!$. The functoriality in $L$ shows by Categories, Remark \ref{categories-remark-pro-category-copresheaves} that we obtain a canonical map $Rg_!(K|_U) \to Rf_!K$ in $\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)$ which is functorial in $K$ by the functoriality of the arrow above in $K$. \end{enumerate} Each of these three constructions gives the same arrow; we omit the details.


duality-remark-covariance-etale-lower-shriek

Let us generalize the covariance of compactly supported cohomology given in Remark \ref{remark-covariance-open-lower-shriek} to \'etale morphisms. Namely, in Situation \ref{situation-shriek} suppose given a commutative diagram $$ \xymatrix{ U \ar[rr]_h \ar[rd]_g & & X \ar[ld]^f \\ & Y } $$ of $\textit{FTS}_S$ with $h$ \'etale. Then there is a canonical morphism $$ Rg_!(h^*K) \longrightarrow Rf_!K $$ functorial in $K$ in $D^b_{\textit{Coh}}(\mathcal{O}_X)$. We define this transformation using the sequence of maps \begin{align*} \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rf_!K, L) & = \Hom_X(K, f^!L) \\ & \to \Hom_U(h^*K, h^*(f^!L)) \\ & = \Hom_U(h^*K, h^!f^!L) \\ & = \Hom_U(h^*K, g^!L) \\ & = \Hom_{\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)}(Rg_!(h^*K), L) \end{align*} functorial in $L$ and $K$. Here we have used Proposition \ref{proposition-duality-compactly-supported} twice, we have used the equality $h^* = h^!$ of Lemma \ref{lemma-shriek-etale}, and we have used the equality $h^! \circ f^! = g^!$ of Lemma \ref{lemma-upper-shriek-composition}. The functoriality in $L$ shows by Categories, Remark \ref{categories-remark-pro-category-copresheaves} that we obtain a canonical map $Rg_!(h^*K) \to Rf_!K$ in $\text{Pro-}D^b_{\textit{Coh}}(\mathcal{O}_Y)$ which is functorial in $K$ by the functoriality of the arrow above in $K$.


duality-remark-covariance-lower-shriek

In Remarks \ref{remark-covariance-open-lower-shriek} and \ref{remark-covariance-etale-lower-shriek} we have seen that the construction of compactly supported cohomology is covariant with respect to open immersions and \'etale morphisms. In fact, the correct generality is that given a commutative diagram $$ \xymatrix{ U \ar[rr]_h \ar[rd]_g & & X \ar[ld]^f \\ & Y } $$ of $\textit{FTS}_S$ with $h$ flat and quasi-finite there exists a canonical transformation $$ Rg_! \circ h^* \longrightarrow Rf_! $$ As in Remark \ref{remark-covariance-etale-lower-shriek} this map can be constructed using a transformation of functors $h^* \to h^!$ on $D^+_{\textit{Coh}}(\mathcal{O}_X)$. Recall that $h^!K = h^*K \otimes \omega_{U/X}$ where $\omega_{U/X} = h^!\mathcal{O}_X$ is the relative dualizing sheaf of the flat quasi-finite morphism $h$ (see Lemmas \ref{lemma-perfect-comparison-shriek} and \ref{lemma-flat-quasi-finite-shriek}). Recall that $\omega_{U/X}$ is the same as the relative dualizing module which will be constructed in Discriminants, Remark \ref{discriminant-remark-relative-dualizing-for-quasi-finite} by Discriminants, Lemma \ref{discriminant-lemma-compare-dualizing}. Thus we can use the trace element $\tau_{U/X} : \mathcal{O}_U \to \omega_{U/X}$ which will be constructed in Discriminants, Remark \ref{discriminant-remark-relative-dualizing-for-flat-quasi-finite} to define our transformation. If we ever need this, we will precisely formulate and prove the result here.


more-morphisms-remark-action-by-derivations

Assumptions and notation as in Lemma \ref{lemma-action-by-derivations}. The action of a local section $\theta$ on $a'$ is sometimes indicated by $\theta \cdot a'$. Note that this means nothing else than the fact that $(a')^\sharp$ and $(\theta \cdot a')^\sharp$ differ by a derivation $D$ which is related to $\theta$ by Equation (\ref{equation-D}).


more-morphisms-remark-special-case

A special case of Lemmas \ref{lemma-difference-derivation}, \ref{lemma-action-by-derivations}, \ref{lemma-sheaf}, and \ref{lemma-action-sheaf} is where $Y = Y'$. In this case the map $A$ is always zero. The sheaf of Lemma \ref{lemma-sheaf} is just given by the rule $$ U' \mapsto \{a' : U' \to Y\text{ over }S\text{ with } a'|_U = a|_U\} $$ and we act on this by the sheaf $\SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/S}, \mathcal{C}_{X/X'})$.


more-morphisms-remark-another-special-case

Another special case of Lemmas \ref{lemma-difference-derivation}, \ref{lemma-action-by-derivations}, \ref{lemma-sheaf}, and \ref{lemma-action-sheaf} is where $S$ itself is a thickening $Z \subset Z' = S$ and $Y = Z \times_{Z'} Y'$. Picture $$ \xymatrix{ (X \subset X') \ar@{..>}[rr]_{(a, ?)} \ar[rd]_{(g, g')} & & (Y \subset Y') \ar[ld]^{(h, h')} \\ & (Z \subset Z') } $$ In this case the map $A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ is determined by $a$: the map $h^*\mathcal{C}_{Z/Z'} \to \mathcal{C}_{Y/Y'}$ is surjective (because we assumed $Y = Z \times_{Z'} Y'$), hence the pullback $g^*\mathcal{C}_{Z/Z'} = a^*h^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'}$ is surjective, and the composition $g^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ has to be the canonical map induced by $g'$. Thus the sheaf of Lemma \ref{lemma-sheaf} is just given by the rule $$ U' \mapsto \{a' : U' \to Y'\text{ over }Z'\text{ with } a'|_U = a|_U\} $$ and we act on this by the sheaf $\SheafHom_{\mathcal{O}_X}(a^*\Omega_{Y/Z}, \mathcal{C}_{X/X'})$.


more-morphisms-remark-tiny-improvement

Lemma \ref{lemma-action-by-derivations-etale-localization} can be improved in the following way. Suppose that we have commutative diagrams as in Lemma \ref{lemma-action-by-derivations-etale-localization} but we do not assume that $X_2 \to X_1$ and $S_2 \to S_1$ are \'etale. Next, suppose we have $\theta_1 : a_1^*\Omega_{X_1/S_1} \to \mathcal{C}_{T_1/T'_1}$ and $\theta_2 : a_2^*\Omega_{X_2/S_2} \to \mathcal{C}_{T_2/T'_2}$ such that $$ \xymatrix{ f_*\mathcal{O}_{X_2} \ar[rr]_{f_*D_2} & & f_*a_{2, *}\mathcal{C}_{T_2/T_2'} \\ \mathcal{O}_{X_1} \ar[rr]^{D_1} \ar[u]^{f^\sharp} & & a_{1, *}\mathcal{C}_{T_1/T_1'} \ar[u]_{\text{induced by }(h')^\sharp} } $$ is commutative where $D_i$ corresponds to $\theta_i$ as in Equation (\ref{equation-D}). Then we have the conclusion of Lemma \ref{lemma-action-by-derivations-etale-localization}. The importance of the condition that both $X_2 \to X_1$ and $S_2 \to S_1$ are \'etale is that it allows us to construct a $\theta_2$ from $\theta_1$.


more-morphisms-remark-geometric-proof-bertini-irreducible

Let us sketch a ``geometric'' proof of a special case of Lemma \ref{lemma-bertini-irreducible}. Namely, say $k$ is an algebraically closed field and $X \subset \mathbf{P}^n_k$ is smooth and irreducible of dimension $\geq 2$. Then we claim there is a hyperplane $H \subset \mathbf{P}^n_k$ such that $X \cap H$ is smooth and irreducible. Namely, by Varieties, Lemma \ref{varieties-lemma-bertini} for a general $v \in V = kT_0 \oplus \ldots \oplus kT_n$ the corresponding hyperplane section $X \cap H_v$ is smooth. On the other hand, by Enriques-Severi-Zariski the scheme $X \cap H_v$ is connected, see Varieties, Lemma \ref{varieties-lemma-connectedness-ample-divisor}. Hence $X \cap H_v$ is smooth and irreducible.


more-morphisms-remark-necessary-condition-slice-smooth

The second condition in Lemma \ref{lemma-slice-smooth-once} is necessary even if $x$ is a closed point of a positive dimensional fibre. An example is the following: Let $k$ be a field of characteristic $p > 0$ which is imperfect. Let $a \in k$ be an element which is not a $p$th power. Let $\mathfrak m = (x, y^p - a) \subset k[x, y]$. This corresponds to a closed point $w$ of $X = \mathbf{A}^2_k$. Set $S = \mathbf{A}^1_k$ and let $f : X \to S$ be the morphism corresponding to $k[x] \to k[x, y]$. Then there does not exist any commutative diagram $$ \xymatrix{ S' \ar[rr]_h \ar[rd]_g & & X \ar[ld]^f \\ & S } $$ with $g$ \'etale and $w$ in the image of $h$. This is clear as the residue field extension $\kappa(f(w)) \subset \kappa(w)$ is purely inseparable, but for any $s' \in S'$ with $g(s') = f(w)$ the extension $\kappa(f(w)) \subset \kappa(s')$ would be separable.


more-morphisms-remark-topologies

In terms of topologies Lemmas \ref{lemma-dominate-etale-neighbourhood-finite-flat} and \ref{lemma-dominate-etale-affine-finite-flat} mean the following. Let $S$ be any scheme. Let $\{f_i : U_i \to S\}$ be an \'etale covering of $S$. There exists a Zariski open covering $S = \bigcup V_j$, for each $j$ a finite locally free, surjective morphism $W_j \to V_j$, and for each $j$ a Zariski open covering $\{W_{j, k} \to W_j\}$ such that the family $\{W_{j, k} \to S\}$ refines the given \'etale covering $\{f_i : U_i \to S\}$. What does this mean in practice? Well, for example, suppose we have a descent problem which we know how to solve for Zariski coverings and for fppf coverings of the form $\{\pi : T \to S\}$ with $\pi$ finite locally free and surjective. Then this descent problem has an affirmative answer for \'etale coverings as well. This trick was used by Gabber in his proof that $\text{Br}(X) = \text{Br}'(X)$ for an affine scheme $X$, see \cite{Hoobler}.


more-morphisms-remark-change-topologies

As a consequence of Lemma \ref{lemma-fppf-ph} we obtain a comparison morphism $$ \epsilon : (\Sch/S)_{ph} \longrightarrow (\Sch/S)_{fppf} $$ This is the morphism of sites given by the identity functor on underlying categories (with suitable choices of sites as in Topologies, Remark \ref{topologies-remark-choice-sites}). The functor $\epsilon_*$ is the identity on underlying presheaves and the functor $\epsilon^{-1}$ associated to an fppf sheaf its ph sheafification. By composition we can in addition compare the ph topology with the syntomic, smooth, \'etale, and Zariski topologies.


more-morphisms-remark-full-specialization-sequence

The proof of Lemma \ref{lemma-closed-point-nearby-fibre} actually shows that there exists a sequence of specializations $$ x \leadsto x_1 \leadsto x_2 \leadsto \ldots \leadsto x_d \leadsto x' $$ where all $x_i$ are in the fibre $X_s$, each specialization is immediate, and $x_d$ is a closed point of $X_s$. The integer $d = \text{trdeg}_{\kappa(s)}(\kappa(x)) = \dim(\overline{\{x\}})$ where the closure is taken in $X_s$. Moreover, the points $x_i$ can be chosen to avoid any closed subset of $X_s$ which does not contain the point $x$.


more-morphisms-remark-alternative-closed-point-nearby-fibre

We can use Lemma \ref{lemma-quasi-finite-quasi-section-meeting-nearby-open} or its variant Lemma \ref{lemma-quasi-finite-quasi-section-meeting-nearby-open-X} to give an alternative proof of Lemma \ref{lemma-closed-point-nearby-fibre} in case $S$ is locally Noetherian. Here is a rough sketch. Namely, first replace $S$ by the spectrum of the local ring at $s'$. Then we may use induction on $\dim(S)$. The case $\dim(S) = 0$ is trivial because then $s' = s$. Replace $X$ by the reduced induced scheme structure on $\overline{\{x\}}$. Apply Lemma \ref{lemma-quasi-finite-quasi-section-meeting-nearby-open-X} to $X \to S$ and $x' \mapsto s'$ and any nonempty open $U \subset X$ containing $x$. This gives us a closed subscheme $x' \in Z \subset X$ a point $z \in Z$ such that $Z \to S$ is quasi-finite at $x'$ and such that $f(z) \not = s'$. Then $z$ is a closed point of $X_{f(z)}$, and $z \leadsto x'$. As $f(z) \not = s'$ we see $\dim(\mathcal{O}_{S, f(z)}) < \dim(S)$. Since $x$ is the generic point of $X$ we see $x \leadsto z$, hence $s = f(x) \leadsto f(z)$. Apply the induction hypothesis to $s \leadsto f(z)$ and $z \mapsto f(z)$ to win.


more-morphisms-remark-perfect-permanence

It is not true that a morphism between schemes $X, Y$ perfect over a base $S$ is perfect. An example is $S = \Spec(k)$, $X = \Spec(k)$, $Y = \Spec(k[x]/(x^2)$ and $X \to Y$ the unique $S$-morphism.


sites-modules-remark-no-extension

In general the functor $g_!$ cannot be extended to categories of modules in case $g$ is (part of) a morphism of ringed topoi. Namely, given any ring map $A \to B$ the functor $M \mapsto B \otimes_A M$ has a right adjoint (restriction) but not in general a left adjoint (because its existence would imply that $A \to B$ is flat). We will see in Section \ref{section-localize} below that it is possible to define $j_!$ on sheaves of modules in the case of a localization of sites. We will discuss this in greater generality in Section \ref{section-lower-shriek-modules} below.


sites-modules-remark-localize-shriek-equal

In the situation of Lemma \ref{lemma-extension-by-zero} the diagram $$ \xymatrix{ \textit{Mod}(\mathcal{O}_U) \ar[r]_{j_{U!}} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}_\mathcal{C}) \ar[d]^{forget} \\ \textit{Ab}(\mathcal{C}/U) \ar[r]^{j^{Ab}_{U!}} & \textit{Ab}(\mathcal{C}) } $$ commutes. This is clear from the explicit description of the functor $j_{U!}$ in the lemma.


sites-modules-remark-localize-presheaves

Localization and presheaves of modules; see Sites, Remark \ref{sites-remark-localize-presheaves}. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let $U$ be an object of $\mathcal{C}$. Strictly speaking the functors $j_U^*$, $j_{U*}$ and $j_{U!}$ have not been defined for presheaves of $\mathcal{O}$-modules. But of course, we can think of a presheaf as a sheaf for the chaotic topology on $\mathcal{C}$ (see Sites, Examples \ref{sites-example-indiscrete}). Hence we also obtain a functor $$ j_U^* : \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}_U) $$ and functors $$ j_{U*}, j_{U!} : \textit{PMod}(\mathcal{O}_U) \longrightarrow \textit{PMod}(\mathcal{O}) $$ which are right, left adjoint to $j_U^*$. Inspecting the proof of Lemma \ref{lemma-extension-by-zero} we see that $j_{U!}\mathcal{G}$ is the presheaf $$ V \longmapsto \bigoplus\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) $$ In addition the functor $j_{U!}$ is exact (by Lemma \ref{lemma-extension-by-zero-exact} in the case of the discrete topologies). Moreover, if $\mathcal{C}$ is actually a site, and $\mathcal{O}$ is actually a sheaf of rings, then the diagram $$ \xymatrix{ \textit{Mod}(\mathcal{O}_U) \ar[r]_{j_{U!}} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}) \\ \textit{PMod}(\mathcal{O}_U) \ar[r]^{j_{U!}} & \textit{PMod}(\mathcal{O}) \ar[u]_{(\ )^\#} } $$ commutes.


sites-modules-remark-j-shriek-tensor

Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf of sets on $\mathcal{C}$ and consider the localization morphism $j : \Sh(\mathcal{C})/\mathcal{F} \to \Sh(\mathcal{C})$. See Sites, Definition \ref{sites-definition-localize-topos}. We claim that (a) $j_!\mathbf{Z} = \mathbf{Z}_\mathcal{F}^\#$ and (b) $j_!(j^{-1}\mathcal{H}) = j_!\mathbf{Z} \otimes_\mathbf{Z} \mathcal{H}$ for any abelian sheaf $\mathcal{H}$ on $\mathcal{C}$. Let $\mathcal{G}$ be an abelian on $\mathcal{C}$. Part (a) follows from the Yoneda lemma because $$ \Hom(j_!\mathbf{Z}, \mathcal{G}) = \Hom(\mathbf{Z}, j^{-1}\mathcal{G}) = \Hom(\mathbf{Z}_\mathcal{F}^\#, \mathcal{G}) $$ where the second equality holds because both sides of the equality evaluate to the set of maps from $\mathcal{F} \to \mathcal{G}$ viewed as an abelian group. For (b) we use the Yoneda lemma and \begin{align*} \Hom(j_!(j^{-1}\mathcal{H}), \mathcal{G}) & = \Hom(j^{-1}\mathcal{H}, j^{-1}\mathcal{G}) \\ & = \Hom(\mathbf{Z}, \SheafHom(j^{-1}\mathcal{H}, j^{-1}\mathcal{G})) \\ & = \Hom(\mathbf{Z}, j^{-1}\SheafHom(\mathcal{H}, \mathcal{G})) \\ & = \Hom(j_!\mathbf{Z}, \SheafHom(\mathcal{H}, \mathcal{G})) \\ & = \Hom(j_!\mathbf{Z} \otimes_\mathbf{Z} \mathcal{H}, \mathcal{G}) \end{align*} Here we use adjunction, the fact that taking $\SheafHom$ commutes with localization, and Lemma \ref{lemma-internal-hom-adjoint-tensor}.


sites-modules-remark-functoriality-principal-parts

Let $\mathcal{C}$ be a site. Suppose given a commutative diagram of sheaves of rings $$ \xymatrix{ \mathcal{B} \ar[r] & \mathcal{B}' \\ \mathcal{A} \ar[u] \ar[r] & \mathcal{A}' \ar[u] } $$ a $\mathcal{B}$-module $\mathcal{F}$, a $\mathcal{B}'$-module $\mathcal{F}'$, and a $\mathcal{B}$-linear map $\mathcal{F} \to \mathcal{F}'$. Then we get a compatible system of module maps $$ \xymatrix{ \ldots \ar[r] & \mathcal{P}^2_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \ar[r] & \mathcal{P}^1_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \ar[r] & \mathcal{P}^0_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \\ \ldots \ar[r] & \mathcal{P}^2_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[r] \ar[u] & \mathcal{P}^1_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[r] \ar[u] & \mathcal{P}^0_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[u] } $$ These maps are compatible with further composition of maps of this type. The easiest way to see this is to use the description of the modules $\mathcal{P}^k_{\mathcal{B}/\mathcal{A}}(\mathcal{M})$ in terms of (local) generators and relations in the proof of Lemma \ref{lemma-module-principal-parts} but it can also be seen directly from the universal property of these modules. Moreover, these maps are compatible with the short exact sequences of Lemma \ref{lemma-sequence-of-principal-parts}.


sites-modules-remark-not-pushforward

Warning: The result of Lemma \ref{lemma-stalk-j-shriek} has no analogue for $j_{U, *}$.