Derived tensor–Hom and inverse/direct image for field sheaves

The inverse image $f^{\ast }=f^{-1}$ is exact and $f^{\ast }⊣f_{\ast }$, and $f_{\ast }$ preserves injective objects. By [F4] there is a natural isomorphism $(f^{-1}G{)}_x\cong G_{f(x)}$ under which the stalk of $f^{-1}\phi$ is ${\phi }_{f(x)}$; since exactness is tested stalkwise, $f^{-1}$ preserves kernels, images and cokernels and is exact. The adjunction bijection of [F4] is $k$-linear, its two directions being given on sections by the canonical $k$-linear maps $G\to f_{\ast }{f}^{-1}G$ and $f^{-1}{f}_{\ast }F\to F$; hence $f^{\ast }$ is left adjoint to $f_{\ast }$ in ${\mathrm{Sh}}_k$. If $I$ is injective, then ${\mathrm{Hom}}_Y(-,f_{\ast }I)\cong {\mathrm{Hom}}_X(f^{\ast }(-),I)$ is a composite of the exact functor $f^{\ast }$ and the exact representable functor ${\mathrm{Hom}}_X(-,I)$, so $f_{\ast }I$ is injective.

The tensor product has the expected stalks, and tensoring is exact in each variable. Define $A{\otimes }_{k}B$ as the sheafification of the sectionwise tensor presheaf of [F5]. By [F5] sheafification preserves stalks, so $(A{\otimes }_{k}B{)}_x$ is the filtered colimit over the open neighbourhoods of $x$; since tensor products of vector spaces commute with filtered colimits, this is $A_x{\otimes }_k{B}_x$, and the stalk of $A\otimes \phi$ is $\mathrm{id}\otimes {\phi }_x$. For fixed $B$, the functor $A\mapsto A{\otimes }_{k}B$ is therefore stalkwise $A_x\mapsto A_x{\otimes }_k{B}_x$, which is exact because every $k$-vector space is flat, and exactness is stalkwise by [F4]; the same holds in the other variable. The restriction to opens is the evident one, so all these maps are compatible with restriction, and no choice is used in this step.

Injective $k$-sheaves are flasque. Let $I$ be injective and let $U\subseteq V$ be open. The canonical map $j_{U!}{k}_U\to j_{V!}{k}_V$ of [F6] is a monomorphism of $k$-sheaves: on stalks it is the identity $k\to k$ on $U$, the monomorphism $0\to k$ on $V\setminus U$, and $0\to 0$ elsewhere. By [F6], ${\mathrm{Hom}}_X(j_{U!}{k}_U,I)\cong {\mathrm{Hom}}_U(k_U,I{\mathrm{\mid }}_U)\cong I(U)$ and similarly with $V$. Injectivity of $I$ extends every morphism $j_{U!}{k}_U\to I$, that is, every section over $U$, along this monomorphism to a morphism $j_{V!}{k}_V\to I$, that is, a section over $V$ restricting to it. Hence the restriction $I(V)\to I(U)$ is surjective, and $I$ is flasque by [F8].

The adjunction $f^{\ast }⊣R{f}_{\ast }$ holds on bounded-below derived categories. Let $F$ be a bounded-below complex on $X$ and let $j:F\to I_F$ be a bounded-below termwise injective replacement, supplied under DC by [F3]; declare $R{f}_{\ast }F:=f_{\ast }{I}_F$, well defined up to canonical isomorphism because any two such replacements are homotopy equivalent by [F3] and $f_{\ast }$ preserves homotopy equivalences. Since $f_{\ast }$ preserves zero objects in each degree, $f_{\ast }{I}_F$ is bounded below, and it is termwise injective by step 1.1; hence it is K-injective by [F3], and for every bounded-below complex $G$ on $Y$ the localization identifies ${\mathrm{Hom}}_{D(Y)}(G,f_{\ast }{I}_F)\cong {\mathrm{Hom}}_{K(Y)}(G,f_{\ast }{I}_F)$. The levelwise adjunction of [F4] identifies this with ${\mathrm{Hom}}_{K(X)}(f^{\ast }G,I_F)$, and because $I_F$ is K-injective and quasi-isomorphic to $F$, this is ${\mathrm{Hom}}_{D(X)}(f^{\ast }G,F)$. All identifications are natural and $k$-linear, and the triangle identities are the triangle identities of the levelwise adjunction, so $f^{\ast }⊣R{f}_{\ast }$ on $D^{+}$.

Flasque $k$-sheaves are acyclic for sections on every open set. Let $F$ be flasque and embed it into an injective $I$ using [F2], with quotient $Q$. Viewed as a short exact sequence of underlying abelian sheaves, the sequence $0\to F\to I\to Q\to 0$ has flasque kernel $F$ and flasque middle term $I$ by step 1.3, so [F8] gives that $I(U)\to Q(U)$ is surjective for every open $U$ and that $Q$ is flasque. Write $H^q(U,-)$ for the right derived functors of the $k$-linear sections functor at $U$ formed with the functorial datum of [F2]; no comparison with abelian sheaf cohomology is yet assumed. By [F7] the family $(H^q(U,-){)}_{q\ge 0}$ is a cohomological delta functor, so the displayed short exact sequence yields a long exact sequence; by [F7] again $H^q(U,I)=0$ for $q>0$ because $I$ is injective. Hence $H^1(U,F)=0$ and, for every $n\ge 1$, $H^{n+1}(U,F)\cong H^n(U,Q)$. Since $Q$ is again flasque, the same argument applies to $Q$, and induction on $n$ gives $H^n(U,F)=0$ for all $n\ge 1$ and every open $U$, uniformly over all flasque $k$-sheaves. To establish the comparison, resolve any $k$-sheaf $G$ by the functorial $k$-injective resolution of [F2]. Forgetting scalars and restricting to $U$ preserves exactness by the stalk criterion [F4]; each resolution term is flasque by step 1.3, so its underlying abelian sheaf on $U$ is acyclic by [F10]. The abelian resolution datum in Sheaf cohomology as right derived global sections is defined on all of $\mathrm{A}\mathrm{b}(U)$, hence includes every kernel in this resolution. The acyclic-resolution theorem of [F10] therefore identifies the cohomology of this section complex naturally with the published $H^q(U,G{\mathrm{\mid }}_U)$. The same section complex computes the $k$-linear derived functors, giving their natural comparison as underlying abelian groups. This uses flasqueness, not injectivity after forgetting scalars; for example ${\mathbb{F}}_p$ is injective as a vector space on a point but not as an abelian group.

Bounded complexes are K-flat. Filter a bounded complex $A$ by its finitely many descending stupid truncations; each successive quotient is one sheaf $A^p$ placed in degree $p$. For any acyclic complex $C$, tensoring this quotient with $C$ gives a shift of $A^p{\otimes }_{k}C$, acyclic by the exactness proved in step 1.2. The filtration sequences are split in each degree, so tensoring preserves them; their long exact cohomology sequences prove by finite induction that $\mathrm{Tot}(A{\otimes }_{k}C)$ is acyclic. This argument permits unbounded $C$, since only finitely many $A$-degrees occur in each total degree. For a quasi-isomorphism $B\to B^{\mathrm{\prime }}$ its cone is acyclic by [F9], and its tensor with $A$ is the cone of the tensor map, with the signed total differential. Hence tensor with $A$ preserves quasi-isomorphisms. Applying this in both variables shows that the signed total tensor of bounded representatives computes $A{\otimes }_k^{L}B$, independently of the representatives.

The sheaf tensor–Hom adjunction holds on sections over each open. For $k_X$-modules $H,A,G$ there is a natural bijection ${\mathrm{Hom}}_X(H{\otimes }_{k}A,G)\cong {\mathrm{Hom}}_X(H,\mathcal{H}om(A,G))$: on sections over an open $U$ it is the vector-space tensor–Hom transposition $g\mapsto (h\mapsto g(h\otimes -))$, with inverse $s\mapsto (h\otimes a\mapsto s(h)(a))$, and these sectionwise bijections are compatible with restriction to smaller opens and with the $k_X$-module structures because both constructions are built from restriction maps and scalar multiplication. Hence for fixed $A$ the functor $\mathcal{H}om(A,-)$ is right adjoint to the exact functor $-{\otimes }_{k}A$ of step 1.2; consequently $\mathcal{H}om(A,J)$ is injective whenever $J$ is injective, its representable functor being the composite of $-{\otimes }_{k}A$ and ${\mathrm{Hom}}_X(-,J)$. The bijection is natural in all three variables.

The internal derived Hom has the injective model. For bounded $A$ and a bounded-below injective replacement $B\to I_B$, put $R\mathcal{H}om(A,B)=\mathcal{H}o{m}^{\bullet }(A,I_B)$, with degree-$n$ term ${\prod }_p\mathcal{H}om(A^p,I_B^{p+n})$ and differential $d\phi =d_I\phi -(-1{)}^n\phi d_A$. The product is finite. Its terms are injective by step 2.4 and it is bounded below, hence K-injective by [F3]. Restriction of an injective sheaf to any open $U$ remains injective, because restriction is right adjoint to exact extension by zero [F6]. Thus the same model over $U$ computes $R{\mathrm{Hom}}_U(A{\mathrm{\mid }}_U,B{\mathrm{\mid }}_U)$ after taking sections. In particular the object represents the tensor–Hom functor $C\mapsto {\mathrm{Hom}}_D(C{\otimes }_k^{L}A,B)$, rather than $C\mapsto {\mathrm{Hom}}_D(C,B)$; it is a sheaf-valued internal Hom, not Hom of stalks.

Bounded-below flasque complexes are acyclic for derived sections. Let $C^{\bullet }$ be a bounded-below complex of flasque $k$-sheaves, with $C^n=0$ for $n<b$, let $\phi :C^{\bullet }\to J^{\bullet }$ be a bounded-below termwise injective replacement supplied by [F3], and fix an open $U$. I first record a lemma: if $K^{\bullet }$ is an acyclic bounded-below complex whose every term is acyclic for the sections functor at $U$, then $\mathrm{\Gamma }(U,K^{\bullet })$ is acyclic. Indeed, put $Z^n=\ker (K^n\to K^{n+1})$; the complex is acyclic, so $0\to Z^n\to K^n\to Z^{n+1}\to 0$ is a short exact sequence for every $n$, and $Z^n=0$ for $n<b$. If $Z^n$ is acyclic for sections at $U$, then the long exact sequence of [F7] for this sequence, together with the acyclicity of $K^n$, shows that $Z^{n+1}$ is again acyclic for sections at $U$; inducting on $n$ upward from $n<b$, every $Z^n$ is acyclic. With all $Z^n$ acyclic, the same sequences give short exact sequences $0\to \mathrm{\Gamma }(U,Z^n)\to \mathrm{\Gamma }(U,K^n)\to \mathrm{\Gamma }(U,Z^{n+1})\to 0$ on sections, so the kernel of $\mathrm{\Gamma }(U,K^n)\to \mathrm{\Gamma }(U,K^{n+1})$ equals the image of $\mathrm{\Gamma }(U,K^{n-1})\to \mathrm{\Gamma }(U,K^n)$: the former is the kernel of $\mathrm{\Gamma }(U,K^n)\to \mathrm{\Gamma }(U,Z^{n+1})$, which is the image of $\mathrm{\Gamma }(U,Z^n)$ because $\mathrm{\Gamma }$ of a short exact sequence is left exact and the sequence is right exact at the quotient, and $\mathrm{\Gamma }(U,K^{n-1})\to \mathrm{\Gamma }(U,Z^n)$ is surjective, so the two images agree. Hence $H^n(\mathrm{\Gamma }(U,K^{\bullet }))=0$ for every $n$. Now apply this lemma to the cone $K^{\bullet }=\mathrm{Cone}(\phi )$ of the cochain map $\phi$ in the convention of Derived category of an abelian category. Its terms $J^n\oplus C^{n+1}$ are direct sums of an injective term, acyclic for sections at $U$ by [F7], and a flasque term, acyclic by step 2.2; a direct sum of two acyclic objects is acyclic by the long exact sequence [F7] applied to $0\to J^n\to J^n\oplus C^{n+1}\to C^{n+1}\to 0$. The cone is acyclic and bounded below, so the lemma gives that $\mathrm{\Gamma }(U,\mathrm{Cone}(\phi ))$ is acyclic. The canonical identification of complexes $\mathrm{\Gamma }(U,\mathrm{Cone}(\phi ))\cong \mathrm{Cone}(\mathrm{\Gamma }(U,\phi ))$ holds because sections over $U$ preserve finite direct sums and the differentials are those of $\phi$; by the cone criterion [F9] the map $\mathrm{\Gamma }(U,C^{\bullet })\to \mathrm{\Gamma }(U,J^{\bullet })$ is therefore a quasi-isomorphism for every open $U$. Since any two bounded-below injective replacements of $C^{\bullet }$ are homotopy equivalent by [F3], the ordinary section complex $\mathrm{\Gamma }(U,C^{\bullet })$ represents the derived sections $R\mathrm{\Gamma }(U,C^{\bullet })$ of the bounded-below flasque complex.

The internal model is independent of representatives. For a quasi-isomorphism $g:A\to A^{\mathrm{\prime }}$ of bounded complexes, put $C=\mathrm{Cone}(g)$. On every open $U$, the restriction $I{\mathrm{\mid }}_U$ of a bounded-below injective complex is K-injective by step 3.1 and [F3], so ${\mathrm{Hom}}_U^{\bullet }(C{\mathrm{\mid }}_U,I{\mathrm{\mid }}_U)$ is acyclic. These are the sections of $\mathcal{H}o{m}^{\bullet }(C,I)$; taking exact filtered stalk colimits shows that this sheaf complex is acyclic. The cone of $\mathcal{H}o{m}^{\bullet }(A^{\mathrm{\prime }},I)\to \mathcal{H}o{m}^{\bullet }(A,I)$ is isomorphic, with the usual Hom signs, to $\mathcal{H}o{m}^{\bullet }(C,I)[1]$, so the map is a quasi-isomorphism by [F9]. In the second variable, any homotopy equivalence between injective replacements is carried by $\mathcal{H}o{m}^{\bullet }(A,-)$ to a homotopy equivalence. The injective-model comparison of [F3] therefore makes the construction a natural bifunctor of the derived objects, compatible with restriction to opens.

The tensor–Hom adjunction and its coherence identities hold on the derived category. For a bounded complex $C$ and a bounded-below complex $I$ of injectives, the currying map $\mathcal{H}o{m}^{\bullet }(A{\otimes }_{k}C,I)\to \mathcal{H}o{m}^{\bullet }(C,\mathcal{H}o{m}^{\bullet }(A,I))$, which sends a morphism $\phi$ to $c\mapsto (a\mapsto (-1{)}^{\mathrm{\mid }c\mathrm{\mid }\mathrm{\mid }a\mathrm{\mid }}\phi (a\otimes c))$ on pure tensor sections and extends by linearity and gluing, is an isomorphism of complexes: over each open it is the vector-space tensor–Hom transposition of step 2.4, bijective on sections, and the Koszul sign makes both sides carry the single differential rule $d\psi =d_I\psi -(-1{)}^n\psi d_{\text{source}}$ of Derived hom in the bounded setting, so it commutes with differentials and with restriction. Taking $I$ to be an injective replacement of the bounded-below complex $B$ and using that $C$ is K-flat and $I$, $\mathcal{H}o{m}^{\bullet }(A,I)$ are K-injective, the $H^0$ transposition identifies ${\mathrm{Hom}}_D(C{\otimes }_k^{L}A,B)\cong {\mathrm{Hom}}_D(C,R\mathcal{H}om(A,B))$ naturally in the bounded $C,A$ and bounded-below $B$; this is the tensor–Hom adjunction, with unit and counit the canonical currying maps. The associativity isomorphism $(A{\otimes }_{k}B){\otimes }_{k}C\cong A{\otimes }_k(B{\otimes }_{k}C)$, the symmetry isomorphism $A{\otimes }_{k}B\cong B{\otimes }_{k}A$ with its Koszul sign, the evaluation morphism $R\mathcal{H}om(A,B){\otimes }_k^{L}A\to B$, and the currying bijection all hold on pure tensor sections, where they are the usual vector-space identities, and extend by linearity, gluing and the comparison maps of step 4.1; being defined on sections they are natural and compatible with restriction to opens, hence internal. The triangle identities are those of step 2.4 at chain level.

The claims are now assembled. Inverse image is exact and $f^{\ast }⊣R{f}_{\ast }$ on $D^{+}$ is step 2.1; the derived tensor of bounded complexes is the signed total tensor by step 2.3; the internal derived Hom is the injective model of step 3.1, well defined by step 4.1; flasque sheaves are acyclic for sections on every open set by step 2.2, and bounded-below flasque complexes compute their derived sections by step 3.2; the tensor–Hom adjunction and the dg associativity, symmetry, evaluation and currying identities are step 5.1. AC is assumed and consumed exactly as follows: AC yields DC by [F1], DC is used by the injective replacement suppliers of [F3] and the delta-functor and injective-acyclicity statements of [F7], the functorial injective resolutions of [F2] use the Grothendieck embedding theorem under AC, and the flasque section lifting of [F8] uses Zorn's lemma under AC; the stalk, tensor, Hom and adjunction verifications are choice-free. No bounded-constructibility claim for the internal Hom is made: that is supplied later on this page by the geometric finite conical and incidence models, which are not prerequisites of this lemma.