Let \(f: X \to Y\) be a finite morphism of noetherian schemes. For any quasi-coherent \(\mco_Y\)-module \(\mcg\), \(\sheafhom_Y(f_* \mco_X, \mcg)\) is a quasi-coherent \(f_* \mco_X\)-module, hence corresponds to a quasi-coherent \(\mco_X\)-module, which we call \(f^{!} \mcg\) (II, Ex. 5.17e).
Show that for any coherent \(\mcf\) on \(X\) and any quasi-coherent \(\mcg\) on \(Y\), there is a natural isomorphism \[ f_* \sheafhom_X(\mcf, f^{!} \mcg) \iso \sheafhom_Y(f_* \mcf, \mcg). \]
For each \(i \geq 0\), construct a natural map \[ \varphi_i: \Ext_X^i(\mcf, f^{!} \mcg) \to \Ext_Y^i(f_* \mcf, \mcg). \]
Now assume that \(X\) and \(Y\) are separated, \(\Coh(X)\) has enough locally frees, and assume that \(f_* \mco_X\) is locally free on \(Y\) (this is equivalent to saying \(f\) is flat — see §9). Show that \(\varphi_i\) is an isomorphism for all \(i\), all \(\mcf\) coherent on \(X\), and all \(\mcg\) quasi-coherent on \(Y\).
Hint
For (c), first construct a map \(\Ext_X^i(\mcf,f^{!}\mcg)\to\Ext_Y^i(f_*\mcf,f_*f^{!}\mcg)\) and then compose with a suitable map \(f_*f^{!}\mcg\to\mcg\). For (d), first do \(i=0\), then \(\mcf=\mco_X\) using (Ex. 4.1), and then \(\mcf\) locally free. Do the general case by induction on \(i\), writing \(\mcf\) as a quotient of a locally free sheaf.
Solution
Put \(\mathcal B=f_*\OO_X\). For a finite morphism, \(f_*\) preserves coherence and is exact on quasi-coherent sheaves, by restriction of scalars on affine opens [@Har10a, Proposition II.5.8 and Exercise II.5.5]. All Ext groups below are computed in the categories of all module sheaves, not only the quasi-coherent subcategories.
1. The construction in (a) defines \(f^!\mcg\), and it carries a natural evaluation morphism\[ \varepsilon_\mcg:f_*f^!\mcg=\sheafhom_Y(\mathcal B,\mcg)\longrightarrow\mcg, \qquad \lambda\longmapsto\lambda(1). \]
For local sections \(b,b'\) of \(\mathcal B\), give sheaf Hom the action \((b\lambda)(b')=\lambda(bb')\). This is a unital associative \(\mathcal B\)-module action compatible with restriction. The sheaf \(\mathcal B\) is coherent, so P-AGH363EXTCOHERENT, in degree zero, makes \(\sheafhom_Y(\mathcal B,\mcg)\) quasi-coherent. The affine-morphism equivalence in P-AGH2517AFFMOR, part (e), therefore gives a corresponding quasi-coherent sheaf \(f^!\mcg\) on \(X\). On \(V=\Spec A\subseteq Y\), write \(f^{-1}(V)=\Spec B\) and \(\mcg|_V=\widetilde N\). The corresponding \(B\)-module is \(\Hom_A(B,N)\) with the displayed action. These descriptions commute with localization because \(B\) is finite over the noetherian ring \(A\) and hence finitely presented. Evaluation at \(1\) is \(\OO_Y\)-linear and commutes with restrictions and morphisms of \(\mcg\), giving \(\varepsilon_\mcg\).
2. Evaluation at \(1\) gives the natural isomorphism in (b); on global sections it is \(\varphi_0\).
On the affine opens of step 1, write \(\mcf|_{f^{-1}(V)}=\widetilde M\) for a finite \(B\)-module \(M\). The relevant module isomorphism is \[ \Hom_B(M,\Hom_A(B,N))\longrightarrow\Hom_A(M,N), \qquad h\longmapsto\bigl(m\mapsto h(m)(1)\bigr). \] Its inverse sends \(a:M\to N\) to the map \[ m\longmapsto\bigl(b\mapsto a(bm)\bigr). \] The latter is \(B\)-linear since replacing \(m\) by \(b'm\) has the same effect as replacing the argument \(b\) by \(bb'\). The first composite is the identity by setting \(b=1\). For the other composite, \(B\)-linearity of \(h\) gives \(h(bm)(1)=(b h(m))(1)=h(m)(b)\). Thus both composites are identities. The formulas commute with localizations and with maps of \(M,N\), so they identify the sheaves and prove (b). On global Hom, the isomorphism takes \(a:\mcf\to f^!\mcg\) to \(\varepsilon_\mcg\circ f_*a\).
3. The natural maps in (c) exist and commute with the long exact Ext sequences in the coherent first variable.
For any quasi-coherent sheaf \(H\) on \(X\), take an injective resolution \(H\to I^\bullet\) in \(\Mod(X)\). One has \(R^qf_*H=0\) for \(q>0\): over an affine open \(V\subseteq Y\), the inverse image is affine, and affine vanishing applies to \(H|_{f^{-1}(V)}\) [@Har10a, Theorem III.3.5]. Consequently \(f_*I^\bullet\) is a resolution of \(f_*H\), although its terms need not be injective.
Choose an injective resolution \(f_*H\to J^\bullet\) on \(Y\). The comparison theorem for injective resolutions gives a chain map \(f_*I^\bullet\to J^\bullet\) extending the identity, unique up to homotopy. Compose it with pushforward on Hom to get \[ \Hom_X(\mcf,I^\bullet)\longrightarrow \Hom_Y(f_*\mcf,f_*I^\bullet)\longrightarrow \Hom_Y(f_*\mcf,J^\bullet). \] Taking cohomology defines the first map in the hint, \[ \Ext_X^i(\mcf,H)\longrightarrow\Ext_Y^i(f_*\mcf,f_*H). \] Comparison maps and homotopies show that it is independent of resolutions and natural in both arguments [@Har10a, Chapter III, §1]. Now take \(H=f^!\mcg\) and follow this map by the map on Ext induced by \(\varepsilon_\mcg\). The result is \(\varphi_i\). Equivalently, use a chain map \(f_*I^\bullet\to J_\mcg^\bullet\) extending \(\varepsilon_\mcg\), with \(J_\mcg^\bullet\) an injective resolution of \(\mcg\).
For a short exact sequence of coherent first arguments, pushforward is exact. Applying Hom into the injective terms \(I^q\) on \(X\) and \(J_\mcg^q\) on \(Y\) gives short exact sequences of complexes and a morphism between them. Their connecting maps therefore commute with \(\varphi_i\). No exactness of \(f_*\) on arbitrary module sheaves is assumed.
4. Under the hypotheses of (d), \(\varphi_i\) is an isomorphism for every \(i\) when the first argument \(E\) is locally free of finite rank.
The sheaf \(f_*E\) is locally free of finite rank on \(Y\). Indeed, over \(V=\Spec A\) as in step 1, \(B\) is finite projective over \(A\), and the module of \(E\) on \(\Spec B\) is finite projective over \(B\). It is a direct summand of some \(B^{\oplus r}\), hence finite projective over \(A\).
For a finite-rank locally free first argument, global Ext is the cohomology of its sheaf Hom [@Har10a, Propositions III.6.3 and III.6.7]. The affine cohomology comparison P-AGH341AFFINEMORPH and step 2 give \[ \begin{aligned} \Ext_X^i(E,f^!\mcg) &\cong H^i(X,\sheafhom_X(E,f^!\mcg))\\ &\cong H^i(Y,f_*\sheafhom_X(E,f^!\mcg))\\ &\cong H^i(Y,\sheafhom_Y(f_*E,\mcg)) \cong\Ext_Y^i(f_*E,\mcg). \end{aligned} \] In particular this proves the \(E=\OO_X\) case suggested by the hint.
3.To check this, use there a chain map \(f_*I^\bullet\to J_\mcg^\bullet\) extending evaluation.It induces a map\[ f_*\sheafhom_X(E,I^\bullet) \longrightarrow\sheafhom_Y(f_*E,J_\mcg^\bullet). \]Both complexes are flasque resolutions of their degree-zero sheaves: tensoring an injective by a finite-rank locally free sheaf preserves injectivity [@Har10a, Lemma III.6.6], direct image preserves flasqueness, and the quasi-coherent sheaf \(\sheafhom_X(E,f^!\mcg)\) has vanishing positive direct images under \(f\).The map on degree-zero sheaves is the isomorphism of step
2.Its global-section map is precisely the complex map defining \(\varphi_i\), so it induces the displayed cohomology isomorphisms.
5. For \(i\ge1\), if \(\varphi_j\) is an isomorphism for all coherent first arguments and all \(0\le j<i\), then \(\varphi_i\) is injective for all coherent first arguments.
Fix \(\mcg\), and abbreviate \(T^j(F)=\Ext_X^j(F,f^!\mcg)\) and \(S^j(F)=\Ext_Y^j(f_*F,\mcg)\). Choose \(0\to K\to E\to F\to0\) with \(E\) locally free of finite rank and \(K\) coherent. The long exact sequences have corresponding segments \[ T^{i-1}(E)\to T^{i-1}(K)\xrightarrow{\partial_T}T^i(F) \to T^i(E)\to T^i(K), \] and the same sequence with \(S\) in place of \(T\); step 3 makes the comparison commute.
4.Thus \(a=\partial_T b\) for some \(b\in T^{i-1}(K)\).The image of \(b\) in \(S^{i-1}(K)\) has zero boundary and comes from \(S^{i-1}(E)\).Lift that element to \(T^{i-1}(E)\), using step 4, and subtract its image from \(b\).The adjusted element still has boundary \(a\) and maps to zero in \(S^{i-1}(K)\).It is zero by the induction hypothesis, so \(a=0\).This proves injectivity for every coherent \(F\) at once.
6. Under the same induction hypothesis, \(\varphi_i\) is also surjective for every coherent first argument.
4.The image of \(a_E\) in \(T^i(K)\) maps to zero in \(S^i(K)\) by exactness.Step 5 already proved injectivity in degree \(i\) for every coherent sheaf, including \(K\), so this image is zero.Hence \(a_E\) lifts to some \(a\in T^i(F)\).
The difference \(b-\varphi_i(a)\) has zero image in \(S^i(E)\) and is the boundary of an element of \(S^{i-1}(K)\). Lift this element to \(T^{i-1}(K)\) by the induction hypothesis and add its boundary to \(a\). The resulting element maps to \(b\), proving surjectivity.
7. Q.E.D.
Steps <1>1–<1>3 prove (a)–(c). Step 2 gives the degree-zero isomorphism for every coherent first argument. Starting with that base case, steps <1>4–<1>6 prove by induction that every \(\varphi_i\) is an isomorphism under the hypotheses of (d). This completes all four parts and identifies the isomorphism with the map constructed in (c).