We can again work locally on \(S\) and \(S'\), say \(S=\operatorname{Spec}(R)\), \(S'=\operatorname{Spec}(R')\) and can then compute the cohomology groups as Čech cohomology groups for a finite affine open cover \({\mathscr U}\) of \(X\) and \({\mathscr U}':=(g')^{-1}({\mathscr U})\) of \(X'\). (Note that \(v\) is an affine morphism since \(S\), \(S'\) are affine, and the property of being an affine morphism is stable under base change.) But \(C^\bullet ({\mathscr U}', (g')^*{\mathscr F}) = C^\bullet ({\mathscr U}, {\mathscr F})\otimes _RR'\). We obtain a homomorphism \(C^\bullet ({\mathscr U}, {\mathscr F})\to C^\bullet ({\mathscr U}', (g')^*{\mathscr F})\) of complexes, and this induces the desired base change homomorphisms.

Now \(g\) is flat, so \(-\otimes _RR'\) commutes with passing to the cohomology objects, and the proposition follows. Instead of assuming that \(f\) be separated it is enough that \(f\) is qcqs. See  [ GW2 ]  Section(23.28).

We can compute the cohomology groups in question using Čech cohomology for a fixed finite affine open cover \({\mathscr U}\) of \(X\), and \(C^\bullet ({\mathscr U}, {\mathscr F}\otimes _RM) \cong C^\bullet ({\mathscr U}, {\mathscr F})\otimes _RM\) since tensor product commutes with finite products. Thus the proposition follows from the following general fact in homological algebra.

We start by constructing finite free \(R\)-modules \(E^i\) by descending induction on \(i\) (together with differentials \(d^i_E\colon E^i\to E^{i+1}\) and maps \(\psi ^i\colon E^i\to C^i\)) such that

1. \(d_E^{i+1}\circ d_E^i = 0\), \(d^i_C\circ \psi _i = \psi ^{i+1}\circ d_E^i\),

2. the induced map \(H^{i+1}(E^\bullet )\to H^{i+1}(C^\bullet )\) is an isomorphism,

3. the induced map \(\operatorname{Ker}(d^i_E)\to H^i(C)\) is surjective.

For \(i\) large, we may set \(E^i= 0\). For the induction step, to construct \(E^i\), choose generators of the finitely generated \(R\)-module \(H^i(C^\bullet )\) and lift them to elements \(x_1,\dots , x_r\in \operatorname{Ker}(d^i_C)\). Also, choose generators \(y_{r+1}, \dots , y_s\) of \((\psi ^{i+1})^{-1}(\mathop{\rm Im}(d^{i}_C)) \cap \operatorname{Ker}(d^{i+1}_E)\) and lift their images in \(\mathop{\rm Im}(d^i_C)\) to elements \(x_{r+1}, \dots , x_s\in C^i\).

Set \(E^i = R^{\oplus s}\) and define \(\psi ^i\) by mapping the standard basis vectors to the \(x_\nu \). Define \(d^i_E\) by mapping \(x_\nu \mapsto 0\) for \(\nu =1,\dots , r\), and \(x_\nu \mapsto y_\nu \) for \(\nu = r+1, \dots , s\). This constructs \(E^i\), \(d^i_E\) and \(\psi _i\) satisfying (a), (b) and (c).

One checks that this gives bounded above complex \(E^\bullet \) of finite free \(R\)-modules, together with a morphism \(E^\bullet \to C^\bullet \) which is a quasi-isomorphism (this is precisely property (b) above).

Next let us show that we can replace \(E^\bullet \) by a bounded complex with the same properties (where we allow finite flat rather than free entries). Say \(C^i = 0\) for all \(i {\lt} i_0\). We replace \(E^i\) by \(0\) for all \(i{\lt} i_0\) and replace \(E^{i_0}\) by \(E^{i_0}/\mathop{\rm Im}(d^{i_0-1})\), so that the new complex has the same cohomology objects as the old one, and still comes with a morphism to \(C^\bullet \), which we again denote by \(\psi \). We need to show that the new \(E^{i_0}\) is flat over \(R\). Let \(M_\psi \) be the mapping cone of \(\psi \colon E^\bullet \to C^\bullet \). Since \(\psi \) is a quasi-isomorphism, the complex \(M_\psi \) is exact. Its terms have the form \(E_i\oplus C_{i-1}\) and its left-most entry is \(E_{i_0}\). We obtain an exact sequence

where all entries, except possibly the left-most one, are flat \(R\)-modules. It follows that \(E_{i_0}\) is flat, as well. (Namely, one can inductively apply the following observation: Given a short exact sequence \(0\to M'\to M\to M''\to 0\) with \(M\) and \(M''\) flat, it follows by looking at the long exact Tor sequences that \(M'\) is flat, as well.)

Finally we check that the induced homomorphisms \(H^i(E^\bullet \otimes _RM)\to H^i(C^\bullet \otimes _RM)\) are isomorphisms for all \(R\)-modules \(M\). Since every \(R\)-module is a filtered direct limit of finitely generated modules, and both sides commute with direct limits, it is enough to consider finitely generated \(M\). Write \(M\) as the quotient \(F/N\) for a finite free \(R\)-module \(F\). We obtain a commutative diagram

of complexes with exact rows (here we use that all terms of \(E^\bullet \), and \(C^\bullet \), respectively, are flat). This induces a similar commutative diagram of the long exact cohomology sequences. In view of (a), the middle vertical arrow induces isomorphisms on the cohomology objects. The claim now follows by descending induction on \(i\), using suitable versions of the 5-lemma: The claim is clear for sufficiently large \(i\) since then both sides vanish. It then follows that the maps in question are surjective. Finally, invoking the 5-lemma again, it follows that they are isomorphisms.

Alternatively, for the final step one can again use the mapping cone, and use that the mapping cone for \(\psi \otimes M\) is \(M_\psi \otimes M\) and that this is again exact because of the flatness.

Let \(E^\bullet \) be as in the lemma. Then for every \(s\in S\),

This proves Part (1).

For Part (2), write \(H^i(E^\bullet \otimes \kappa (s)) = \operatorname{Coker}(E^{i-1}\otimes \kappa (s)\to \operatorname{Ker}(d^i_{E\otimes \kappa (s)}))\). Its dimension is

which is upper semicontinuous.

It is clear that (i) \(\Rightarrow \) (ii) \(\Rightarrow \) (iii), and that (iv) \(\Rightarrow \) (i).

(iii) \(\Rightarrow \) (iv). We may assume that \(R\) is local with residue class \(\kappa \) and that \(s\in \operatorname{Spec}(R)\) is the closed point. (For the reduction to this situation, note that \({\mathscr E}/\iota ({\mathscr U})\) is of finite presentation, so if its stalk at \(s\) is free over \({\mathscr O}_{S,s}\), then \({\mathscr E}/\iota ({\mathscr U})\) is free in a neighborhood of \(s\) (Algebraic Geometry 2, Proposition 2.17; or  [ GW1 ] Proposition 7.27). In this case, it follows that \(\iota ({\mathscr U})\) is locally free in a neighborhood of \(s\), and hence that \(\operatorname{Ker}(\iota )\subseteq {\mathscr U}\) is a direct summand. Therefore \(\operatorname{Ker}(\iota )\) is of finite type in a neighborhood of \(s\), so its support is closed. Since \(s\) is not contained in the support, it follows that the sheaf \(\operatorname{Ker}(\iota )\) is zero on some neighborhood of \(s\).)

Then it is enough to show the following: Let \(i\colon N\to M\) be a homomorphism of \(R\)-modules with \(N\) finitely generated, \(M\) projective and such that the map \(i\otimes \kappa \colon N\otimes \kappa \to M\otimes \kappa \) is injective. Then \(i\) is injective and \(i(N)\) is a direct summand of \(M\).

Let \(r_0\) be a retraction of \(i\otimes \kappa \). The homomorphism \(M\to M\otimes \kappa \xrightarrow {r_0}N\otimes \kappa \) factors as \(M\xrightarrow {r’} N\to N\otimes \kappa \) since \(M\) is projective. Then \(r'\circ i\) is an endomorphism of \(N\) which induces the identity after \(-\otimes \kappa \). By the lemma of Nakayama it is surjective and hence, as a surjective endomorphism of a finitely generated module, bijective ( [ M2 ] Theorem 2.4). We obtain \((r'\circ i)^{-1}\circ r' \circ i = \operatorname{id}_N\), so \(i\) admits a retraction.

(iii) \(\Leftrightarrow \) (v). We already know that (iii) implies that \(\iota \) is injective, and \({\mathscr U}\) is locally free around \(s\). Thus \(\iota ^\vee \otimes \kappa (s) = (\iota \otimes \kappa (s))^\vee \), so the injectivity of \(\iota \otimes \kappa (s)\) is equivalent to the surjectivity of \(\iota ^\vee \otimes \kappa (s)\), and by the lemma of Nakayama, to the surjectivity of \(\iota ^\vee \).

(i) \(\Rightarrow \) (ii). We need to show that the localization \(\operatorname{Coker}(d)_{\mathfrak p}\) at the prime ideal \(\mathfrak p\) corresponding to \(s\) is free. This means that we may assume that \(R\) is local with closed point \(s\), and then show that \(\operatorname{Coker}(d)\) is free. In this case, \(M\) and \(N\) are free, of ranks \(m\) and \(n\), say. We write \(\kappa =\kappa (s)\) and \(r = \operatorname{rk}(d\otimes \kappa )\). We will show that \(\mathop{\rm Im}(d)\subseteq N\) is a direct summand of rank \(r\); this implies our claim.

To this end, by Lemma 4.6, it is enough to prove that \(\bar{d}\otimes \kappa \) is injective, where \(\bar{d}\colon M/\operatorname{Ker}(d)\to N\) is the injection induced by \(d\). But we can lift a basis of \(\operatorname{Ker}(d\otimes \kappa )\) along the surjection \(\operatorname{Ker}(d)\to \operatorname{Ker}(d)\otimes \kappa \to \operatorname{Ker}(d\otimes \kappa )\) to elements \(x_\bullet \) in \(\operatorname{Ker}(d)\), and extend it by elements \(y_\bullet \) in \(M\) whose residue classes are a basis of the quotient \((M\otimes \kappa )/\operatorname{Ker}(d\otimes \kappa )\). Since \(M\) is free of rank \(\dim _\kappa (M\otimes \kappa )\), Nakayama’s lemma implies that in this way we obtain a basis of \(M\). The residue classes of the \(y_\bullet \) then generate \(M/\operatorname{Ker}(d)\). Since the images of the \(y_\bullet \) are linearly independent in \(N\otimes \kappa \), this proves the desired injectivity.

(ii) \(\Rightarrow \) (i). Given (ii), after replacing \(R\) by \(R_f\), the short exact sequence \(0\to \mathop{\rm Im}(d)\to N\to \operatorname{Coker}(d)\to 0\) splits, so \(\mathop{\rm Im}(d)\) is projective. Hence the short exact sequence \(0\to \operatorname{Ker}(d)\to M\to \mathop{\rm Im}(d)\to 0\) splits, too, and thus the formation of \(\operatorname{Ker}(d)\) commutes with arbitrary tensor product \(-\otimes _RK\). In particular (i) holds (and the map there is even an isomorphism).

The final assertion is clear.

Part (1) holds since all \(E^i\) are flat over \(R\) by assumption. Since \(T^i\) is an \(R\)-linear functor, we have a map

so we obtain (2). For (3), note that \(T^i\) commutes with arbitrary direct sums, so that \(\beta ^i(M)\) is always an isomorphism for every free \(R\)-module \(M\). If \(T^i\) is right exact, we get the same for arbitrary \(R\)-modules \(M\) by considering a presentation \(F'\to F\to M\to 0\) with \(F\), \(F'\) free. Conversely, use that the functor \(T^i(R)\otimes -\) is right exact.

After shrinking \(S\), we may assume that all \(E^i\) are finite free \(R\)-modules. By the assumptions, \(\operatorname{Ker}(d^i)\otimes \kappa (s) \to \operatorname{Ker}(d^i\otimes \kappa (s))/\mathop{\rm Im}(d^{i-1}\otimes \kappa (s))\) is surjective. Since \(\mathop{\rm Im}(d^{i-1})\subseteq \operatorname{Ker}(d^i)\) and the maps \(\mathop{\rm Im}(d^{i-1})\to \mathop{\rm Im}(d^{i-1})\otimes \kappa (s)\to \mathop{\rm Im}(d^{i-1}\otimes \kappa (s))\) are surjective, we even have that the map \(\operatorname{Ker}(d^i)\otimes \kappa (s) \to \operatorname{Ker}(d^i\otimes \kappa (s))\) is surjective. Thus by Lemma 4.7, after localizing \(R\), if necessary, \(\mathop{\rm Im}(d^i)\) and \(\operatorname{Coker}(d^i)\) are free.

Since \(\mathop{\rm Im}(d^i)\) is free, \(\operatorname{Ker}(d^i)\) is a direct summand of \(E^i\), so its formation commutes with \(-\otimes M\). Thus the same is true for \(H^i(E^\bullet \otimes M) = \operatorname{Coker}(E^{i-1}\to \operatorname{Ker}(d^i))\).

For the final statement, use Lemma 4.8.

By Proposition 4.9, after shrinking \(S\), \(\beta ^i(M)\) is an isomorphism for every \(M\) and \(T^i(-) = H^i(E^\bullet )\otimes -\) is right exact. It is left exact if and only if \(T^{i-1}\) is right exact since \((T^i)_i\) is a \(\delta \)-functor (Lemma 4.8).

Now if \(\beta ^{i-1}(\kappa (s))\) is surjective, then (after shrinking \(S\) again, if necessary), also \(T^{i-1}\) is right exact, so \(T^i\) is exact. But this means that \(H^i(E)\) is flat, and being finite over the noetherian ring \(R\), it is locally free.

Conversely, if \(H^i(E)\) is flat, then \(T^i\) is exact, so \(T^{i-1}\) is right exact and (using Lemma 4.8 once again) \(\beta ^{i-1}(M)\) is an isomorphism for all \(M\), and in particular (i) holds.

We may assume that \(S\) is affine. Let \(E^\bullet \) be as above. We may assume that all \(E^i\) are finite free \(R\)-modules. In view of 5, it follows that the maps

being lower semicontinuous with constant sum, are constant on \(S\). Since \(S\) is reduced by assumption, this implies that \(\operatorname{Coker}(d^i)\) and \(\operatorname{Coker}(d^{i-1})\) are locally free \({\mathscr O}_S\)-modules ( [ GW1 ] Corollary 11.19 or  [ Mu ] §5 Lemma 1). This implies that the exact sequence

splits for every quasi-coherent \({\mathscr O}_S\)-module \({\mathscr G}\). In particular \(\mathop{\rm Im}(d^i)\) is locally free, and \(\mathop{\rm Im}(d^i\otimes {\mathscr G})\cong \mathop{\rm Im}(d^i)\otimes {\mathscr G}\) (since the same holds for the cokernel). From this, considering the short exact sequence

we obtain that \(H^i(E)\) is locally free and that \(\beta ^i\) is an isomorphism of functors. By Theorem 4.3, \(\beta ^{i-1}\) is an isomorphism, as well.

Problem sheet 12.