4 Cohomology and base change

General references: [ GW2 ] Ch. 23, in particular Sections (23.24)–(23.28), [ H ] Ch. III.12, [ Mu ] §5.

(4.1) The base change homomorphism

Jan. 8,

We consider a cartesian diagram

            X' \ar[r, "g'"] \ar[d, "f'"] & X \ar[d, "f"]\\
            S' \ar[r, "g"] & S

where \(f\) is quasi-compact and quasi-separated.

For every quasi-coherent \({\mathscr O}_X\)-module \({\mathscr F}\) and \(i\ge 0\) we have a natural homomorphism of \({\mathscr O}_{S'}\)-modules, called the base change homomorphism,

\[ g^* R^if_*{\mathscr F}\longrightarrow R^i f'_* (g')^*{\mathscr F}, \]

and these are functorial in \({\mathscr F}\). We are interested in finding conditions when these homomorphisms are isomorphisms.

For \(i=0\), the natural homomorphism \(g^*f_*\to f'_* (g')^*\) of functors corresponds, by adjunction, to a morphism \(f_*\to g_*f'_*(g')^* = f_* g'_* (g')^*\), which we obtain by applying \(f_*\) to the unit \(\operatorname{id}\to g'_*(g')^*\) of the adjunction between \((g')^*\) and \(g'_*\). To handle the case of \(i {\gt} 0\), we use the following remark. (Cf.  [ GW2 ] Section (23.28) for a construction in the framework of derived categories.)

Remark 4.1
(Construction of the base change homomorphism) We may work locally on \(S\) and \(S'\) (and in the general case, use gluing). If \(S=\operatorname{Spec}(R)\) and \(S'=\operatorname{Spec}(R')\) are affine, then in view of Proposition 3.46 we need to construct a natural map
\[ H^i(X, {\mathscr F})\otimes _RR'\to H^i(X', (g')^*{\mathscr F}) = H^i(X, {\mathscr F}\otimes _RR'), \]
where the final identification uses Corollary 3.47 (2). Since cohomology is a functor, the natural \(R\)-module homomorphism \({\mathscr F}\to {\mathscr F}\otimes _RR'\), \(f\mapsto f\otimes 1\), induces a homomorphism \(H^i(X, {\mathscr F})\to H^i(X, {\mathscr F}\otimes _RR')\). Since the right hand side is an \(R'\)-module, we obtain the desired map \(H^i(X, {\mathscr F})\otimes _RR'\to H^i(X, {\mathscr F}\otimes _RR')\).

Proposition 4.2
If in the above situation \(g\) is flat and \(f\) is separated, then for every \(i\) the base change homomorphism
\[ g^* R^if_*{\mathscr F}\longrightarrow R^i f'_* (g')^*{\mathscr F}, \]
is an isomorphism.


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).

Often, the base change homomorphism is most interesting, however, if \(g\) is not flat. A typical example is when \(g\) is the natural morphism \(\operatorname{Spec}\kappa (s)\to S\) for a point \(s\in S\).

(4.2) Proper base change for coherent modules

From now on we will focus on the situation

(S) \(S=\operatorname{Spec}(R)\) is noetherian, \(f\) proper, \({\mathscr F}\) a coherent \({\mathscr O}_X\)-module which is flat over \(S\).

For \(s\in S\), we write \(X_s := X\times _S\operatorname{Spec}\kappa (s)\) for the fiber, and \({\mathscr F}_s := {\mathscr F}_{|X_s}\) for the restriction of \({\mathscr F}\) to \(X_s\) (i.e., the pullback of \({\mathscr F}\) along the projection morphism \(X_s\to X\)).

For the results below, we need to use that in situation (S) for the coherent \({\mathscr O}_X\)-module \({\mathscr F}\) the higher direct images \(R^if_*{\mathscr F}\) are again coherent for all \(i\). This is true (for \(S\) noetherian and \(f\) proper and any coherent \({\mathscr O}_X\)-module \({\mathscr F}\)), but in Algebraic Geometry 2 we proved it only for \(f\) projective. See  [ GW2 ] Ch. 23 for a proof in the general case.

Theorem 4.3

( [ GW2 ] Theorem 23.139, Theorem 23.140)

Consider situation (S).

  1. The Euler characteristic \(s\mapsto \chi _{{\mathscr F}}(s):= \sum _i (-1)^i \dim _{\kappa (s)}H^i(X_s, {\mathscr F}_s)\) is locally constant on \(S\).

  2. The map \(S\to \mathbb {Z}_{\ge 0}\), \(s\mapsto \dim _{\kappa (s)} H^i(X_s, {\mathscr F}_s)\) is upper semicontinuous, i.e., for every \(n\in \mathbb {Z}\) the set

    \[ \{ s\in S;\ \dim _{\kappa (s)} H^i(X_s, {\mathscr F}_s) \ge n \} \]

    is closed in \(S\).

  3. Let \(s\in S\). For \(i\in \mathbb {Z}_{\ge 0}\), the following are equivalent.

    1. The base change homomorphism \(\beta ^i(\kappa (s))\) for \(S'=\operatorname{Spec}\kappa (s)\to S\) is surjective.

    2. After replacing \(S\) by an open neighborhood \(U\) of \(s\), the base change homomorphism is an isomorphism for every \(S'\to S\) (and the fixed sheaf \({\mathscr F}\) and degree \(i\)).

  4. Let \(s\in S\) such that \(\beta ^i(\kappa (s))\) is surjective. The following are equivalent.

    1. The homomorphism \(\beta ^{i-1}(\kappa (s))\) is surjective.

    2. There exists an open neighborhood \(V\) of \(s\) such that the \({\mathscr O}_V\)-module \(R^if_*{\mathscr F}_{|V}\) is finite locally free.

    In this case, the function \(s\mapsto \dim _{\kappa (s)}H^i(X_s, {\mathscr F}_s)\) is locally constant on \(V\).

As the proof will show, in Part (4) of the theorem it is allowed to take \(i=0\), in which case condition (i) is automatically satisfied (by convention; or from a derived category point of view where we would define the functor \(R^if_*- = H^i(Rf_*-)\) which makes sense for all \(i\in \mathbb {Z}\) and vanishes for \(i{\lt}0\), if applied to a complex concentrated in non-negative degrees).

All statements of the theorem are local on \(S\), so we may assume that \(S=\operatorname{Spec}(R)\) for a noetherian ring \(R\).

In the next proposition, we write \({\mathscr F}\otimes _RM\) for the quasi-coherent \({\mathscr O}_X\)-module \({\mathscr F}\otimes _{{\mathscr O}_X}f^*\widetilde{M}\). Equivalently, this is the \({\mathscr O}_X\)-module with sections \({\mathscr F}(U)\otimes _RM\) on every affine open \(U\subseteq X\). If \(M=R'\) is an \(R\)-algebra, then we are in the situation of the base change homomorphism considered previously. A particularly important case is when \(s\in \operatorname{Spec}(r)\) and \(M=R'=\kappa (s)\) – then we obtain \(H^i(X, {\mathscr F}\otimes \kappa (s)) = H^i(X_s, {\mathscr F}_s)\).

Jan. 10,

Proposition 4.4

( [ GW2 ] Corollary 23.137, [ Mu ] §5, [ H ] III Proposition 12.2)

In the above situation, there exists a bounded complex \(E^\bullet \) concentrated in degrees \(\ge 0\) of finitely generated projective \(R\)-modules and isomorphisms, functorial in the \(R\)-module \(M\),

\[ H^i(X, {\mathscr F}\otimes _RM)\overset {\sim }{\to }H^i(E^\bullet \otimes _RM), \]

for all \(i\ge 0\).


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.

Lemma 4.5
Let \(R\) be a noetherian ring and let \(C^\bullet \) be a complex of flat \(R\)-modules such that \(H^i(C^\bullet )\) is finitely generated for every \(i\). Assume that \(C^\bullet \) is bounded, concentrated in degrees \([a, b]\), say. Then there exists a bounded complex \(E^\bullet \) concentrated in degrees \([a, b]\) of finitely generated flat \(R\)-modules and a morphism \(E^\bullet \to C^\bullet \) such that for every \(R\)-module \(M\) and every \(i\) the induced homomorphism
\[ H^i(E^\bullet \otimes M)\to H^i(C^\bullet \otimes M) \]
is an isomorphism.


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

\[ 0\to E_{i_0}\to M_{\psi , i_0+1}\to \cdots \to 0 \]

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

                0\ar[r] & E^\bullet\otimes_R N\ar[r]\ar[d] & E^\bullet\otimes_R F\ar[r]\ar[d] & E^\bullet\otimes_R M\ar[r]\ar[d] & 0\\
                0\ar[r] & C^\bullet\otimes_R N\ar[r] & C^\bullet\otimes_R F\ar[r] & C^\bullet\otimes_R M\ar[r] & 0

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.

We can now prove the first two parts of the theorem.

Proof of Theorem 4.3, Parts (1), (2)

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

\begin{align*} \chi _{\mathscr F}(s) & = \sum _i (-1)^i \dim _{\kappa (s)} H^i(E^\bullet \otimes \kappa (s))\\ & = \sum _i (-1)^i \dim _{\kappa (s)} E^i\otimes \kappa (s) = \sum _i (-1)^i \operatorname{rk}E^i. \end{align*}

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

\begin{equation} \label{eq:dim-Hi} \dim (E^i\otimes \kappa (s)) - \operatorname{rk}(d^i\otimes \kappa (s)) - \operatorname{rk}(d^{i-1}\otimes \kappa (s)), \end{equation}

which is upper semicontinuous.

Jan. 15,

The next two lemmas are crucial ingredients from linear algebra that go into the proof of parts (3) and (4) of the theorem.

Lemma 4.6
( [ GW1 ] Proposition 8.10) Let \(S\) be a scheme and let \(\iota \colon {\mathscr U}\to {\mathscr E}\) be a homomorphism of \({\mathscr O}_S\)-modules, where \({\mathscr U}\) is of finite type and \({\mathscr E}\) is locally free of finite rank. Let \(s\in S\). The following are equivalent.
  1. In an open neighborhood of \(S\), there exists a retraction \(\pi \) of \(\iota \) (i.e., \(\pi \circ \iota = \operatorname{id}\)).

  2. After replacing \(S\) with an open neighborhood of \(s\), for every scheme morphism \(f\colon T\to S\), the homomorphism \(f^*{\mathscr U}\to f^*{\mathscr E}\) is injective.

  3. The homomorphism \(\iota \otimes \kappa (s)\colon {\mathscr U}(s)\to {\mathscr E}(s)\) between the fibers is an injective homomorphism of \(\kappa (s)\)-vector spaces.

  4. In an open neighborhood of \(S\), the homomorphism \(\iota \) is injective and the quotient \({\mathscr E}/{\mathscr U}\) is locally free.

  5. After replacing \(S\) with an open neighborhood of \(s\), the \({\mathscr O}_S\)-module \({\mathscr U}\) is locally free and the dual \(\iota ^\vee \) is surjective.


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 \(\iota ^\vee \).

Lemma 4.7
( [ GW2 ] Lemma 23.122) Let \(R\) be a ring, and let \(d\colon M\to N\) be a homomorphism of finitely generated projective \(R\)-modules. Let \(s\in \operatorname{Spec}(R)\). The following are equivalent.
  1. The natural map \(\operatorname{Ker}(d)\otimes \kappa (s)\to \operatorname{Ker}(d\otimes \kappa (s))\) is surjective.

  2. There exists \(f\in R\) with \(s\in D(f)\) such that \(\operatorname{Coker}(d)\otimes _RR_f\) is a projective \(R_f\)-module.

In this case, \(\mathop{\rm Im}(d\otimes _RR_f)\) is a direct summand of \(N\otimes _RR_f\).


(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.

Jan. 17,

Parts (3) and (4) of Theorem 4.3 follow from the following two propositions. For these propositions, the morphism \(f\) and the sheaf \({\mathscr F}\) do not directly play any role; we only need the complex \(E^\bullet \) from Proposition 4.4. Given \(E^\bullet \), we define functors \(T^i\) from the category of \(R\)-modules to itself by

\[ T^i(M) := H^i(E^\bullet \otimes _R M). \]

Lemma 4.8
( [ GW2 ] , Lemma 23.120) Let \(E^\bullet \) be a complex of finitely generated projective \(R\)-modules.
  1. The collection \((T^i)_i\) is a \(\delta \)-functor.

  2. For every \(R\)-module \(M\), we have the base change homomorphism

    \[ \beta ^i(M) = T^i(R)\otimes _RM\to T^i(M). \]
  3. The morphism \(\beta ^i\) of functors is an isomorphism (i.e., \(\beta ^i(M)\) is an isomorphism for all \(M\)) if and only if \(T^i\) is right exact.


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

\[ M= \operatorname{Hom}_R(R, M)\to \operatorname{Hom}_R(T^i(R), T^i(M)), \]

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.

With \(E^\bullet \) as in Proposition 4.4, the functor \(T^i\) can be written as \(T^i(M) = H^i(X, {\mathscr F}\otimes _RM)\), and \(\beta ^i(M)\) is the base change homomorphism of Theorem 4.3.

Proposition 4.9

(cf.  [ GW2 ] Proposition 23.123)

Let \(R\) be a noetherian ring, and let \(E^\bullet \) be a bounded complex of finite projective \(R\)-modules. Fix \(i\in \mathbb {Z}\) and \(s\in \operatorname{Spec}R\) such that the base change map

\[ \beta ^i(\kappa (s))\colon H^i(E^\bullet )\otimes \kappa (s)\to H^i(E^\bullet \otimes \kappa (s)) \]

is surjective.

Then there exists \(f\in R\) such that \(s\in D(f)\) and such that after replacing \(R\) by \(R_f\), \(\beta ^i(M)\) is an isomorphism for every \(R\)-module \(M\), and the functor \(T^i\) 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.

Proposition 4.10

( [ GW2 ] Proposition 23.127)

Let \(S=\operatorname{Spec}R\) be a noetherian affine scheme and let \(E^\bullet \) be a bounded complex of coherent flat \({\mathscr O}_S\)-modules. Let \(s\in S\) such that \(\beta ^i(\kappa (s))\) is surjective.

Then \(\beta ^i(M)\) is an isomorphism for every \(R\)-module \(M\). Moreover, the following are equivalent.

  1. \(\beta ^{i-1}(\kappa (s))\) is surjective,

  2. \(H^i(E^\bullet )\) is locally free.


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.

(4.3) Cohomology and base change, corollaries and applications

Corollary 4.11
(Grauert’s theorem) Consider situation (S), assume that \(S\) is reduced and that the function \(s\mapsto \dim _{\kappa (s)}H^i(X_s, {\mathscr F}_s)\) is locally constant on \(S\). Then \(R^if_*{\mathscr F}\) is a locally free \({\mathscr O}_S\)-module (of finite rank), and the base change homomorphisms \(\beta ^j\) are isomorphisms for \(j= i, i-1\).


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

\[ s\mapsto \dim \operatorname{Coker}(d^i\otimes \kappa (s)),\quad s\mapsto \dim \operatorname{Coker}(d^{i-1}\otimes \kappa (s)), \]

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

\[ 0\to \mathop{\rm Im}(d^i)\otimes {\mathscr G}\to E^{i+1}\otimes {\mathscr G}\to \operatorname{Coker}(d^i)\otimes {\mathscr G}\to 0 \]

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

\[ 0\to H^i(E)\otimes {\mathscr G}\to \operatorname{Coker}(d^{i-1})\otimes {\mathscr G}\to \mathop{\rm Im}(d^i)\otimes {\mathscr G}\to 0 \]

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.

Corollary 4.12
Let \(k\) be a field. Let \(f\colon X\to S\) be a flat projective surjective morphism of \(k\)-schemes of finite type. Assume that all fibers of \(f\) are geometrically reduced and geometrically connected (i.e., denoting by \(\overline{\kappa (s)}\) an algebraic closure of \(\kappa (s)\), the scheme \(X_s\otimes _{\kappa (s)}\overline{\kappa (s)}\) is reduced and connected for every \(s\in S\)). Then the natural homomorphism \({\mathscr O}_S\to f_*{\mathscr O}_X\) is an isomorphism.


Problem sheet 12.

(Conversely, it follows from the “Theorem on Formal Functions” that a proper morphism \(f\colon X\to S\) such that \({\mathscr O}_S\cong f_*{\mathscr O}_X\) has geometrically connected fibers, see, e.g., [ GW2 ] Corollary 24.50.)