04.03.07 · algebraic-geometry / cohomology

Higher direct images and base change

shipped3 tiersLean: none

Anchor (Master): Grothendieck-Dieudonné *EGA III* (Publ. Math. IHÉS 11, 17, 1961–63) §1 (cohomology of a sheaf on a scheme), §6 (proper morphisms and finiteness), §7 (cohomology and base change); Hartshorne *Algebraic Geometry* §III.8–§III.12; Mumford *Abelian Varieties* §5 (cohomology and base change for an abelian variety); Hartshorne *Residues and Duality* Ch. II; Illusie *Cohomologie des intersections complètes* SGA 6 Exposé II

Intuition [Beginner]

Imagine a family of varieties parametrised by a base scheme $S$ — for each point $s$ of $S$ there is a fibre $X_{s}$ sitting over $s$ , and as $s$ moves the fibres deform continuously. A natural question is how the cohomology of a sheaf on a fibre varies with the parameter. Does $H^{q} (X_{s}, F_{s})$ have the same dimension for every $s$ ? If so, do the cohomology groups assemble into a vector bundle over $S$ ? If not, where do the dimensions jump and by how much?

The higher direct image $R^{q} f_{*} F$ is the construction that organises these fibre-by-fibre cohomologies into a single sheaf on the base $S$ . Concretely, it is the sheaf on $S$ whose sections over an open $U$ record the $q$ -th cohomology of $F$ restricted to the preimage of $U$ . Under good hypotheses the stalk of $R^{q} f_{*} F$ at a point $s$ equals the cohomology of the fibre $X_{s}$ , identifying the global construction with the pointwise data we started with.

The cohomology-and-base-change theorem records exactly when this stalk-equals-fibre identification holds. The hypotheses are mild — the morphism should be proper and flat, the sheaf should be flat over the base, and one needs a regularity condition at the point $s$ . When all these hold, the dimensions $dim H^{q} (X_{s}, F_{s})$ are locally constant on $S$ and the higher direct image is a vector bundle. When they fail, the dimensions can jump on a closed subvariety, and the semicontinuity theorem records that the jump is upward.

Visual [Beginner]

A schematic shows a base scheme $S$ as a horizontal line with three marked points, and above each marked point a fibre $X_{s}$ drawn as a curve. A sheaf $F$ is indicated by shading on the total space $X$ , with restrictions $F_{s}$ to the fibres. To the right of the diagram, three vector spaces labelled $H^{q} (X_{s}, F_{s})$ are drawn over the corresponding base points, with the two outer fibres having the same dimension and the middle fibre showing a jump in dimension — the semicontinuous behaviour.

Worked example [Beginner]

Take the structure morphism $f : P_{S}^{1} \to S$ from projective space over a base $S = Spec k$ where $k$ is a field, and the line bundle $F = O_{P^{1}} (2)$ . Here $S$ is a point, so the higher direct image $R^{q} f_{*} F$ on $S$ is a $k$ -vector space, the cohomology $H^{q} (P^{1}, O (2))$ computed in the cohomology-of-projective-space unit.

Step 1. Compute the candidate fibre cohomologies. From the projective-space cohomology table, $H^{0} (P^{1}, O (2))$ has dimension $(1 1 + 2) = 3$ (the three monomials $x_{0}^{2}, x_{0} x_{1}, x_{1}^{2}$ ), and $H^{1} (P^{1}, O (2)) = 0$ .

Step 2. Compute the higher direct images on the base. Because $S$ is a point, $f_{*} O (2) = H^{0} (P^{1}, O (2)) = k^{3}$ as a $k$ -module, and $R^{1} f_{*} O (2) = H^{1} (P^{1}, O (2)) = 0$ . The higher direct images on $S$ are exactly the global cohomology groups.

Step 3. Generalise to a higher-dimensional base. Replace $S = Spec k$ with $S = Spec A$ for $A$ a noetherian ring. The morphism is now $f : P_{A}^{1} \to Spec A$ and the structure sheaf carries the standard $A$ -module structure. The higher direct image is the quasi-coherent sheaf on $Spec A$ associated to the $A$ -module $H^{q} (P_{A}^{1}, O (2))$ . By the Čech computation on the standard two-piece affine cover, $H^{0} = A x_{0}^{2} \oplus A x_{0} x_{1} \oplus A x_{1}^{2}$ is free of rank $3$ and $H^{1} = 0$ . The higher direct image $R^{0} f_{*} O (2)$ is the free rank- $3$ sheaf $O_{S}^{\oplus 3}$ , and $R^{1} f_{*} O (2) = 0$ .

What this tells us. The higher direct images of $O (d)$ along the structure morphism of relative projective space are free sheaves on the base, with rank given by the projective-space dimension formula. The fibre cohomology is constant — every fibre is the same projective space and the cohomology dimension never jumps — and the cohomology-and-base-change identification holds at every point of the base. This is the simplest substantive example of a higher direct image, and the dimension count $dim H^{0} (P^{n}, O (d)) = (n n + d)$ from the projective-space unit governs every component of the answer.

Check your understanding [Beginner]

Exercise (easy, true-false).

True or false: for the structure morphism $f : P_{S}^{n} \to S$ over a noetherian base, the higher direct image $R^{1} f_{*} O_{P^{n}}$ is the zero sheaf on $S$ for every $n \geq 1$ .

Hint

Look up $H^{1} (P^{n}, O)$ in the projective-space cohomology table from the cohomology-of-projective-space unit. The dimension is zero for every $n \geq 1$ because $d = 0$ is in the vanishing range of the middle-degree cohomology.

Answer

True. From the table $H^{p} (P^{n}, O (d))$ , the value at $p = 1$ and $d = 0$ is zero unless $n = 1$ and $d \leq - 2$ , neither of which applies. The higher direct image $R^{1} f_{*} O_{P^{n}}$ on the base is the sheafification of the assignment $U \mapsto H^{1} (P_{U}^{n}, O)$ , and each summand vanishes, so the sheaf vanishes. The next non-vanishing higher direct image is $R^{n} f_{*} O_{P^{n}} (- n - 1) = O_{S}$ , the duality companion of $f_{*} O_{P^{n}} = O_{S}$ .

Formal definition [Intermediate+]

Let $f : X \to S$ be a morphism of ringed spaces (in practice, of schemes) and $F$ a sheaf of abelian groups on $X$ .

Definition (direct image). The direct image $f_{*} F$ is the sheaf on $S$ given on an open $V \subseteq S$ by $$ (f_* \mathcal{F})(V) := \mathcal{F}(f^{-1}(V)) $$ with restriction maps inherited from $F$ . The assignment $F \mapsto f_{*} F$ is a left-exact additive functor from $Sh (X)$ to $Sh (S)$ (and similarly from $O_{X}$ -modules to $O_{S}$ -modules when $f$ is a morphism of ringed spaces).

Definition (higher direct image). The $q$ -th higher direct image of $F$ along $f$ is the right-derived functor $$ R^q f_* \mathcal{F} := R^q (f_*) (\mathcal{F}), $$ computed by choosing an injective resolution $F \to I^{∙}$ in $Sh (X)$ (the category of sheaves of abelian groups on $X$ has enough injectives, Hartshorne III.2.2), applying $f_{*}$ termwise to obtain a complex $f_{*} I^{∙}$ in $Sh (S)$ , and taking cohomology in degree $q$ . Equivalently, $R^{q} f_{*} F$ is the sheaf on $S$ associated to the presheaf $$ V \mapsto H^q\big(f^{-1}(V), \mathcal{F}\big|{f^{-1}(V)}\big). $$ For $q = 0$ this recovers $f* \mathcal{F} $b y l e f t e x a c t n ess . T h ec a t e g or y o f$ \mathcal{O}X $- m o d u l es a l so ha se n o ug hinj ec t i v es, an d t h er es u l t in g$ R^q f* $o n$ \mathcal{O}_X$-modules agrees with the abelian-sheaves version by a comparison theorem (Hartshorne III.8.1).

Definition (proper morphism). A morphism $f : X \to S$ is proper if it is separated, of finite type, and universally closed. Properness is the algebraic-geometric analogue of compactness of fibres in the topological setting; under it, finiteness theorems for cohomology become available.

Definition (flat morphism). A morphism $f : X \to S$ is flat if for every $x \in X$ the local ring $O_{X, x}$ is a flat $O_{S, f (x)}$ -module. A coherent sheaf $F$ on $X$ is flat over $S$ if for every $x$ , the stalk $F_{x}$ is a flat $O_{S, f (x)}$ -module. Flatness is the algebraic version of a continuously-varying family — the fibres deform without sudden collapse.

Definition (base change). Given $f : X \to S$ and a morphism $g : S^{'} \to S$ , the base change is the cartesian square $$ \begin{array}{ccc} X' & \xrightarrow{g'} & X \ \downarrow f' & & \downarrow f \ S' & \xrightarrow{g} & S \end{array} $$ with $X^{'} := X \times_{S} S^{'}$ the fibre product and $f^{'}, g^{'}$ the projections. For a sheaf $F$ on $X$ , there is a canonical base-change morphism $$ g^* R^q f_* \mathcal{F} \longrightarrow R^q f'* (g'^* \mathcal{F}) $$ in $\mathcal{O}{S'} $- m o d u l es, co n s t r u c t e df r o m t h e u ni t o f t h e a d j u n c t i o n$ g^* \dashv g_*$ and the universal property of cohomology. The base-change morphism is an isomorphism under hypotheses recorded in the Key theorem below.

Definition (fibre). For a point $s \in S$ with residue field $k (s)$ , the fibre of $f$ over $s$ is the base change $X_{s} := X \times_{S} Spec k (s)$ , a scheme over $Spec k (s)$ . For a sheaf $F$ on $X$ , the restriction to the fibre is $F_{s} := F \otimes_{O_{X}} O_{X_{s}}$ , a sheaf on $X_{s}$ .

Counterexamples to common slips

The base-change morphism $g^{*} R^{q} f_{*} F \to R^{q} f_{*}^{'} (g^{' *} F)$ is canonical but not always an isomorphism. A standard failure: take $f : X \to S$ with $S = Spec k [t]$ , $X$ the family of plane cubics degenerating to a nodal cubic at $t = 0$ , $F = O_{X}$ . The dimensions of $H^{1} (X_{s}, O_{X_{s}})$ jump at the singular fibre, the base-change map is not surjective there, and the higher direct image $R^{1} f_{*} O_{X}$ is not free on $S$ .
Higher direct images of a non-proper morphism can fail to be coherent. The structure morphism $f : A_{k}^{1} \to Spec k$ is not proper; $f_{*} O_{A^{1}} = k [t]$ is not a finitely generated $k$ -module. Properness is essential for the coherence half of the Grothendieck finiteness theorem.
The Leray spectral sequence $H^{p} (S, R^{q} f_{*} F) \Rightarrow H^{p + q} (X, F)$ holds for any morphism of ringed spaces. The Grothendieck composite-functor spectral sequence is the version with two arbitrary left-exact functors composed; the Leray spectral sequence is the specialisation to $Γ_{S} \circ f_{*} = Γ_{X}$ .
Flatness of $F$ over $S$ is a stalkwise condition; it does not imply that $F$ is locally free. A torsion-free coherent sheaf on a one-dimensional regular base is flat but generically locally free; on a higher-dimensional base, flat and torsion-free can fail to be locally free even on the regular locus.

Key theorem with proof [Intermediate+]

Theorem (cohomology and base change; Grothendieck EGA III §7, Hartshorne III.12.11 ^[pending]). Let $f : X \to S$ be a proper morphism of noetherian schemes, $F$ a coherent sheaf on $X$ flat over $S$ , and $q \geq 0$ an integer. Consider the function $$ \varphi^q(s) := \dim_{k(s)} H^q(X_s, \mathcal{F}_s), \qquad s \in S. $$

(a) Semicontinuity. $φ^{q}$ is upper semicontinuous on $S$ : the set ${s \in S : φ^{q} (s) \geq n}$ is closed for every $n$ .

(b) Grauert constancy. If $φ^{q}$ is locally constant on $S$ , then $R^q f_ \mathcal{F} $i s l oc a l l y f r eeo f r ank$ \varphi^q $, an df or e v er y$ s \in S$ the natural base-change map* $$ (R^q f_* \mathcal{F}) \otimes_{\mathcal{O}_S} k(s) \longrightarrow H^q(X_s, \mathcal{F}_s) $$ is an isomorphism.

(c) Cohomology and base change. If the base-change map at $s$ is surjective, then it is an isomorphism, and the map at every $s^{'}$ in a neighbourhood of $s$ is an isomorphism. Furthermore the base-change map at $s$ in degree $q - 1$ is then surjective if and only if $R^q f_ \mathcal{F} $i s l oc a l l y f r ee n e a r$ s$.*

Proof. The argument proceeds via a local representing complex for the cohomology of the fibres, then deduces the three statements from properties of this complex.

Step 1: local representing complex. Because $S$ is noetherian and the question is local, restrict to an affine open $Spec A = U \subseteq S$ and the preimage $X_{U} = f^{- 1} (U)$ . The cohomology of a coherent sheaf on the proper $A$ -scheme $X_{U}$ is computed by the Čech complex on a finite affine cover of $X_{U}$ , which is a complex of $A$ -modules $$ C^\bullet \colon C^0 \to C^1 \to \cdots \to C^N $$ with each $C^{p}$ a finitely generated free $A$ -module (the cover is finite, the intersections are affine, and on each intersection the sections of the flat $F$ form a flat $A$ -module which one replaces, by a standard reduction, with a free $A$ -module of finite rank; alternatively, replace the Čech complex by a quasi-isomorphic complex of finite free $A$ -modules using that the cohomology is finitely generated over $A$ by the finiteness half of Grothendieck's theorem). The cohomology of $C^{∙}$ computes $H^{q} (X_{U}, F ∣_{X_{U}})$ , and base-changing along $A \to k (s)$ gives a complex $C^{∙} \otimes_{A} k (s)$ whose cohomology is $H^{q} (X_{s}, F_{s})$ . The local statement to be proved becomes: for a complex of finite free $A$ -modules $C^{∙}$ on a noetherian local ring $A$ with residue field $k$ , study how $dim_{k} H^{q} (C^{∙} \otimes_{A} k)$ depends on $A$ and on the morphism $A \to k$ . This is a linear-algebraic / commutative-algebra problem on a fixed complex.

Step 2: semicontinuity from rank-jump on a linear map. For each $q$ , write the two adjacent differentials as $$ d^{q-1} \colon C^{q-1} \to C^q, \qquad d^q \colon C^q \to C^{q+1}. $$ The cohomology $H^{q} (C^{∙} \otimes_{A} k (s)) = ker (d^{q} \otimes k (s)) / im (d^{q - 1} \otimes k (s))$ . As $s$ varies, the dimension of the kernel of $d^{q} \otimes k (s)$ is upper semicontinuous (kernels of linear maps over a varying ring jump up on closed subschemes — this is the rank lemma: $rk (d^{q} \otimes k (s))$ is lower semicontinuous, hence $dim ker$ is upper semicontinuous), and the dimension of the image of $d^{q - 1} \otimes k (s)$ is lower semicontinuous by the same lemma applied to $d^{q - 1}$ . The cohomology dimension is $dim ker - dim im$ ; an upper-semicontinuous minus a lower-semicontinuous function is upper semicontinuous, proving statement (a).

Step 3: Grauert constancy. If $φ^{q}$ is locally constant on $S$ near $s$ , then both $rk (d^{q})$ and $rk (d^{q - 1})$ are locally constant. A linear map between finite free $A$ -modules with constant rank on $Spec A$ has both a kernel and a cokernel that are locally free $A$ -modules — this is the rank-constancy lemma for matrices over a noetherian ring. Hence $ker (d^{q})$ and $coker (d^{q - 1})$ are locally free $A$ -modules in the relevant range, and the cohomology $H^{q} (C^{∙})$ is locally free over $A$ of rank $φ^{q}$ . The base change $H^{q} (C^{∙}) \otimes_{A} k (s) \to H^{q} (C^{∙} \otimes_{A} k (s))$ is an isomorphism because tensor product commutes with kernel-and-image taking when the relevant modules are flat — flat base change for a complex of free modules of constant rank. This is statement (b).

Step 4: base-change isomorphism from surjectivity. Statement (c) refines (b): instead of requiring local constancy of $φ^{q}$ , require surjectivity of the base-change map at a single point. The base-change map factors $$ H^q(C^\bullet) \otimes_A k(s) \xrightarrow{\alpha} H^q(C^\bullet \otimes_A k(s)) $$ and the cokernel of $α$ measures the failure of cohomology to commute with base change. The cokernel is supported on the locus where the rank of $d^{q}$ drops — a closed subset. If $α$ is surjective at $s$ , then $s$ is not in this locus, hence not in a neighbourhood, and $α$ is an isomorphism on a neighbourhood. The lower-degree statement about $q - 1$ follows from the dual rank analysis applied to $d^{q - 1}$ , exchanging the roles of kernel and image. $□$

Bridge. The cohomology-and-base-change theorem builds toward 04.04.01 (Riemann-Roch theorem for curves) in the relative setting: for a flat family of smooth proper curves $f : X \to S$ with a relative line bundle $L$ , the higher direct images $R^{q} f_{*} L$ are locally free with ranks given by the Euler characteristic $χ (L_{s}) = de g L_{s} + 1 - g_{s}$ , and base change holds at every point of $S$ . The construction appears again in 04.03.04 (cohomology of line bundles on projective space) where the fibre-by-fibre formula $dim H^{q} (P_{k (s)}^{n}, O (d)) = (n n + d)$ (for $q = 0$ , $d \geq 0$ ) is identical for every $s$ , so the relative higher direct image $R^{0} f_{*} O (d)$ of the structure morphism $P_{S}^{n} \to S$ is the free sheaf $O_{S}^{\oplus (n n + d)}$ . The foundational reason is that the Čech complex of $O (d)$ on the standard cover is already a complex of free $A$ -modules — Grothendieck's relative construction reduces to bookkeeping in this case.

The central insight is that the cohomology of a fibre is the cohomology of a complex of finitely generated free modules over the base, and the dependence of cohomology on the point reduces to how kernels and images of linear maps between free modules vary — a question controlled by semicontinuity of rank. The absolute-to-relative passage is therefore a universal coefficient principle: a flat family's cohomology is computed by a single complex of finite free modules on the base, and every fibre-wise cohomology question becomes a linear-algebra question on that complex. This is exactly what generalises to the derived-category formulation in 04.03.06, where $R f_{*}$ replaces the family ${R^{q} f_{*}}$ and the base-change isomorphism becomes a single coherent statement in $D_{coh}^{b} (S)$ . Relative Serre duality lands in the same picture: the relative dualising complex $ω_{X / S}^{∙}$ realises $R^{q} f_{*} F ≅ Hom_{O_{S}} (R^{n - q} f_{*} (F^{\lor} \otimes ω_{X / S}), O_{S})^{*}$ , the relative version of the absolute Serre duality recorded in 04.03.04.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

For the structure morphism $f : P_{S}^{n} \to S$ over a noetherian base and any integer $d$ , compute the rank of the locally-free sheaf $R^{p} f_{*} O_{P^{n}} (d)$ on $S$ for every $0 \leq p \leq n$ . Express the answer using binomial coefficients.

Hint

The fibre cohomology $H^{p} (P_{k (s)}^{n}, O (d))$ is constant in $s$ because the structure morphism is the same projective space at every point. Apply the Grauert constancy half of cohomology and base change to deduce that $R^{p} f_{*} O (d)$ is free with rank equal to the fibre dimension.

Answer

The fibre cohomology is the projective-space cohomology table from 04.03.04: $dim H^{0} (P^{n}, O (d)) = (n n + d)$ for $d \geq 0$ and zero for $d < 0$ ; $dim H^{n} (P^{n}, O (d)) = (n - d - 1)$ for $d \leq - n - 1$ and zero for $d \geq - n$ ; $dim H^{p} (P^{n}, O (d)) = 0$ for $0 < p < n$ and every $d$ . Because the structure morphism is proper and flat (smooth, in fact) and the dimensions are constant on $S$ (the formula does not depend on $s$ ), Grauert's constancy statement gives $R^{p} f_{*} O_{P^{n}} (d)$ locally free of the corresponding rank, and the base-change map is an isomorphism at every point.

So: $R^{0} f_{*} O (d) = O_{S}^{\oplus (n n + d)}$ for $d \geq 0$ and zero otherwise; $R^{n} f_{*} O (d) = O_{S}^{\oplus (n - d - 1)}$ for $d \leq - n - 1$ and zero otherwise; the middle direct images $R^{p} f_{*} O (d) = 0$ for $0 < p < n$ regardless of $d$ . Rubric: full credit for all three cases and the citation to constancy of fibre dimension; partial credit for $R^{0}$ alone or for the table without the rank-constancy argument.

Exercise 4 (medium, short-answer).

State and prove the Leray spectral sequence for a morphism $f : X \to S$ of ringed spaces and a sheaf $F$ on $X$ : there is a convergent spectral sequence $$ E_2^{p, q} = H^p(S, R^q f_* \mathcal{F}) \Rightarrow H^{p + q}(X, \mathcal{F}). $$

Hint

This is the Grothendieck composite-functor spectral sequence applied to $Γ_{X} = Γ_{S} \circ f_{*}$ . Use that $f_{*}$ takes injectives to $Γ_{S}$ -acyclics (flasque sheaves, which are $Γ$ -acyclic on any topological space).

Answer

The functor $Γ_{X} : Sh (X) \to Ab$ factors as $Γ_{X} = Γ_{S} \circ f_{*}$ because a global section of a sheaf on $X$ is the same data as a global section of its pushforward on $S$ : $Γ (X, F) = Γ (f^{- 1} (S), F) = (f_{*} F) (S) = Γ (S, f_{*} F)$ .

Both $f_{*}$ and $Γ_{S}$ are left-exact additive functors between abelian categories with enough injectives. The hypothesis for the Grothendieck composite-functor spectral sequence is that $f_{*}$ takes injective sheaves on $X$ to $Γ_{S}$ -acyclic sheaves on $S$ . Verification: an injective sheaf $I$ on $X$ is flasque (every restriction map is surjective, Hartshorne III.2.4), the pushforward $f_{*} I$ is then flasque on $S$ (flasque is preserved by pushforward — direct check on opens), and flasque sheaves are $Γ$ -acyclic on every topological space (Godement-Tohoku). So $f_{*} I$ is $Γ_{S}$ -acyclic.

The Grothendieck spectral sequence then reads $$ E_2^{p, q} = R^p \Gamma_S (R^q f_* \mathcal{F}) \Rightarrow R^{p + q} \Gamma_X (\mathcal{F}), $$ which on substitution becomes $$ E_2^{p, q} = H^p(S, R^q f_* \mathcal{F}) \Rightarrow H^{p + q}(X, \mathcal{F}), $$ the Leray spectral sequence. Rubric: full credit for the factorisation, the acyclicity verification via flasque preservation, and the application of the Grothendieck spectral sequence; partial credit for the spectral sequence statement alone.

Exercise 5 (medium, short-answer).

State the Grothendieck composite-functor spectral sequence in general: for additive left-exact functors $F : A \to B$ and $G : B \to C$ between abelian categories with enough injectives, with the hypothesis that $F$ takes injectives to $G$ -acyclics, there is a convergent spectral sequence $$ E_2^{p, q} = R^p G (R^q F (A)) \Rightarrow R^{p + q} (G \circ F)(A) $$ for every $A \in A$ . Then specialise to the two settings $G \circ F = Γ_{S} \circ f_{*}$ (Leray) and $G \circ F = Hom (M, -) \circ H o m (F, -)$ (local-to-global Ext).

Hint

The general statement is built from the double complex obtained by applying $G$ to a Cartan-Eilenberg resolution of $F (I^{∙})$ for an injective resolution $A \to I^{∙}$ . The two spectral sequences of the double complex give the composite-functor spectral sequence and a degenerating dual.

Answer

General statement. Choose an injective resolution $A \to I^{∙}$ in $A$ . Apply $F$ to get $F (I^{∙})$ , a cochain complex in $B$ . Choose a Cartan-Eilenberg resolution of $F (I^{∙})$ : a double complex $J^{∙, ∙}$ of injectives in $B$ such that each column is an injective resolution of $F (I^{p})$ , and the induced cocycle, coboundary, and cohomology data in each row is exact except in the $A$ -augmentation degree. Apply $G$ termwise to obtain a double complex $G (J^{∙, ∙})$ in $C$ .

The associated total complex has two spectral sequences. First spectral sequence (row first). The $E_{1}$ page reads $E_{1}^{p, q} = R^{q} G (F (I^{p}))$ . By hypothesis, $F (I^{p})$ is $G$ -acyclic, so $E_{1}^{p, q} = 0$ for $q \geq 1$ , and $E_{1}^{p, 0} = G (F (I^{p}))$ . The $E_{2}$ page is $E_{2}^{p, 0} = R^{p} (G \circ F) (A)$ via the explicit complex $G \circ F (I^{∙})$ , concentrated in a single row. The spectral sequence converges to the cohomology of the total complex, identified with $R^{p + q} (G \circ F) (A)$ .

Second spectral sequence (column first). The $E_{1}$ page reads $E_{1}^{p, q} = G (R^{q} F (A))$ in column $q$ , since the column-cohomology of the Cartan-Eilenberg resolution is the cohomology of $F (I^{∙})$ which is $R^{q} F (A)$ . The $E_{2}$ page is $E_{2}^{p, q} = R^{p} G (R^{q} F (A))$ . The spectral sequence converges to the same total cohomology, giving the stated composite-functor spectral sequence.

Leray specialisation. $A = Sh (X)$ , $B = Sh (S)$ , $C = Ab$ , $F = f_{*}$ , $G = Γ_{S}$ . The acyclicity hypothesis is that $f_{*}$ of an injective is $Γ_{S}$ -acyclic, verified via flasqueness. The spectral sequence is $E_{2}^{p, q} = H^{p} (S, R^{q} f_{*} F) \Rightarrow H^{p + q} (X, F)$ .

Local-to-global Ext specialisation. $A = B = O_{X}$ -Mod, $C = Ab$ , $F = H o m_{O_{X}} (F, -)$ , $G = Γ_{S}$ . The acyclicity hypothesis is that $H o m_{O_{X}} (F, I)$ is flasque for $I$ injective (Hartshorne III.6.3). The spectral sequence is $E_{2}^{p, q} = H^{p} (X, Ext^{q} (F, G)) \Rightarrow Ext^{p + q} (F, G)$ , the local-to-global Ext spectral sequence from 04.03.06.

Rubric: full credit for the general spectral sequence with the double-complex justification and both specialisations; partial credit for the general statement alone.

Exercise 6 (medium, short-answer).

Prove that for a proper morphism $f : X \to S$ of noetherian schemes and a coherent sheaf $F$ on $X$ , the higher direct image $R^{q} f_{*} F$ is a coherent sheaf on $S$ for every $q$ . (Grothendieck finiteness; EGA III §3.)

Hint

The question is local on $S$ ; reduce to $S$ affine. Use the Stein factorisation and the fact that for a projective morphism — the case $f : X \subseteq P_{S}^{N} \to S$ — the Čech complex on the affine cover restricted from $P_{S}^{N}$ computes $R^{q} f_{*}$ , and is a complex of finitely generated $O_{S}$ -modules. Reduce general proper to projective via Chow's lemma.

Answer

The question is local on $S$ , so assume $S = Spec A$ for a noetherian ring $A$ , and the statement reduces to: $H^{q} (X, F)$ is a finitely generated $A$ -module for every $q$ . The cohomology is computed by Čech on a finite affine cover; if $f$ is projective (factored through a closed immersion $X ↪ P_{S}^{N}$ ), the standard cover restricted from the $(N + 1)$ -piece cover of $P_{S}^{N}$ to $X$ is finite affine, and the Čech complex consists of finitely generated $A$ -modules in each degree (cohomology of $O_{X}$ -modules on an affine open is a finitely generated $A$ -module by Hartshorne II.5.4 plus coherence of $F$ ). The cohomology of a complex of finitely generated $A$ -modules over a noetherian ring is finitely generated, proving the projective case.

For general proper $f$ , Chow's lemma produces a projective $f^{'} : X^{'} \to S$ and a birational morphism $X^{'} \to X$ over $S$ ; the higher direct images of $F$ on $X$ are related to higher direct images on $X^{'}$ by a spectral sequence whose terms are coherent by the projective case, and a noetherian-induction argument on the support of $F$ deduces coherence of $R^{q} f_{*} F$ for all $q$ . The vanishing $R^{q} f_{*} F = 0$ for $q > dim_{m a x} (fibres)$ is a consequence of Grothendieck vanishing for cohomology on a finite-dimensional space (Hartshorne III.2.7), combined with the fibre dimension bound for a proper morphism. Rubric: full credit for the projective case with the Čech argument plus the Chow's-lemma reduction to projective; partial credit for the projective case alone.

Exercise 7 (hard, short-answer).

State the Mumford constancy theorem in the form used in Abelian Varieties: for a proper flat morphism $f : X \to S$ of noetherian schemes with $S$ reduced and connected, and a coherent sheaf $F$ on $X$ flat over $S$ , the function $φ^{q} (s) = dim_{k (s)} H^{q} (X_{s}, F_{s})$ is locally constant on $S$ if and only if $R^{q} f_{*} F$ is a locally free sheaf on $S$ . Apply this to the morphism $f : X \to S$ where $X$ is an abelian variety, $S$ is its dual, and $F$ is the Poincaré line bundle, recovering the cohomology computation that gives the seesaw and theorem of the cube.

Hint

The forward direction is Grauert constancy. The reverse uses that if $R^{q} f_{*} F$ is locally free of rank $r$ , then by base change the fibre $H^{q} (X_{s}, F_{s})$ has dimension at least $r$ at every $s$ and equals $r$ on an open dense; semicontinuity plus reducedness force equality everywhere.

Answer

Statement. Under the hypotheses, $φ^{q}$ is locally constant on $S$ if and only if $R^{q} f_{*} F$ is locally free of rank $φ^{q}$ on $S$ and the base-change map $(R^{q} f_{*} F) \otimes k (s) \to H^{q} (X_{s}, F_{s})$ is an isomorphism for every $s$ .

Proof. The forward direction is the Grauert constancy half of cohomology and base change recorded in the Key theorem above: local constancy of $φ^{q}$ on a noetherian base gives a constant-rank linear map between free modules in the representing complex, hence local freeness of the cohomology and the base-change isomorphism.

The reverse direction: assume $R^{q} f_{*} F$ is locally free of rank $r$ on a neighbourhood $U ∋ s$ . The base-change map $(R^{q} f_{*} F) \otimes k (s^{'}) \to H^{q} (X_{s^{'}}, F_{s^{'}})$ is surjective on a dense open by Grothendieck's generic flatness for a coherent module on a reduced noetherian scheme. On the dense open, the base-change map is an isomorphism (statement (c) of the Key theorem), so $φ^{q} (s^{'}) = r$ there. Upper semicontinuity (statement (a)) forces $φ^{q} (s) \geq r$ at every point; reducedness of $S$ prevents the inequality from being strict (a coherent sheaf on a reduced scheme whose rank exceeds the local-free rank generically must be supported on the non-reduced locus, which is empty here). Hence $φ^{q}$ is constant equal to $r$ on $U$ .

Application to abelian varieties. Let $X$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ , $X$ its dual, and $P$ the Poincaré line bundle on $X \times X$ . The projection $f : X \times X \to X$ is the structure morphism of $X$ over $X$ , proper and flat. By Mumford's calculation in Abelian Varieties §13, $H^{q} (X, P_{\overset{x}{^}}) = 0$ for every $\overset{x}{^} \in X ∖ {0}$ and every $q$ , with $H^{q} (X, P_{0}) = H^{q} (X, O_{X}) = (q g) k$ . So $φ^{q}$ is not locally constant — it jumps at $0$ — and the higher direct image $R^{q} f_{*} P$ is supported at ${0} \subset X$ with multiplicity $(q g)$ . Mumford uses this calculation to establish the seesaw principle on $X \times Y$ and the theorem of the cube on $X^{3}$ : the Poincaré bundle is non-zero in $Pic (X \times Y)$ iff its restriction to one fibre is non-zero, and the cube theorem follows from the cohomology calculation plus the structure of $X$ as the moduli of degree-zero line bundles.

Rubric: full credit for both directions of the constancy theorem and the abelian-variety application; partial credit for the statement and forward direction alone.

Exercise 8 (hard, short-answer).

For the morphism $f : X \to S$ where $X$ is a flat family of plane cubic curves over $S = A_{k}^{1} = Spec k [t]$ with smooth fibres for $t \neq = 0$ and a nodal cubic at $t = 0$ , compute the higher direct image $R^{1} f_{*} O_{X}$ near $t = 0$ and verify the failure of cohomology-and-base-change.

Hint

A smooth elliptic curve has $H^{1} (E, O_{E}) = k$ (one-dimensional). A nodal cubic has arithmetic genus $1$ and geometric genus $0$ , but $H^{1}$ of the structure sheaf is still one-dimensional by the arithmetic-genus computation. The base-change phenomenon to inspect is whether $R^{1} f_{*} O_{X}$ is locally free near $t = 0$ .

Answer

For each $t \neq = 0$ , the fibre $X_{t}$ is a smooth elliptic curve of arithmetic genus $1$ , and $H^{1} (X_{t}, O_{X_{t}}) = k (t)$ , one-dimensional. For $t = 0$ , the fibre $X_{0}$ is a nodal cubic; the arithmetic genus is still $1$ (computed from the embedding in $P^{2}$ via the standard $p_{a} = (d - 1) (d - 2) /2 = 1$ for $d = 3$ ), so $H^{1} (X_{0}, O_{X_{0}}) = k$ , also one-dimensional.

The fibre cohomology dimension $φ^{1} (t) = 1$ is constant on $S$ . By Grauert constancy, $R^{1} f_{*} O_{X}$ is locally free of rank $1$ on $S$ , and the base-change map is an isomorphism at every $t$ . The cohomology-and-base-change identification does hold in this example, and the example by itself does not exhibit a base-change failure.

The standard base-change failure example replaces $O_{X}$ by the ideal sheaf of a section that becomes the node — say $F = I_{Σ}$ where $Σ$ is the section through the node of the nodal fibre and a smooth point on each smooth fibre. The arithmetic genus of $X_{t}$ minus a smooth point is the same as the genus of $X_{t}$ for $t \neq = 0$ (no change), but at $t = 0$ the section is at the node and the cohomology of the ideal sheaf jumps. This is the cleaner example of base-change failure: $dim H^{0} (X_{t}, F_{t})$ is upper-semicontinuous but jumps at $t = 0$ , the higher direct image $R^{0} f_{*} F$ fails to be locally free near $t = 0$ , and the base-change map $(R^{0} f_{*} F) \otimes k (0) \to H^{0} (X_{0}, F_{0})$ has nonzero cokernel.

The cleanest base-change failure on plane cubics is in higher cohomology: take a family where one fibre becomes more singular (a cuspidal cubic at $t = 0$ , smooth elsewhere), and consider an ample line bundle. The dimension of $H^{0}$ of the ample bundle jumps at $t = 0$ because the cuspidal singularity adds a new section, and the base-change map fails to be an isomorphism at the cuspidal fibre. Rubric: full credit for the constancy on smooth-versus-nodal cubics with the arithmetic-genus calculation and the observation that the canonical example with $O_{X}$ does not exhibit failure; partial credit for either half.

Lean formalization [Intermediate+]

The named higher direct image functor $R^{q} f_{*}$ on quasi-coherent sheaves of a morphism of noetherian schemes is not yet packaged in Mathlib. The intended statement, in the Lean 4 / Mathlib idiom expected when the API arrives, is the following.

import Mathlib.AlgebraicGeometry.Morphisms.Proper
import Mathlib.Algebra.Homology.DerivedFunctor
import Mathlib.CategoryTheory.Sheaf

namespace AlgebraicGeometry

variable {X S : Scheme} (f : X ⟶ S) [IsProper f]
variable (F : CoherentSheaf X) [IsFlat F f]

/-- The $q$-th higher direct image of a coherent sheaf along a proper morphism. -/
def higherDirectImage (q : ℕ) : CoherentSheaf S :=
  sorry  -- defined as R^q (f.pushforward) F via the right-derived-functor API

/-- Cohomology and base change: if the fibre-dimension function is locally
    constant, the higher direct image is locally free and the base-change map
    is an isomorphism at every point. -/
theorem cohomology_and_base_change (q : ℕ)
    (h_const : LocallyConstant S (fun s ↦ Module.rank (k s) (H q (Xₛ s) (Fₛ s)))) :
    IsLocallyFree (higherDirectImage f F q) ∧
    ∀ s : S, IsIso (baseChangeMap f F q s) := by
  sorry

end AlgebraicGeometry

The Mathlib gap is the higherDirectImage definition and the cohomology_and_base_change theorem; the surrounding category-theoretic infrastructure (IsProper, IsFlat, CoherentSheaf, the derived-functor API on abelian categories with enough injectives) is partially present. See the frontmatter lean_mathlib_gap for the full list of missing components.

Advanced results [Master]

Theorem (Grothendieck composite-functor spectral sequence; EGA III §0.12.2 ^[pending]). Let $A, B, C$ be abelian categories with enough injectives and $F : A \to B$ , $G : B \to C$ left-exact additive functors. Assume $F$ sends injectives in $A$ to $G$ -acyclic objects in $B$ . For every $A \in A$ there is a first-quadrant convergent spectral sequence $$ E_2^{p, q} = R^p G \big( R^q F (A) \big) \Rightarrow R^{p + q} (G \circ F) (A), $$ natural in $A$ and in compatibly varying pairs $(F, G)$ .

The spectral sequence is constructed from the Cartan-Eilenberg resolution of $F (I^{∙})$ for an injective resolution $A \to I^{∙}$ . Two specialisations recur. Leray spectral sequence: $F = f_{*}$ for $f : X \to S$ , $G = Γ_{S}$ , giving $E_{2}^{p, q} = H^{p} (S, R^{q} f_{*} F) \Rightarrow H^{p + q} (X, F)$ — the Leray construction of 1946 in modern dress, with $f_{*}$ of an injective sheaf flasque and hence $Γ_{S}$ -acyclic. Local-to-global Ext spectral sequence: $F = H o m_{O_{X}} (F, -)$ , $G = Γ_{X}$ , giving $E_{2}^{p, q} = H^{p} (X, Ext_{O_{X}}^{q} (F, G)) \Rightarrow Ext_{O_{X}}^{p + q} (F, G)$ from 04.03.06, with $H o m$ of an injective flasque (Hartshorne III.6.3) and hence $Γ_{X}$ -acyclic.

Theorem (proper base change for a flat square; EGA III §7.7 ^[pending]). Consider the cartesian square $$ \begin{array}{ccc} X' & \xrightarrow{g'} & X \ \downarrow f' & & \downarrow f \ S' & \xrightarrow{g} & S \end{array} $$ with $f$ proper and $g$ flat. For every coherent sheaf $F$ on $X$ and every $q \geq 0$ , the base-change morphism $$ g^* R^q f_* \mathcal{F} \xrightarrow{\sim} R^q f'_* (g'^* \mathcal{F}) $$ is an isomorphism of coherent sheaves on $S^{'}$ .

The flat-base-change theorem is the foundational statement of relative cohomology — it asserts that under flatness the higher direct images commute with base change in the categorical sense, without the cohomology-and-base-change machinery of jump loci. The proof reduces, as in the absolute case, to a complex-of-free-modules representation of cohomology on an affine slice; flat base change preserves the cokernel and kernel structure of a complex of free modules, giving the isomorphism. The cohomology-and-base-change theorem of the Key theorem is the more refined statement where $g$ is not assumed flat — typically $g$ is the inclusion of a fibre $Spec k (s) \to S$ , which is not flat unless $s$ is a generic point or the cohomology happens to be locally free at $s$ .

Theorem (Grauert direct-image theorem; Grauert 1960 ^[pending]). Let $f : X \to S$ be a proper morphism of complex-analytic spaces and $F$ a coherent analytic sheaf on $X$ . The higher direct images $R^q f_ \mathcal{F} $a r eco h er e n t ana l y t i cs h e a v eso n$ S$, and the semicontinuity, constancy, and cohomology-and-base-change theorems hold in the analytic category.*

Grauert's 1960 paper Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen (Publ. Math. IHÉS 5) is the analytic ancestor of EGA III, and the analytic and algebraic theorems agree under GAGA (Serre 1956) on projective varieties over $C$ . The analytic version is the tool of choice for the moduli theory of complex-analytic objects — Kuranishi families, deformations of compact complex manifolds, period maps — where the algebraic-scheme framework is unavailable.

Theorem (relative duality on a proper morphism; Hartshorne Residues and Duality Ch. VI ^[pending]). Let $f : X \to S$ be a proper morphism of noetherian schemes of constant relative dimension $n$ , with $ω_{X / S}^{∙}$ the relative dualising complex. For every $F \in D_{coh}^{b} (X)$ there is a natural isomorphism in $D_{coh}^{b} (S)$ $$ Rf_* R\mathcal{H}om_{\mathcal{O}X}(\mathcal{F}, \omega{X/S}^\bullet) \xrightarrow{\sim} R\mathcal{H}om_{\mathcal{O}S}(Rf* \mathcal{F}, \mathcal{O}_S). $$

Relative duality on a proper morphism is the relative form of the Serre-duality identity from 04.03.04, with the dualising sheaf $ω_{X}$ replaced by the relative dualising complex $ω_{X / S}^{∙}$ and the duality landing in $D_{coh}^{b} (S)$ rather than in a $k$ -vector space. When $f$ is smooth, $ω_{X / S}^{∙}$ is concentrated in degree $- n$ and equals the relative canonical sheaf $ω_{X / S} = det Ω_{X / S}^{1}$ ; for a smooth projective family this is the wedge-power of the cotangent bundle. When $f$ is proper but not smooth — Gorenstein, Cohen-Macaulay, or worse — $ω_{X / S}^{∙}$ is more subtle, with non-zero cohomology in multiple degrees encoding the singularity structure of the fibres.

Theorem (cohomology and base change via the Tate complex on an abelian variety; Mumford Abelian Varieties §13 ^[pending]). Let $X$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ , $X$ the dual abelian variety, and $P$ the Poincaré line bundle on $X \times X$ . The projection $π : X \times X \to X$ is proper and flat, and $$ R^q \pi_* \mathcal{P} = 0 \text{ for } 0 \le q < g, \qquad R^g \pi_* \mathcal{P} = k(0), $$ the skyscraper sheaf at the origin of $X$ with $g$ -dimensional fibre $H^{g} (X, O_{X})$ .

The Mumford calculation is the prototype of a cohomology-and-base-change failure: the fibre cohomology $H^{q} (X, P_{\overset{x}{^}})$ is zero for every $\overset{x}{^} \neq = 0$ and every $q$ , but jumps to $(q g)$ at $\overset{x}{^} = 0$ . The higher direct image $R^{q} π_{*} P$ is therefore a torsion sheaf supported at the origin of $X$ , and the base-change map is not an isomorphism at the origin. Mumford uses this calculation to prove the seesaw principle (a line bundle on $X \times Y$ that restricts to $O$ on every horizontal fibre $X \times {y}$ is the pullback of a line bundle on $Y$ ), the theorem of the cube (a line bundle on $X^{3}$ that restricts to $O$ on each pairwise product of axes is itself $O$ ), and the symmetry-and-anti-symmetry properties of the Poincaré bundle that ground the duality between $X$ and $X$ .

Theorem (semicontinuity in characteristic $p$ and Hodge spectral sequence; Deligne-Illusie 1987 ^[pending]). Let $X / S$ be a proper smooth morphism in characteristic $p$ and assume $X$ admits a smooth lift to $W_{2} (k)$ (the second-order Witt vectors). Then the Hodge-to-de-Rham spectral sequence $$ E_1^{p, q} = R^q f_* \Omega^p_{X/S} \Rightarrow R^{p + q} f_* \Omega^\bullet_{X/S} $$ degenerates at $E_{1}$ .

The Deligne-Illusie result extends classical Hodge theory from complex projective geometry to the modular setting via a finite-characteristic detour, and is the foundation of $p$ -adic Hodge theory. The proof uses a base-change argument on the Cartier isomorphism and the cohomology-and-base-change machinery to lift the degeneration from $W_{2}$ to $C$ via reduction modulo $p$ . The Hodge spectral sequence is itself a Leray-type spectral sequence whose abutment is the de Rham cohomology of the family.

Synthesis. The higher direct image functor $R^{q} f_{*}$ is the relative form of cohomology: a single proper morphism $f : X \to S$ replaces the standalone proper scheme $X / k$ , and the cohomology groups $H^{q} (X, F)$ become coherent sheaves $R^{q} f_{*} F$ on the base $S$ . The central insight is that the cohomology of a flat family is computed by a single complex of finitely generated free modules on the base, and the dependence of cohomology on the base point is therefore a linear-algebra question about kernels and images of varying linear maps between free modules. The foundational reason for the semicontinuity-and-constancy dichotomy is the rank lemma for matrices over a noetherian ring: the rank of a linear map between free modules is a lower semicontinuous function of the parameter, hence the kernel dimension is upper semicontinuous, hence the cohomology dimension (kernel minus image) is upper semicontinuous; equality on a neighbourhood forces local freeness of the kernel and image and hence of the cohomology. This is exactly the bridge between fibre-wise data — the function $φ^{q} (s) = dim H^{q} (X_{s}, F_{s})$ — and global-on-base data — the higher direct image $R^{q} f_{*}$ as a coherent sheaf.

The Grothendieck composite-functor spectral sequence builds toward the Leray spectral sequence and the local-to-global Ext spectral sequence simultaneously, identifying both as instances of one categorical mechanism. The cohomology-and-base-change theorem is dual to the Grothendieck finiteness theorem: finiteness says $R^{q} f_{*} F$ is a coherent sheaf, base change says how its stalks compare to the fibre cohomology. Putting these together, the relative cohomology framework — higher direct images, base-change comparison, semicontinuity, the Grothendieck composite-functor spectral sequence — is the unified machinery in which Riemann-Roch in families, the seesaw and theorem of the cube on abelian varieties, the proper base change theorem in étale cohomology, and the modular degeneration of the Hodge spectral sequence all live as instances of one categorical mechanism. The synthesis is that the absolute case studied in 04.03.04 (a single $P^{n}$ over a field) extends, with the same Čech-and-binomial-coefficient bookkeeping, to the relative case of $P_{S}^{n} \to S$ over an arbitrary noetherian base, and that the EGA III finiteness-and-base-change package generalises the absolute projective-space calculation to every proper morphism. The bridge to 04.03.06 (derived functors and Ext) is direct: $R^{q} f_{*}$ is a right-derived functor in the same abelian-category framework, with $f_{*} ⊣ f^{*}$ replacing the bare $Γ ⊣ const$ pair, and the cohomology-and-base-change theorem is the derived-category statement about flat tensor product preserving the cohomology of a complex of finite free modules.

The relative formalism generalises in three directions. First, to the derived-category formulation $R f_{*} : D_{coh}^{b} (X) \to D_{coh}^{b} (S)$ as a triangulated functor between bounded derived categories of coherent sheaves, with the Grothendieck six-functor formalism $(R f_{!}, R f_{*}, L f^{*}, R f^{!}, R H o m, \otimes^{L})$ packaging the relative-cohomology operations into a coherent algebraic structure (Lipman-Hashimoto for Noether-coherent schemes; Grothendieck-Verdier for the topos-theoretic version). Second, to étale cohomology and the proper base change theorem in characteristic $p$ , where the same flat-base-change formalism with $R^{q} f_{*}$ on étale sheaves yields the Lefschetz trace formula for varieties over finite fields and the rationality of the zeta function (Grothendieck-Deligne). Third, to the analytic category via Grauert's coherence theorem, where the same machinery powers the moduli theory of compact complex manifolds, the variation of Hodge structures along a smooth proper family (Griffiths 1968), and the analytic-side proof of Kodaira-Spencer deformation theory. The synthesis is that the relative-cohomology framework is the carrier of all geometry-of-families information, the categorical extension of absolute cohomology in the same sense that a family of varieties is a categorical extension of a single variety, and the foundation on which moduli theory, deformation theory, and the modern theory of motives are built.

Full proof set [Master]

Proposition (Leray spectral sequence; existence and convergence). Let $f : X \to S$ be a morphism of ringed spaces, $F$ a sheaf of $O_{X}$ -modules. Then there is a convergent spectral sequence $$ E_2^{p, q} = H^p(S, R^q f_* \mathcal{F}) \Rightarrow H^{p + q}(X, \mathcal{F}). $$

Proof. The functor $Γ (X, -) = Γ (S, -) \circ f_{*}$ factorises as the composite of two left-exact additive functors between abelian categories with enough injectives. To apply the Grothendieck composite-functor spectral sequence, verify that $f_{*}$ takes injective $O_{X}$ -modules to $Γ (S, -)$ -acyclic $O_{S}$ -modules. Let $I$ be an injective $O_{X}$ -module. Then $I$ is flasque: for any open $V \subseteq X$ and any section $s \in I (V)$ , the section $s$ extends to a section on all of $X$ by the lifting property along the monomorphism $j_{!} O_{V} ↪ O_{X}$ (Hartshorne III.2.4). The pushforward $f_{*} I$ on $S$ is then flasque: a section $t$ over an open $W \subseteq S$ is by definition a section over $f^{- 1} (W) \subseteq X$ , and by flasqueness of $I$ on $X$ it extends to $X$ , restricting to a section over $S = f^{- 1} (S)$ . A flasque sheaf is $Γ$ -acyclic on every topological space (the proof uses the long exact sequence in cohomology associated to the inclusion $V ↪ S$ and downward induction, or alternatively Godement's standard argument).

So the hypothesis of the Grothendieck composite-functor spectral sequence holds, yielding $$ E_2^{p, q} = R^p \Gamma(S, R^q f_* \mathcal{F}) \Rightarrow R^{p + q} \Gamma(X, \mathcal{F}), $$ i.e. $E_{2}^{p, q} = H^{p} (S, R^{q} f_{*} F) \Rightarrow H^{p + q} (X, F)$ . Convergence is first-quadrant convergence, which is automatic for a non-negatively-graded composite-functor spectral sequence on bounded complexes. $□$

Proposition (representability of cohomology by a complex of finite free modules over a noetherian local ring). Let $f : X \to S = Spec A$ be a proper morphism with $A$ a noetherian ring, and $F$ a coherent sheaf on $X$ flat over $A$ . There is a bounded complex $$ C^\bullet \colon C^0 \xrightarrow{d^0} C^1 \xrightarrow{d^1} \cdots \xrightarrow{d^{N-1}} C^N $$ of finitely generated free $A$ -modules such that for every $A$ -module $M$ , $$ H^q(X, \mathcal{F} \otimes_A M) \cong H^q(C^\bullet \otimes_A M) $$ naturally in $M$ . Such a complex is called a representing complex for $f_{*} F$ .

Proof. Choose a finite affine cover $U = {U_{i}}_{i = 0}^{N}$ of $X$ . The Čech complex $\overset{ˇ}{C}^{∙} (U, F)$ is a bounded complex of $A$ -modules with $\overset{ˇ}{C}^{p} (U, F) = \prod_{i_{0} < \dots < i_{p}} F (U_{i_{0} \dots i_{p}})$ . Each term is the value of a coherent sheaf on a noetherian affine scheme over $A$ , hence a finitely generated $A$ -module. Flatness of $F$ over $A$ implies that each $\overset{ˇ}{C}^{p} (U, F)$ is a flat $A$ -module.

The complex $\overset{ˇ}{C}^{∙}$ is quasi-isomorphic to a bounded complex $C^{∙}$ of finitely generated free $A$ -modules — this is the standard replacement of a bounded complex of finitely generated flat modules over a noetherian ring by a quasi-isomorphic complex of finitely generated frees (the argument: take a finite free cover of the lowest-degree term, lift the differential to a chain map of free complexes, and iterate; finiteness comes from noetherianity of $A$ and finite generation of each term). For every $A$ -module $M$ , $$ H^q(X, \mathcal{F} \otimes_A M) = H^q(\check C^\bullet \otimes_A M) = H^q(C^\bullet \otimes_A M), $$ the first equality by the projection formula for a quasi-coherent sheaf on a proper morphism over $Spec A$ (which uses flatness of $F$ to commute tensor product past Čech cohomology), and the second by quasi-isomorphism. $□$

Proposition (semicontinuity of cohomology dimension). With the hypotheses of the cohomology-and-base-change theorem, the function $φ^{q} (s) = dim_{k (s)} H^{q} (X_{s}, F_{s})$ on $S$ is upper semicontinuous: ${s \in S : φ^{q} (s) \geq n}$ is closed for every integer $n$ .

Proof. By the representing-complex proposition, on an affine neighbourhood of $s$ there is a bounded complex of finitely generated free $A$ -modules $C^{∙}$ with $H^{q} (X_{s^{'}}, F_{s^{'}}) = H^{q} (C^{∙} \otimes_{A} k (s^{'}))$ for every $s^{'} \in Spec A$ . The differential $d^{q} \otimes k (s^{'}) : C^{q} \otimes k (s^{'}) \to C^{q + 1} \otimes k (s^{'})$ is a $k (s^{'})$ -linear map of finite-dimensional $k (s^{'})$ -vector spaces; its rank is the rank of the matrix of $d^{q}$ over $A$ reduced modulo the maximal ideal corresponding to $s^{'}$ .

The rank of a matrix over a varying point of a noetherian scheme is lower semicontinuous: ${s^{'} : rk (d^{q} \otimes k (s^{'})) \leq r}$ is closed (cut out by the vanishing of $(r + 1) \times (r + 1)$ minors), hence its complement ${s^{'} : rk (d^{q} \otimes k (s^{'})) > r}$ is open. Equivalently, $dim_{k (s^{'})} ker (d^{q} \otimes k (s^{'}))$ is upper semicontinuous: ${s^{'} : dim ker \geq n}$ is closed (the rank dropping is the kernel growing).

The cohomology dimension is $$ \varphi^q(s') = \dim \ker(d^q \otimes k(s')) - \dim \mathrm{im}(d^{q-1} \otimes k(s')) = \dim \ker(d^q \otimes k(s')) - \mathrm{rk}(d^{q-1} \otimes k(s')). $$ The first term is upper semicontinuous; the second term is lower semicontinuous (as a rank). Their difference is upper semicontinuous, proving the claim. $□$

Proposition (Grauert constancy half of cohomology and base change). With the hypotheses of the Key theorem, if $φ^{q}$ is constant on an open neighbourhood $U$ of $s \in S$ , then $R^{q} f_{*} F$ is locally free on $U$ of rank $φ^{q} (s)$ , and the base-change map $(R^{q} f_{*} F) \otimes_{O_{S}} k (s^{'}) \to H^{q} (X_{s^{'}}, F_{s^{'}})$ is an isomorphism for every $s^{'} \in U$ .

Proof. By the representing-complex proposition, on a possibly smaller affine open $U = Spec A$ inside the original neighbourhood the cohomology is computed by a bounded complex $C^{∙}$ of finitely generated free $A$ -modules: $H^{q} (X_{s^{'}}, F_{s^{'}}) = H^{q} (C^{∙} \otimes_{A} k (s^{'}))$ . Local constancy of $φ^{q}$ on $U$ says $dim ker (d^{q} \otimes k (s^{'})) - rk (d^{q - 1} \otimes k (s^{'})) = φ^{q}$ is constant. Both summands are upper semicontinuous (kernel-dimension) and lower semicontinuous (rank), and their difference is constant, so each summand is locally constant (a function that is both upper and lower semicontinuous and differs from a constant by a locally constant function is itself locally constant). Hence $rk (d^{q} \otimes k (s^{'}))$ and $rk (d^{q - 1} \otimes k (s^{'}))$ are locally constant on $U$ .

A linear map between finite free $A$ -modules whose rank on $Spec A$ is locally constant has a kernel and a cokernel that are locally free $A$ -modules of constant rank — the rank-stratification of a matrix is open-closed when the rank is locally constant. Hence $ker d^{q}$ and $coker d^{q - 1}$ are locally free $A$ -modules near $s$ , and the cohomology $H^{q} (C^{∙}) = ker d^{q} / im d^{q - 1}$ is locally free of rank $φ^{q}$ .

For base change, the map $$ \alpha \colon H^q(C^\bullet) \otimes_A k(s') \to H^q(C^\bullet \otimes_A k(s')) $$ is induced by the comparison of taking cohomology before and after tensoring. Because $ker d^{q}$ and $im d^{q - 1}$ are both locally free, tensoring with $k (s^{'})$ preserves the short exact sequence $0 \to im d^{q - 1} \to ker d^{q} \to H^{q} \to 0$ , and the quotient on the right is $H^{q} (C^{∙}) \otimes k (s^{'})$ , identified with $H^{q} (C^{∙} \otimes k (s^{'}))$ . Hence $α$ is an isomorphism. $□$

Proposition (base-change failure under non-flatness). There exist a proper flat morphism $f : X \to S$ of noetherian schemes, a coherent sheaf $F$ on $X$ flat over $S$ , an integer $q$ , and a point $s \in S$ such that the base-change map $(R^{q} f_{*} F) \otimes_{O_{S}} k (s) \to H^{q} (X_{s}, F_{s})$ is not surjective. (Witness: Mumford abelian-variety Poincaré bundle calculation.)

Proof. Take $X = A \times A$ for $A$ an abelian variety of dimension $g \geq 1$ over an algebraically closed field, $A$ the dual, $P$ the Poincaré bundle on $A \times A$ , and $f = π_{2} : A \times A \to A$ the second projection. Then $f$ is proper (projective, in fact) and flat (smooth product projection). The Poincaré bundle is flat over $A$ because it is locally a product in the analytic topology and flatness is local (for an analytic morphism with smooth source and target, local-product structure on a sheaf implies flatness).

By Mumford's calculation in Abelian Varieties §13, $H^{q} (A, P ∣_{A \times {\overset{a}{^}}}) = 0$ for every $\overset{a}{^} \neq = 0$ and every $q$ , while $H^{q} (A, P ∣_{A \times {0}}) = H^{q} (A, O_{A}) = \land^{q} H^{1} (A, O_{A}) = \land^{q} k^{g}$ of dimension $(q g)$ . So $φ^{q}$ is zero everywhere except at $\overset{a}{^} = 0$ where it equals $(q g) > 0$ for $0 \leq q \leq g$ .

The higher direct image $R^{q} π_{2 *} P$ on $A$ has zero fibre at every $\overset{a}{^} \neq = 0$ by Grauert constancy on the open complement of ${0}$ , hence vanishes on this complement. By coherence, $R^{q} π_{2 *} P$ is supported at ${0}$ . The base-change map at $\overset{a}{^} = 0$ is $0 \otimes k (0) \to k^{(q g)}$ , which is not surjective for $0 \leq q \leq g$ . By Mumford's analysis, $R^{q} π_{2 *} P$ is in fact zero for $q < g$ and $R^{g} π_{2 *} P = k (0)$ (the residue field of $0 \in A$ ), so the base-change failure is concentrated in the top degree and the higher direct image is a non-zero torsion sheaf supported at one point. $□$

Connections [Master]

Sheaf cohomology 04.03.01. The higher direct image $R^{q} f_{*}$ specialises to ordinary sheaf cohomology when $S$ is a point: $R^{q} f_{*} F = H^{q} (X, F)$ for $f : X \to Spec k$ . Every property of sheaf cohomology — long exact sequence, finiteness on a proper scheme, vanishing in degrees above the dimension — extends to the relative setting via the higher direct image construction. The Leray spectral sequence connects the absolute cohomology of $X$ to the absolute cohomology of $S$ with coefficients in the higher direct images, providing the categorical bridge.
Cohomology of line bundles on projective space 04.03.04. The fibre-by-fibre calculation of $H^{q} (P_{k (s)}^{n}, O (d))$ that anchors the absolute projective-space cohomology table extends without change to the relative setting: $R^{q} f_{*} O_{P_{S}^{n}} (d)$ on the structure morphism $f : P_{S}^{n} \to S$ is the free $O_{S}$ -module of the corresponding rank. This is the prototypical example where the cohomology-and-base-change identification holds at every point, the higher direct image is locally free, and the dimension table from the absolute case transports verbatim to the relative case.
Derived functors and Ext 04.03.06. The higher direct image $R^{q} f_{*}$ is a right-derived functor of the left-exact additive functor $f_{*}$ , computed via injective resolutions in the abelian category of $O_{X}$ -modules. The Grothendieck composite-functor spectral sequence yielding the Leray spectral sequence is the same machine that produces the local-to-global Ext spectral sequence in 04.03.06 — both are specialisations of one categorical mechanism on $Γ_{X} = G \circ F$ factorisations with $F$ taking injectives to $G$ -acyclics. The derived-category formulation $R f_{*} : D_{coh}^{b} (X) \to D_{coh}^{b} (S)$ is the modern setting where higher direct images live alongside derived pullback $L f^{*}$ , derived tensor product $\otimes^{L}$ , and the six-functor formalism.
Coherent sheaf 04.06.02. Grothendieck's finiteness theorem (the coherence half of the EGA III package) says $R^{q} f_{*} F$ is a coherent sheaf on $S$ when $f$ is proper and $F$ is coherent on $X$ . This is the relative form of the absolute Serre finiteness theorem (cohomology of coherent on a noetherian projective scheme is a finitely generated module). The proper-morphism hypothesis is essential — without it, the higher direct image can fail to be coherent (the structure morphism $A^{1} \to pt$ has $f_{*} O_{A^{1}} = k [t]$ , not finitely generated). The coherent-sheaf framework is therefore the natural target of the higher direct image construction in the relative setting.
Serre's vanishing and finiteness theorems 04.03.05. The absolute Serre vanishing theorem $H^{q} (P^{n}, F (d)) = 0$ for $q \geq 1$ and $d ≫ 0$ has a relative form: $R^{q} f_{*} (F (d)) = 0$ for $q \geq 1$ and $d ≫ 0$ on a relatively very-ample line bundle, with the threshold $d_{0}$ a coherent function of $F$ (the relative Castelnuovo-Mumford regularity). The relative version powers the construction of Hilbert schemes and Quot schemes — for $d ≫ 0$ , the projective scheme $Proj (⨁_{d} f_{*} F (d))$ behaves well, and the higher direct images vanish.

Historical & philosophical context [Master]

The systematic development of higher direct images and the cohomology-and-base-change theorem begins with Jean Leray's 1946 paper L'anneau d'homologie d'une représentation in Comptes Rendus 222, where Leray defined the spectral sequence of a continuous map and proved its convergence to the cohomology of the total space ^[pending]. Leray was a prisoner of war in Oflag XVIIA from 1940 to 1945 and developed sheaf cohomology and the spectral sequence in the camp; his 1946 announcement is the original construction, with the technical details filled out by Henri Cartan in his Séminaire 1948–51 and by Serre in his 1951 thesis Homologie singulière des espaces fibrés. The Leray spectral sequence in its modern sheaf-theoretic form — $E_{2}^{p, q} = H^{p} (S, R^{q} f_{*} F) \Rightarrow H^{p + q} (X, F)$ — is the topological ancestor of the Grothendieck composite-functor spectral sequence.

Alexander Grothendieck's Éléments de Géométrie Algébrique III, published in two parts in Publications Mathématiques de l'IHÉS volumes 11 (1961) and 17 (1963), is the comprehensive scheme-theoretic treatment ^[pending]. EGA III.1 (1961) develops the cohomology of a sheaf on a scheme, the higher direct image $R^{q} f_{*}$ on quasi-coherent sheaves, and the finiteness theorem for proper morphisms of noetherian schemes; EGA III.2 (1963) treats the cohomology-and-base-change theorem, the semicontinuity statement, and the application to flat families. Grothendieck's framework axiomatised the relative-cohomology machinery into a categorical formalism — proper morphisms in place of compact spaces, coherent sheaves in place of locally constant sheaves, derived functors of $f_{*}$ in place of the Leray spectral sequence — and established the foundations on which modern algebraic geometry, étale cohomology, and the theory of motives all rest.

Hans Grauert's 1960 paper Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen (Publ. Math. IHÉS 5, 233–292) ^[pending] established the analytic counterpart: for a proper morphism of complex-analytic spaces and a coherent analytic sheaf, the higher direct images are coherent analytic sheaves, and the semicontinuity, constancy, and cohomology-and-base-change theorems hold. Grauert's framework was the technical foundation of Kuranishi's local theory of deformations of compact complex manifolds (1962) and of the analytic-side moduli theory developed by Schlessinger, Artin, and others through the 1960s. The analytic and algebraic theorems agree under the GAGA principle (Serre 1956) on projective varieties over $C$ .

David Mumford's Abelian Varieties (Tata Institute lecture notes, 1968; Oxford University Press 1970) ^[pending] used cohomology and base change as the central tool in the theory of abelian varieties. Mumford's §5 develops the cohomology-and-base-change theorem with the proof via a representing complex of finite free modules over the base, and §13 applies the theorem to the Poincaré bundle on $X \times X$ to prove the seesaw principle, the theorem of the cube, and the symmetry of the Poincaré bundle. The Mumford calculation of $R^{q} π_{*} P$ — vanishing for $q < g$ , skyscraper at the origin for $q = g$ — is the prototype of a base-change failure that nevertheless yields the correct duality between $X$ and $X$ , and is the analytic-algebraic bridge that connects abelian-variety theory to the broader cohomology-and-base-change framework.

Robin Hartshorne's Algebraic Geometry (Springer GTM 52, 1977) ^[pending] §III.8–§III.12 gives the canonical textbook treatment of higher direct images, flat morphisms, and the cohomology-and-base-change theorem, distilling EGA III into a form accessible to graduate students. Hartshorne's §III.12.11 is the standard reference for the cohomology-and-base-change statement; the proof he gives via the representing complex of finite free modules over the base is the modern pedagogical standard. The derived-category reformulation of higher direct images as a triangulated functor $R f_{*} : D_{coh}^{b} (X) \to D_{coh}^{b} (S)$ is developed in Hartshorne's Residues and Duality (Springer LNM 20, 1966) ^[pending] and in Verdier's 1967 thesis, with the relative-duality theorem $R f_{*} R H o m (F, ω_{X / S}^{∙}) ≅ R H o m (R f_{*} F, O_{S})$ as the central statement of the Grothendieck duality formalism on a proper morphism.

Bibliography [Master]

@article{LerayCR1946,
  author    = {Leray, Jean},
  title     = {L'anneau d'homologie d'une repr{\'e}sentation},
  journal   = {Comptes Rendus de l'Acad{\'e}mie des Sciences},
  volume    = {222},
  year      = {1946},
  pages     = {1366--1368}
}

@article{EGAIII1,
  author    = {Grothendieck, Alexander and Dieudonn{\'e}, Jean},
  title     = {{\'E}l{\'e}ments de G{\'e}om{\'e}trie Alg{\'e}brique III: {\'E}tude cohomologique des faisceaux coh{\'e}rents, Premi{\`e}re partie},
  journal   = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume    = {11},
  year      = {1961},
  pages     = {5--167}
}

@article{EGAIII2,
  author    = {Grothendieck, Alexander and Dieudonn{\'e}, Jean},
  title     = {{\'E}l{\'e}ments de G{\'e}om{\'e}trie Alg{\'e}brique III: {\'E}tude cohomologique des faisceaux coh{\'e}rents, Seconde partie},
  journal   = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume    = {17},
  year      = {1963},
  pages     = {5--91}
}

@article{Grauert1960,
  author    = {Grauert, Hans},
  title     = {Ein Theorem der analytischen Garbentheorie und die Modulr{\"a}ume komplexer Strukturen},
  journal   = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume    = {5},
  year      = {1960},
  pages     = {233--292}
}

@book{MumfordAbelianVarieties,
  author    = {Mumford, David},
  title     = {Abelian Varieties},
  publisher = {Oxford University Press / Tata Institute of Fundamental Research},
  year      = {1970}
}

@book{HartshorneAG,
  author    = {Hartshorne, Robin},
  title     = {Algebraic Geometry},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {52},
  year      = {1977}
}

@book{HartshorneResiduesDuality,
  author    = {Hartshorne, Robin},
  title     = {Residues and Duality},
  publisher = {Springer-Verlag},
  series    = {Lecture Notes in Mathematics},
  volume    = {20},
  year      = {1966}
}

@incollection{IllusieSGA6,
  author    = {Illusie, Luc},
  title     = {Cohomologie des intersections compl{\`e}tes},
  booktitle = {SGA 6: Th{\'e}orie des intersections et th{\'e}or{\`e}me de Riemann-Roch},
  publisher = {Springer-Verlag},
  series    = {Lecture Notes in Mathematics},
  volume    = {225},
  year      = {1971},
  pages     = {Expos{\'e} II}
}

@article{DeligneIllusie1987,
  author    = {Deligne, Pierre and Illusie, Luc},
  title     = {Rel{\`e}vements modulo $p^2$ et d{\'e}composition du complexe de de Rham},
  journal   = {Inventiones Mathematicae},
  volume    = {89},
  year      = {1987},
  pages     = {247--270}
}

@article{Griffiths1968,
  author    = {Griffiths, Phillip A.},
  title     = {Periods of integrals on algebraic manifolds, I, II},
  journal   = {American Journal of Mathematics},
  volume    = {90},
  year      = {1968},
  pages     = {568--626, 805--865}
}

@book{Kuranishi1965,
  author    = {Kuranishi, Masatake},
  title     = {New proof for the existence of locally complete families of complex structures},
  publisher = {Proceedings of the Conference on Complex Analysis, Minneapolis 1964; Springer},
  year      = {1965},
  pages     = {142--154}
}

@book{Liu2002,
  author    = {Liu, Qing},
  title     = {Algebraic Geometry and Arithmetic Curves},
  publisher = {Oxford University Press},
  series    = {Oxford Graduate Texts in Mathematics},
  volume    = {6},
  year      = {2002}
}

Prerequisites

04.03.01
04.03.04
04.03.06
04.06.02

Tier anchors

beginner: Hartshorne *Algebraic Geometry* §III.8 informal; Mumford *Abelian Varieties* §5 informal — how a family of cohomology groups varies as the base point moves
intermediate: Hartshorne *Algebraic Geometry* §III.8 (higher direct images), §III.9 (flat morphisms), §III.12 (cohomology and base change); Mumford *Abelian Varieties* §5
master: Grothendieck-Dieudonné *EGA III* (Publ. Math. IHÉS 11, 17, 1961–63) §1 (cohomology of a sheaf on a scheme), §6 (proper morphisms and finiteness), §7 (cohomology and base change); Hartshorne *Algebraic Geometry* §III.8–§III.12; Mumford *Abelian Varieties* §5 (cohomology and base change for an abelian variety); Hartshorne *Residues and Duality* Ch. II; Illusie *Cohomologie des intersections complètes* SGA 6 Exposé II

References

TODO_REF
Hartshorne — Algebraic Geometry · §III.8 higher direct images of a morphism; §III.9 flat morphisms and flat families of sheaves; §III.12 the cohomology-and-base-change theorem, semicontinuity and the Grauert constancy theorem
TODO_REF
Grothendieck-Dieudonné — Éléments de Géométrie Algébrique III · Publications Mathématiques de l'IHÉS 11 (1961) and 17 (1963); §1 derived functors of $f_*$ on quasi-coherent sheaves, §6 finiteness for proper morphisms of noetherian schemes, §7 cohomology and base change with the semicontinuity statement
TODO_REF
Mumford — Abelian Varieties · Tata Institute / Oxford University Press 1970; §5 cohomology and base change on an abelian variety, with the application to the seesaw and theorem-of-the-cube proofs
TODO_REF
Hartshorne — Residues and Duality · Springer LNM 20, 1966; Ch. II derived-category formulation of higher direct images $Rf_*$ and the base-change isomorphism for a flat base-change square
TODO_REF
Illusie — Cohomologie des intersections complètes · SGA 6 Exposé II (1971); the cohomology and base change framework for a perfect complex on a proper morphism, with the relative-duality refinement
TODO_REF
Leray — L'anneau d'homologie d'une représentation · Comptes Rendus 222 (1946) 1366-1368; originator of the Leray spectral sequence for a continuous map, the topological ancestor of the Grothendieck composite-functor spectral sequence

Reviewer

TBD

Estimated time

beginner: 18m
intermediate: 45m
master: 80m