Draft lesson.

Higher direct images and the Leray spectral sequence

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

The sections of a direct image are sections upstairs on inverse images of opens. After deriving, the corresponding cohomology groups first form a presheaf; its associated sheaf is the higher direct image. Leray then assembles the cohomology of those sheaves into the cohomology upstairs. Both the sheafification and the higher differentials can change the answer.

Let \(f:(X,\mathcal O_X)\to(Y,\mathcal O_Y)\) be a morphism of ringed spaces, with unital rings and left modules. Commutativity is not needed. Fix universes and use choice. Derived categories and functors in this lesson are bounded below. For a sheaf in degree zero all the displayed spectral sequences are first quadrant. No separation, properness or flatness hypothesis on \(f\) is assumed.

Prerequisites are Godement resolutions and flabby sheaf cohomology, Sections 1–9, for enough injectives, open restriction and flabby acyclicity; Universal cohomology and acyclic resolutions, Section 5, for the canonical derived-composition maps and their associativity; Postnikov towers and Grothendieck spectral sequences, Sections 1–5, for derived spectral maps and finite convergence; and Filtered complexes and Cartan–Eilenberg resolutions, Sections 1–5, for the complete double-resolution construction, comparisons and sign conventions. The finite calculations use Finite-poset sheaves and canonical homological resolutions, Sections 1–3 and 6. The existing finite-poset direct-image formulas remain available in Derived direct images and base change on finite posets.

1. The section functor and its derived object

Define \((f_*F)(V)=F(f^{-1}V)\). A scalar \(a\in\mathcal O_Y(V)\) acts through \(f^\#(a)\in\mathcal O_X(f^{-1}V)\). Restriction of sections and scalars makes this an \(\mathcal O_Y\)-module sheaf: preimages take an open cover and its intersections to the corresponding cover upstairs, so the sheaf gluing equations are exactly those for \(F\). Maps are applied on these same sections. Limits of module sheaves are computed sectionwise, so \(f_*\) preserves all limits and in particular is additive and left exact.

Choose an injective resolution \(F\to I^\bullet\) on \(X\). Then \(Rf_*F\) is represented by \(f_*I^\bullet\), and \(R^qf_*F=\mathcal H^q(f_*I^\bullet)\). For a bounded-below complex use its ordinary bounded-below injective replacement in the same way. The prerequisite comparison theorem gives maps for every derived arrow, not only every original sheaf map. Two comparisons are homotopic before applying \(f_*\), and additivity preserves that homotopy. Thus all these objects and maps have the canonical choice comparisons already proved there.

2. The associated presheaf and every stalk map

For \(q\ge0\), form the presheaf of \(\mathcal O_Y\)-modules

using the usual cohomology restriction map. Restricting an injective on \(X\) to an open remains injective, so (2.1) is the presheaf cohomology of \(V\mapsto I^\bullet(f^{-1}V)\). A section cycle defines a section of the cohomology sheaf. It is zero there exactly when it is locally a section boundary. Conversely a section of the cohomology sheaf is locally represented by section cycles: first lift through its quotient sheaf locally, and use the sectionwise kernel description for cycles. These two local assertions show that the induced map from the associated presheaf is an isomorphism:

Equivalently, exact associated-sheaf formation takes the presheaf kernel/image quotient to the sheaf kernel/image quotient. Neither argument identifies the presheaf with its associated sheaf before this operation.

At \(y\in Y\) the actual isomorphism is

The map sends a cohomology class on one inverse neighborhood to the germ of its image in (2.2). To see that it is an isomorphism directly, the stalk complex of \(f_*I^\bullet\) is the filtered colimit of its section complexes on neighborhoods. Filtered colimits of abelian groups are exact: finite representative equalities hold at a common later stage, which proves preservation of kernels and images. Their cohomology therefore commutes with this colimit. The action of \(\mathcal O_{Y,y}\) is computed by choosing a common neighborhood for a scalar germ and a cohomology representative; the resulting action agrees with the stalk action. This proves (2.3) as a module isomorphism, with all its maps.

For an open \(V\subseteq Y\), write \(X_V=f^{-1}V\) and \(f_V:X_V\to V\). Restriction gives an equality of complexes \((f_*I^\bullet)|_V=(f_V)_*(I^\bullet|_{X_V})\). The restricted source complex is an injective replacement. Consequently there are canonical isomorphisms \((Rf_*F)|_V\simeq R(f_V)_*(F|_{X_V})\), compatible with smaller opens, arrows and every cohomology group. These are open-restriction comparisons; they assert no replacement of inverse neighborhoods by literal point fibres.

3. Flabby computation without a false injectivity assumption

If \(A\) is flabby on \(X\), then \(f_*A\) is flabby on \(Y\): restriction along \(W\subseteq V\) is exactly \(A(f^{-1}V)\to A(f^{-1}W)\), which is onto. Also \(A\) is \(f_*\)-acyclic. The flabby theorem gives \(H^q(f^{-1}V,A)=0\) for every open \(V\) and \(q>0\); (2.2) then gives \(R^qf_*A=0\).

Thus the canonical Godement resolution \(F\to J^\bullet(F)\) computes the entire derived direct image by \(f_*J^\bullet(F)\), with the same explicit cofaces, differential signs and functorial maps as in its defining lesson. Its terms are \(f_*\)-acyclic, and the proved acyclic-resolution comparison identifies this actual complex with \(Rf_*F\). Over general coefficient rings these terms need not be injective.

The higher direct images agree with those of the underlying abelian sheaf. Indeed a module-injective resolution is an exact abelian-sheaf resolution whose terms are flabby. The preceding argument, applied to abelian sheaves, makes those terms \(f_*\)-acyclic there as well. The natural acyclic-resolution comparison therefore computes both derived images with the same underlying complex \(f_*I^\bullet\). It proves the comparison for bounded-below complexes too, together with all maps. It does not assume that forgetting scalars preserves injectivity.

4. Coherent derived composition

For an injective \(I\) on \(X\), the sheaf \(f_*I\) is flabby and hence acyclic for sections on every open of \(Y\). Apply the derived-composition theorem to \(f_*\) followed by \(\Gamma(Y,-)\). Its hypothesis is precisely this acyclicity, and it gives the canonical isomorphism

for every bounded-below \(K\), after restriction of scalars from \(\mathcal O_X(X)\) to \(\mathcal O_Y(Y)\). Restriction of scalars is exact because it leaves the underlying groups and maps unchanged. More concretely, on an injective replacement \(I_K\), both sides are computed by the identical section complex \(I_K(X)\): the terms of \(f_*I_K\) are section-acyclic. The isomorphism is the inverse, in the displayed direction, of the canonical map \(R(\Gamma_Y f_*)\to R\Gamma_Y Rf_*\) of the composition prerequisite. Its augmentation and all derived-map comparisons are therefore specified.

If \(g:Y\to Z\) is another ringed-space morphism, \(f_*I\) is also \(g_*\)-acyclic by Section 3, since it is flabby on \(Y\). The same theorem gives

on \(D^+(\operatorname{Mod}(\mathcal O_X))\). The map on a source replacement is the canonical \(g_*\)-replacement transformation applied to \(f_*I_K\). Thus it is the actual derived-composition comparison. For three morphisms the two resulting composites agree by the associativity identity of that theorem, including their replacement-choice comparisons. Apply the same section argument on \(V\subseteq Y\) to obtain \(R\Gamma(V,Rf_*K)\simeq R\Gamma(f^{-1}V,K)\); its restriction maps are the literal restrictions of these section complexes.

5. Both Leray sequences, with finite convergence

Put \(L=Rf_*K\). Apply the derived hypercohomology construction of the Cartan–Eilenberg prerequisite to the left exact functor \(\Gamma(Y,-)\) and the bounded-below complex \(L\). Together with (4.1), it gives

For a sheaf \(F\), \(\mathcal H^q(Rf_*F)=R^qf_*F\) and \(p,q\ge0\). For a complex whose cohomology begins in degree \(m\), one has \(p\ge0,q\ge m\), because a bounded-below replacement starts at that same lower bound and \(f_*\) is left exact.

For explicit signs use a Cartan–Eilenberg grid with resolution degree \(p\) horizontal and the original complex degree \(q\) vertical, and total differential

This is the transpose of the prerequisite's grid, identified with it by multiplication by \((-1)^{pq}\). Direct substitution shows that this scaling commutes with both differentials and their total signs. Its column filtration gives the derived sequence (5.1) with the ordinary \(H^p(Y,H^qL)\) second-page identifications of that prerequisite. In particular \(d_r\) has bidegree \((r,1-r)\). Every total degree has finitely many summands, even when no upper bound exists. The proved finite convergence theorem gives a finite filtration on each abutment, with

No splitting of that filtration is asserted. The Postnikov construction in the other spectral-sequence prerequisite likewise provides the intrinsic truncation and edge maps; the double-grid convention (5.2) fixes the computations here.

Replacing \(\Gamma_Y\) by \(g_*\), the same construction and (4.2) give the relative sequence of \(\mathcal O_Z\)-module sheaves

For a sheaf it has \(E_2^{p,q}=R^pg_*(R^qf_*F)\). All maps of \(K\), including derived arrows, induce maps on every page and on the filtered abutment. This follows from the full Cartan–Eilenberg comparison theorem: different identity comparisons give the same maps from page two, respect composition, and induce the canonical derived map. Equations (4.1)–(4.2) transport those abutments compatibly. Exact restriction to an open commutes with cohomology and the displayed source replacement; hence these page and abutment maps commute with the open-restriction comparisons of Section 2. All section-group maps are linear over the target section ring through the given scalar maps.

6. Edge maps and the five-term sequence

For a sheaf \(F\), the bottom edge \(H^n(Y,f_*F)\to H^n(X,F)\) is induced by \(f_*F[0]\to Rf_*F\), followed by (4.1). The top edge \(H^n(X,F)\to\Gamma(Y,R^nf_*F)\) is obtained from the canonical quotient \(\tau_{\le n}L\to H^nL[-n]\). Its source is identified with \(H^nR\Gamma L\): the omitted tail has lower cohomological bound \(n+1\), and right-derived sections preserve this lower bound. These maps are exactly the corresponding edge maps of the derived construction, with their augmentation conventions.

In low degrees the resulting sequence is

Here is its exactness argument. The position \((1,0)\) has no incoming or outgoing differential from page two and is the bottom filtration piece of \(H^1\). The first-column position \((0,1)\) has no incoming differential; its only possible outgoing differential is \(d_2\), so its stable value is \(\ker d_2\), the quotient of \(H^1\) by that bottom piece. Finally \((2,0)\) has no outgoing differential and only the incoming \(d_2\) from \((0,1)\); its stable value is \(\operatorname{coker}d_2\), embedded as the last filtration piece of \(H^2\). The edge descriptions give precisely (6.1).

The arrow \(d_2\) is the actual second differential, not a freely chosen map making dimensions agree. In (5.2), represent its source by a vertical cocycle \(x_0\) of bidegree \((0,1)\). Being a second-page class means that one can correct it by \(x_1\) of bidegree \((1,0)\) with \(d_hx_0=d_vx_1\). Then \(D(x_0+x_1)=d_hx_1\), so \(d_2[x_0]=[d_hx_1]\). Different corrections change this by a target boundary, as proved by the filtered-page formulas. When quotient representatives require epimorphic lifts, the same kernel/image formulas of that prerequisite define the morphism. Exercise 4 computes this sign explicitly.

7. Exact local tests and degeneration criteria

For a fixed \(q>0\), \(R^qf_*F=0\) if and only if every class \(c\in H^q(f^{-1}V,F)\) restricts to zero after shrinking \(V\) around each of its points. This is exactly the zero-germ criterion in the filtered colimit (2.3). A sufficient condition for all higher images to vanish is a basis of target opens whose inverse images are \(F\)-acyclic. The converse basis assertion does not follow just from vanishing of the associated sheaf: local death of each class need not mean that its original open has zero cohomology.

If \(R^qf_*F=0\) for \(q>0\), (5.1) has only its bottom row, and its actual bottom edges give \(H^n(Y,f_*F)\simeq H^n(X,F)\) for all \(n\). If instead every \(R^qf_*F\) is acyclic for sections on \(Y\), only the first column remains, and the top edges give \(H^n(X,F)\simeq\Gamma(Y,R^nf_*F)\). These are identifications via the specified maps, since a one-piece filtration introduces no extension choice.

8. Four exercises with full solutions

8.1 A nonflat coefficient map on a point

Exercise 1 (basic). Let \(f:(\{*\},\mathbb F_p)\to(\{*\},\mathbb Z)\) use reduction modulo \(p\). Compute all its derived direct images. Show why direct image of an injective need not be injective in this situation.

Solution. Direct image is restriction of scalars from vector spaces to abelian groups. It is exact, since it changes no underlying groups or maps. Thus \(R^0f_*V=V\) with its \(\mathbb Z\)-action and \(R^qf_*V=0\) for \(q>0\). Every vector space is injective by extension from a basis. But \(f_*\mathbb F_p\) is not injective as an abelian group: across \(\mathbb Z\xrightarrow{\times p}\mathbb Z\), the map from the domain sending one to \(1\in\mathbb F_p\) cannot extend, since any extension would require \(p a=1\). Its sections are nevertheless flabby, because the space has one point. The inverse coefficient functor \(\mathbb F_p\otimes_{\mathbb Z}-\) is not exact: the displayed injection becomes zero. Thus replacing the flabby acyclicity argument by unconditional preservation of injectives would be false. All positive section and direct-image groups on these point spaces vanish, consistently with Leray.

8.2 The identity map and a nonzero cohomology presheaf

Exercise 2 (intermediate). On the four-point upper poset \(b_0,b_1<c_0,c_1\), take the constant field sheaf \(k\) and \(f=1_X\). Exhibit a nonzero section of \(P_f^1(k)\) which has zero germ everywhere. Compute its associated sheaf.

Solution. The normalized cochain complex has \(C^0=k^4\), \(C^1=k^4\) indexed by the four strict edges, and no higher terms. Its differential sends vertex values to their endpoint differences. Its kernel is the constant tuples, so its rank is three and \(H^1(X,k)=k\). The cochain equal to one on \(b_0<c_0\) and zero on the other three edges is not a boundary: pairing an edge cochain with \([b_0,c_0]-[b_0,c_1]-[b_1,c_0]+[b_1,c_1]\) kills every endpoint difference and takes value one on this cochain.

Each minimal open has a least point, so its section functor is the exact evaluation at that point. Thus its positive cohomology vanishes. Explicitly the displayed edge class restricts on \(\{b_0,c_0,c_1\}\) to the boundary of vertex values \((0,1,0)\); on \(\{b_1,c_0,c_1\}\) it is already zero. These two opens cover \(X\). Hence the nonzero element of \(P_f^1(k)(X)\) has zero germ at every point. In fact all positive cohomology presheaves have zero associated sheaf: \(R^q(1_X)_*k=0\) for \(q>0\), because the identity functor is exact, and (2.2) applies. This proves concretely why the presheaf in that formula cannot be used before sheafification.

8.3 A Sierpinski product over an arbitrary base

Exercise 3 (advanced). Let \((Z,\mathcal O)\) be any ringed space and \(S=\{0<1\}\) with its upper-set topology. Describe module sheaves on \(Z\times S\), and compute every derived direct image of the projection \(\pi:Z\times S\to Z\). Show that this derived image need not detect nonzero sheaves.

Solution. An open is uniquely a pair of opens \(W_0\subseteq W_1\) of \(Z\), representing \((W_0\times\{0\})\cup(W_1\times\{1\})\). Indeed the product-open generators are \(V\times S\) and \(V\times\{1\}\), and their unions have exactly this form. Give the product its pulled-back ring sheaf, described on such a pair by \(\mathcal O(W_1)\), with its usual restrictions. Its map from \(\mathcal O(V)\) on \(V\times S\) is the identity; the stalk ring at either \((z,i)\) is \(\mathcal O_z\).

A module sheaf determines sheaves \(A_0(V)=F(V\times S)\), \(A_1(V)=F(V\times\{1\})\) and their restriction arrow \(a:A_0\to A_1\). Conversely an arrow of \(\mathcal O\)-module sheaves defines

Scalar multiplication uses restriction from \(\mathcal O(W_1)\) in the first component. The sheaf condition follows by gluing the two components, which retain their compatibility. For an original \(F\), the cover by \(W_0\times S\) and \(W_1\times\{1\}\), with intersection \(W_0\times\{1\}\), identifies it with this fibre product. Conversely evaluating (8.1) at \((V,V)\) and \((\varnothing,V)\) recovers \(A_0\), \(A_1\) and \(a\). Maps are the corresponding commuting pairs, so these are inverse equivalences. Kernels and cokernels of arrows are computed at each component, giving an abelian equivalence.

Now \(\pi_*F(V)=F(V,V)=A_0(V)\). Evaluation at the first component is exact, hence \(R^q\pi_*F=0\) for \(q>0\). The bottom edge gives \(H^n(Z\times S,F)=H^n(Z,A_0)\) in every degree. For any nonzero \(A\), the arrow \(0\to A\) reconstructs a nonzero sheaf supported on \(Z\times\{1\}\), but its entire derived direct image is zero. Its cohomology on the product is consequently zero too. Exactness of a direct-image functor does not imply that it detects nonzero objects.

8.4 A punctured six-point sphere and a signed nonzero differential

Exercise 4 (advanced). Let \(Y\) have three levels \(a_0,a_1\), \(b_0,b_1\), \(c_0,c_1\), with every point of an earlier level below every point of a later level. Put \(X=Y\setminus\{a_0\}\) and let \(j:X\hookrightarrow Y\) be the open inclusion. For the constant field sheaf, compute all terms and the nonzero differential in its Leray sequence, using (5.2).

Solution. The source \(X\) has least point \(a_1\), so its section functor is exact and \(H^0(X,k)=k\), with zero positive groups. The inverse minimal neighborhood at \(a_0\) is the four-point circle \(L=\{b_0,b_1,c_0,c_1\}\) from Exercise 2. Every other inverse minimal neighborhood has its original point as least element. Formula (2.3), whose neighborhood system has this least open, therefore gives \(j_*k=k_Y\), \(R^1j_*k=S_{a_0}\), and \(R^qj_*k=0\) for \(q>1\). Here \(S_{a_0}\) is \(k\) at \(a_0\) and zero elsewhere, with zero outgoing maps. It is the injective point direct image from the finite-poset prerequisite, since \(a_0\) is minimal. Thus \(H^0(Y,S_{a_0})=k\) and its positive groups vanish.

The normalized constant cochain spaces on \(Y\) have dimensions \(6,12,8\) in degrees \(0,1,2\). The first differential has rank five because its kernel consists of constant tuples. For a two-cochain \(z=(z_{ijk})\), define \(\Lambda(z)=\sum_{i,j,k=0}^1(-1)^{i+j+k}z_{ijk}\). It kills every one-cochain boundary: each face term is independent of one of the three indices, whose two signs cancel. Conversely use coefficients on edges \(a_i<b_j\) to subtract \(z_{ij0}\) from both corresponding triangle coordinates. The remaining values are zero for \(k=0\), and \(d_{ij}=z_{ij1}-z_{ij0}\) for \(k=1\). Coefficients on \(a_i<c_1\) eliminate \(d_{i0}\), leaving only \(d_{i1}-d_{i0}\); a coefficient on \(b_1<c_1\) eliminates the value at \(i=0\). The sole remainder is \(d_{11}-d_{10}-d_{01}+d_{00}\) at triangle \((1,1,1)\), the negative of \(\Lambda(z)\). Thus the boundary image is exactly \(\ker\Lambda\), of dimension seven. It follows that \(H^0(Y,k)=k\), \(H^1(Y,k)=0\), \(H^2(Y,k)=k\), and higher groups vanish. Let \(e_{000}\) be the two-cochain supported at \(a_0<b_0<c_0\); its cohomology class has \(\Lambda=1\).

The second page consequently has only \(E_2^{0,0}=k\), \(E_2^{0,1}=k\), \(E_2^{2,0}=k\). Its \(d_2:E_2^{0,1}\to E_2^{2,0}\) must be invertible because the positive abutment groups are zero. We now compute its coefficient rather than infer only its rank.

Take the field-stalk Godement injective resolution \(J_X^\bullet\), and put \(K=j_*J_X^\bullet\). For a diagram \(N\) on \(Y\), the normalized resolution functors \(J_Y^p(N)\) of the finite-poset lesson are exact and take every diagram to an injective diagram: their terms are products of point injectives on vector spaces. Applying them to \(K\) therefore also resolves its vertical boundaries, cycles and cohomology by injectives, since they are exact. The resulting grid is a Cartan–Eilenberg resolution. Its section double complex is

The horizontal differential deletes vertices, with the last face using the diagram transition; the vertical differential is the Godement differential. This is the model of (5.2). The transpose scaling \((-1)^{pq}\) is one at both positions \((0,1)\) and \((2,0)\), so it changes neither displayed generator. Each \(K^q\) is flabby, hence section-acyclic, so its horizontal row also identifies the total abutment with the original section cohomology on \(X\).

For constant coefficients, sections of \(J_X^q\) on an upper open have coordinates indexed by weak chains of length \(q+1\) there. In degree zero, \((d_vs)_{x\le y}=s_y-s_x\); in degree one, \((d_vt)_{x\le y\le z}=t_{yz}-t_{xz}+t_{xy}\). Let \(z\) on \(L\) be one on weak chain \((b_0,c_0)\) and zero on all others, including repeated chains. It is a cycle by that formula and generates \(H^1(L,k)\) by the four-point pairing of Exercise 2.

Let \(x_0\in B^{0,1}\) have value \(z\) at \(a_0\) and zero at every other vertex. Its horizontal differential has just one nonzero edge component: at \(a_0<b_0\) it is \(-z\) restricted to \(\{b_0,c_0,c_1\}\). Define \(x_1\in B^{1,0}\) to be zero except at that same edge, where its three coordinate values are zero at \(b_0\), minus one at \(c_0\) and zero at \(c_1\). The degree-zero formula gives \(d_vx_1=d_hx_0\). Its horizontal differential is minus one at \(a_0<b_0<c_0\) and zero at every other strict triangle. Therefore

Both classes have their specified unit pairing, so this is coefficient \(-1\), nonzero in every field. In characteristic two it equals \(+1\). This gives an actual signed second differential and explains the cancellation required by Leray, together with every higher direct image and abutment in the example.

Further reading

The Stacks Project gives freely readable proofs of the locality of higher direct images and the Leray and relative Leray spectral sequences.