Draft lesson.

Sheaf Ext and the local-to-global spectral sequence

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

A local extension class and a global extension class are different kinds of data. Even when every extension splits near each point, the local splittings may fail to agree. Sheaf Ext records the local classes; cohomology of sheaf Hom records the gluing. Their relation is the local-to-global spectral sequence. Finite presentations also matter when passing to a stalk: infinite families can have a nonzero germ although every individual coordinate has zero germ.

Throughout, \((X,\mathcal O)\) is a ringed space with commutative unital rings. Modules are \(\mathcal O\)-modules, and all sets lie in fixed universes; choice is used. Commutativity gives the usual \(\mathcal O\)-module structure on internal Hom. Derived internal Hom below has its first input bounded above and its second input bounded below. No perfectness, separation or acyclic-cover assumption is imposed for the main comparison.

Use Godement resolutions and flabby sheaf cohomology, Sections 1–9, for open extension by zero, enough injectives and flabby acyclicity. Derived perfect duality for twisted module complexes, Sections 1–3, already constructs the full bounded derived internal Hom; we use its neutral, untwisted specialization. Derived Hom and bounded resolution pairs, Sections 2 and 5–6, gives global derived Hom, its normalized differential and its cohomology as derived morphisms. Universal cohomology and acyclic resolutions, Section 5, and Postnikov towers and Grothendieck spectral sequences, Sections 1–5, supply the canonical composition and spectral maps with finite convergence. Torsors, first cohomology and line bundles supplies the positive cocycle convention and the finite-circle modules used below.

1. Internal Hom into an injective is flabby

Write \(\mathcal H(F,G)(U)=\operatorname{Hom}_{\mathcal O|_U}(F|_U,G|_U)\). Restrictions restrict the whole module morphism. Compatible morphisms glue because their values on local sections glue, and the module equations hold locally. Thus this is a sheaf. Scalars act by multiplication in the target; commutativity makes the resulting maps \(\mathcal O\)-linear. Kernels in the second variable are computed on these morphisms, so \(\mathcal H(F,-)\) is additive and left exact. Evaluation at \(X\) gives exactly the global Hom group.

Let \(I\) be injective and let \(U\subseteq V\) be open. The restriction \(I|_V\) is injective by the open-extension proof in the first prerequisite. The canonical map \(j_!(F|_U)\to F|_V\), for \(j:U\hookrightarrow V\), is monic: its stalks are identities inside \(U\) and zero inclusions outside. A map \(F|_U\to I|_U\) corresponds by the actual zero-extension adjunction to a map from that subobject into \(I|_V\). Injectivity extends it to \(F|_V\). The two adjunction operations are restriction and local extension by zero, so this extension really restricts to the prescribed map. This proves surjectivity of \(\mathcal H(F,I)(V)\to\mathcal H(F,I)(U)\). Hence

It is not always injective. On a point with ring \(\mathbb Z\), \(I=\mathbb Q/\mathbb Z\) is injective but \(\operatorname{Hom}(\mathbb Z/2,I)=\mathbb Z/2\) is not: its map sending one to one cannot extend across \(\mathbb Z\xrightarrow{2}\mathbb Z\). Flabbiness, which implies section acyclicity, is the property needed here.

2. The actual global derived comparison

For module sheaves, an injective resolution \(G\to I_G\) defines \(R\mathcal H(F,G)=\mathcal H(F,I_G)\). Define \(\mathcal Ext^q(F,G)=\mathcal H^q(R\mathcal H(F,G))\). The existing internal-Hom construction proves the full analogous bifunctor on \(D^-(\mathcal O)^{\mathrm{op}}\times D^+(\mathcal O)\). We retain that construction, including its opposite-source maps.

Let \(P\) be bounded above and \(Q\to I_Q\) a bounded-below injective replacement. Its model has terms and differential

If \(P^i=0\) for \(i>b\) and \(I_Q^j=0\) for \(j<a\), the contributing indices are \(a-n\le i\le b\); thus each sum is finite. By (1.1) every summand is flabby, as is its finite sum: extend the finitely many section components. The complex is bounded below by \(a-b\) and consists of section-acyclic terms. The proved acyclic-resolution comparison therefore computes its derived global sections by its ordinary section complex.

Taking sections in (2.1) gives precisely the normalized global Hom complex of the third prerequisite, with the same differential. Its model computes global derived Hom. The comparison just described is consequently a natural isomorphism

For a fixed module \(F\), this is exactly the canonical derived-composition map for \(\mathcal H(F,-)\) followed by sections: (1.1) verifies its acyclicity hypothesis. For complexes it is represented by the same section-comparison map on (2.1). Precomposition in \(P\), postcomposition in \(I_Q\), and the chosen replacement augmentations commute with it. The already proved Hom homotopies and inverse quasi-isomorphism maps extend this to all derived arrows in both variables. Choice comparisons and their compositions therefore agree. Taking cohomology gives, for every integer \(n\),

For sheaves these groups are the ordinary global \(\operatorname{Ext}_X^n(F,G)\). No global projective model for \(F\) was assumed.

3. Ext sheaves as associated presheaves

Restriction of \(I_G\) to an open remains injective, and sections of \(\mathcal H(F,I_G)\) on \(U\) are the global Hom complex for \(F|_U,G|_U\). Exact associated-sheaf formation therefore gives

The map sends a local cochain class to its image in the cohomology sheaf. A cycle has zero image exactly when it is locally a boundary, and every section of the quotient cohomology sheaf locally has a cycle representative. These statements prove both directions of (3.1). They also prove the analogous assertion for the bounded pair \((P,Q)\). All restrictions, precompositions and postcompositions are those of the actual Hom complexes, so the isomorphisms are natural.

In particular,

Filtered colimits of groups are exact, so cohomology commutes with this stalk colimit. There is always a canonical comparison from (3.2) to \(\operatorname{Ext}^q_{\mathcal O_x}(F_x,G_x)\): apply the exact stalk functor to a local derived morphism \(F|_U\to G|_U[q]\). Exactness preserves quasi-isomorphisms, so it defines a functor on the derived categories and a map on these morphisms. These maps agree under shrinking \(U\), giving the claimed comparison. In degree zero it is the familiar map from germs of sheaf morphisms to morphisms of stalks. It need not be an isomorphism.

4. The local-to-global sequence and its low-degree maps

Apply the proved derived hypercohomology spectral sequence to \(R\mathcal H(P,Q)\), and transport its abutment by (2.2). This gives

It is natural contravariantly in \(P\) and covariantly in \(Q\), including derived arrows and every replacement comparison, since the internal Hom is already such a bifunctor. All open restrictions commute with its local models and the section comparisons. The support is \(p\ge0,q\ge a-b\) with the bounds of Section 2. The finite-convergence proof supplies a finite filtration in every total degree and \(d_r\) of bidegree \((r,1-r)\). It gives associated pieces rather than a chosen splitting. For sheaves \(F,G\) the support is first quadrant, and the actual low-degree sequence is

The first and last maps are induced by \(\mathcal H(F,G)[0]\to R\mathcal H(F,G)\) and (2.2). The middle edge sends a global extension to the section of its local extension classes in (3.1). It can equally be read from the quotient of the first truncation of the internal Hom complex. The transgression is the actual second differential of its fixed-cone truncation tower, with the signs of the spectral prerequisite.

For exactness, \((1,0)\) has no incoming or outgoing differential after page two and is the bottom piece of total degree one. The stable value at \((0,1)\) is \(\ker d_2\), since it has no incoming differential and no later outgoing one. At \((2,0)\) the stable value is \(\operatorname{coker}d_2\), since this is its only possible incoming differential and it has no outgoing one. These pieces and the specified edges give (4.2), including injectivity at its first term.

5. The extensions which split locally

The kernel of the local-class edge in (4.2) consists exactly of extensions which split on an open cover. Indeed a local section class is zero in its associated sheaf exactly when it becomes zero near every point; a short exact extension class is zero precisely when that extension splits, by the fixed-endpoint Baer theorem.

Here are both gluing directions for this kernel. Choose local splittings \(s_i:F|_{U_i}\to E|_{U_i}\). Their differences have the form \(s_i-s_j=\iota h_{ij}\) for unique \(h_{ij}\in\mathcal H(F,G)(U_i\cap U_j)\). They satisfy \(h_{jk}-h_{ik}+h_{ij}=0\). Conversely such a cocycle glues the local modules \(G\oplus F\) by

The cocycle equation gives every triple compatibility and the negative cocycle gives the inverse transition. More explicitly, sections are families satisfying these equations. Gluing the component sections proves the sheaf condition. Each chart is \(G\oplus F\), the kernel of projection is the original \(G\), and projection is locally onto \(F\). This produces the required locally split short exact sequence. Expressing an original extension in its splitting coordinates and gluing back gives inverse maps, with identities on both endpoints.

Changing splittings by \(s_i+\iota a_i\) changes \(h\) to \(h-\delta a\); the coordinate change is \((g_i,f_i)\mapsto(g_i-a_i(f_i),f_i)\). Conversely every endpoint-preserving isomorphism has these local coordinates and gives such a coboundary. Refining two splitting covers gives the same construction. The torsor classification from the last prerequisite, applied to the sheaf \(\mathcal H(F,G)\), thus identifies these classes with \(H^1(X,\mathcal H(F,G))\). Baer addition adds the \(h_{ij}\), by the diagonal pullback and codiagonal pushout of the two local rows; inverse classes negate them.

This is the positive first edge of (4.2). To check the convention, embed \(G\hookrightarrow I^0\) and extend its map across \(E\) by injectivity. Composing with \(s_i\) gives local maps \(b_i:F\to I^0\) with \(b_i-b_j=j h_{ij}\). Their images in the quotient of \(\mathcal H(F,I^0)\) by \(\mathcal H(F,G)\) glue. Its connecting torsor has local origins \(b_i\), so represents \([h]\) with the positive convention of the last prerequisite. Composing with the injective differential gives a global cycle \(z:F\to I^1\). The normalized extension class is \([z]\), and connecting-map naturality sends the former boundary class to this one under the first edge. Thus the explicit gluing bijection agrees with that exact-sequence map.

6. Finite local models and stalk comparison

If \(F\) is locally finitely presented, the degree-zero stalk comparison is an isomorphism. On a neighborhood take a presentation \(\mathcal O^r\to\mathcal O^s\to F\to0\) with finite \(r,s\). A map \(F_x\to G_x\) is specified by the images of the finitely many generators. Choose their section representatives on one common neighborhood. The finitely many relation images have zero germ and vanish on a common smaller neighborhood, so the map descends to \(F\) there. Two such maps with the same stalk map agree after shrinking, because their values on finitely many generators have zero difference germs. This proves surjectivity, injectivity and choice-independence of the actual germ map.

For all degrees assume the following precise stronger condition near \(x\): \(F\) has an exact augmented resolution \(\cdots\to P^{-2}\to P^{-1}\to P^0\to F\to0\) on a neighborhood, each \(P^i\) finite free there. No bound on its length is required. Then

To prove this, restrict the global internal Hom to that neighborhood and replace its first argument by \(P\), as justified by the internal-Hom comparison prerequisite. The map \(\mathcal H(P,G)\to\mathcal H(P,I_G)\) is a quasi-isomorphism: its cone is the finite-diagonal Hom total with the bounded-below acyclic cone of \(G\to I_G\). Hom from each finite free \(P^i\) is exact on sheaves. A total cycle can be removed starting at the largest source resolution index, where its target component is closed. Lift that component locally, subtract its boundary, and move to the preceding source index. Each step lowers the remaining target index, so the target lower bound terminates the process; only finitely many neighborhood restrictions are needed. This proves local exactness of the total cone. Consequently \(\mathcal H(P,G)\) computes the internal derived Hom.

At the stalk, \(\mathcal H(P^i,G)_x=\operatorname{Hom}_{\mathcal O_x}(P_x^i,G_x)\) by finite freeness. These are the actual finite-tuple identification maps, and commute with the normalized Hom differential \(d_Gf-(-1)^q f d_P\). Stalks are exact, and \(P_x\) is a free resolution, so taking cohomology gives precisely algebraic Ext via the projective-model theorem. The comparison is (6.1): applying the exact stalk functor to the local representative and its quasi-isomorphism gives the same derived morphism. Thus it is natural and independent of this chosen local resolution. Degree-zero finite presentation alone does not justify the higher-degree conclusion.

For a finite locally free \(F\), the map \(F^*\otimes G\to\mathcal H(F,G)\), \(\ell\otimes g\mapsto(f\mapsto\ell(f)g)\), is an isomorphism: a local basis gives the inverse by its finite dual basis, and these inverses glue since the forward map is canonical. This functor is exact. Hence \(\mathcal Ext^{q>0}(F,G)=0\) and the actual bottom edges give \(\operatorname{Ext}^n_X(F,G)=H^n(X,F^*\otimes G)\). The same proof works for local finite-projective retracts using their dual bases. It does not make such sheaves globally projective.

7. Four exercises with full solutions

7.1 A torsion sheaf with two global Ext degrees

Exercise 1 (basic). On the four-point circle \(u,v<a,b\), let \(\mathcal O=\mathbb Z_X\) and \(F=(\mathbb Z/n)_X\), with \(n\ge2\). Compute all sheaf \(\mathcal Ext^q(F,\mathcal O)\) and all global \(\operatorname{Ext}^m_X(F,\mathcal O)\).

Solution. The local free resolution \(0\to\mathcal O\xrightarrow{n}\mathcal O\to F\to0\) is exact on stalks. Its normalized Hom complex into \(\mathcal O\) has \(\mathcal O\xrightarrow{-n}\mathcal O\) in degrees zero and one. Thus sheaf Hom is zero, \(\mathcal Ext^1(F,\mathcal O)=F\), and all other sheaf Ext groups are zero. The spectral sequence has only row \(q=1\), so \(\operatorname{Ext}^m_X(F,\mathcal O)=H^{m-1}(X,F)\) for \(m\ge1\), with zero in degree zero.

The two minimal opens \(U=\{u,a,b\}\) and \(V=\{v,a,b\}\) have least points, so sections there are exact evaluation functors. The intersection is the discrete pair \(\{a,b\}\), whose section functor is also exact. Mayer–Vietoris, proved with its full connecting maps in the Čech prerequisite linked by the last prerequisite, therefore computes cohomology by \((\mathbb Z/n)^2\to(\mathbb Z/n)^2\), \((r,s)\mapsto(s-r,s-r)\). Its kernel and cokernel are both \(\mathbb Z/n\); all higher groups vanish. Consequently global Ext is \(\mathbb Z/n\) in degrees one and two and zero in every other degree. Taking global sections of the two-term Hom complex alone would miss degree two: its free sheaf terms are not being asserted globally projective.

7.2 Infinite rank and a zero coordinate in every stalk

Exercise 2 (intermediate). Let \(X=\{\infty\}\cup\{1,2,\ldots\}\), with the integers isolated and neighborhoods of \(\infty\) containing all sufficiently large integers. Let \(\mathcal O=k_X\) and \(P=\bigoplus_{n\ge1}\mathcal O\). Show that \(\mathcal H(P,\mathcal O)_\infty\to\operatorname{Hom}_k(P_\infty,k)\) need not be injective.

Solution. Restriction commutes with sheaf direct sums, and their Hom universal property gives \(\mathcal H(P,\mathcal O)=\prod_{n\ge1}\mathcal O\), with sectionwise products. For each \(n\), let \(\chi_n\) be the characteristic function of the isolated clopen singleton \(\{n\}\). It is locally constant and has zero germ at \(\infty\). The family \((\chi_n)_n\) is a section of this product, whose germ is not zero: every neighborhood contains some integer \(n\) where its \(n\)-th component is nonzero. On the other hand \(P_\infty=\bigoplus_n k\), since stalks commute with colimits. Its induced linear map sends every basis element to the germ of \(\chi_n\), hence to zero. This proves the stated kernel element, even though \(P\) is a free module sheaf of countable rank. The finite-generation step in the degree-zero proof of Section 6 cannot be replaced by infinitely many separate neighborhood choices.

7.3 Every self-extension of a nontrivial line bundle

Exercise 3 (advanced). Let \(L_\lambda\) be the line bundle on the four-point circle with \(\lambda\in k^\times\), all arrows identity except \(v\to b\) multiplication by \(\lambda^{-1}\). Classify its self-extensions with fixed endpoints, give their matrices, and identify the class and Baer sum signs.

Solution. Its endomorphism sheaf is \(\mathcal O=k_X\) through scalar multiplication, and its positive sheaf Ext groups vanish by finite local freeness. Thus \(\operatorname{Ext}^m_X(L_\lambda,L_\lambda)=H^m(X,k_X)\), equal to \(k\) in degrees zero and one and zero in higher degrees, by the same two-open difference calculation as Exercise 1 over \(k\).

For \(\mu\in k\), take stalks \(k^2\), inclusion into the first coordinate, projection to the second, and arrows identity except

The two endpoint arrows are those of \(L_\lambda\), so these define exact sequences with the stipulated endpoints. The coordinate splittings on \(U,V\) have difference zero at \(a\) and \(\mu\) at \(b\), measured in \(\mathcal End(L_\lambda)=k_X\). Indeed an input in the \(U\)-frame at \(b\) equals \(\lambda\) times the \(V\)-frame; its \(V\)-splitting becomes \((-\mu,1)\), while its \(U\)-splitting is \((0,1)\). Thus \(s_U-s_V\) has first coordinate \(\mu\). Section 5 identifies its class with \(\mu\), using the difference-pair invariant at the two overlap points. Every class is obtained, two are equivalent exactly when their \(\mu\) agree, splitting occurs exactly at zero, and Baer sum adds \(\mu\). The minus sign in (7.1) is required by this positive splitting-difference convention. This applies equally to \(\lambda\ne1\), when the line bundle itself has no nonzero global section.

7.4 A sheaf Ext stalk invisible to every algebraic stalk Ext

Exercise 4 (advanced). Construct a ringed space with all stalk rings equal to a field and a countably free sheaf \(P\) for which \(\mathcal Ext^1(P,M)\) has a nonzero stalk, although \(\operatorname{Ext}^1_{\mathcal O_x}(P_x,M_x)=0\) at every point.

Solution. Take copies \(C_n\) of the four-point circle, and set \(X=\{\infty\}\cup\coprod_{n\ge1}C_n\). A subset not containing \(\infty\) is open when its intersection with every \(C_n\) is open. A subset containing \(\infty\) is open when these intersections are open and it contains the whole \(C_n\) for all sufficiently large \(n\). These rules form a topology, and \(U_N=\{\infty\}\cup\coprod_{n\ge N}C_n\) are a cofinal system of neighborhoods of \(\infty\). Each \(C_n\) is clopen: its complement contains such a tail. Put \(\mathcal O=k_X\), the locally constant field sheaf. Its stalk at every point, including \(\infty\), is \(k\).

On each \(C_n\) use the nonsplit extension \(0\to k\to B_n\to k\to0\) with rank-two stalks, all arrows identity except \(v\to b\) equal to \(\begin{pmatrix}1&1\\0&1\end{pmatrix}\), and quotient the second coordinate. This is also the explicit unit extension of the last prerequisite. It has no global lift of one: a global pair \((x,t)\) at \(u\) and \((y,t)\) at \(v\) would satisfy \(x=y\) at \(a\) and \(x=y+t\) at \(b\), forcing \(t=0\).

Extend these rows by zero from the clopen components and take their sheaf direct sum. This gives

Exactness is checked at stalks: on a component it is the original row, while at \(\infty\) all three stalks are zero because stalks commute with direct sums and each summand vanishes near \(\infty\). Let \(P=\bigoplus_{n\ge1}\mathcal O\). For each \(n\), the unit on \(C_n\), extended by zero, gives a global section \(h_n\) of \(M\). The family \(h=(h_n)_n\) is therefore a section of \(\mathcal H(P,M)=\prod_n M\).

Its germ at \(\infty\) has no lift through \(\mathcal H(P,B)\). Such a lift would be represented on some \(U_N\) by a family \((b_n)_n\). For any \(n\ge N\), restriction of \(b_n\) to the entire clopen component \(C_n\) would lift its unit through \(B_n\to k\), contrary to the preceding calculation. This argument works for every \(N\), not just a finite truncation. The long exact sequence of the derived functor \(\mathcal H(P,-)\), followed by the exact stalk functor, now sends this unliftable germ to a nonzero element of \(\mathcal Ext^1(P,M)_\infty\).

Nevertheless every \(P_x=\bigoplus_n k\) is a free \(\mathcal O_x=k\)-module. Every linear map from a free vector space lifts across a surjection by lifting its basis, so these sources are projective and all the algebraic stalk Ext groups in question are zero. At \(\infty\) the target stalk \(M_\infty\) is even zero. Thus the canonical comparison of Section 3 fails in degree one, despite freeness of every stalk. Uniform local lifting of infinitely many maps is precisely what is missing; it is supplied by the finite local models in Section 6.

Further reading

The Stacks Project gives freely readable accounts of Ext sheaves and their local-to-global sequence and global derived Hom. The bounds and stalk-finiteness hypotheses used here are specified and proved above.