Interval cohomology and homotopy invariance for sheaves

Written by GPT-6.1 Sol (OpenAI), October 2026. Original text: CC0.

An interval admits arbitrarily fine finite chains of closed subintervals. Consecutive members meet at one point, so their closed-cover resolution has only two terms. This controls arbitrary sheaf cohomology on the interval. The same chains, used as slabs in a cylinder, prove the derived projection identity and then homotopy invariance for locally constant coefficients.

Fix a unital ring \(R\), possibly noncommutative, universes and choice. Sheaves are left modules over the constant ring sheaf \(R_X\). Let \(I=[0,1]\). The base spaces below are arbitrary topological spaces. Derived functors are bounded below; the coefficient sheaves in the statements are in degree zero. No unbounded replacement or general proper base-change theorem is assumed.

Use Finite closed covers and sheaf cohomology, Sections 1–3 for closed restrictions, their cohomology and the exact signed resolution; Godement resolutions and flabby sheaf cohomology, Sections 1–2 and 6 for exact stalks, enough injectives and restriction to opens; Higher direct images and the Leray spectral sequence, Sections 2–4 for inverse-neighborhood stalks and coherent derived composition; and Injective objects in locally module stacks, Sections 6–7 for the bounded-below inverse/direct-image adjunction. Locally constant module sheaves and transition maps, Sections 1–2 gives constant models and all-morphism descent. That lesson's finite-generation theorem concerns arbitrary spaces; the connected-factor argument here also permits infinitely generated modules.

1. Local vanishing and fine interval chains

Lemma 1.1. If \(q>0\), every class in \(H^q(Y,F)\) is zero after restriction to a sufficiently small neighborhood of any specified point of \(Y\).

Proof. Represent the class by a cocycle in \(\Gamma(Y,J^q)\), for an injective resolution \(F\to J^\bullet\). Exactness of that sheaf resolution in degree \(q>0\) gives a primitive for its germ at the point. Represent the primitive locally and shrink until its differential equals the chosen cocycle. Restriction of the resolution to this open is still injective, so the restricted class is zero. This argument also makes finitely many positive-degree classes zero on one common neighborhood, by intersecting their neighborhoods. \(\square\)

For any open cover of \(I\), there is a positive number \(\lambda\) such that every nonempty subset of diameter less than \(\lambda\) lies in a cover member. Here is the finite argument. For each \(t\), choose \(\epsilon_t>0\) whose relative interval of radius \(2\epsilon_t\) is in a cover member. Compactness selects finitely many of the relative intervals of radius \(\epsilon_t\) covering \(I\). Let \(\lambda\) be the minimum of their radii. Given a point of a subset of diameter less than \(\lambda\), choose one of these smaller intervals containing it. The whole subset lies in its corresponding doubled interval, by the triangle inequality. Thus this \(\lambda\) has the required property.

Choose \(n\ge2\), with \(h=1/n<\lambda/3\), and put \(t_j=jh\). The closed intervals \(K_j=[t_{j-1},t_j]\), for \(1\le j\le n\), refine the cover, meet consecutively only at \(t_j\), and have no triple intersection. We shall also use the relative open intervals

Their diameters are at most \(3h/2<\lambda\), so each refines a cover member. They cover \(I\); only consecutive intervals meet, and those overlaps are nonempty connected intervals. The intersection of \(W_j\) with the union of its predecessors is exactly \(W_{j-1}\cap W_j\). These assertions follow directly from the gap \(h-2(h/4)>0\) between nonconsecutive intervals. Both kinds of chain may therefore be chosen subordinate to any prescribed open cover.

2. Locally constant sheaves on an interval

First note that on a nonempty connected space \(Y\), for arbitrary modules \(M,N\),

Indeed a locally constant function on a connected space is constant: a nonempty proper fibre and its complement would be disjoint nonempty open sets. For a sheaf map, the image of each constant section \(m\) is consequently a constant section with some value \(a(m)\). The map \(a\) is \(R\)-linear because the sheaf map respects addition and constant scalars. Its values determine the map on every locally constant input section by restriction and gluing. Conversely \(a\) defines that sheaf map, and the two constructions are inverse. Composition and identities are preserved. An isomorphism in (2.1) is therefore exactly an isomorphism of the modules. This argument uses connectedness of the whole domain of the map and needs no finite generation.

Proposition 2.1. A locally constant module sheaf \(L\) on \(I\) is constant. For every \(t\in I\), evaluation is a canonical isomorphism \(\Gamma(I,L)\to L_t\).

Proof. Choose the open chain (1.1) subordinate to a trivializing cover, with models \(M_j\). Start with a trivialization on \(W_1\). Suppose the sheaf on the union of the first \(j-1\) intervals has been identified with the constant model \(M_1\). Its overlap with \(W_j\) is nonempty and connected. By (2.1), the two coordinate systems there differ by one actual module isomorphism \(M_j\to M_1\). Apply that isomorphism to the whole \(W_j\) chart. The two maps now agree on the overlap and glue, as do their inverses, by the morphism-sheaf axiom. Induction supplies a global isomorphism \(L\simeq(M_1)_I\). For this model, every global section is constant and evaluation at any \(t\) is bijective onto \(M_1\). Transporting through the isomorphism proves that the original evaluation map is an isomorphism. Its definition is intrinsic, so the conclusion does not depend on the chosen trivialization. \(\square\)

Write \(e_t\) for this evaluation. The canonical transports are

They are natural in all module maps, have identity at \(s=t\), and satisfy \(\tau_{t,u}\tau_{s,t}=\tau_{s,u}\), by cancelling the middle evaluation isomorphisms. For a specified constant model \(M_I\), every transport is identity on \(M\).

3. Cohomology of arbitrary sheaves on the interval

For a closed subset \(Z\) of a space \(Y\), retain \(F[Z]=i_{Z*}i_Z^{-1}F\). This is closed restriction followed by pushforward, not sections with support. The closed chain of Section 1 gives the short exact sequence

The component of \(\delta\) at \(t_j\) is the restriction from \(K_{j+1}\) minus that from \(K_j\). This is the finite closed-cover resolution, with no higher intersection terms. Closed pushforward identifies the cohomology of each term with cohomology on its indicated closed subset. In particular the point terms have no positive cohomology, since sections of module sheaves on a point are the identity module functor and are exact.

Theorem 3.1. For every module sheaf \(F\) on \(I\), one has \(H^q(I,F)=0\) for \(q>1\). If \(\Gamma(I,F)\to F_t\) is surjective for every \(t\), then \(H^q(I,F)=0\) for every \(q>0\).

Proof. Fix a positive-degree class \(c\). Lemma 1.1 gives an open cover on which its restrictions vanish. Choose the closed chain subordinate to that cover. Its restriction to every \(K_j\) is zero too, since restriction to that closed interval factors through the open on which it vanishes. Therefore the image of \(c\) in \(H^q(I,A)\) is zero. The long exact sequence of (3.1) makes \(c\) the boundary of an element of \(H^{q-1}(I,B)\). If \(q>1\), this latter group is zero by the point calculation. Thus \(c=0\).

For \(q=1\) under the additional hypothesis, the map \(\Gamma(I,A)\to\Gamma(I,B)=\bigoplus_j F_{t_j}\) is onto. To realize prescribed values \(a_j\), start with a zero section on \(K_1\). Recursively, after choosing a section \(s_j\) on \(K_j\), choose a global section of \(F\) whose germ at \(t_j\) is \(a_j+s_j(t_j)\), and restrict it to \(K_{j+1}\) to obtain \(s_{j+1}\). The difference at that joint is \(a_j\), and later choices do not change earlier components. All stalks of closed restrictions at the joint are the original stalk \(F_{t_j}\), as proved in the closed-cover prerequisite. Thus this recursion realizes the whole tuple. Its connecting homomorphism is zero, so the same boundary description gives \(c=0\). \(\square\)

By Proposition 2.1 the extra hypothesis holds for any locally constant \(L\) on \(I\). Consequently

The second map is the actual evaluation-induced derived restriction to the point: it is the isomorphism of Proposition 2.1 in degree zero, with zero groups in other degrees. For \(L=M_I\), the augmentation \(M[0]\to R\Gamma(I,M_I)\) and every evaluation compose to identity on \(M[0]\).

4. The compact interval projection for arbitrary coefficients

Let \(p:I\times X\to X\), and let \(i_t:X\to I\times X\) be \(x\mapsto(t,x)\). Inverse image is exact and \(p^{-1}R_X=R_{I\times X}\), so it is the ordinary module-sheaf inverse image. Its local representatives and stalk formula are proved in Relative pullback, Section 1. The derived-adjunction prerequisite applies with central base \(\mathbb Z_X\) and the single possibly noncommutative algebra model \(R_X\). We shall prove

for every \(R_X\)-module sheaf, with its actual adjunction unit. The proof uses only finite closed slabs and the inverse-neighborhood stalk formula.

Degree zero. The unit \(F\to p_*p^{-1}F\) has a left inverse by restricting sections to \(i_0\). We check stalk surjectivity. Let a section \(s\) of \(p^{-1}F\) be defined on \(I\times V\), with \(x\in V\). By the local-representative construction of inverse image, near each \((t,x)\) it is the pullback of a section of \(F\) on a neighborhood of \(x\). Shrink those neighborhoods to product rectangles. Compactness of \(I\) selects finitely many whose interval factors cover \(I\).

The restriction of \(p^{-1}F\) to the fibre \(I\times\{x\}\) is the constant sheaf with value \(F_x\). Indeed the composite map from this interval to \(X\) is constant at \(x\); its inverse-image presheaf on every nonempty open has colimit \(F_x\), and its sheafification is the constant sheaf described in Section 1 of the locally constant prerequisite. The restriction of \(s\) to this connected fibre is therefore constant. All the finitely many chosen base sections have the same germ at \(x\). Intersect their domains and shrink again until these finitely many germ equalities are actual section equalities. They now give one section of \(F\) on an open \(U\ni x\). Its pullback equals \(s\) on every selected interval rectangle over \(U\), hence on all of \(I\times U\) by the sheaf axiom. This proves stalk surjectivity, and therefore the degree-zero unit is an isomorphism. Its inverse is evaluation at any endpoint, since each such evaluation is a left inverse.

Positive degrees and closed slabs. By the higher-direct-image stalk formula, it is enough to show that every class

dies after shrinking \(V\) about a specified \(x\in V\). Lemma 1.1 gives product neighborhoods of the points \((t,x)\) on which this class is zero. Choose finitely many interval factors covering \(I\), a fine closed chain \((K_j)\) subordinate to them, and one common base neighborhood \(U\ni x\) contained in all the selected base neighborhoods. The class on \(I\times U\) then restricts to zero on every closed slab \(K_j\times U\).

These slabs are closed in \(I\times U\). Their consecutive intersections are the closed copies \(\{t_j\}\times U\), and there are no higher intersections. Apply (3.1) on this cylinder, with \(E=p^{-1}(F|_U)\), slabs in place of intervals, and joints in place of points. On a joint the restricted sheaf is \(F|_U\), since \(pi_{t_j}=1\). The boundary part of its long exact sequence says that the class is the image of a tuple

For \(q=1\), the section difference map onto these joint sections is surjective without any assumption on \(F\). For a tuple \(a_j\in F(U)\), put \(b_1=0\) and \(b_{j+1}=b_j+a_j\). Pull \(b_j\) back from \(U\) to the \(j\)-th slab. Its difference with the next slab section at the joint is exactly \(a_j\). Hence the connecting map is zero and the class already vanishes on \(I\times U\).

For \(q>1\), the components of (4.3) have positive degree. Lemma 1.1 makes all of them zero on one smaller neighborhood of \(x\), since there are only finitely many. Restricting the closed-slab short exact sequence to this smaller open cylinder preserves that sequence and its differential signs. Naturality of its connecting maps makes the original class zero there. This proves the required stalk vanishing in every positive degree.

The negative derived groups vanish because a sheaf's right-derived direct image is bounded below in degree zero. Combining the positive-degree vanishing with the degree-zero calculation proves (4.1) as the stated unit isomorphism. No claim that an arbitrary direct-image stalk equals the cohomology of its literal fibre was used.

Derived pullback maps. For any continuous \(f:X\to Y\), the bounded-below derived adjunction and coherent composition from the prerequisites define

It is obtained by applying \(R\Gamma(Y,-)\) to \(G\to Rf_*f^{-1}G\), then using the section-composition isomorphism. It is natural in \(G\), and the unit of a composite adjunction is the composite of the two units after transposition. Therefore for composable \(f,g\), with the canonical inverse-image identifications, \((gf)^\sharp=f^\sharp g^\sharp\), and the identity map gives identity. These equations hold in the derived category: the comparisons are the coherent derived-adjunction and composition comparisons, not arbitrary chain lifts.

Applying \(R\Gamma\) to (4.1) and using derived composition shows that \(p^\sharp\) is an isomorphism for every sheaf \(F\). Since \(pi_t=1\),

for the coefficient \(p^{-1}F\), identifying \(i_t^{-1}p^{-1}F\) with \(F\). In particular these maps are the same for every \(t\).

5. Uniform locally constant cylinder descent

We first need a connected-factor observation. If \(J\) is nonempty and connected and \(V\) is any space, every locally constant function \(J\times V\to N\) is constant on each vertical fibre. Evaluation at one fixed \(t\in J\) consequently identifies it with a locally constant function on \(V\); the latter is locally constant because evaluation is composition with the continuous section \(x\mapsto(t,x)\). Pullback is the inverse. Applying this on every open subset of \(V\) and to the images of each constant section \(m\in M\) gives an inverse pair

The forward map is inverse image under the projection and the reverse map is restriction to \(t\). To check that they recover an entire sheaf map, use its values on constant sections over \(J\times U\), for every open \(U\subseteq V\). Those images are constant along the connected \(J\) fibres by the preceding observation. An arbitrary locally constant input is locally constant with one value \(m\), so the recovered maps agree on all inputs and all smaller opens. This also proves naturality under restriction in \(V\), compatibility with composition, and the assertion for inverse maps. No finite-generation hypothesis on either module is needed. The maps on the base in (5.1) are sheaf maps, and are not assumed to be locally constant families of elements of an abstract Hom group.

Theorem 5.1. If \(L\) is locally constant on \(I\times X\), then \(A=p_*L\) is locally constant on \(X\), and the ordinary counit

is an isomorphism. Inverse image and direct image give inverse equivalences on the categories of locally constant module sheaves, including all their maps.

Proof. Fix \(x\in X\). Each point of \(I\times\{x\}\) has a product rectangle on which \(L\) is a constant model. Choose finitely many interval factors covering \(I\), a fine open chain \((W_j)\) subordinate to them, and a common base neighborhood \(V\ni x\). On \(W_j\times V\) we have a model \((M_j)_{W_j\times V}\). Every consecutive overlap is the product of a nonempty connected interval with \(V\). Formula (5.1) says that its transition map is pulled back from one sheaf isomorphism between \((M_j)_V\) and \((M_{j+1})_V\). Its inverse is the pullback of the inverse base map.

Let \(g_{j,j+1}\) convert the next chart's coordinates to the preceding chart's coordinates. Set \(a_1=1\) on \((M_1)_V\) and recursively \(a_{j+1}=a_jg_{j,j+1}\). Pull these base sheaf isomorphisms back and apply them to the corresponding charts. Adjacent charts now have identity transitions into the same constant model \((M_1)_{I\times V}\). There are no nonconsecutive overlaps, so the maps and their inverses glue to

For every open \(U\subseteq V\), the connected-factor observation identifies sections of this model over \(I\times U\) with sections of \((M_1)_V\) over \(U\). Thus \(A|_V\simeq(M_1)_V\), proving local constancy. Under these local identifications the counit evaluates the pulled-back sections and is identity on the constant model. Hence it is an isomorphism globally by the stalk criterion.

For a locally constant \(F\) on \(X\), its pullback is locally constant: the inverse image of \(M_U\) is the constant model on \(p^{-1}U\). The unit is the degree-zero isomorphism of Section 4. Together with the counit (5.2), these are the original adjunction's unit and counit and retain its triangle identities. They therefore give the asserted equivalence on objects and on all morphisms. The argument allows arbitrary modules; it never replaces a base sheaf automorphism by one element of a discrete automorphism group. \(\square\)

Let \(\epsilon_t:A\to i_t^{-1}L\) be (5.2) restricted to the \(t\)-slice. It is an isomorphism and defines natural sheaf transports

These obey the identity and composition laws of (2.2), now as sheaf maps on \(X\). For a specified constant model on the whole cylinder, the identifications with that model make every \(\Theta_{s,t}\) identity.

6. The cohomological homotopy theorem

For a locally constant \(L\) on the cylinder, every endpoint restriction

is an isomorphism, and these maps satisfy

To prove both assertions, use (5.2) to replace \(L\) by \(p^{-1}A\). For this coefficient, (4.5) gives the same inverse to \(p^\sharp\) on every slice. Naturality of (4.4) under the sheaf isomorphism (5.2) gives (6.1) and (6.2), with exactly the maps \(\epsilon_t\) used in (5.4). Thus the comparisons are the given restriction maps and transports, not just abstract isomorphisms between cohomology groups.

Let \(H:I\times X\to Y\) be a homotopy from \(f_0\) to \(f_1\), and let \(G\) be locally constant on \(Y\). Inverse image preserves local constancy: pulling back a constant chart gives a constant model on its inverse open. This follows directly from the canonical map for constant presheaves, whose stalk maps identify the same module with itself, followed by the stalk isomorphism criterion. Thus \(L=H^{-1}G\) is locally constant. Apply (5.4) to obtain a natural isomorphism of sheaves

Functoriality of the pullback maps gives \(f_t^\sharp=i_t^\sharp H^\sharp\). Combining this with (6.2) proves the homotopy identity

For constant coefficients \(G=M_Y\), its inverse image is canonically \(M_{I\times X}\). The normalization after (5.4) says that \(\alpha_H\) is identity on \(M_X\). Consequently \(f_0^\sharp=f_1^\sharp\) under those canonical constant-coefficient identifications. For general local coefficients the transport is part of the statement and need not be identity.

Corollary 6.1. A homotopy equivalence \(f:X\to Y\) induces an equivalence of locally constant module-sheaf categories by inverse image. For locally constant \(G\) its actual map \(f^\sharp\) is an isomorphism on derived sections.

Proof. Choose a homotopy inverse \(g\). The sheaf transports (6.3) for \(fg\simeq1_Y\) and \(gf\simeq1_X\) identify both composite inverse-image functors with identity. They are therefore quasi-inverse equivalences of the locally constant categories.

For the assertion about the actual derived map, put \(F=f^{-1}G\), \(G'=g^{-1}F\), and write \(a=f_G^\sharp\), \(b=g_F^\sharp\), \(a'=f_{G'}^\sharp\). The homotopies and (6.4) give coefficient isomorphisms \(\theta:G'\to G\) and \(\psi:f^{-1}G'\to F\) such that

Set \(c=R\Gamma(\theta)b\), so \(ca=1\). Naturality of \(f^\sharp\) in \(\theta\) gives \(aR\Gamma(\theta)=R\Gamma(f^{-1}\theta)a'\). The second equation of (6.5) consequently makes

an automorphism of \(R\Gamma(X,F)\). Also \((ac)^2=a(ca)c=ac\). An invertible idempotent is identity, so \(ac=1\) as well. Thus \(c\) is a two-sided inverse to the specified map \(a\). No coherence between arbitrary chosen homotopies was assumed. \(\square\)

In particular a nonempty contractible space has only constant locally constant module sheaves up to isomorphism. For such a sheaf \(L\), evaluation at any point is an isomorphism \(R\Gamma(X,L)\to L_x[0]\): the inclusion of that point is itself a homotopy equivalence, so apply Corollary 6.1 to its restriction map. All positive cohomology vanishes. Combining this with the finite closed-cover theorem gives a further useful conclusion: if a finite closed cover has every nonempty finite intersection contractible, then a locally constant coefficient sheaf is acyclic on all those intersections. Its cohomology is computed by the actual closed-intersection section complex. The restriction maps in that complex still belong to the given coefficient sheaf; no arbitrary identification of all fibres with one fixed module is imposed.

7. Four exercises with complete solutions

Exercise 1 — the interval bound is sharp

Introductory. For \(j:(0,1)\hookrightarrow I\) and a nonzero \(R\)-module \(M\), compute the cohomology of \(F=j_!M_{(0,1)}\). Does Theorem 3.1 force its first cohomology to vanish?

Solution. Extension by zero has stalk \(M\) in the open interval and zero at the endpoints, as proved in the Godement prerequisite. There is a stalkwise exact sequence

The first map extends the locally vanishing sections by zero; the last map is the pair of closed restrictions. In the interior its target stalk is zero and its kernel stalk is \(M\). At an endpoint it is identity from \(M\) to that endpoint's only surviving copy. These checks prove exactness. The last two sheaves are acyclic by (3.2) and the point calculation. Their section map is the diagonal \(M\to M\oplus M\). Its kernel is zero, and its cokernel is \(M\) by \((a,b)\mapsto b-a\). The long exact sequence therefore gives \(H^0(I,F)=0\), \(H^1(I,F)\simeq M\), and zero in every higher degree. The global-to-stalk map at an interior point is \(0\to M\), not surjective. Thus the additional hypothesis for degree one fails, and the general bound cannot be reduced to zero.

Exercise 2 — a cylinder over a non-Hausdorff finite space

Intermediate. Let \(X\) be the six-point cycle with upper-set topology from Exercise 2 of Finite closed covers and sheaf cohomology, and take its constant coefficient diagram with module \(M\). Compute the cohomology of its inverse image on \(I\times X\), and compare the endpoint maps. Does acyclicity of every interval fibre imply acyclicity of the whole cylinder?

Solution. The cited exercise proves over arbitrary \(R\)-modules that the coefficient sheaf \(F\) has \(H^0(X,F)=M\), \(H^1(X,F)=M\), and zero in higher degrees, by its explicit three-member closed cover. The projection unit (4.1), followed by the proved derived-composition map, identifies its entire derived section object with that of \(p^{-1}F\). Thus the cylinder has exactly those same cohomology groups, and the identification is the actual \(p^\sharp\).

Both endpoints give its inverse by (4.5), so they are the same map after the canonical coefficient identifications. Each individual interval fibre has the constant coefficient \(F_x\) and no positive cohomology by (3.2). The cylinder nevertheless retains the nonzero degree-one group of the base when \(M\ne0\). The projection theorem identifies the derived direct image with \(F\); it does not erase the subsequent cohomology on \(X\). This example is included in the theorem despite the absence of a Hausdorff hypothesis on the base.

Exercise 3 — a loop can give nonidentity coefficient transport

Advanced local coefficients. On \(Y=\mathbb R/\mathbb Z\), let \(L_T\) be the explicitly glued sheaf of Exercise 1 in Locally constant module sheaves and transition maps. Choose \(W_+=(0,1/2)\), \(W_-=(1/2,1)\), viewed in the circle, so the transition from \(V\)-coordinates to \(U\)-coordinates is identity on \(W_+\) and \(T\) on \(W_-\). Calculate (6.3) for the loop \(\gamma(t)=t\bmod\mathbb Z\), regarded as a homotopy between two maps from a point to \(Y\).

Solution. Use the \(V\)-frame at the common endpoint \(0\bmod\mathbb Z\). Start with value \(m\) there. Along the initial \(V\)-chart, transport keeps its coordinate \(m\). On entering the \(U\)-chart through \(W_+\), the transition is identity, so its \(U\)-coordinate is still \(m\). On returning to \(V\) through \(W_-\), the equation between coordinates is \(m_U=T m_V\). Its new \(V\)-coordinate must therefore be \(T^{-1}m\), and it stays so until the endpoint. These chart computations define the unique global section of the pulled-back locally constant sheaf on \(I\), by Proposition 2.1. Its endpoint evaluation is exactly the transport constructed in Section 5. Hence \(\alpha_\gamma=T^{-1}\) in the chosen frame.

For nonidentity \(T\), this transport is not identity, although the two endpoint maps from a point are the same. There is no conflict with (6.4). Its derived identity is supplied by that theorem; in degree zero it says that \(T^{-1}\) fixes the image of evaluation on global sections of \(L_T\). The cited exercise identifies that image with \(\ker(T-1)\), on which \(T^{-1}\) is indeed identity. For constant coefficients \(T=1\), the whole coefficient transport is identity, as required by the normalization. This calculation concerns the given loop and gluing and uses no classification by a fundamental group.

Exercise 4 — noncompact fibres do not determine direct-image stalks

Advanced boundary case. Let \(q:\mathbb R_t\times\mathbb R_x\to\mathbb R_x\) be projection, let \(Z=\{(t,x):tx=1\}\), and let \(E=i_*M_Z\) for its closed inclusion \(i\) and a nonzero module \(M\). Compare \((q_*E)_0\) with sections of \(E\) on the literal fibre over zero. Locate the failure of a uniform-neighborhood argument.

Solution. The set \(Z\) is closed as the inverse image of \(\{1\}\) under the continuous multiplication map. Projection identifies it with \(\mathbb R\setminus\{0\}\), with inverse \(x\mapsto(1/x,x)\). Over an interval \((-\epsilon,\epsilon)\), it has two connected components, each an interval. Sections of the constant sheaf on those components are independently constant values. Thus \((q_*E)((-\epsilon,\epsilon))=M\oplus M\), with identity restriction maps as \(\epsilon\) decreases. These intervals are cofinal at zero, so

Every point \((t,0)\) is outside the closed set \(Z\). The stalk formula for closed pushforward makes \(E\) zero at each such point; exact inverse image therefore makes its restriction to the fibre the zero sheaf. Its fibre sections are zero, so they do not recover (7.2).

For a concrete neighborhood, the open set \(\{(t,x):|tx|<1/2\}\) contains the entire fibre over zero. It contains no product \(\mathbb R\times V\) with \(V\) a neighborhood of zero: choose a nonzero \(x\in V\) and \(t=1/x\). This is the uniform shrinking step for which compactness of \(I\) supplied finitely many interval rectangles. The example concerns an arbitrary coefficient sheaf for a noncompact projection. It is not a counterexample to the proved compact interval identity (4.1), which also specifies the pulled-back coefficient.

8. Further reading

Pierre Schapira's freely accessible An Introduction to Sheaves on Grothendieck Topologies, October 2026, Sections 3.5–3.6, treats interval cohomology and homotopy invariance. The argument here uses the finite closed-cover resolution and local vanishing of positive-degree classes, including in the cylinder projection proof. For general direct images, retain the inverse-neighborhood stalk formula and the hypotheses of any additional fibre comparison being used.