Abelian gerbes and second sheaf cohomology

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

Local objects may exist without admitting compatible identifications. A torsor records the failure to choose one local origin consistently; a gerbe records the failure to choose both objects and their identifications consistently. For a fixed abelian sheaf of automorphisms, this second obstruction is exactly second sheaf cohomology. We prove the classification by turning arrows of a gerbe into an exact row of abelian sheaves and by turning such a row back into a stack.

Let \(C\) be a small site and \(E=\mathsf{Sh}(C)\) its sheaf topos. Write \(1\) for its terminal sheaf, \(A\) for an abelian group sheaf, and \(\mathbb Z=\mathbb Z[1]\) for the free abelian sheaf on \(1\). All object, arrow and relation sets lie in fixed ambient universes. The site need not be subcanonical or have enough points. We work over arbitrary base sheaves using Stacks over sheaves and epimorphic descent, Theorems 2.1 and 3.1. Thus intersections below are products in \(E\), including when they are not represented in \(C\).

Module sheaves on the element site, Lemma 7.1 and Proposition 7.2 supplies sheafification, its finite local descriptions, local exactness and the Grothendieck category of abelian sheaves. The entire injective construction supplies enough injectives. Higher extensions, roofs and Yoneda composition, Sections 1–4 proves classification by exact rows, their endpoint-preserving comparisons and the sign of their derived classes. Stackification by cartesian sections supplies stackification with unchanged morphism sheaves and its functor universal property. These are the proof prerequisites. Basic references for the notion of a gerbe and its cocycle obstruction are [Stacks, Gerbes] and [Stacks, Second cohomology].

1. Fixed bands and local equivalences

An \(A\)-banded gerbe \(\mathsf G\) is a stack in groupoids with objects locally over \(1\), with any two objects locally isomorphic, and with specified identifications

They commute with restriction. For every arrow \(r:x\to y\), they satisfy

A cartesian functor of banded gerbes respects these identifications. Equivalence classes always use this fixed band: changing \(A\) by an automorphism is not silently allowed. The sheaf \(\mathcal Isom(x,y)\) is an \(A|_T\)-torsor, with action \(r+a=r\iota_x(a)=\iota_y(a)r\). Indeed it has local sections by local connectedness; two arrows differ by one unique automorphism, and (1.2) identifies its two actions. Composition adds the two torsor coordinates.

Lemma 1.1. Every functor between \(A\)-banded gerbes that respects the band is an equivalence.

Proof. Its map on each isomorphism sheaf is an equivariant map of \(A\)-torsors. Locally choose one arrow; both torsors become \(A\), and the map is a translation. It is consequently an isomorphism of sheaves. On sections over any base this proves full faithfulness. For an object \(y\) in the target, locally choose source objects \(x_i\) and identifications \(F x_i\simeq y\). The comparisons between their images lift uniquely by full faithfulness. They satisfy the cocycle equation because their images do. Effective descent in the source glues the \(x_i\), and descent of morphisms glues their identifications with \(y\). Hence \(y\) is in the essential image over its original base. \(\square\)

Let \(BA\) denote the stack of \(A\)-torsors and equivariant isomorphisms. Sheaves, their actions and the assertion that the action is simply transitive descend: first descend the underlying sheaves, then descend the action maps, and check their equations locally. The sheaf-category descent used here is Stacks over sheaves, Lemma 4.1. Local torsor origins show that \(BA\) is an \(A\)-banded gerbe. Its automorphism corresponding to \(a\) is translation by \(a\).

Lemma 1.2. A gerbe has an object over \(1\) if and only if it is band-preservingly equivalent to \(BA\).

Proof. Given \(x\in\mathsf G(1)\), send \(y\) to the torsor \(\mathcal Isom(x,y)\). On arrows use postcomposition. This is a band-preserving functor, hence an equivalence by Lemma 1.1. Conversely the image of the trivial torsor under an equivalence \(BA\simeq\mathsf G\) is a global object. More explicitly, the inverse transports \(x\) by a torsor: choose local torsor origins, use their differences as automorphisms of \(x\), and apply effective descent. A change of origins changes the descent datum by its specified comparison and therefore gives the same equivalence. \(\square\)

A gerbe with a global object is called neutral. A neutral gerbe can have many nonisomorphic global objects; neutrality is an existence assertion.

2. An exact row made from objects and arrows

Choose an epimorphism \(U\to1\) and \(x\in\mathsf G(U)\). Such a choice exists: take a small family of local objects and combine them over the coproduct of their bases by descent. Define a sheaf \(R\) of arrows. A section over \(T\) consists of \(u,v:T\to U\) and an isomorphism \(r:u^*x\to v^*x\). Its source and target are \(u,v\); composition is composition of these isomorphisms. Morphism descent proves that \(R\) is a sheaf, and local connectedness says \(R\to U\times U\) is epic.

For a sheaf \(S\), \(\mathbb Z[S]\) means the sheafification of the presheaf of free abelian groups on \(S(T)\). A section is locally a finite integer combination of sections of \(S\). An equality holds exactly when, locally, it is an equality of the corresponding finite combinations. This follows from the local representative and equality assertions of the sheafification prerequisite. Its free property follows from extending a map on generators and then sheafifying.

Put \(F=\mathbb Z[U]\). Let \(M\) be the quotient of \(\mathbb Z[R]\) by the subsheaf generated by

The first relation ranges over composable arrows.

Equivalently, the relation subsheaf is the image of the map from \(\mathbb Z[R\times_U R]\oplus\mathbb Z[U]\) sending its generators to the two expressions in (2.1). Thus its local sections, and local witnesses for quotient equalities, use finitely many such relations.

Define

Composition telescopes sources and targets, so \(d\) respects (2.1), and \(pd=0\). The sign in (2.2) is part of the convention.

There is a canonical map \(i:A\to M\). Locally choose \(u\) and send \(a\) to \([\iota_{x_u}(a)]\). For \(r:u\to v\), relation (1.2) and (2.1) give

After any two choices, such an \(r\) exists locally. Thus the local maps agree and descend to \(i\). Composition of automorphisms proves additivity.

Theorem 2.1. These actual maps form an exact row

Proof: finite local normal forms. Every assertion of exactness is local. An element, an equality in a free abelian sheaf, or a relation in its quotient has a finite witness after refinement. Only finitely many objects and arrows occur in such a witness. Choose one local base object \(u_0\), and, for each of those objects \(v\), choose an arrow \(q_v:u_0\to v\), taking \(q_{u_0}=1\). This requires only a finite successive refinement of epimorphisms, not a choice of arrows for every section of \(U\). For every arrow \(r:v\to w\) in the witness, define \(a_r\in A\) by

Transport of the band shows \(a_{r_2r_1}=a_{r_2}+a_{r_1}\), and \(a_{1_v}=0\). Thus replacing every arrow in a finite relation witness by its coordinate \(a_r\) preserves each relation (2.1). If \(i(a)=0\), use a finite witness for this equality and choose the indicated \(q_v\). Its coordinate equation is \(a=0\). This proves that \(i\) is monic. It does not assume that all arrow torsors can be trivialized simultaneously.

For an element \(m=\sum n_r[r]\) with \(dm=0\), refine to finite representatives. Refine further so the equality in \(\mathbb Z[U]\) is a literal equality of finite free abelian combinations. In (2.5) the coefficient of \([q_v]\) is the total target coefficient at \(v\) minus the total source coefficient there. These coefficients vanish because \(dm=0\); when labels agree, choose the same \(q_v\) for that label. Consequently \(m=i(\sum n_r a_r)\). Also \(di=0\). This proves \(\ker d=\operatorname{im}i\).

For \(\sum n_v[v]\in\ker p\), choose \(u_0,q_v\) for its finite representatives. Locally its integer sum is zero. Then

Thus \(\operatorname{im}d=\ker p\). Finally a local integer \(n\) lifts to \(n[u_0]\), so \(p\) is epic. These local assertions give exactness of sheaves. Empty covers and empty finite combinations obey the same sheafification rules; no stalk criterion was used. \(\square\)

Presentation independence. If a second chosen object is identified with the pullback of \(x\) over a refined cover, conjugating arrows by that identification gives a map of the two groupoid presentations. The free maps respect (2.1), (2.2) and the band. They give a commuting map of exact rows with identity on \(A\) and \(\mathbb Z\).

For two unrelated presentations \(x\) over \(U\) and \(y\) over \(V\), take over \(U\times V\) the sheaf of isomorphisms between their pullbacks. It is epic over \(U\times V\), hence its total sheaf \(W\) covers \(1\). Its tautological comparison gives a common presentation mapping to both. Therefore the two rows are equivalent as two-extensions. The same construction for a band-preserving functor of gerbes gives the same extension class: present the source by \(x\) and the target by its image. Lemma 1.1 identifies the arrow torsors.

3. Recover the gerbe and identify its cohomology class

Start with any exact row \(e\) of the form (2.4). Define a prestack \(\mathsf P_e\). Over \(T\), its objects are sections \(f\in F(T)\) with \(p(f)=1\). An arrow from \(f\) to \(f'\) is a section \(m\in M(T)\) such that

Compose by addition; the identity is zero and the inverse is negation. All restriction maps are those of the sheaves. Morphisms form sheaves. Exactness of \(p\) gives objects locally, and exactness at \(F\) gives arrows locally between any two objects. The automorphism \(i(a)\) gives the band \(A\). Stackify this prestack and call the result \(\mathsf G_e\). Stackification retains its morphism sheaves and local objects, so it is an \(A\)-banded gerbe.

An endpoint-preserving row map gives a band-preserving functor \(\mathsf G_e\to\mathsf G_{e'}\), hence an equivalence by Lemma 1.1. Thus equivalent two-extensions give equivalent gerbes.

Proposition 3.1. The two constructions are mutually inverse on equivalence classes.

Proof. Starting with a gerbe, its presentation groupoid maps to \(\mathsf P_{e(\mathsf G,x)}\) by

Equation (2.2) is exactly (3.1). Composition and identities follow from (2.1), and the band map is the identity. Stackifying the presentation recovers \(\mathsf G\): its morphism sheaves are the original ones, and every object is locally isomorphic to a restriction of \(x\). The same descent proof as in Lemma 1.1 gives an equivalence. After stackification, (3.2) is therefore an equivalence of the two gerbes by that lemma.

Conversely present \(\mathsf G_e\) over an epimorphism \(U\to1\) by actual objects of \(\mathsf P_e\). This is possible after refinement because every stackified object locally comes from the prestack. Let \(f_u\) be its section of \(F\). Each presentation arrow is an actual section \(m_r\) of \(M\), since stackification did not change morphism sheaves. The assignments

respect the composition relations, commute with \(d,p\), and are the identity on the kernel \(A\). On \(\mathbb Z\) they are the identity because \(p(f_u)=1\). Thus (3.3) is an endpoint-preserving map from the reconstructed two-extension to \(e\). The extension classes agree. Presentation independence proves all choices harmless. \(\square\)

Evaluation at \(1\) gives

Indeed a map from the free abelian sheaf on \(1\) is determined by the image of its generator, and any global section supplies that map. Apply (3.4) termwise to an injective resolution \(A\to I^\bullet\). Bounded-below injective complexes compute derived Hom by the injective prerequisite. Consequently \(\operatorname{Ext}^2(\mathbb Z,A)=H^2(E,A)\). The higher-extension prerequisite identifies this Ext group with equivalence classes of the exact rows (2.4).

Theorem 3.2. There is a canonical bijection

It uses band-preserving equivalences and applies to every small site, without enough points or an acyclicity hypothesis on covers.

Proof. Compose Theorem 2.1, presentation independence, Proposition 3.1 and (3.4) with the proven two-extension classification. These are inverse classifications, so (3.5) is both surjective and injective. Its definition uses only the gerbe and its fixed band; any presentation gives the same class. In particular the collection of equivalence classes is a set in the stated universe, because it is in bijection with this cohomology group. \(\square\)

Here is the precise sign and a concrete cohomology representative. Write \(j:A\to I^0\), choose \(v:M\to I^0\) extending \(j\), and factor \(d_Iv\) through \(\operatorname{im}d=\ker p\). Injectivity of \(I^1\) extends that factor to \(w:F\to I^1\). Thus \(wd=d_Iv\). There is a unique global degree-two cocycle \(z\) with

To prove the sign, use the augmented complex \(A\to M\to F\) in degrees \(-2,-1,0\). Its normalized extension numerator is \(-1_A\), as proved in the higher-extension prerequisite. After mapping to \(I^\bullet[2]\), compare that numerator with the map \(-zp\) in degree zero. The degree-minus-one homotopy has components \(-v:M\to I^0\) and \(w:F\to I^1\). Its three differential equations are \(-vi=-j\), \(-d_Iv+wd=0\), and \(d_Iw=zp\). This proves (3.6), including the minus sign.

For example, a global cocycle \(a\in I^2\) has the explicit representative

The last map is locally surjective by exactness of the resolution at \(I^2\). Its kernel is the image of \(I^0\), and the preceding kernel is \(A\). Taking \(v=1\) and \(w(t,n)=t\) in (3.6) gives class \([a]\). If \(a'=a+d_Ib\), the map \((t,n)\mapsto(t-nb,n)\), together with the identity on \(I^0\), is a row map from (3.7) for \(a\) to that for \(a'\). The resulting gerbes are equivalently banded. This gives actual inverse objects and actual comparisons for cohomology representatives.

4. Neutralization, coefficient changes and addition

Theorem 4.1. A gerbe is neutral if and only if its class is zero.

Proof. The zero two-extension can be represented by \(M=A\), \(F=\mathbb Z\), \(d=0\), \(p=1\). Its lifting prestack has one object, with automorphism sheaf \(A\); its stackification is \(BA\). Proposition 3.1 identifies its class with zero. Theorem 3.2 and Lemma 1.2 now give both implications. \(\square\)

For a homomorphism \(\phi:A\to B\), extend each arrow torsor of a presentation by contracted product with \(B\). In local coordinates this sends the relation \((r+a,b-\phi(a))\sim(r,b)\) to its \(B\)-coordinate. Composition adds coordinates and hence descends; identities and inverses descend too. Stackification gives the change of band \(\phi_*\mathsf G\). Common presentation refinements show independence of the presentation.

The row for this changed gerbe receives a map from the pushout row whose first middle sheaf is

The maps from the two summands are its new band map and the old arrow map. They agree on the indicated relation; their row map is identity on \(B\) and \(\mathbb Z\). Thus the extension classes agree, and the endpoint naturality of derived Ext gives

The product of two gerbes has band \(A\oplus A\). Change that band by addition. This defines their contracted sum. Its class is the sum of their classes. Here is the extension comparison establishing the assertion. Present the two gerbes over one common cover. The product presentation maps into the direct sum of their middle rows, pulled back at the last term along \(\mathbb Z\to\mathbb Z\oplus\mathbb Z\). On each object generator it gives the pair of its object generators; on each arrow it gives the pair of arrow classes. The kernel map is identity on \(A\oplus A\), and the terminal map is identity. Push out by addition as in (4.1). To compute its class, use one injective resolution of \(A\), and sum the two maps \(v,w\) of (3.6). They descend through the pushout because they add the two kernel coordinates. Their degree-two cocycle is \(z_1+z_2\); (3.6) gives \(-[z_1+z_2]=-[z_1]-[z_2]\). This is also the usual direct-sum, diagonal-pullback and addition-pushout definition of the Baer sum.

The neutral element is \(BA\). Changing the band by \(-1_A\) gives the inverse. Associativity and commutativity on equivalence classes follow either from the just-proved class calculation and injectivity of (3.5), or from the corresponding tuple quotients. Thus (3.5) is an isomorphism of abelian groups for these operations. All coefficient maps and restriction comparisons are the ones induced by the actual local arrow formulas, not choices of unrelated group structures.

For completeness, restriction to a base sheaf commutes with this classification. The inverse image into its slice topos is exact and preserves colimits by the module-sheaf prerequisite. It therefore preserves the free abelian sheaves, relation quotients and exact rows above. Pulling back \(x\) gives the pulled-back presentation, with the same bands and arrow formulas. Applying the exact inverse-image functor to the normalized extension roof is the canonical map on \(\operatorname{Ext}^2(\mathbb Z,A)\), and hence on sheaf cohomology under (3.4). This proves the restriction assertion without assuming that inverse image preserves injective objects.

5. Actual cocycles, coboundaries and refinement

Fix an epimorphism \(U\to1\), \(x\in\mathsf G(U)\), and an actual arrow \(\lambda_{01}:x_1\to x_0\) over \(U\times U\). Number subscripts denote pullback along the coordinate projections. The arrow is additional data: local connectedness alone does not give a global section of its arrow torsor. We may normalize its diagonal to the identity. Indeed its diagonal is a section \(h\in A(U)\); multiplying \(\lambda_{01}\) by \(\iota_{x_0}(-h_0)\) removes that diagonal. Hereafter \(\Delta^*\lambda=1_x\).

The full Čech nerve of this cover has \(U^{n+1}\) in degree \(n\). A cochain is a section of \(A\) there; its differential is the alternating sum of coordinate-deletion pullbacks. For an indexed family, use \(U=\coprod U_i\). Equal index labels do not identify two coordinates in \(U_i\times U_i\). Normalization in this section means pullback along actual diagonal maps, not setting all repeated-index components to zero.

Define \(c\in A(U^3)\) by

Comparing the two composites of three arrows and using (1.2) gives

The identity on the diagonal gives \(c(u,u,v)=0=c(u,v,v)\). No condition \(c(u,v,u)=0\) is imposed. Values such as these mean pullback of the section along the indicated maps into \(U^3\).

For \(b\in A(U^2)\) with \(\Delta^*b=0\), replace the arrow by \(\lambda'_{01}=\iota_{x_0}(b_{01})\lambda_{01}\). The normalized cocycle changes by

This follows by substituting the two modified arrows into (5.1) and transporting the middle scalar by (1.2).

Conversely take any normalized two-cocycle \(c\in A(U^3)\), where normalization is the two adjacent-diagonal conditions just stated. Make an internal groupoid with object sheaf \(U\). Its arrows \(v\to u\) form the trivial \(A\)-torsor on \(U^2\), and composition is

The two associative products agree by (5.2). The identity is \((u,u,0)\), by the adjacent-diagonal conditions. The inverse is

Indeed (5.2) on \((u,v,u,v)\) gives \(c(u,v,u)=c(v,u,v)\), since its other two terms are adjacent-degenerate. Both inverse composites are therefore their identity arrows. Every formula is a sheaf map and preserves the additive band. Stackification gives an \(A\)-gerbe \(\mathsf G_c\), whose tautological arrow \((u,v,0)\) has cocycle exactly \(c\).

For \(c'=c+\delta b\) with diagonal-zero \(b\), the map

is a functor from the first presentation to the second. In its composition equation, the three \(b\)-terms in (5.3) cancel to the single term \(-b(u,w)\). The diagonal-zero condition preserves identities; preservation of inverses then follows from uniqueness of an inverse. The reverse map adds \(b\). These maps and their stackifications are inverse band-preserving equivalences.

For a refinement \(f:V\to U\), with \(V\to1\) epic, use \(f^{n+1}\) on the full nerves. The induced groupoid functor maps objects by \(f\) and keeps the arrow coefficient. It is band-preserving; the stackified gerbes are equivalent by Lemma 1.1. No epimorphism \(V\to U\) is required. Thus the construction is compatible with actual cover-refinement maps.

Every two-cocycle has a normalized representative. Put \(h(u)=c(u,u,u)\). The cocycle equations on \((u,u,u,v)\) and \((u,v,v,v)\) give respectively \(c(u,u,v)=h(u)\) and \(c(u,v,v)=h(v)\). For the one-cochain \(t(u,v)=h(u)\) we have

satisfies both adjacent-diagonal normalization conditions and represents the same Čech cohomology class. This explicit degree-two normalization requires neither an ordering of the cover sheaf nor an alternating-index reduction.

6. Match the cocycle with derived cohomology

Here is a canonical Čech comparison on the full nerve of \(\mathcal U=(U\to1)\). Put \(C_n=\mathbb Z[U^{n+1}]\), with boundary the alternating coordinate-deletion maps. The augmentation \(C_0\to\mathbb Z\) is a resolution: locally choose a section of the cover, and prepend that section to a tuple. This operation contracts the augmented complex by \(\partial h+h\partial=1\), verified by separating the first deletion from the remaining deletions. Exactness is local, so this proves exactness globally without points.

For an injective resolution \(A\to I^\bullet\), set \(D^{p,q}=\operatorname{Hom}(C_p,I^q)=I^q(U^{p+1})\). Applying \(\operatorname{Hom}(-,I^q)\) to the exact augmented resolution gives exact augmented rows: factor each differential through its image and use injectivity on the resulting short exact sequences. Give the first-quadrant total complex differential

There are chain maps \(a:\Gamma(I^\bullet)\to\operatorname{Tot}D\) by restriction in column zero and \(b:C^\bullet(\mathcal U,A)\to\operatorname{Tot}D\) by \(A\to I^0\) in row zero. The exact augmented rows make \(a\) a rowwise quasi-isomorphism from the double complex concentrated in column zero. The finite-diagonal comparison in Filtered complexes, Sections 1–3 makes its total map a quasi-isomorphism. Thus

Injective-resolution comparisons commute with \(a,b\); their homotopies give the same cohomology map. Refinement acts by the coordinate maps of the full nerves and commutes with the augmentations. Hence (6.2) is independent of the resolution and natural for refinement.

Theorem 6.1. For the presentation and actual arrow of Section 5,

Proof. Use its presentation row (2.4). There is the tautological section \(f\) of \(F\) over \(U\), sending \(u\) to \([u]\), and the section \(e=[\lambda]\) of \(M\) over \(U^2\). Their exact equations are

Choose \(v,w,z\) as in (3.6). In the double complex let \(b_1=ve\) have bidegree \((1,0)\), and let \(b_0=wf\) have bidegree \((0,1)\). Equations (6.4) and \(wd=d_Iv\) give

Consequently \(D(b_0+b_1)=a(z)+b(c)\), so \(H(b)[c]=-H(a)[z]\). Equations (3.6) and (6.2) prove (6.3). This checks the degree-two normalization against the source-minus-target arrow convention on the full Čech nerve. \(\square\)

For an arbitrary site, (6.3) concerns the stated actual presentation and arrow; it does not supply a global arrow section on an arbitrary chosen cover. The classification of all gerbes is Theorem 3.2, proved independently using two-extensions.

Now let \(X\) be paracompact Hausdorff and take indexed open covers. Their members are subterminal sheaves: a repeated open factor in an intersection is the same open. In precisely this open-cover setting the full tuple and increasing alternating complexes agree by the integral proof in Čech comparison, Section 4. The complete all-open-cover theorem gives

Its proof is Čech comparison, Sections 2–3. These are the same augmentation and coefficient comparison maps as (6.2). The two common total-differential conventions are related by multiplying bidegree \((p,q)\) by \((-1)^{pq}\), which is identity on both axes and hence leaves that comparison unchanged.

Every cover cohomology class has a normalized representative by (5.6), and in this open-cover case it also has an alternating representative by the cited integral comparison. Theorems 3.2 and 6.1 imply that every \(A\)-gerbe on such a space is equivalent to some \(\mathsf G_c\) after choosing an open cover. Two such gerbes are equivalent exactly when their cohomology classes agree after a common open-cover refinement, by (6.6). This conclusion uses the proved refinement-limit isomorphism, rather than a simultaneous choice of all overlap arrows.

7. Scalar twisting and the locally ringed hypothesis

Let \(R\) be a commutative ring sheaf and set \(A=R^\times\), now written multiplicatively. Given an \(A\)-banded gerbe, its presentation arrow torsor from \(u\) to \(v\) gives an associated rank-one module \(L_{v,u}\). It is the sheaf quotient of pairs \((r,t)\), where \(r:u\to v\) and \(t\in R\), by \((ra,t)\sim(r,at)\). A local choice of \(r\) gives a basis, so this is locally free of rank one. Composition of arrows induces

The relation and commutativity of \(R\) make it well-defined. In local bases it is the usual product \(R\otimes_RR\to R\), hence an isomorphism. Composition proves associativity and the identity arrow gives the unit. Twisted modules, Theorem 2.1 therefore constructs a stack \(\mathsf S_{\mathsf G}\) of all modules with this twisting. Its refinement equivalences and line-change comparisons are proved in that lesson's Sections 9–10.

When actual arrows \(\lambda_{ij}\) as in Section 5 are chosen, their associated bases \(l_{ij}\) satisfy

This is precisely the scalar twisting law. Equations (5.3), (5.5) and (6.3) identify its changes of bases and its \(H^2(R^\times)\) class.

For a full classification of locally ordinary module stacks, assume now that \(X\) is a locally ringed space with structure sheaf \(R\). A tensor-invertible module is then locally free of rank one, by the complete proof in Torsors and line bundles, Section 7.

For an \(R\)-linear stack \(\mathsf S\) locally equivalent to \(\mathsf{Mod}(R)\), let \(\mathsf L(\mathsf S)\) be the groupoid of objects that become locally free rank-one modules in local module descriptions, using only isomorphisms as arrows. This condition is independent of the description: transition equivalences are tensoring with invertible modules by Local module categories, Theorem 4.1, and these modules are locally free rank one by the preceding hypothesis. Rank-one objects and isomorphisms descend, so this is an \(R^\times\)-banded gerbe. Its band is the actual scalar action.

There is a sign in recovering this gerbe from (7.1). For a presentation object \(u\), the corresponding local twisted line \(H^u\) has component \(L_{v,u}\) at \(v\); its action is (7.1). An arrow \(r:u\to u'\) gives \(H^u\to H^{u'}\) by composing on the right with \(r^{-1}\). On an automorphism \(a\) this is multiplication by \(a^{-1}\). Thus this functor induces the inverse band and, by Lemma 1.1 after that band change, gives

Conversely take the opposite gerbe of \(\mathsf L(\mathsf S)\), keeping the same scalar identifications on automorphisms. Inversion of arrows identifies its class with the negative of the original one. Present its objects by local rank-one objects \(H_u\) of \(\mathsf S\). Its associated line \(L_{v,u}\) is

Indeed an isomorphism \(r:H_v\to H_u\) sends the associated-line element \([r,t]\) to \(tr\). In a local choice of \(r\), this is the basis isomorphism \(R\to\mathcal Hom(H_v,H_u)\), so it is an isomorphism of sheaves and respects composition.

For \(K\in\mathsf S\), put \(M_u=\mathcal Hom_{\mathsf S}(H_u,K)\). Composition gives the required twisted action \(L_{v,u}\otimes M_u\to M_v\). Locally choose a rank-one basis for \(H_u\). This functor is then the ordinary module equivalence \(\mathcal Hom(H_u,-)\), with inverse \(N\mapsto H_u\otimes N\). The arbitrary-module tensor–Hom and restriction comparisons are proved in Tensor and Hom for twisted module stacks, Lemma 1.1.

These local equivalences give a global equivalence: morphisms lift uniquely because the maps on their sheaves are locally isomorphisms; local inverse objects and their uniquely lifted comparisons satisfy descent and therefore glue. The same argument applies to all module maps, including noninvertible ones. It is compatible with restrictions and scalar actions. A band-preserving equivalence of gerbes gives isomorphisms of their arrow torsors, hence of their associated lines and multiplication maps; it therefore gives an equivalence of the constructed module stacks. An equivalence of module stacks preserves rank-one objects and their scalar bands. Thus both constructions respect the equivalence relations as well as recovering every object.

Corollary 7.1. On a locally ringed space, equivalence classes of \(R\)-linear stacks locally equivalent to \(\mathsf{Mod}(R)\) are classified by

For a scalar presentation (7.2), this is \(\sigma^2[c]\). The neutral class consists exactly of stacks equivalent to \(\mathsf{Mod}(R)\).

Proof. Equations (7.3)–(7.4) and the just-proved equivalences give inverse classifications by module stacks and gerbes with the indicated inverse band. Theorem 3.2 supplies (7.5); (6.3) and (7.3) give its scalar sign. The neutral gerbe constructs ordinary module descent, whose equivalence with modules is Twisted modules, Lemma 9.1. Conversely an ordinary module stack has the global rank-one object \(R\), so its line gerbe is neutral by Lemma 1.2. \(\square\)

On a general ringed site the full Morita presentation by tensor-invertible modules remains valid, as proved in the earlier lesson. Such modules need not admit local bases on that site. The construction (7.1) and the gerbe classification remain valid there, but (7.5) is asserted with the locally ringed hypothesis just used. The distinction between tensor-invertible modules and locally free line bundles, including a counterexample, is proved in the torsor lesson's Section 7 and Exercise 4.

8. Four graded exercises with full solutions

Exercise 1 (foundation: a neutral class has many objects). For an abelian sheaf \(A\), compute the automorphism sheaf of an object \(P\) of \(BA\). Show directly how a global object of an \(A\)-gerbe identifies all its objects with torsors. Decide whether a zero class forces the gerbe to have a unique global object up to isomorphism.

Solution. Locally choose an origin \(p\) of \(P\). An equivariant automorphism is \(p+a\mapsto p+a+b\), determined by one section \(b\) of \(A\). Changing the origin leaves this translation unchanged because \(A\) is abelian. Hence these local identifications glue to \(\mathcal Aut(P)=A\), with global automorphism group \(\Gamma(E,A)\).

For \(x\in\mathsf G(1)\), the assignment \(y\mapsto\mathcal Isom(x,y)\) takes values in torsors. An arrow \(y\to z\) acts by postcomposition. Conversely, local origins of an \(A\)-torsor supply copies of \(x\); their translations are automorphisms of \(x\), and their cocycle glues these copies. The morphism-sheaf condition shows that both composites give the original objects and arrows. This is the specified equivalence of Lemma 1.2. Thus global isomorphism classes are the isomorphism classes of \(A\)-torsors. Whenever a nontrivial torsor exists, zero cohomology class of the gerbe does not force uniqueness of its global object. Exercise 3 gives an explicit example.

Exercise 2 (calculation: a nonzero cocycle with zero class). Over a point, take the indexed cover with three copies of that point. For an abelian group \(A\) and \(a\in A\), let the alternating two-cochain have \(c_{012}=a\), with repeated-index values zero. Prove it is a cocycle, find a one-cochain whose coboundary is \(c\), and write the band-preserving equivalence from the zero-cocycle gerbe to \(\mathsf G_c\).

Solution. The full tuple differential preserves alternating cochains: swapping adjacent positions negates each term, while the two terms deleting one of two equal positions cancel and every other term has a repeated argument. Thus its values on repeated tuples are zero. Every four-tuple of three indices has a repeated index, so \(\delta c=0\).

Put \(b_{01}=a\), \(b_{02}=b_{12}=0\), extend by \(b_{ji}=-b_{ij}\), and set \(b_{ii}=0\). Then \((\delta b)_{012}=b_{12}-b_{02}+b_{01}=a\). Both sides are alternating and zero on repeats, so \(\delta b=c\) on every tuple. Formula (5.5) is the functor

Its composition check is

exactly the image of the zero-cocycle composite. Its inverse adds \(b_{ij}\), so the presentation functors and their stackifications are mutually inverse. The band is unchanged. The index-zero object is already defined over the point and is a global object. Also sections on a point form an exact functor on abelian groups, so \(H^2(\mathrm{pt},A)=0\). A nonzero value \(a\) in a chosen presentation therefore does not itself imply a nonzero gerbe class.

Exercise 3 (hypotheses: chosen objects can lack overlap arrows). Let \(X\) be the finite topological space with upper-open topology for the order having minimal elements \(u,v\), maximal elements \(p,q\), and all four relations from a minimal to a maximal element. Let \(A\) be the constant \(\mathbb Z/2\)-sheaf. Construct an \(A\)-torsor \(P\) with no global origin. In the neutral gerbe \(BA\), use the cover with two copies of \(X\) and choose the objects \(A,P\). Explain why the overlap arrow needed in Section 5 cannot be chosen, and how refinement repairs that problem.

Solution. Put \(U=\{u,p,q\}\) and \(V=\{v,p,q\}\). Glue \(A|_U\) to \(A|_V\) by translation zero at \(p\) and translation one at \(q\). Gluing sheaves and actions gives a torsor. A section of the constant sheaf on \(U\) is constant: a neighborhood of its least point \(u\) is all of \(U\). The same holds on \(V\). A global origin would consequently give \(b_U,b_V\in\mathbb Z/2\) with both \(b_U=b_V\) and \(b_U=b_V+1\), which is impossible.

An equivariant map \(A\to P\) is determined by the image of zero, and every section of \(P\) supplies one. Thus \(\mathcal Isom(A,P)=P\); an isomorphism \(A\to P\) on the full overlap \(X\) would give the nonexistent origin. Nevertheless \(BA\) is neutral, with the global object \(A\), so its gerbe class is zero.

Refine both indexed copies of \(X\) by \(U,V\). The chosen trivializations of \(P\) there give local comparisons \(\phi_i:A\to x_i\) to every chosen object. Set \(\lambda_{ij}=\phi_i\phi_j^{-1}\). Their composites obey \(\lambda_{ij}\lambda_{jk}=\lambda_{ik}\), so their cocycle is identically zero. Local existence of objects and neutral cohomology class therefore did not provide arrows between the initially chosen objects on the initially chosen overlap. Actual arrows required the refinement just constructed.

Exercise 4 (advanced: dimension shifting gives every obstruction). Let \(X\) be any topological space, let \(0\to A\to I\xrightarrow{\pi}B\to0\) be an exact sequence of abelian sheaves with \(I\) injective, and let \(P\) be a \(B\)-torsor. A lift of \(P\) is an \(I\)-torsor \(Q\) with an identification \(\pi_*Q\simeq P\). Prove that their stack is an \(A\)-banded gerbe with class \(\partial[P]\in H^2(X,A)\), for the positive connecting convention of the extension prerequisite. Prove it is neutral exactly when \(P\) is trivial, and explain why every \(A\)-gerbe has a representative of this form.

Solution. A local origin \(p\) of \(P\) gives the lift \(Q=I\), with identification \(t\mapsto p+\pi(t)\). Two such local lifts have an arrow when their origins differ by a section of \(\operatorname{im}\pi\), which holds locally by the exact sequence. An automorphism is translation by a section of \(I\) mapping to zero in \(B\), hence is a unique section of \(A\). These identifications are unchanged by conjugation. Torsors, their equivariant maps and their specified identifications descend, so the lift category is an \(A\)-banded gerbe \(\mathsf L(P)\).

The complete first-cohomology and extension classification in Torsors and line bundles, Sections 3–5 identifies \(P\) with the fibre over one of an exact row

whose positive extension class is \([P]\). Splice it with \(0\to A\to I\to B\to0\). Its lifting prestack has objects \(p\in P\) and arrows \(t:p\to p'\) satisfying \(\pi(t)=p-p'\). An equivariant map between the two trivial lifts is \(s\mapsto s+t\); compatibility with their maps to \(P\) is precisely that equation. Every lifted torsor is locally trivial, so stackification of this prestack is \(\mathsf L(P)\).

The normalized splicing theorem in Higher extensions, Theorem 4.1 identifies the class of the spliced two-extension with the positive connecting map applied to \([P]\). Theorem 3.2 therefore gives \(\operatorname{cl}\mathsf L(P)=\partial[P]\), with no sign left unspecified.

Injectivity gives \(H^1(X,I)=H^2(X,I)=0\), because the one-term injective resolution computes them. The long exact sequence makes \(\partial:H^1(X,B)\to H^2(X,A)\) an isomorphism. Consequently \(\mathsf L(P)\) is neutral exactly when \([P]=0\), equivalently when \(P\) has a global origin. Directly, every \(I\)-torsor is trivial by the same first-cohomology classification, so a global lift can only map to a trivial \(B\)-torsor; a trivial \(P\) has the displayed trivial lift. Finally every class in \(H^2(X,A)\) is \(\partial[P]\) for some \(B\)-torsor, and Theorem 3.2 identifies its gerbe with \(\mathsf L(P)\). This supplies a concrete representative for every obstruction.

References

• The Stacks Project, Gerbes, Section 8.11, for the stack and abelian-band notions.
• The Stacks Project, Second cohomology and gerbes, Section 21.11, for the relation between overlap torsors, cocycle obstructions and objects.
• The internal proof prerequisites are the stackification and arbitrary-base descent theorems, the module-sheafification and injective constructions, the normalized higher-extension comparison, and the Čech comparison linked above. The local finite relation proof, inverse constructions, sign calculation and exercises are given here. Original exposition and solutions are CC0; linked works retain their own licences.