Draft lesson.

Torsors, first cohomology and line bundles

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

A torsor has local origins but no specified global origin. First sheaf cohomology measures precisely the obstruction to choosing one. This interpretation also turns unit-valued transition functions into line bundles. On a general ringed space, however, a tensor inverse need not supply a local basis; the distinction requires an explicit hypothesis.

Work on a topological space \(X\), in fixed universes and with choice. Initially \(A\) is a sheaf of abelian groups, written additively. For module statements \(\mathcal O\) is a sheaf of unital rings and modules are left modules. In the line-bundle statements \(\mathcal O\) is commutative. A line bundle here means an \(\mathcal O\)-module locally isomorphic to \(\mathcal O\). A tensor-invertible module means one with a tensor inverse; these definitions are kept distinct.

We use Čech complexes, refinements and acyclic covers, Sections 1–7, for all cocycle maps, injective Čech acyclicity and the actual total-complex comparison; Godement resolutions and flabby sheaf cohomology, Sections 1–9, for enough injectives, flabby acyclicity and its natural derived comparison; Short exact extensions and the Baer group, Sections 1–4, for fixed-endpoint extension classes and Baer addition; and Dualizable sheaves, finite covers and global projectivity, Sections 1–2, for local finite-projective retracts, dual bases and tensor–Hom. No separation or acyclic-cover hypothesis on \(X\) is assumed.

1. Local origins and their differences

An \(A\)-torsor is a sheaf of sets \(T\) with an \(A\)-action such that locally it has a section and the map \(A\times T\to T\times T\), \((a,t)\mapsto(t+a,t)\), is an isomorphism of sheaves. Equivalently, on every open with a section, the action on its sections is simply transitive. Local differences are unique and agree on overlaps, so they glue; this proves the equivalence of the formulations. In particular, \(t-t'\) denotes the unique section of \(A\) with \(t=t'+(t-t')\).

Any equivariant map of torsors is an isomorphism: after choosing local origins it is a translation of \(A\), whose inverse is the opposite translation. These local inverses agree by uniqueness and glue. A global section \(t\) gives an isomorphism \(A\to T\), \(a\mapsto t+a\). Conversely an isomorphism with \(A\) gives a global section by taking the image of zero. Every automorphism of \(T\) is translation by a unique element of \(A(X)\): its local translation differences agree on overlaps, since \(A\) is abelian. Thus \(\operatorname{Aut}_A(T)=A(X)\) canonically, even when \(T(X)\) is empty.

Choose local sections \(t_i\in T(U_i)\) on a cover. Use the convention

Then \(g_{jk}-g_{ik}+g_{ij}=0\) on triple intersections, so \(g\) is a degree-one cocycle for the ordered-tuple complex of the first prerequisite. Changing origins to \(t_i+a_i\) changes \(g\) to \(g-\delta a\). Restricting origins gives the stated refinement maps, so its cohomology class is independent of these choices.

2. Gluing every cocycle back to a torsor

Given such a cocycle \(g\), define \(T_g(W)\) to consist of families \(f_i\in A(W\cap U_i)\) satisfying

Restriction acts on every component. Gluing component sections proves the sheaf condition, because the equations are local. The action adds the restrictions of one section of \(A(W)\) to all components. On \(U_i\), any chosen value of \(f_i\) determines \(f_j=f_i+g_{ij}\); the cocycle equation verifies every other compatibility. Conversely every family has this form. Thus \(T_g|_{U_i}\simeq A|_{U_i}\), and its local origin with \(i\)-coordinate zero has difference (1.1). For an existing torsor, sending \(t\) to its coordinates \(t=t_i+f_i\) identifies it with (2.1): the coordinate equation follows from (1.1), and the inverse glues the sections \(t_i+f_i\). Both operations are inverse and equivariant.

If \(g'=g-\delta a\), the isomorphism \(T_g\to T_{g'}\) sends \(f_i\) to \(f_i-a_i\). Conversely any equivariant isomorphism, expressed in the chosen local coordinates, is translation by a local \(a_i\), giving precisely such a coboundary relation. For origins on different covers, pass to the intersection cover and apply the same argument. Thus no chosen trivializing cover is part of the torsor itself.

3. The complete first-cohomology classification

Choose a monomorphism \(j:A\hookrightarrow I\) into an injective abelian sheaf, and put \(Q=I/jA\). The ordinary long exact sequence gives

with isomorphism induced by the positive connecting map \(\partial:Q(X)\to H^1(X,A)\); injectivity of \(I\) gives \(H^1(X,I)=0\). For \(q\in Q(X)\), let \(T_q(W)=\{z\in I(W):\bar z=q|_W\}\). This is a sheaf of sets. The quotient is an epimorphism of sheaves, so \(q\) lifts locally; two lifts differ by a unique section of \(jA\). Hence \(T_q\) is an \(A\)-torsor.

Every torsor is of this form. For a cocycle (1.1), injective Čech acyclicity supplies \(b_i\in I(U_i)\) with \(b_j-b_i=-j(g_{ij})\). Their images in \(Q\) agree and glue to \(q\in Q(X)\). The local maps \(t_i+a\mapsto b_i+j(a)\) glue: since \(t_i=t_j+g_{ij}\), the right sides satisfy the same equality. They give an equivariant isomorphism \(T\simeq T_q\).

Moreover \(T_q\simeq T_{q'}\) if and only if \(q'-q\) comes from \(I(X)\). A lift \(b\) gives the translation \(z\mapsto z+b\). Conversely an equivariant isomorphism has local differences \(\phi(z)-z\in I\); replacing \(z\) by \(z+j(a)\) leaves the difference unchanged. Thus these differences are independent of the local origin and glue to a global \(b\), with \(\bar b=q'-q\). This proves both directions and all inverse checks in the bijection

The bijection is canonical. Changing the \(b_i\) while retaining their differences changes them by a global section of \(I\), which does not change (3.1). Changing origins adds \(j(a_i)\) to \(b_i\), and leaves \(q\) unchanged. For another injective embedding \(A\hookrightarrow I'\), injectivity extends the identity on \(A\) to \(I\to I'\); the induced map on quotients sends the construction to the same class by naturality of the connecting map. Thus the result does not depend on the embedding, cover or origins. The neutral class consists exactly of torsors with a global section.

This convention agrees with the actual Čech comparison \(\sigma\) of the first prerequisite. Extend \(I\) to an injective resolution of \(A\), with \(Q\hookrightarrow I^1\); the differential of the local \(b_i\) glues to the global cocycle \(z\) representing \(\partial q\). In its total complex, the local degree-zero cochain has differential \(-b(g)+a(z)\), because \(\delta b=-j g\). Thus \([b(g)]=[a(z)]\), and

Using instead \(t_j-t_i\) would negate (3.3). No acyclicity of the chosen torsor-trivializing opens was used; exactness of the injective Čech row suffices.

4. Addition, coefficient changes and connecting fibres

The contracted sum \(T\mathbin{+_A}T'\) is the associated sheaf of the sectionwise quotient of \(T\times T'\) by \((t+a,t'-a)\sim(t,t')\). Its action adds to the first coordinate. Locally chosen origins identify this quotient with \(A\) by addition, so it is a torsor. Its transition differences are \(g_{ij}+g'_{ij}\). In the representatives of Section 3, addition of lifts induces an isomorphism \(T_q+_A T_{q'}\simeq T_{q+q'}\). It follows that (3.2) preserves the actual contracted sum and the group law in cohomology.

The neutral torsor is \(A\), and the inverse of \(T\) is the same sheaf with action \(a\star t=t-a\). The difference map on two copies of \(T\) identifies their contracted sum, with the second action reversed, with \(A\). Associativity and symmetry are induced by rearranging tuples before passing to the quotient; all resulting maps agree locally with the corresponding sums in \(A\), so they agree globally. This proves the group operations on isomorphism classes directly.

For \(f:A\to B\), form \(f_*T\) as the associated sheaf of \((T\times B)/((t+a,b-f(a))\sim(t,b))\). In local coordinates it is a \(B\)-torsor with differences \(f(g_{ij})\). It represents \(H^1(f)(\operatorname{class}T)\): extend \(A\to B\hookrightarrow I_B\) across \(A\hookrightarrow I_A\), take the induced quotient map, and use naturality of the connecting homomorphism in (3.1). This also proves independence of the extension map. Identity and successive coefficient changes induce the same local formulas, so they compose correctly. The constructions commute with restriction to every open by the same formulas and natural restriction of cohomology.

For a short exact sequence \(0\to A\to B\to C\to0\) and \(c\in C(X)\), the sheaf of its local lifts in \(B\) is an \(A\)-torsor. Its class is exactly the positive boundary \(\partial c\). Indeed extend \(A\hookrightarrow I\) to \(B\to I\) by injectivity, producing \(C\to Q=I/A\). The lift torsor maps equivariantly and locally isomorphically to the fibre over the image of \(c\) in \(Q(X)\). Connecting-map naturality and (3.2) give the asserted equality, including its sign.

5. Module extensions and the fibre over one

For a module sheaf \(M\), evaluation at one identifies \(\operatorname{Hom}_{\mathcal O}(\mathcal O,M)\) with \(M(X)\); the inverse sends a section \(m\) to \(r\mapsto r m\) on each open. On an injective resolution these are identical complexes, so naturally

These are also the cohomology groups of the underlying abelian sheaf. To verify this without assuming the forgetful functor preserves injectives, take a module-injective resolution. Its underlying abelian resolution is exact and all its terms are flabby by the Godement prerequisite. They are therefore acyclic as abelian sheaves, by that lesson with ring \(\mathbb Z_X\). Its natural acyclic-resolution comparison identifies the same section complex with abelian sheaf cohomology. This proves the comparison, with all maps.

For an extension \(0\to M\to E\xrightarrow{p}\mathcal O\to0\), the fibre of \(p\) over \(1\in\mathcal O(X)\) is an \(M\)-torsor. Its class is \(\partial(1)\), hence the extension class under (5.1): the fixed-endpoint boundary morphism of the Baer prerequisite, evaluated at one, is this same connecting map. Consequently (3.2), (5.1) and the Baer classification give mutually inverse classifications by such extensions and by additive \(M\)-torsors. Pullback along the diagonal and pushout along addition make the fibre of a Baer sum the contracted sum of the two fibres: a local pair of lifts of one has difference in \(M\oplus M\), which is sent to its sum. Thus the group laws agree by actual local maps, not only by counting classes.

6. Unit torsors and line bundles

Assume \(\mathcal O\) commutative and put \(G=\mathcal O^\times\), an abelian group sheaf written multiplicatively. For a line bundle \(L\), let \(\operatorname{Fr}(L)(U)\) be the set of module isomorphisms \(\mathcal O|_U\to L|_U\), or equivalently their generating sections. Compatible frames glue to a frame, since both a module map and its inverse can be checked locally. Any two frames differ by a unique unit. Local freeness supplies local frames, so this is a \(G\)-torsor.

Conversely take a unit torsor, choose local origins \(t_i\), and write \(t_i=g_{ij}t_j\). On an open \(W\), define a module of tuples \(r_i\in\mathcal O(W\cap U_i)\) with \(r_j=g_{ij}r_i\) on overlaps. Addition and scalar multiplication preserve these equations, and gluing component sections proves the sheaf condition. On \(U_i\) its \(i\)-coordinate gives an isomorphism with \(\mathcal O|_{U_i}\), whose inverse assigns \(r_j=g_{ij}r_i\). These are inverse because of the cocycle equation. Call the resulting line bundle \(L_T\). Its local frame \(e_i\) with \(i\)-coordinate one satisfies \(e_i=g_{ij}e_j\). Therefore \(t_i a\mapsto a e_i\) gives a well-defined equivariant isomorphism \(T\to\operatorname{Fr}(L_T)\), checked on every overlap.

Starting with \(L\), its local frame construction identifies the glued module with \(L\) by \((r_i)\mapsto r_i e_i\); the sections agree on overlaps and the local inverse is the coordinate map. An equivariant map between frame torsors extends uniquely to a module isomorphism by sending each local basis to its image. Conversely every module isomorphism acts this way on frames. All choices of local origins, covers and refinements are thus connected by these unique compatible local maps. We have proved both inverse constructions, including morphisms.

Tensoring two local frames gives a frame of \(L\otimes L'\). Its transitions are the products of the two unit cocycles, so the induced map from the contracted product of frame torsors is an isomorphism. The unit is \(\mathcal O\). The dual \(L^*=[L,\mathcal O]\) has reciprocal transition units, and evaluation \(L\otimes L^*\to\mathcal O\) is an isomorphism in each local basis, hence globally. Thus line-bundle classes form a group \(\operatorname{Pic}_{\mathrm{line}}(X,\mathcal O)\), and the preceding constructions and (3.2) give the natural group isomorphism

It respects restriction to opens. For a morphism of commutative ring sheaves \(\mathcal O\to\mathcal O'\) on \(X\), scalar extension preserves the local rank-one bases. Its frame torsor is the coefficient pushout along \(\mathcal O^\times\to\mathcal O'^\times\). Section 4 therefore proves the corresponding naturality of (6.1), without a flatness assumption.

7. When tensor-invertible really means a line bundle

Every line bundle is tensor-invertible by evaluation above. Conversely, suppose all stalk rings are local and \(P\otimes Q\simeq\mathcal O\). Tensoring by \(P\) and \(Q\) gives inverse equivalences. Here is the needed duality check. Let \(e:Q\otimes P\to\mathcal O\) be the given isomorphism after symmetry, and initially let \(c\colon\mathcal O\to P\otimes Q\) be its symmetric inverse. The composite \(u=(1_P\otimes e)(c\otimes1_P)\) is an automorphism. Replace \(c\) by \(c'=(u^{-1}\otimes1_Q)c\); then its \(P\)-triangle is the identity. For the other composite \(v=(e\otimes1_Q)(1_Q\otimes c')\), expanding the two composites and interchanging the two evaluations gives \(e(v\otimes1_P)=e(1_Q\otimes1_P)=e\). Cancel the isomorphism \(e\); tensoring by \(P\) is faithful because it is an equivalence, so \(v=1_Q\). This proves both dual triangles. The local-duality prerequisite now supplies local finite-projective retract presentations and dual bases for \(P\).

At a local stalk ring \(A\), write the inverse image of \(1\) under \(P_x\otimes Q_x\simeq A\) as a finite sum \(\sum p_i\otimes q_i\). The sum of their evaluations is one, so at least one evaluation \(u\) is a unit: otherwise every term belongs to the maximal ideal. Its pair \((p,q)\) gives a split inclusion \(A\to P_x\), \(a\mapsto a p\), with retraction \(z\mapsto u^{-1}\operatorname{ev}(z,q)\). Write \(P_x=A\oplus K\). Tensoring by \(Q_x\) decomposes \(A\) as \(Q_x\oplus(K\otimes Q_x)\). A nonzero local ring has no nontrivial idempotents: for \(e^2=e\), one of \(e,1-e\) is a unit, forcing the other to be zero. Thus its module \(A\) cannot be a direct sum of two nonzero submodules. Since \(Q_x\ne0\), the second summand vanishes. Tensoring with the inverse of \(Q_x\) gives \(K=0\), proving \(P_x\simeq A\).

This stalk basis extends to a basis on a neighborhood, as follows. The stalk retraction can be represented by a local functional from the sheaf dual, using the finite-projective presentation and its dual basis. Choose local representatives \(p,\ell\); their pairing is a unit near \(x\), since its germ is a unit. Normalize to \(\ell(p)=1\), giving a local decomposition \(P=\mathcal O\oplus K\). The remaining summand is a finite-projective retract and is locally finitely generated. Its stalk at \(x\) is zero. Choose finitely many local generators; their zero germs vanish on a common neighborhood, so \(K\) vanishes there. Hence \(P\) is locally free of rank one. This proves that on a locally ringed space (6.1) classifies all tensor-invertible module sheaves. On a general ringed space it classifies the line bundles defined at the outset; Exercise 4 shows the distinction is necessary.

8. Four exercises with full solutions

8.1 Additive torsors on a finite circle

Exercise 1 (basic). Use the four-point circle \(u,v<a,b\) and its two minimal opens \(U=\{u,a,b\}\), \(V=\{v,a,b\}\). For the constant diagram of an abelian group \(A_0\), classify all torsors and their sums and automorphisms.

Solution. Every torsor has an origin on \(U\): a local origin near its least point \(u\) is defined on an open containing the minimal neighborhood \(U\), and restricts there. Similarly it has an origin on \(V\). Sections of the constant diagram on each minimal open are \(A_0\), and on the intersection they are \(A_0^2\). Thus a torsor is specified by the difference pair \((c_a,c_b)\). Changing its two origins subtracts a diagonal pair, so the unique invariant is \(\lambda=c_b-c_a\). Every \(\lambda\) is realized by \((0,\lambda)\) through Section 2. An equivariant isomorphism exists exactly when these invariants agree, by its two local translations. Contracted sum adds \(\lambda\), the inverse negates it, and a global origin exists exactly when \(\lambda=0\). Global sections of the constant coefficient sheaf are the diagonal \(A_0\); hence every torsor's automorphism group is \(A_0\). The classification gives \(H^1(X,A)=A_0\), with the cocycle convention (1.1).

8.2 All line bundles on the finite circle

Exercise 2 (intermediate). For the constant field ring sheaf \(k\) on the same circle, classify all line bundles, their tensor products and their global sections.

Solution. A local frame at a least vertex trivializes the entire minimal open, as in Exercise 1. Unit transitions are a pair \((c_a,c_b)\in(k^\times)^2\), modulo multiplication by a common unit from changing the two frames. The invariant is \(\lambda=c_b/c_a\), realized by \((1,\lambda)\). Therefore \(\operatorname{Pic}_{\mathrm{line}}(X,k)=k^\times\), tensor product multiplies \(\lambda\), and duality inverts it. In diagram form one representative has every stalk \(k\), all arrows identity except \(v\to b\), which is multiplication by \(\lambda^{-1}\). Its minimal-open frames satisfy \(e_U/e_V=(1,\lambda)\), agreeing with the prescribed convention. A global section is a pair of chart coefficients \((r,s)\) with \(r=s\) at \(a\) and \(r=\lambda^{-1}s\) at \(b\). If \(\lambda=1\), its section space is \(k\); if \(\lambda\ne1\), it is zero. Thus local rank one does not guarantee a nonzero global section or a global frame.

8.3 A quotient section and an actual nonsplit extension

Exercise 3 (advanced). Let \(F=k_X\) on this circle and \(I=TF\), the first Godement term. Give a global section of \(Q=I/F\) that does not lift globally, construct its torsor and pullback extension, and compute all the extension's stalk maps.

Solution. At either minimal vertex \(x\), \(I_x=k^3\) has coordinates \((f_x,f_a,f_b)\); at \(a,b\) it is \(k\), with coordinate projections as arrows. The unit embeds \(F_x\) diagonally. Thus \(Q_x=k^2\), with quotient \((f_a-f_x,f_b-f_x)\), and \(Q_a=Q_b=0\). Take \(q_u=(0,0)\), \(q_v=(0,1)\). These define a global section of \(Q\), since its outgoing maps are zero. A global lift would have \(f_a=f_b=f_u\) and \(f_a=f_v, f_b=f_v+1\), which is impossible. Its torsor \(T_q\) consequently has nonzero class. Local lifts on \(U,V\) with their minimal coordinates zero have overlap difference \((0,-1)\); thus its additive label from Exercise 1 is \(-1\).

Define \(E\subseteq I\oplus k_X\) by the equation \(\bar f=t q\). Its projection to \(t\) has kernel \(F\), and is locally onto since \(q\) lifts locally, giving \(0\to F\to E\to k_X\to0\). Its fibre over one is exactly \(T_q\). At \(u\), parameters are \((x,t)\) with \(f_a=f_b=x\); at \(v\), parameters are \((y,t)\) with \(f_a=y,f_b=y+t\). At each maximal vertex use coordinates \((f_a,t)\) or \((f_b,t)\). Every stalk is \(k^2\); all arrows are identities except \(v\to b\), whose matrix is \(\begin{pmatrix}1&1\\0&1\end{pmatrix}\). The kernel embedding is \(z\mapsto(z,0)\), and the quotient is \((z,t)\mapsto t\). A splitting would give a global lift of one, contradicting the displayed equations. Thus the extension is genuinely nonsplit, with precisely the nonzero class and fibre specified by Section 5.

8.4 A tensor inverse without a local basis

Exercise 4 (advanced). On a one-point space put \(R=\mathbb Z[s]/(s^2+5)\) and \(J=(2,1+s)\). Prove that its module sheaf is tensor-invertible but not a line bundle. Compare with \(H^1(X,R^\times)\).

Solution. Embed \(R\) in \(\mathbb C\) by \(s=i\sqrt5\); every element has the unique form \(a+bs\), so this is a domain. Evaluation \(s\mapsto-1\) gives surjections to both \(\mathbb F_2\) and \(\mathbb F_3\), with distinct maximal kernels, so \(R\) is not local. Direct multiplication gives \(J^2=(2)\). Its three generating products are \(4,2(1+s),(1+s)^2=-4+2s\), all divisible by two, and \(2=2(1+s)-(1+s)^2-4\) proves the reverse inclusion. Put \(K=J/2\) in the fraction field. Then \(JK=R\), and an explicit unit decomposition is

Both first factors belong to \(J\), both second factors belong to \(K\), and every product of one element from each ideal is in \(R\). Multiplication \(J\otimes_R K\to R\) has inverse \(r\mapsto r\sum_i x_i\otimes y_i\), with the two pairs of (8.1). Its composite on \(R\) is the identity by that equation. On a tensor \(x\otimes y\), the other composite is \(\sum_i x_i\otimes y_i x y=\sum_i x_i(y_i x)\otimes y=x\otimes y\), since \(y_i x\in R\) and \(\sum_i x_i y_i=1\). These are both inverse checks, so \(J\) is tensor-invertible.

It is not free of rank one. If \(J=\alpha R\), its square would give \(\alpha^2R=2R\), hence \(\alpha^2=2u\) for a unit \(u\): writing each generator as a multiple of the other and cancelling in the domain gives the unit equation. The multiplicative norm \(N(a+bs)=a^2+5b^2\) sends every unit to one, because it is a positive integer on nonzero elements and norms of inverses multiply to one. Taking norms would give \(N(\alpha)=2\), which is impossible: for \(b=0\) an integer square is not two, and for \(b\ne0\) the norm is at least five. Thus \(J\) is not principal. An isomorphism \(R\to J\) would make it principal by the image of one, so no such isomorphism exists. A neighborhood of the point is the whole space, and local rank one there would be this same isomorphism. Finally \(H^1(X,R^\times)=0\), since sections of abelian sheaves on a point are exact. It classifies the trivial line-bundle group on this space, while the tensor-invertible class of \(J\) is nontrivial. This is the exact boundary excluded by the local-ring hypothesis in Section 7.

Further reading

The Stacks Project gives freely accessible treatments of first cohomology and torsors and first cohomology and invertible sheaves, the latter with an explicit local-stalk-ring hypothesis.