Draft lesson.

Twisted modules from coherent gluing

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

An ordinary module sheaf glues by maps whose composite on a triple overlap is the direct transition map. A twisted module allows a controlled modification of that law. The modification itself must be coherent on quadruple overlaps. The result is a category of modules that is ordinary locally, even when no global trivialization exists.

The prerequisites are Representing objects from local data and Local module categories and Morita equivalence. We also assume the definitions of a site, a sheaf and a covering. We recall the part of stack descent used in the construction. Basic references are [D'Agnolo–Polesello], [Stacks], and [Schapira, Sheaves].

1. From a cover to a multiplication law

Work in the topos \(\mathcal E\) of sheaves on a small site. Let \(R\) be a sheaf of commutative unital rings. All products below are products in \(\mathcal E\); thus the construction also works when the site has no object representing a particular product. For a sheaf \(V\), module sheaves over \(V\) mean modules in the slice topos \(\mathcal E/V\), over the restriction of \(R\).

Let \(\pi:U\to1\) be an epimorphism. It is a cover of the terminal sheaf. A covering family of site objects gives one by taking the coproduct of their associated sheaves. Epimorphisms of sheaves have sections locally: a section of their target lifts after a covering. Consequently the pulled-back cover \(U\times V\to V\) has a section locally on \(V\). The local-lifting criterion is proved in Sieves, local isomorphisms, and presheaf localization, Section 3. Apply it to the unique terminal section and pair each lift with the restriction of the chosen base section.

A line in this lesson is an invertible \(R\)-module: a module \(L\) with a tensor inverse \(L^*\). We do not assume every such module is locally free on the given site. For a locally ringed space, it is a line bundle in the familiar sense.

Choose an invertible module \(L\) over \(U\times U\), together with an isomorphism over \(U^3\)

Subscripts indicate pullback along the relevant projection. On \(U^4\), require

Both sides take \(L_{12}\otimes L_{23}\otimes L_{34}\) to \(L_{14}\). Parentheses are identified by the canonical tensor associator.

There is a canonical unit \(R_U\simeq\Delta^*L\), where \(\Delta:U\to U^2\) is the diagonal. Here is why. Restrict (1.1) to the triple diagonal to obtain an associative isomorphism \(m:D\otimes D\to D\) on the invertible module \(D=\Delta^*L\). Tensoring with \(D\) is an equivalence. There is therefore a unique map \(e:R\to D\) with \(m(e\otimes\mathrm{id})=\mathrm{id}_D\). Associativity implies \(m(\mathrm{id}\otimes e)=\mathrm{id}_D\): compare the two maps after tensoring on the left with \(D\), use the first unit identity, then cancel that equivalence. Since \(m\) and tensoring with \(D\) are isomorphisms, \(e\) is an isomorphism. Restricting (1.2) to repeated coordinates gives the same unit identities for \(L(x,y)\). We will use this canonical unit without additional choices.

For a topological open cover \((U_i)\), these data become lines \(L_{ij}\) on \(U_i\cap U_j\) and associative isomorphisms \(L_{ij}\otimes L_{jk}\simeq L_{ik}\). Invertibility and (1.1) identify \(L_{ji}\) as a tensor inverse of \(L_{ij}\).

2. The twisted category

For any \(V\in\mathcal E\), define \(\mathsf T(V)\) as follows. An object is an \(R\)-module \(M\) over \(U\times V\) and an isomorphism over \(U^2\times V\)

such that over \(U^3\times V\)

A morphism \((M,\theta)\to(N,\psi)\) is a module map \(a:M\to N\) with

Pullback in \(V\) defines the restriction functors. They carry the canonical coherence isomorphisms of pullback, so this is a prestack. We use stack for a prestack in categories whose morphisms are sheaves and whose descent data for objects are effective. Morphisms inside a category need not be invertible. The comparison maps used to glue objects are invertible.

The diagonal restriction of \(\theta\) is the identity after identifying \(\Delta^*L\) with \(R\). In fact (2.2) on the diagonal says that its invertible endomorphism \(h\) satisfies \(h^2=h\). Multiplying by \(h^{-1}\) gives \(h=\mathrm{id}\).

Theorem 2.1. The prestack \(\mathsf T\) is an \(R\)-linear stack and is locally equivalent to the stack of \(R\)-modules.

Proof of descent. Let \((V_a\to V)\) be a cover, and suppose objects of \(\mathsf T(V_a)\) have isomorphisms on pairwise overlaps satisfying the usual cocycle law. Their underlying module sheaves glue over \(U\times V\). One explicit description of the glued sheaf is compatible local sections after pulling back the cover to each test object; restrictions and addition act coordinatewise. The sheaf axiom gives uniqueness, and the local module actions glue because their compatibility is one of the local equalities. The maps \(\theta_a\) glue to \(\theta\), since (2.3) says they agree under the gluing maps. Its inverse glues too. Equation (2.2) is an equality of sheaf maps and can be checked after the cover, where it holds. This proves effectiveness for objects. The same compatible-section description glues morphisms uniquely and preserves (2.3). Hence the morphism presheaves are sheaves. Linearity is inherited from module morphisms. \(\square\)

3. Trivialize using a section of the cover

We finish the proof of Theorem 2.1 by constructing the local equivalence explicitly. Suppose the cover over \(V\) has a section \(s:V\to U\). For an ordinary module \(N\) over \(V\), put

This notation abbreviates pullback of \(L\) by \((x,v)\mapsto(x,s(v))\), tensor pullback of \(N\) by \(U\times V\to V\). It does not require points of the topos. The transition isomorphism is

Equation (1.2) is exactly (2.2) for this object. Define \(E_s(M,\theta)=s^*M\), pulling back along \(v\mapsto(s(v),v)\). The unit of \(L\) gives \(E_s\Phi_s(N)\simeq N\). Conversely, restricting \(\theta\) to \((x,s(v),v)\) gives

It is a morphism in \(\mathsf T\) by (2.2). All these maps are natural and commute with further restriction. Thus \(E_s,\Phi_s\) are inverse equivalences of stacks over \(V\). Sections \(s\) exist locally, proving local equivalence everywhere. \(\square\)

Two choices \(s,t\) differ by an ordinary invertible module:

Thus a change of local description is a Morita equivalence. It need not be the identity functor.

4. Every locally ordinary module stack has this form

Theorem 4.1. Let \(\mathsf S\) be an \(R\)-linear stack locally equivalent to \(\mathsf{Mod}(R)\). After choosing a cover and local equivalences, it is equivalent to a stack \(\mathsf T\) constructed above.

Proof. Choose a cover \(U\to1\) and an equivalence \(F:\mathsf S|_U\to\mathsf{Mod}(R)|_U\) with an adjoint inverse. On \(U^2\), the change from the second trivialization to the first is \(F_1F_2^{-1}\). By Theorem 4.1 of Local module categories and Morita equivalence, it is tensoring with

The two scalar actions agree because the functor is \(R\)-linear. Its Morita inverse is consequently an ordinary tensor inverse of this \(R\)-module.

On \(U^3\), cancel the middle inverse equivalence in \(F_1F_2^{-1}F_2F_3^{-1}\). Evaluating the resulting natural isomorphism to \(F_1F_3^{-1}\) at \(R\) gives \(\mu:L_{12}\otimes L_{23}\simeq L_{13}\). On \(U^4\), cancel the two middle pairs in either order. The two counits act in distinct positions, so naturality makes the two cancellations equal. The associativity constraints for composition give (1.2); the adjunction triangle identities give the diagonal unit.

An object of \(\mathsf S(V)\), restricted to \(U\times V\) and sent through \(F\), gives \(M\). Its canonical descent isomorphism, expressed in the two trivializations, gives \(\theta\). The cocycle law for that descent isomorphism gives (2.2) by the construction of \(\mu\). A morphism becomes (2.3).

Conversely, apply \(F^{-1}\) locally to \(M\). Equation (2.2), with the same cancellation isomorphisms, is precisely the ordinary cocycle law for these local objects of \(\mathsf S\). They glue because \(\mathsf S\) is a stack. The sheaf condition for morphisms gives full faithfulness. These procedures are inverse up to the chosen equivalences and their units and counits, and are compatible with restriction. \(\square\)

There is also a canonical identification

Locally on an ordinary module category, a natural endomorphism of the identity is multiplication by a section of \(R\). To see this, evaluate it at \(R\); naturality for \(R\to M\), \(1\mapsto m\), determines it on every local section of every module. It is invertible exactly for a unit. Tensoring with an invertible module preserves that scalar action, so the local identifications agree and glue. No chosen trivialization remains in (4.1).

Theorem 4.2. Every \(R\)-linear autoequivalence \(H\) of \(\mathsf S\) is tensoring with an invertible \(R\)-module. That module is uniquely determined up to isomorphism.

Proof. Let \(P=\mathcal Nat(\mathrm{id}_{\mathsf S},H)\), the sheaf of natural transformations of these stack functors. It is an \(R\)-module sheaf. Its components glue because morphisms in \(\mathsf S\) are sheaves, and naturality is a local equality of maps.

In an ordinary module description, the sheaf Morita theorem writes \(H\) as \(P_0\otimes_R-\) for an invertible module \(P_0\). Evaluation at \(R\) identifies \(\mathcal Nat(\mathrm{id},P_0\otimes_R-)\) with \(P_0\): a section \(p\) acts by \(m\mapsto p\otimes m\), and naturality for every local map \(R\to M\) determines all its components. Thus \(P\) is locally invertible. The local evaluation maps \(P\otimes_R P^*\to R\), where \(P^*=\mathcal Hom_R(P,R)\), are isomorphisms; hence \(P\) is globally invertible.

Evaluation of transformations, followed by the tensor universal property of Corollary 3.1 in Representing objects from local data, gives \(P\otimes_R K\to H(K)\). In each ordinary module description it is the tensor isomorphism just obtained, so it is an isomorphism everywhere. Conversely, if \(H\simeq Q\otimes_R-\), the same local calculation identifies its sheaf of natural transformations from the identity with \(Q\). This proves uniqueness. \(\square\)

5. Scalar cocycles and their changes

On a topological open cover, suppose each \(L_{ij}\) is given a basis. Then \(\mu\) is multiplication by a unit \(c_{ijk}\in R^\times(U_i\cap U_j\cap U_k)\). Equation (1.2) becomes

The twisted module maps \(t_{ij}:M_j\to M_i\) satisfy

All identities involving a repeated index are normalized by the unit. This is a concrete example of the preceding construction, rather than an assumption that all lines on all sites have bases.

If units \(b_{ij}\) change the transition maps to \(t'_{ij}=b_{ij}t_{ij}\), then

Thus a coboundary change gives an equivalent twisted category. For an example, choose arbitrary units \(a_i\) on \(U_i\) and put \(b_{ij}=a_i a_j^{-1}\). The resulting coboundary is one, so these changes preserve \(c\). They act by changing local frames, without changing the coherence law.

Here is a numerical coherence check. Cover a point by four labeled copies of itself and take \(R=\mathbb Q\). Set \[ b_{01}=2,\quad b_{12}=3,\quad b_{02}=5,\quad b_{23}=7,\quad b_{03}=11,\quad b_{13}=13, \] with \(b_{ii}=1\) and \(b_{ji}=b_{ij}^{-1}\). Define \(c_{ijk}=b_{ij}b_{jk}b_{ik}^{-1}\). The four factors on the overlap labeled \((0,1,2,3)\) are \[ c_{012}=\frac65,\quad c_{023}=\frac{35}{11},\quad c_{123}=\frac{21}{13},\quad c_{013}=\frac{26}{11}. \] Both products in (5.1) are \(42/11\). Taking \(M_i=\mathbb Q\) and \(t_{ij}=b_{ij}\) gives a twisted module with these nonidentity factors. The twist is nevertheless a coboundary, as its definition shows; this example demonstrates coherence, without claiming a nontrivial cohomology class.

The Čech description depends on the cover and the existence of the chosen bases. A classification by sheaf cohomology requires a further comparison with gerbes and refinements. Equations (5.1)–(5.3) alone do not establish that comparison. The full fixed-abelian-band gerbe classification is Abelian gerbes and second sheaf cohomology, Theorem 3.2; its Theorem 6.1 proves the actual Cech comparison and refinement statement. The classification of locally ordinary module stacks on locally ringed spaces, with its scalar convention, is Corollary 7.1. The distinction between tensor-invertible modules and locally free line bundles on a general ringed site is retained.

6. Exercises with solutions

Exercise 6.1 (foundation). If \(L\) is the trivial line and \(\mu\) its ordinary multiplication, identify \(\mathsf T\).

Solution. Equation (2.1) becomes an ordinary descent isomorphism \(M_2\to M_1\) over the cover \(U\times V\to V\). Equation (2.2) is its cocycle law. Effective descent for module sheaves glues \(M\) to a unique module over \(V\), up to its canonical isomorphism; morphisms glue uniquely as well. Hence \(\mathsf T\simeq\mathsf{Mod}(R)\).

Exercise 6.2 (calculation). In the scalar setting, compose two changes \(b_{ij}\) and \(d_{ij}\). Check the resulting coherence factor and the inverse change.

Solution. The new maps are \(d_{ij}b_{ij}t_{ij}\). Their factor is \[ (d_{ij}d_{jk}d_{ik}^{-1})(b_{ij}b_{jk}b_{ik}^{-1})c_{ijk}. \] Commutativity allows the factors to regroup as the coboundary of \((d_{ij}b_{ij})\) times \(c\). Replacing \(b\) by \(b^{-1}\) gives the inverse. This also verifies that the equivalence on morphisms leaves (2.3) intact.

Exercise 6.3 (obstruction). Suppose a scalar-twisted module is locally free of constant positive rank \(r\). What does taking determinants in (5.2) imply? What is special about rank one?

Solution. Write \(d_{ij}=\det t_{ij}\) after choosing local bases. Then \(d_{ij}d_{jk}=c_{ijk}^{r}d_{ik}\). Thus \(c^r\) is the coboundary of the unit cochain \(d\). In particular the Čech class of \(c\) has order dividing \(r\). For rank one, \(c\) itself is a coboundary, so rescaling transitions turns this particular twist into ordinary descent. This is a necessary condition for an object of the stated rank; no existence of such an object follows from a torsion condition alone.

Exercise 6.4 (synthesis). For sections \(s,t,u\) of the covering over \(V\), compare \(E_u\Phi_t E_t\Phi_s\) with \(E_u\Phi_s\). Identify the map that makes this comparison coherent for four sections.

Solution. Formula (3.3) gives the first functor as tensoring with \(L(u,t)\otimes L(t,s)\), and the second as tensoring with \(L(u,s)\). Their comparison is \(\mu(u,t,s)\). With four sections, the two comparisons multiply three line factors in opposite parenthesizations. Equation (1.2) makes these maps equal. Hence the local equivalences carry the full composition coherence, in addition to being pairwise equivalences.

References

• [D'Agnolo–Polesello] Andrea D'Agnolo and Pietro Polesello, Stacks of twisted modules and integral transforms, in Geometric Aspects of Dwork Theory, Walter de Gruyter, 2004. Open preprint.
• [Stacks] The Stacks Project Authors, The Stacks Project: Stacks and Modules on Sites.
• [Schapira, Sheaves] Pierre Schapira, An Introduction to Sheaves on Grothendieck Topologies, lecture notes, version of 1 August 2026, Sections 1.9 and 2.5–2.6, for glueing, modules over sheaves of rings and ringed sites.

7. Module sheaves on the element site

The global twisted category needs more than local equivalences: a generator must be an actual global twisted object. We first establish the module-sheaf facts used to construct one. All sites and section sets in this section are small in the fixed universe.

For a sheaf \(V\), its element site has objects \((T,v)\), with \(T\) an object of the original site and \(v\in V(T)\). An arrow \((T',v')\to(T,v)\) is an arrow \(f: T'\to T\) with \(v'=v|_f\). Covering sieves are induced by covering sieves on \(T\). The identity and composition laws follow from restriction in \(V\). This is a small site because its objects and arrows are indexed by the small original site and its small section sets.

A sheaf in the slice over \(V\) is the same as a sheaf on this element site. For a sheaf \(A\to V\), take the fibre of \(A(T)\to V(T)\) over \(v\). Conversely, from fibre sheaves \(N(T,v)\), form the disjoint union of these sets over \(v\in V(T)\). A matching family first glues its labels in \(V\), and then glues its entries in the fibre sheaf. Thus this disjoint-union presheaf is a sheaf, and the two constructions are inverse, on maps as well as objects. A module over the restriction of \(R\) is accordingly a sheaf of \(R(T)\)-modules \(N(T,v)\), with semilinear restrictions. All subsequent section formulas have this interpretation; they use generalized sections, not points of the topos.

Lemma 7.1 (sheafification and local exactness). Presheaves of modules on a small ringed site have a sheafification functor. Its sections are locally represented by presheaf sections, and two such representatives have the same image exactly when they agree locally. Sheafification is exact. In module sheaves a kernel is computed sectionwise, a cokernel is the sheafification of the sectionwise cokernel, and an epimorphism is locally surjective on sections.

Proof. Here is the construction, including the two local assertions. For a presheaf \(P\) and object \(T\), a matching family over a covering sieve \(S\) assigns \(x_f\in P(T')\) to each \(f: T'\to T\) in \(S\), compatibly with further restriction. Define \[ P^+(T)=\underset{S\text{ covering }T}{\operatorname{colim}} \{\text{matching families in }P\text{ on }S\}. \tag{7.1} \] Transitions restrict to smaller covering sieves. Intersections are covering, so any two representatives have a common comparison sieve. Equality means equality after a further covering refinement. Addition is coordinatewise on a common refinement; multiplication by \(r\in R(T)\) uses its restriction on each coordinate. These operations descend, satisfy the module laws and commute with pullback of the covering sieve. Thus \(P^+\) is a module presheaf. The canonical map \(P\to P^+\) uses the family of all restrictions of a section.

Every section of \(P^+\) is locally in this image: restrict a represented matching family along its own covering sieve, and use its component at that arrow. Two sections of \(P\) have the same image if and only if they agree on a covering sieve, directly by the comparison rule in (7.1).

The presheaf \(P^+\) is separated. If two of its sections agree after a cover, choose a covering comparison on each member of that cover where the represented families become equal. Composites of those comparison covers form a covering sieve by the transitivity axiom of a topology. Intersect it with the two representative sieves. On the resulting covering sieve the ordinary matching families agree, so their classes in (7.1) are equal.

If \(P\) is separated, then \(P^+\) is a sheaf. To glue a matching family of its sections on a covering sieve, choose on each member a covering family of representatives in \(P\). These yield representatives on a covering sieve of composites. Whenever the same composite arrow has two such representatives, their images in \(P^+\) agree by compatibility of the original family. The map \(P\to P^+\) is injective for separated \(P\), by the local equality rule; hence those representatives agree already in \(P\). They therefore form a well-defined matching family in \(P\) on the composite sieve. Its class glues the given family. Uniqueness follows from separatedness of \(P^+\).

Set \(aP=P^{++}\). The preceding two paragraphs show that this is a sheaf. A section of \(aP\) locally lifts to \(P^+\) and then, after a further cover, to \(P\). Transitivity combines the covers. Equality of the images of two original representatives similarly reduces after a covering refinement to equality in \(P\). For any map \(P\to M\) to a sheaf, glue the images of each matching family to define a unique map \(P^+\to M\), then do the same a second time. The local representative rule makes both extensions unique. This proves the sheafification adjunction, with all module operations preserved.

A kernel of a map of module sheaves is a sheaf because compatible sections in the kernel glue in the source and their images glue to zero. The sectionwise cokernel, followed by \(a\), has the cokernel universal property: a map killing the original map factors sectionwise, then extends uniquely by the adjunction. Its zero sections are precisely those locally in the image of the original map, by the local equality rule. Thus a map is epic exactly when every target section is locally in its image.

Finally suppose a sequence of module presheaves is exact sectionwise. A section of the middle sheafified term which maps to zero is locally represented by an original section. Refine until its image in the original last term is zero, using the equality rule. Sectionwise exactness supplies a lift from the preceding presheaf there. This proves exactness in the middle. If the original first arrow is injective, an original representative whose image becomes locally zero is itself locally zero, so the sheafified first arrow is monic. If the original last arrow is surjective, local representatives in its target have original lifts, so the sheafified arrow is epic. These arguments prove exactness of \(a\). \(\square\)

Proposition 7.2. Module sheaves on a small ringed site form a locally small Grothendieck abelian category. Pullback along a map of base sheaves, with the scalar ring pulled back from \(R\), is exact and preserves small colimits. A tensor product of module sheaves is the sheafification of their sectionwise tensor presheaf; its sections are locally finite sums of pure tensors.

Proof. Morphisms form a subset of the product of sectionwise map sets over the small site, so form a small set. Sums and zero maps are sectionwise; finite direct sums are both products and coproducts. By Lemma 7.1, kernels and cokernels exist. The ordinary coimage-to-image comparison is locally the comparison in modules: a section of its image is locally represented in the source, and two source representatives have the same image exactly when their difference locally belongs to the kernel. The local rules of Lemma 7.1 therefore make that comparison an isomorphism. This proves abelianity and the stated local test for exactness.

For any small diagram, take the colimit of its module presheaves sectionwise and sheafify. The presheaf colimit universal property followed by the sheafification adjunction is exactly the universal property of its sheaf colimit. For a filtered diagram of short exact rows of sheaves, sectionwise left exactness is immediate. At the middle and last terms, lifts of sections exist locally by Lemma 7.1. A section of a sectionwise filtered colimit has a stage representative, and equality to zero holds at a later stage, by the complete element calculation in Hom, tensor, and the exactness of module limits, Section 3. Consequently the colimit row is exact by the same local test. This proves exactness of filtered colimits; it does not assume that a sheaf epimorphism is sectionwise surjective.

For an object \(S\) of the ringed site, let \(H_S\) be the sheafification of the module presheaf which on \(T\) is free over \(R(T)\) on the arrows \(T\to S\). Restrictions send a basis arrow to its composite and restrict its coefficient. For every module sheaf \(N\), evaluation at the basis vector of \(\operatorname{id}_S\) gives \[ \operatorname{Hom}(H_S,N)\simeq N(S). \tag{7.2} \] The inverse sends the coefficient \(r\) at an arrow \(f: T\to S\) to \(r(n|_f)\), then extends by sheafification. These constructions are inverse and natural. The small coproduct \(G=\bigoplus_S H_S\) is a generator. If two sheaf maps differ, they differ on some section over some \(S\). The corresponding map from \(H_S\), extended by zero on the other summands, separates them. Together with the colimit and filtered-exactness proofs this is the Grothendieck assertion.

For a base map \(h: V\to W\), restriction to the element site of \(V\) sends \(N(T,w)\) to \(N(T,hv)\). Covering sieves at \((T,v)\) are exactly the original covering sieves on \(T\), and every restriction of its label is fixed. Thus this restriction commutes with the matching-family construction (7.1), hence with \(a\). It preserves sectionwise kernels and presheaf colimits, so is exact and preserves sheaf colimits. This is the claimed pullback statement for modules in these slice topoi.

The sectionwise tensor presheaf is formed from the free module on pairs, modulo the two additive relations and the scalar-balancing relation. A bilinear map to a sheaf factors uniquely through this presheaf and then through its sheafification. This proves the sheaf tensor universal property. Original tensor classes are finite sums of pure tensors; the local representative rule for \(a\) proves the last assertion. Its associative and unit identifications are the maps on pure tensors, and their identities follow because those tensors locally generate. \(\square\)

This argument also applies to the element site of \(U\), so it supplies the exact module category used next. Further reading for the ordinary module-sheaf assertions is the Stacks section on the abelian category of sheaves of modules.

8. A free twisted module and a global generator

Write \(\mathcal D=\mathsf{Mod}(R_U)\), and let \(\mathsf{For}:\mathsf T(1)\to\mathcal D\) forget the transition isomorphism. The unit of \(L\) from §1 is denoted \(e_x:R\to L(x,x)\). Products and pullbacks of base sheaves use their element sites as in §7.

Theorem 8.1 (the free twisted module). The forgetful functor has an \(R\)-linear left adjoint \(F\). For \(N\in\mathcal D\), its underlying sheaf is the sheafification on the element site of \(U\) of \[ P_N(T,x)= \bigoplus_{z\in U(T)} L(T,x,z)\otimes_{R(T)}N(T,z). \tag{8.1} \] Its transition map on the summand labelled \(z\) is multiplication of the two line factors by \(\mu(x,y,z)\).

Proof. Formula (8.1) defines a module presheaf. For \(f: T'\to T\), restrict \(x,z\), the line section and the module section; send the summand labelled \(z\) to that labelled \(z|_f\). Distinct labels may merge, in which case their terms add. Each input has finite support, so the rule is defined without an infinite sum. The two restriction composites agree because the original restrictions do; identities are preserved. A map of \(N\)'s acts on its tensor factor in each summand, giving all functor laws before sheafification.

Multiplication of line sections gives natural maps over pairs of labels. Pullback commutes with sheafification by Proposition 7.2, and tensor with a sheaf is characterized by the proved bilinear universal property. Therefore these maps define \[ \theta^F_{xy}:L(x,y)\otimes (FN)_y\longrightarrow(FN)_x, \qquad \ell\otimes(m\otimes n)_z \longmapsto (\mu(\ell\otimes m)\otimes n)_z. \tag{8.2} \] The formula specifies a sheaf map: sections of \(FN\) locally have finite representatives in (8.1), and tensor sections locally have finite pure representatives. Independence of representatives follows from bilinearity and refinement, so the local formulas glue uniquely. On three line factors, the two composites in (2.2) are the two sides of (1.2), in each summand. They therefore agree on local generators and everywhere. The diagonal transition is identity because the canonical unit multiplies each line identically.

Each map (8.2) is an isomorphism, not merely an action map. The inverse of the line multiplication \(L(x,y)\otimes L(y,x)\simeq L(x,x)\simeq R\) supplies a map \(d_{xy}:R\to L(x,y)\otimes L(y,x)\). An inverse to \(\theta^F_{xy}\) is \[ (FN)_x\xrightarrow{d_{xy}\otimes1} L(x,y)\otimes L(y,x)\otimes(FN)_x \xrightarrow{1\otimes\theta^F_{yx}} L(x,y)\otimes(FN)_y. \tag{8.3} \] One composite is identity by the just-proved transition composition and multiplication followed by its inverse. For the other, the equality \((1_{L(x,y)}\otimes\mu_{yxy})(d_{xy}\otimes1_{L(x,y)})=1_{L(x,y)}\) is the line triangle identity. To check it directly, tensor on the right with \(L(y,x)\), multiply the final \(L(x,y)\otimes L(y,x)\) to \(R\), and use associativity (1.2). The resulting two maps are the same inverse multiplication followed by multiplication. Tensoring with the invertible line is faithful, so the original maps agree. Substitution into (8.3) proves the other composite identity. Thus \(FN\) is an object of \(\mathsf T(1)\).

Define \(\nu_N:N\to\mathsf{For}(FN)\) on a section \(n\in N(T,x)\) by the term \(e_x(1)\otimes n\) in the summand \(z=x\). This respects restriction, so defines a sheaf map. For a map \(a: N\to\mathsf{For}(M)\), send the summand \(z\) of (8.1) to \(M(T,x)\) by \(\ell\otimes n\mapsto\theta^M_{xz}(\ell\otimes a_z(n))\). It is natural and bilinear, hence extends uniquely to a sheaf map \(FN\to M\). Equivariance follows from (2.2) for \(M\), since the multiplication in (8.2) is the same \(\mu\). Its composite with \(\nu_N\) is \(a\), by the diagonal unit identity.

Conversely, an equivariant map \(b: FN\to M\) is determined by \(a=\mathsf{For}(b)\nu_N\). A generator \(\ell\otimes n\) in summand \(z\) is \(\theta^F_{xz}(\ell\otimes\nu_N(n))\), because multiplication by \(e_z\) is identity on \(L(x,z)\). Equivariance determines its image to be precisely the preceding formula. Local finite generators determine all sheaf maps, proving uniqueness. We have therefore constructed a natural bijection \[ \operatorname{Hom}_{\mathsf T(1)}(FN,M) \simeq\operatorname{Hom}_{\mathcal D}(N,\mathsf{For}(M)). \tag{8.4} \] It respects sums and scalars. The arrow construction at the start, and this unique extension, prove identities, composition and naturality in both variables. Thus it is the asserted linear adjunction with unit \(\nu\). \(\square\)

Theorem 8.2 (global abelian and injective structure). For each fixed base sheaf \(V\), the category \(\mathsf T(V)\) is locally small, abelian, has all small colimits, has exact filtered colimits and has a generator. It consequently has ordinary functorial injective embeddings and injective envelopes. Kernels and cokernels, and all colimits, are created by its underlying-module functor.

Proof. First work over the terminal base. The forgetful functor is faithful, and its Hom sets are subgroups of the small Hom sets of \(\mathcal D\). Equivariance is a linear equation, so sums and scalar multiples of equivariant maps are equivariant. Zero and finite direct sums receive the componentwise transition maps and are the zero object and biproducts.

The tensor functor of an invertible line is an equivalence, with inverse tensoring by its inverse line. It therefore preserves kernels, cokernels and all colimits by their universal properties. The two pullbacks to the double cover are exact and preserve colimits by Proposition 7.2. For an equivariant map, take its underlying kernel. The commuting transition square uniquely induces a map from the line tensor of its pulled-back kernel to the other pulled-back kernel; both are kernels of the same corresponding map, so this induced map is an isomorphism. The triple-overlap equation follows by the kernel uniqueness property, or by composing it with the kernel inclusion and cancelling that monomorphism. A kernel factor of an equivariant map is equivariant by the same cancellation. This is the kernel in the twisted category. The cokernel construction is dual: its induced transition is an isomorphism, coherence follows after its epic projection, and every cokernel factor is equivariant.

Thus the underlying functor preserves and creates kernels and cokernels. The coimage-to-image comparison has the ordinary underlying comparison in \(\mathcal D\), which is invertible. The inverse of an equivariant isomorphism is equivariant, by composing its equivariance equation with the inverses on both sides. Hence the comparison is invertible in the twisted category, proving abelianity. The same constructions show that exactness is detected on underlying objects.

For a small diagram of twisted objects, take the colimit of the underlying objects. Pullback and tensor with the line commute with this colimit, so the colimit of the transition isomorphisms gives an isomorphism between the required two sheaves. Its triple and unit equations can be checked after every colimit structure map; there they are the equations already satisfied at that stage. The colimit structure maps are jointly determining, so coherence holds. The underlying universal map to another twisted object is equivariant for the same reason. This proves that the underlying colimit is the actual twisted colimit. For a filtered diagram of short exact rows, its underlying colimit is exact by Proposition 7.2; detection of exactness proves exactness of the twisted colimit.

Take the generator \(G\) of \(\mathcal D\) constructed in Proposition 7.2, and set \[ \Gamma=F(G). \tag{8.5} \] This is a global twisted object by Theorem 8.1. If \(f,g: M\to N\) differ, their underlying arrows differ by faithfulness. A map \(a: G\to\mathsf{For}(M)\) separates them because \(G\) is a generator. Its adjoint map \(\widetilde a: \Gamma\to M\) separates \(f,g\), by naturality of (8.4). Thus \(\Gamma\) is a generator. All Grothendieck hypotheses have now been proved. Applying the full functorial injective construction and envelope proof gives the asserted injective embeddings and envelopes in this category.

For a general fixed \(V\), use the small element site of \(V\) and the epic cover \(U\times V\to V\). Replace \(L\) by its pulled-back line, and perform exactly the constructions (7.1)–(8.5) on that site. Their section, restriction, adjunction and exactness proofs are unchanged; the covering and quadruple equation are preserved by pullback. This proves all assertions for \(\mathsf T(V)\). The injective statements concern each of these ordinary categories for a fixed slice. \(\square\)

Example 8.3 (a point with two covering labels). Take \(R=k\), a field, cover a point by labels \(0,1\), and let every line and multiplication be the ordinary scalar line. An underlying module is a pair \((N_0,N_1)\). Formula (8.1) gives the same space \(N_0\oplus N_1\) at both labels, with identity transition. Its adjunction is the actual bijection \[ \operatorname{Hom}_k(N_0\oplus N_1,M) \simeq\operatorname{Hom}_k(N_0,M)\times \operatorname{Hom}_k(N_1,M). \tag{8.6} \] The two unit components are the two inclusions. Evaluation at label zero identifies the twisted category with vector spaces. The generator \(G=(k,k)\) of the underlying product category yields \(\Gamma\simeq k^2\) in that equivalence. Its endomorphism ring in the twisted category is the full matrix ring \(M_2(k)\); the endomorphism ring of \(G\) in the underlying product category is \(k\times k\). The off-diagonal matrix maps are allowed after applying the free twisted functor. Every vector space is injective by the basis-extension proof in the linked injectivity lesson; the Grothendieck theorem is consistent with this explicit case, without requiring a minimal generator.

Example 8.4 (the numerical scalar twist). Use the four labels and units \(b_{ij}\) of §5, with \(c_{ijk}=b_{ij}b_{jk}b_{ik}^{-1}\). For underlying spaces \(N_z\), the free object has \(\bigoplus_zN_z\) at every label \(x\). Its transition from label \(y\) to \(x\), on the summand labelled \(z\), is multiplication by \(c_{xyz}\). Identify the space at label \(x\) with a fixed \(W=\bigoplus_zN_z\) by multiplying that summand by \(b_{xz}\). Conjugating the transition gives multiplication by \(b_{xy}\) on every summand, since \(b_{xz}c_{xyz}b_{yz}^{-1}=b_{xy}\). These are precisely the scalar-twisted transitions in §5.

For example, take \(N_2=\mathbb Q\) and the other three spaces zero. The original transition from label one to label zero is multiplication by \(c_{012}=6/5\). The two identifications multiply by \(b_{02}=5\) and \(b_{12}=3\). Their conjugated transition is \[ 5\cdot\frac65\cdot\frac13=2=b_{01}. \tag{8.7} \] The diagonal unit at label two is identity because \(b_{22}=1\). The calculation identifies the free construction, its unit and its transition maps; it also displays why the raw transition coefficient depends on the summand label until the chosen identifications are applied.

9. Descent and refinement of the covering

Lemma 9.1 (effective module descent). Let \(p:Y\to X\) be an epimorphism of base sheaves. Module sheaves over \(X\) are equivalent to module sheaves \(N\) over \(Y\) with an isomorphism \(\delta_{y',y}:N_y\to N_{y'}\) over \(Y\times_XY\), normalized on the diagonal and satisfying \(\delta_{y'',y'}\delta_{y',y}=\delta_{y'',y}\). Morphisms are the maps commuting with \(\delta\). Pullback along \(p\) detects isomorphisms.

Proof. We give the glued module on the element site of \(X\), avoiding any assumption that \(X,Y\) are represented by objects of the original site. For \(x\in X(T)\), a section of \(M(T,x)\) is a family \(n_{g,y}\), indexed by all \(g: T'\to T\) and \(y\in Y(T')\) with \(p(y)=x|_g\). The entry lies in \(N(T',y)\). Require compatibility with further restriction and the equality \(n_{g,y'}=\delta_{y',y}(n_{g,y})\). These are small products of section sets with specified equations. Componentwise addition and multiplication by restricted scalars give an \(R(T)\)-module. Restricting a family means composing its indexing maps with the given base arrow; this gives identities and composition of restrictions.

This presheaf is a sheaf. Given compatible such families on a covering sieve of \(T\), fix an indexing pair \((g,y)\) over \(T\). Pull back that sieve along \(g\). Its members give compatible local sections of \(N\) over \((T',y)\), by evaluating the local families at the restricted label \(y\). Glue them uniquely in \(N\). The resulting \(n_{g,y}\)'s satisfy restriction and transition equations, because these equations hold locally and equality of sheaf sections is local. They form the required global family. Their uniqueness follows from uniqueness in \(N\) at every indexing pair.

If \(y\in Y(T)\), evaluation of a family at \((\operatorname{id}_T,y)\) gives \(M(T,p(y))\to N(T,y)\). Its inverse sends \(n\) to the family \(n_{g,y'}=\delta_{y',y|_g}(n|_g)\). Restriction compatibility follows from naturality of \(\delta\), and compatibility with changing \(y'\) follows from its cocycle equation. The normalized diagonal makes evaluation followed by this inverse identity; the transition and restriction requirements make the other composite identity as well. Thus \(p^*M\simeq N\), with exactly the specified transition.

For a module already over \(X\), its sections give these families by restriction. Conversely, an epic sheaf map has local lifts: choose a covering sieve on which \(x\) lifts to \(Y\). The corresponding entries of a family agree over that cover by its transition rule, so glue uniquely to a section over \((T,x)\). These operations are inverse. A map compatible with \(\delta\) acts on the family entries and produces the unique glued map. This proves the equivalence and its full faithfulness. Finally, if a pulled-back map is invertible, its local inverse on each lift of \(x\) agrees on overlaps, by uniqueness of inverse maps. The inverse sheaf map glues. Pullback therefore detects isomorphisms. \(\square\)

Let \(a:V\to U\) be a map over the terminal sheaf such that \(V\to1\) is epic. Its pulled-back line cocycle is \(L^a(v,w)=L(a(v),a(w))\). Write \(\mathsf T_U^L\) and \(\mathsf T_V^{L^a}\) for the two stacks.

Theorem 9.2 (refinement equivalence). Pullback along \(a\) gives an equivalence of \(R\)-linear stacks \(R_a:\mathsf T_U^L\to\mathsf T_V^{L^a}\). No surjectivity of \(a\) onto \(U\) is required.

Proof. We construct its inverse over the terminal base; replacing everything by its pullback over any base gives the same construction and its restriction comparisons. For \(N\in\mathsf T_V^{L^a}(1)\), put on \(U\times V\) \[ K(x,v)=L(x,a(v))\otimes N(v). \tag{9.1} \] Over \(U\times V\times V\), define the ordinary descent map from the \(w\)-copy to the \(v\)-copy by \[ \begin{gathered} L(x,a(w))\otimes N(w)\\ \xrightarrow{\mu^{-1}\otimes1} L(x,a(v))\otimes L(a(v),a(w))\otimes N(w)\\ \xrightarrow{1\otimes\theta^N_{vw}} L(x,a(v))\otimes N(v). \end{gathered} \tag{9.2} \] The triple equation for these maps follows by applying associativity of \(\mu\) to the two line decompositions, then the transition equation for \(N\) to its two successive transitions. Their diagonal is identity by the two unit identities. Thus they are actual ordinary descent data. The map \(U\times V\to U\) is epic, being the pullback of \(V\to1\). Lemma 9.1 glues (9.1) to a module \(Q_aN\) over \(U\).

Multiplication supplies maps \(L(x,y)\otimes K(y,v)\to K(x,v)\). They commute with (9.2): on the same three line factors this is precisely associativity of \(\mu\), followed by \(\theta^N\). They therefore descend to the required transitions on \(Q_aN\). These are isomorphisms since they are isomorphisms after the epic cover, where they are multiplication of invertible lines. Their triple and diagonal equations hold after that cover and hence globally. This constructs the twisted object and its maps; all identities and compositions follow from the entrywise formulas and full faithfulness of descent.

There are natural isomorphisms in both directions. For \(M\in\mathsf T_U^L(1)\), the transition \(L(x,a(v))\otimes M(a(v))\to M(x)\) is compatible with (9.2) by its triple equation. It descends to \(Q_aR_aM\simeq M\), respecting every transition. For \(N\), restrict the descent identification to \(x=a(v)\) and choose the auxiliary label \(v\) itself. The diagonal line unit gives \(R_aQ_aN\simeq N\). Comparing two labels uses exactly (9.2), so this is an isomorphism of twisted objects, not merely of underlying modules. Naturality follows from the same formulas. They prove the equivalence. The constructions over each base commute with restriction: the family formula of Lemma 9.1 and the formulas (9.1)–(9.2) pull back to the corresponding families and maps. They thus prove the stated equivalence of stacks. \(\square\)

For successive refinements, the two pullback functors agree under the canonical pullback identifications: their section and line formulas both substitute the composite map into \(L\) and \(M\). These identifications satisfy the identity and three-refinement laws because substitution does. Inverse equivalences can be compared uniquely once their isomorphisms after pullback are specified: full faithfulness of \(R_a\) lifts each such comparison, and faithfulness forces its equations. This also gives coherence for the inverse comparisons. Two covers have the common refinement \(U\times V\); its map to the terminal sheaf is epic by epic base change and composition. Consequently different covering presentations can always be compared on a common cover, compatibly with further refinement.

Example 9.3 (a refinement which omits old labels). Cover a point by the four labels of §5, and refine by the single label zero. The map from the singleton to the four-element cover is not surjective, but the singleton still covers the point. The pulled-back line is the diagonal unit. Its twisted category is ordinary modules. The inverse in (9.1) is \(M_x=L(x,0)\otimes N\). In scalar bases, its transition from \(y\) to \(x\) is multiplication by \(c_{xy0}\). Formula (5.1) gives its triple equation, and the unit gives its diagonal identity. This is the actual refinement equivalence, including its inverse, despite the omission of three old labels.

10. Line changes, natural transformations and neutralization

Let \((L,\mu^L)\) and \((B,\mu^B)\) be two line cocycles on the same cover \(U\). Subscripts denote generalized cover coordinates. A line change from \(L\) to \(B\) consists of an invertible module \(P\) on \(U\) and isomorphisms \[ \gamma_{ij}:B_{ij}\otimes P_j\longrightarrow P_i\otimes L_{ij}. \tag{10.1} \] They respect the diagonal units and obey \[ \begin{gathered} (1_{P_i}\otimes\mu^L_{ijk})(\gamma_{ij}\otimes1_{L_{jk}}) (1_{B_{ij}}\otimes\gamma_{jk})\\ =\gamma_{ik}(\mu^B_{ijk}\otimes1_{P_k}). \end{gathered} \tag{10.2} \] Both sides take \(B_{ij}\otimes B_{jk}\otimes P_k\) to \(P_i\otimes L_{ik}\); the order of these factors fixes the directions.

Theorem 10.1 (all equivalences and transformations). A line change induces an \(R\)-linear equivalence of stacks \(A_P:\mathsf T_U^L\to\mathsf T_U^B\), with underlying module \(P\otimes M\). Every \(R\)-linear equivalence of these stacks is naturally isomorphic to one obtained this way. All natural transformations between two such functors are given by module maps \(\tau:P\to P'\) satisfying \[ (\tau_i\otimes1_{L_{ij}})\gamma_{ij} =\gamma'_{ij}(1_{B_{ij}}\otimes\tau_j). \tag{10.3} \] The transformation is invertible exactly when \(\tau\) is invertible. Composition of line changes is tensor product with its indicated comparisons.

Proof. Give \(P\otimes M\) the transition which first applies \(\gamma_{ij}\otimes1_{M_j}\), then \(1_{P_i}\otimes\theta^M_{ij}\). Its triple equation follows by substituting (10.2) and then the triple equation for \(M\); its diagonal equation follows from the stated units. Tensoring an equivariant map with \(P\) is equivariant. Tensor with its inverse line supplies the inverse functor: invert (10.1), tensor by \(P_i^{-1}\) on the left and \(P_j^{-1}\) on the right, and cancel the two inverse pairs. This gives \(L_{ij}\otimes P_j^{-1}\to P_i^{-1}\otimes B_{ij}\). The resulting transitions compose with the original ones to the identity under line evaluation, since the only remaining maps are \(\gamma\) and its inverse. Evaluation is a natural isomorphism of twisted objects. Thus \(A_P\) is an equivalence, with its inverse and both comparison maps constructed.

To prove exhaustiveness, let \(A\) be an \(R\)-linear equivalence of stacks. Over \(U\), the diagonal section trivializes each of the two stacks by §3. In these two descriptions, \(A\) is an \(R\)-linear equivalence of ordinary module stacks. The complete sheaf Morita reconstruction theorem identifies it with tensoring by an invertible module \(P\). Its two scalar actions agree by \(R\)-linearity, so this is an ordinary invertible \(R\)-module, not a different bimodule convention.

On \(U^2\), changing the source description from coordinate \(j\) to coordinate \(i\) is tensoring with \(L_{ij}\), while changing the target description is tensoring with \(B_{ij}\), by (3.3). Compatibility of \(A\) with restriction therefore compares the functors \(B_{ij}\otimes P_j\otimes-\) and \(P_i\otimes L_{ij}\otimes-\). A natural transformation of these module-stack tensor functors is determined by its value at the regular module: on each slice, naturality for \(R\to N\), \(1\mapsto n\), determines it on every local pure tensor; such tensors locally generate by Proposition 7.2. Evaluation at \(R\) consequently gives exactly (10.1). The comparison for three successive descriptions gives (10.2), because the two composites are the comparisons for the same stack functor. The identity comparison gives the diagonal condition.

The local tensor identifications are compatible on overlaps precisely by these equations. For each twisted object their maps glue, using the sheaf condition for morphisms, to a natural isomorphism \(A_P\simeq A\). Conversely the explicitly constructed transition above realizes those same local comparisons. This proves exhaustiveness on objects and functors with their restriction data.

For transformations, evaluation at \(R\) in the local descriptions gives \(\tau:P\to P'\), including when it is not invertible. The preceding local-pure-tensor argument makes its component at \(N\) be \(\tau\otimes1_N\). Compatibility on overlaps is exactly (10.3). That equation also directly verifies that these components are equivariant on every twisted object, so they give and uniquely determine the global natural transformation. A natural isomorphism has an invertible value at \(R\); conversely an invertible \(\tau\) has an inverse satisfying the reversed equation. This proves the last transformation assertion.

If \(P\) changes \(L\) to \(B\) and \(Q\) changes \(B\) to \(C\), the composite underlying module is \(Q\otimes P\otimes M\). Its comparison is \[ \begin{gathered} C_{ij}\otimes Q_j\otimes P_j \longrightarrow Q_i\otimes B_{ij}\otimes P_j\\ \longrightarrow Q_i\otimes P_i\otimes L_{ij}. \end{gathered} \tag{10.4} \] The two arrows are the two supplied line changes. Substitution of their two equations (10.2) proves the equation for their composite. The identity change uses \(P=R_U\) and the unit maps. Threefold composition has the tensor associator as comparison; its coherence and compatibility with all transitions follow on pure tensors. Compositions of transformations are their ordinary component compositions and tensor products, with the interchange law inherited from module maps. This supplies all composition data, completing the theorem. \(\square\)

Together with Theorem 9.2 and the presentation theorem in §4, this describes locally ordinary module stacks, their linear equivalences and their natural transformations by line cocycles and line changes on common covers. Further refinement preserves the entire description and its composition laws.

Corollary 10.2 (neutralization by a twisted line). The stack \(\mathsf T_U^L\) is equivalent to \(\mathsf{Mod}(R)\) as an \(R\)-linear stack if and only if it has an object \(H\) whose underlying module on \(U\) is invertible. Equivalently, there are invertible modules \(H_i\) and coherent identifications \[ L_{ij}\simeq H_i\otimes H_j^{-1} \tag{10.5} \] under which \(\mu\) is the contraction of the middle inverse pair. An explicit equivalence sends \(M\) to the ordinary descent module locally represented by \(H_i^{-1}\otimes M_i\).

Proof. For an invertible twisted \(H\), its transition \(L_{ij}\otimes H_j\to H_i\) gives (10.5) by tensoring with \(H_j^{-1}\). Its triple and diagonal equations are exactly the claimed contraction and unit equations. Take \(B_{ij}=R\), \(P_i=H_i^{-1}\) in Theorem 10.1. The transition of \(H\) supplies \(\gamma\), and the induced ordinary descent is effective by Lemma 9.1. Its inverse is tensoring locally by \(H_i\). This proves the asserted equivalence and its formula.

Conversely, an equivalence to ordinary modules, written as the line change of Theorem 10.1, gives \(L_{ij}\simeq P_i^{-1}\otimes P_j\). Put \(H_i=P_i^{-1}\). The comparison equation yields its twisted transition law. Equivalently, apply the inverse stack equivalence to the regular module; in each local module description its image is an invertible module. Thus it is the required twisted object. The two formulations and both directions are proved. \(\square\)

The criterion uses invertible module sheaves on the given site. If bases exist after a refinement, its line equations become the scalar coboundary equations of §5. Identifying all such classes with sheaf cohomology remains a separate comparison theorem; the line-change and refinement assertions above require no such identification.

Example 10.3 (an invertible module without a frame on the given site). Take the one-object site with only identity coverings, and the ring \(R=\mathbb Z[s]\), \(s^2=-5\). Let \(I=(2,1+s)\), considered inside the field \(\mathbb Q(s)\). Its square has generators \(4,2+2s,-4+2s\). They are all multiples of two, and the difference of the second and third is six; subtracting four gives two. Hence \[ I^2=(2),\qquad J=\tfrac12I,\qquad IJ=R. \tag{10.6} \] Multiplication \(I\otimes_RJ\to R\) is an isomorphism. Choose a finite expression \(1=\sum u_iv_i\) with \(u_i\in I\), \(v_i\in J\), available from \(IJ=R\). The inverse sends \(1\) to \(\sum u_i\otimes v_i\). On \(p\otimes q\), its other composite is \(\sum u_i\otimes(pv_i)q=\sum(pv_i)u_i\otimes q=p\otimes q\); each \(pv_i\) lies in \(R\). Thus \(I\) is an invertible module.

It has no global basis. The norm \(N(a+bs)=a^2+5b^2\) is multiplicative, so the only units of \(R\) are \(1,-1\). If \(I=(d)\), (10.6) would give \(d^2=2u\) for a unit \(u\): equality of the two principal ideals supplies mutually inverse coefficient factors. Taking norms would give \(N(d)=2\), but no integers satisfy \(a^2+5b^2=2\). This is impossible. On this site there is no smaller covering on which to choose a frame. Tensoring with \(I\) is nevertheless an \(R\)-linear autoequivalence of ordinary modules, with inverse tensoring with \(J\), by (10.6). The line-change description includes it without replacing invertible modules by scalar transition data.