Draft lesson.

Ringed sieve sites and extension of scalars

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

The inverse image of an underlying sheaf and the pullback of a module over a new ring are different operations. The first transports algebraic operations exactly. The second also extends scalars and can lose injectivity. We construct both operations and their adjunctions, then give precise hypotheses for tensor and internal-Hom comparisons.

Let \(\mathcal F,\mathcal E\) be categories of set-valued sheaves on small sieve sites. Fix a geometric morphism \(f:\mathcal E\to\mathcal F\), written \(f^{-1}\dashv f_*\), where \(f^{-1}\) preserves finite limits. In particular it preserves finite products, and as a left adjoint it preserves small colimits. The preceding sieve-site construction supplies such adjunctions with their actual maps. No points or subcanonical presentation are assumed.

Let \(R\) be an associative unital ring sheaf in \(\mathcal F\), set \(A=f^{-1}R\), and let \(S\) be an associative unital ring sheaf in \(\mathcal E\), with a unital ring map \(\phi:A\to S\). Modules are left modules unless a right action is specified. Commutativity is imposed explicitly for the scalar internal-Hom comparison in Section 7, not for the module adjunction or the balanced-tensor theorem.

Prerequisites are the exact module sheafification and local exactness proof in Twisted modules, Lemma 7.1; the local tensor and Hom constructions in Tensor and Hom, Section 1; the complete sheaf inverse-image adjunction; and the generalized slices, exponentials and base-change maps in Slice sites and dependent products.

1. Transporting algebraic operations and exact sequences

Apply \(f^{-1}\) to addition, zero, negation, multiplication and the unit, using its canonical finite-product comparisons. Every ring identity is an equality between maps from a finite product, so remains an equality. This constructs \(A\) as a ring sheaf. The same argument transports any \(R\)-module \(M\) to an \(A\)-module \(f^{-1}M\), and transports all module maps. Applying \(f^{-1}\) to the formula for the sum of two maps proves additivity on maps. The functor \(f_*\), which preserves products, similarly transports algebraic operations.

The set-sheaf adjunction restricts to an adjunction on these algebraic maps. Here is why no compatibility is missing. Under the product identifications, the unit at a product has components the units at its factors, by naturality at both projections. Its component at the terminal object is the unique map. The counit has the analogous properties. Consequently transposing a map preserves an addition, scalar-action or unit equation if and only if its transpose preserves that equation. For scalars, the coefficient map on the direct image is \(R\xrightarrow\eta f_*A\). This gives, for an \(A\)-module \(N\), the full natural bijection

where \(f_{R,*}N\) uses that coefficient map. The factors are the original set adjunction factors restricted to the equations; their inverse equations and both naturalities therefore remain valid. The argument also applies to right modules, rings and maps with mixed algebraic arguments.

Lemma 1.1. The functor \(f^{-1}:\operatorname{Mod}_R\to\operatorname{Mod}_A\) is exact.

Proof. Kernels are finite limits of underlying sheaves with their zero maps, so are preserved. For \(d:M\to N\), its module cokernel is also the coequalizer in sheaves of sets of

Indeed the sectionwise additive quotient, followed by module sheafification, is the cokernel. Any map of sheaves of sets from \(N\) invariant under the displayed translations descends to that quotient: represent a quotient section locally by \(n\); two representatives differ locally by a value \(d(m)\), so their images agree locally, then glue uniquely in the target sheaf. This proves the entire set coequalizer property, without requiring an algebraic structure on that target. Its induced additive and scalar operations are the module quotient operations.

The inverse image preserves this set coequalizer and the product in (1.2). It therefore gives exactly the same quotient construction for \(f^{-1}d\), with the transported operations. Thus it preserves cokernels as well as kernels. Additivity and these two comparisons prove exactness. The same kernel and quotient proof, using only the additive operations, proves exactness on abelian sheaves. \(\square\)

2. Generalized localization and local verification

For \(V\in\mathcal F\), there is an induced geometric morphism between the slices over \(f^{-1}V\) and \(V\). Its inverse image sends \((X\to V)\) to \((f^{-1}X\to f^{-1}V)\). Its right adjoint sends \((Y\xrightarrow q f^{-1}V)\) to

using \(\eta_V\) and \(f_*q\). To check the adjunction, a map \(f^{-1}X\to Y\) is over \(f^{-1}V\) exactly when its transpose \(a:X\to f_*Y\) satisfies \(f_*q\,a=\eta_Vp_X\). This is exactly the condition for its unique factor \((p_X,a)\) through (2.1). Projection and the original adjunction give the inverse factor. Their equations and naturalities are the original ones together with pullback uniqueness. The unit is \((p_X,\eta_X)\); the counit is \(f^{-1}\) of projection to \(f_*Y\), followed by \(\epsilon_Y\). Its structure equation and both triangles reduce, on those projections, to the original naturality and triangle equations. Finally the inverse image preserves finite slice limits because these are constructed from the corresponding finite limits and structure maps. Thus it is a geometric morphism.

The restricted ring over \(V\) is \(V\times R\to V\), and its inverse image is \(f^{-1}V\times A\to f^{-1}V\). Restricting \(\phi\) gives the corresponding ring map over \(f^{-1}V\). These identifications commute with the units just specified, so also with their transposed coefficient maps. This gives the ringed localization, including the compatibility of its two descriptions of the scalar map.

We will use local presentations on a small jointly epimorphic family \(V_i\to1\). Exactness and isomorphisms of module sheaves can be detected after restriction to all these slices. In fact a section of the terminal sheaf locally lifts to a section of some \(V_i\): use the local lifting criterion for \(\coprod_iV_i\to1\) and the local representative rule for that coproduct. A kernel or cokernel section which vanishes after all the restrictions therefore vanishes on such a cover, hence vanishes as a sheaf section. Restriction itself is exact by Lemma 1.1 applied to the slice localization morphism from the preceding lesson. This proves the asserted detection of exactness.

The family \(f^{-1}V_i\to1\) is again jointly epimorphic. A left adjoint preserves epimorphisms: transpose two maps whose composites with the image of an epimorphism are equal, cancel the original epimorphism, then transpose back. Apply this to the coproduct map, using preservation of coproducts and of the terminal object. The local verification just proved therefore applies on both sides of the inverse image.

3. The balanced-tensor comparison

Let \(P\) be a right \(R\)-module and \(M\) a left \(R\)-module. Tensor sheaves are associated sheaves of the sectionwise balanced tensor products. Their universal property describes maps to any abelian sheaf as additive maps in both arguments satisfying the balance equation.

Proposition 3.1. There is a canonical natural isomorphism

Proof. Apply \(f^{-1}\) to the universal balanced pairing and use its product comparison to obtain the displayed map. For any abelian sheaf \(Q\) on \(\mathcal E\), the algebraic adjunction of Section 1 identifies maps from its right side to \(Q\) with maps \(P\otimes_RM\to f_*Q\), hence with balanced bilinear maps \(P\times M\to f_*Q\). The product comparison and the same adjunction identify those with balanced bilinear maps \(f^{-1}P\times f^{-1}M\to Q\). Additivity is tested on products of three factors, and balance on \(P\times R\times M\); the equation correspondence in Section 1 proves each condition in both directions. The tensor universal property identifies the result with maps from the left side to \(Q\).

Every factor is natural, and the resulting bijection is precomposition by (3.1), since both start with the transported universal pairing. A map which gives bijections into every target is invertible: use its target and source to construct and check the two inverse composites, as in the preceding localization lesson. This proves (3.1). Additional bimodule actions are preserved by the same equation test. No flatness was used. \(\square\)

In particular tensor commutes with restriction to every generalized slice, by applying this proposition to its localization morphism. A right \(A\)-module which is flat after restriction to a jointly epimorphic family is flat globally: apply this tensor comparison and the local exactness detection of Section 2 to each tensor image of a short exact sequence. Here flat means that its tensor functor on left \(A\)-modules is exact.

4. The ringed module adjunction

The coefficient map \(\phi\) is equivalent by the algebraic adjunction to a ring map \(\phi^\sharp:R\to f_*S\). Define

For a left \(S\)-module \(N\), there are actual natural inverse factors

The first sends \(h\) to \(m\mapsto h(1\otimes m)\); its inverse sends an \(A\)-linear \(k\) to \(s\otimes m\mapsto s\,k(m)\). The balance equation makes this inverse well-defined, and \(S\)-linearity and the unit prove both inverse equations on locally generating pure tensors. The second is (1.1) with the indicated restricted action. Its factors and all naturalities were proved there. Consequently (4.2) proves the full module adjunction.

The unit is the geometric unit followed by the direct image of \(m\mapsto1\otimes m\). The counit is

The product-unit compatibility in Section 1 makes \(\epsilon_N\) \(A\)-linear for this action, so (4.3) descends. Transposing the two identity maps through the explicit inverse factors (4.2) gives exactly these unit and counit maps. Applying the inverse factors twice returns the identity maps; these are the two adjunction triangle equations. Equivalently the triangle on \(f^*M\) returns \(s\otimes m\) by the geometric triangle and \(1\otimes m\), while the triangle on \(f_*^{\phi}N\) returns its section by the other geometric triangle and multiplication by one.

5. Exactness and injective objects

The left adjoint \(f^*\) preserves small colimits and is right exact. The right adjoint \(f_*^{\phi}\) is left exact. These are the actual kernel and cokernel comparisons obtained by transposing the corresponding universal factors.

If \(S\) is flat as a right \(A\)-module, then \(f^*\) is exact: first use exactness of \(f^{-1}\) from Lemma 1.1, then exactness of \(S\otimes_A-\). This condition may be verified on a jointly epimorphic family of generalized slices by Section 3. The condition is substantive; an arbitrary extension of scalars need not be exact.

Under this flatness hypothesis \(f_*^{\phi}\) sends injective \(S\)-modules to injective \(R\)-modules. For a monomorphism \(M'\hookrightarrow M\), a map \(M'\to f_*^{\phi}I\) transposes to \(f^*M'\to I\). Exactness makes \(f^*M'\to f^*M\) monic, so injectivity of \(I\) supplies an extension. Transpose it back; naturality of (4.2) gives the required extension over \(M\). Separately, \(f_{R,*}\) in (1.1) preserves injectives without a scalar-extension hypothesis, because its left adjoint \(f^{-1}\) is always exact. Exercise 1 shows why the analogous unconditional assertion for \(f_*^{\phi}\) would be false.

6. Composition and its coefficient maps

Let \(g:\mathcal H\to\mathcal E\) be another geometric morphism, and let \(\psi:g^{-1}S\to T\) be a ring map. The composite coefficient map is

where \(f g\) denotes \(f\circ g\). Its direct-image scalar map is the same map under the composite adjunction: naturality of the units and the product compatibility of Section 1 identify both transposes.

Use (3.1) and the unit-respecting geometric composition comparison from the sieve-site lesson. The comparison between the iterated and composite module inverse images reduces to

It sends \(t\otimes(b\otimes m)\) to \(t\psi(b)\otimes m\), with inverse \(t\otimes m\mapsto t\otimes(1\otimes m)\). The balance relations move \(b\) between the factors and prove both inverse equations. The equations for \(g^{-1}A\) use exactly the composite map in (6.1). These maps are natural, and do not require flatness. The corresponding direct-image functors compose with the scalar action prescribed by (6.1).

For three maps, both associativity composites multiply the same coefficient factors in the same order and keep the same last module element. Associativity of ring multiplication proves equality on pure tensors; these locally generate, so proves equality as sheaf maps. Identity maps use multiplication by the ring unit. The comparison also respects the units in (4.2), on their generators \(1\otimes m\), and hence the counits by their inverse transpose factors. This proves the full composition coherence.

7. Internal Hom and its precise finiteness hypotheses

For arbitrary \(R\), define \(\mathcal Hom_R(M,N)\) as the subsheaf of the exponential \(N^M\) consisting of maps preserving addition, zero and the \(R\)-action on every restriction. It is an abelian sheaf under pointwise addition. This is an equation subsheaf: compare the two addition maps into \(N^{M\times M}\), the two scalar maps into \(N^{R\times M}\), and the value at zero into \(N\); take their equalizers. The exponentials and all these maps exist by the slice lesson. For commutative \(R\), pointwise multiplication by \(R\) also preserves the equations, making this an \(R\)-module. We do not impose that extra module structure for noncommutative \(R\).

Restriction to a generalized slice preserves this internal Hom. The dependent-product base-change theorem identifies restriction of \(N^M\) with the exponential of the two restricted objects; the equalizer equations are the same transported addition and scalar equations. Thus their equalizer is exactly the restricted module Hom. This also recovers the full local-morphism gluing construction in the prerequisite.

Applying \(f^{-1}\) to evaluation and currying gives a canonical map of abelian sheaves

It lands in the stated Hom because the scalar and addition equations transport as in Section 1. Call \(M\) locally finitely presented if on some small jointly epimorphic family of slices it has a presentation \(R^p\to R^q\to M\to0\), with finite \(p,q\) allowed to vary between those slices.

Theorem 7.1. If \(M\) is locally finitely presented, (7.1) is an isomorphism.

Proof. First take a global finite presentation. The actual internal-Hom sheaf is

This kernel is taken in abelian sheaves; it needs no \(R\)-module structure on Hom when \(R\) is noncommutative. Evaluation on the finitely many basis vectors identifies Hom from \(R^q\) with \(N^q\). A tuple belongs to the kernel exactly when it kills the relation map \(R^p\to R^q\), so factors uniquely through \(M\). This factor criterion holds on every slice, whose restriction is exact by Section 2. It therefore proves (7.2) as a sheaf identity with its actual inclusion. Exactness of \(f^{-1}\) and preservation of finite powers turn this kernel into the analogous kernel for the transported presentation \(A^p\to A^q\to f^{-1}M\to0\). Under these identifications (7.1) is the identity on the basis evaluations, so is an isomorphism.

For local presentations, use the induced slice geometric morphisms of Section 2. Restriction of internal Hom commutes with localization as just proved; restriction of inverse image is that induced slice inverse image by its definition. Thus the preceding proof applies over each \(f^{-1}V_i\). These objects are jointly epimorphic, so the local isomorphisms prove that the original canonical map is invertible. No flatness of an additional coefficient ring is involved in this assertion. \(\square\)

Now assume \(R,S\) commutative. Combining (7.1) with scalar extension gives

On local tensors it sends \(s\otimes h\) to the map \(t\otimes m\mapsto st\otimes h(m)\). Here \(h\) belongs to the transported Hom, and evaluation is (7.1). Commutativity makes this map \(S\)-linear in its input. The formula respects all balance relations and all restrictions.

Theorem 7.2. Map (7.3) is an isomorphism if \(M\) is locally finitely presented and \(S\) is flat over \(A\). It is also an isomorphism, without flatness, when \(M\) is locally a retract of a finite free module.

Proof. For a finite presentation, the right side is the kernel of \((f^*N)^q\to(f^*N)^p\), because right exactness transports that presentation to one by \(S^p,S^q\). The left side is the scalar extension of the kernel (7.2), after exact geometric inverse image. Flatness preserves that kernel, and tensor commutes with finite powers, which are finite direct sums in modules. Its two maps are exactly the transported relation matrix. Thus the comparison is the identity on this kernel. Local presentations reduce to this case by Section 2 and slice compatibility of tensor and Hom; this proves the first assertion.

For the second, the finite free case is already an identity on finite tuples and uses no flatness. A local retract of \(R^q\) has inclusion and retraction whose composite is its identity. All functors and comparisons here respect those maps and their composites. Naturality makes the finite-free comparison intertwine the resulting idempotents. Its inverse therefore also intertwines them, and restricts to inverse maps on their retracts. This gives the required local isomorphism, which is global by local detection. The retract hypothesis does not assert the existence of a local basis. \(\square\)

8. Four exercises with full solutions

8.1 Reduction modulo a prime

Exercise 1 (basic). For the identity one-point topos and \(\mathbb Z\to\mathbb F_p\), compute both inverse-image operations and their unit/counit. Show that the module pullback is not exact and the module direct image need not preserve injectives. Give a finite-presentation failure of (7.3).

Solution. The geometric inverse image is identity. The module pullback is \(M\mapsto\mathbb F_p\otimes_{\mathbb Z}M=M/pM\), and the direct image forgets to the underlying abelian group. The tensor identification sends \(\bar a\otimes m\) to \(am\bmod pM\), with inverse \(m\bmod pM\mapsto1\otimes m\). Its relations are exactly the tensor balance and the relation \(p=0\). The unit is reduction modulo \(p\), and the counit \(\mathbb F_p\otimes N\to N\) is scalar multiplication; applying these maps returns every generating tensor or vector, giving both triangles.

The injection \(\mathbb Z\xrightarrow{p}\mathbb Z\) becomes the zero map \(\mathbb F_p\to\mathbb F_p\), so the module pullback is not exact. The vector space \(\mathbb F_p\) is injective over the field: extend a basis of a subspace to a basis and assign arbitrary, for example zero, values on the extra basis vectors. But the group map \(p\mathbb Z\to\mathbb F_p\) sending \(p\) to \(1\) cannot extend to \(\mathbb Z\), since every such extension would send \(p\) to \(p\) times its value at one, hence to zero. Thus direct image fails to preserve this injective. Finally take \(M=\mathbb Z/p\mathbb Z\), \(N=\mathbb Z\). Then \(\operatorname{Hom}_{\mathbb Z}(M,N)=0\), while both pullbacks are \(\mathbb F_p\) and their Hom is \(\mathbb F_p\). The comparison is \(0\to\mathbb F_p\), so finite presentation alone does not replace flatness in the first assertion of Theorem 7.2.

8.2 Coefficients on a fork site

Exercise 2 (intermediate). Let the discrete two-leaf site map into the fork site with both leaves covering the vertex, as in the inverse-image lesson. Start with the pair of rings \((\mathbb Z,\mathbb Z)\), and on the fork extend its inverse-image ring to the ring sheaf with leaf rings \(\mathbb Z[1/2]\), \(\mathbb F_2\). Compute the module adjunction and locate the exactness failure. What changes if the second ring is \(\mathbb Z[1/3]\)?

Solution. The geometric inverse image is the equivalence taking a pair to the fork sheaf with that pair at its leaves and their product at the vertex. Thus the inverse-image ring has vertex \(\mathbb Z\times\mathbb Z\), and the new ring has vertex \(\mathbb Z[1/2]\times\mathbb F_2\). A sheaf of modules is exactly a pair of leaf modules, its vertex the product of those modules with the componentwise ring action. Indeed the sheaf matching comparison is that product, and its scalar compatibility is exactly the componentwise action.

The module pullback of \((M,N)\) has leaf modules \((M[1/2],N/2N)\). The direct image forgets the two coefficient extensions. Their adjunction is (4.2) separately on the two components, with units the fraction and quotient maps and counits multiplication. At the vertex this also agrees with tensor over the product ring: the two complementary idempotents split every module into its two components; cross tensors vanish because the idempotents multiply to zero. Thus the same formula gives all vertex maps and restrictions.

The map \((0,\mathbb Z)\xrightarrow{(0,2)}(0,\mathbb Z)\) is monic before pullback and fails to be monic on the second leaf afterward. The two represented leaf objects jointly cover the terminal sheaf, so local detection identifies the obstruction on that leaf. If the second coefficient ring is \(\mathbb Z[1/3]\), both component functors are exact localizations, by Exercise 3, and hence the sheaf pullback is exact. This example uses the associated-sheaf inverse image: the raw set Kan extension has empty vertex value and cannot by itself carry the transported ring unit.

8.3 Localization and nonuniform local torsion

Exercise 3 (advanced). Let \(R\) be a commutative ring sheaf on a small sieve site and \(s\) a global section. Let \(S=R[s^{-1}]\) be the associated sheaf of sectionwise localizations. Prove that \(S\otimes_R-\) is exact, describe the kernel of \(M\to S\otimes_RM\), and give a section in that kernel which no single power of \(s\) kills globally.

Solution. The associated sheaf of \(U\mapsto M(U)[s|_U^{-1}]\) represents \(S\otimes_RM\). To see the universal map, send \((r/s^i,m)\) to \(rm/s^i\); fraction equality and balance make it well-defined before sheafification. Conversely an \(R\)-linear map from \(M\) to an \(S\)-module extends uniquely by \(m/s^i\mapsto s^{-i}m\) followed by the map. These inverse rules and the sheafification adjunction give the tensor identification on all restrictions.

Ordinary localization is exact by a direct fraction check. An injection remains injective because a fraction whose image is zero has a numerator killed by some power of \(s\). At a middle term, a fraction mapping to zero has, after multiplying its numerator by a power, numerator in the original kernel, which lifts from the preceding module. A surjection remains surjective by lifting a numerator. These are full fraction proofs of preservation of a short exact sequence. For sheaves a short exact sequence need only be locally exact; nevertheless the same argument applies on a covering refinement: represent a section of a localized term locally by a fraction, clear its vanishing denominator locally, then use local exactness to lift its numerator. The analogous injection and surjection arguments use the same local representatives. The exact sheafification and local equality rules therefore prove exactness of the displayed sheaf functor, without assuming surjectivity on each original section set. In particular \(S\) is flat over \(R\).

The unit sends \(m\) to \(m/1\). It is zero exactly when, on a covering sieve, each restriction of \(m\) is killed by some power of the restricted \(s\), with the exponent allowed to depend on the member. This is the fraction zero criterion followed by the precise local equality rule for sheafification.

For the example, take the discrete space \(\{1,2,\ldots\}\), its sheaf of integer-valued functions, \(s=2\), and the sheaf with section module \(M(W)=\prod_{n\in W}\mathbb Z/2^n\mathbb Z\). On each singleton its localization is zero, so its localized sheaf is zero. The section \(m=(1\bmod2^n)_n\) therefore belongs to the kernel. No fixed \(2^k\) kills it: choose \(n>k\). In fact its fraction remains nonzero in the ordinary localization \(M(X)[1/2]\) of the global section module. Thus taking a sectionwise localization and then taking its associated sheaf are distinct steps, even though the sheaf localization functor is exact.

8.4 Flat extension and an infinite Hom obstruction

Exercise 4 (challenge). For \(\mathbb Z\to\mathbb Q\), take \(M=\bigoplus_{n\ge1}\mathbb Z\), \(N=\mathbb Z\). Determine the image of the canonical Hom comparison and exhibit a point outside it. Compare this with finite free modules.

Solution. A map from a direct sum is determined by arbitrary values on its basis, so the comparison is

Its image consists exactly of rational sequences with a common nonzero integer denominator. A tensor is a finite sum of rational coefficients times integer sequences; multiplying their finitely many denominators gives such a common denominator. Conversely \((a_n/d)_n\) is the image of \((1/d)\otimes(a_n)_n\). The map is injective: identify its domain with localization of the product of integer groups; a represented integer sequence which is zero in every rational coordinate is the zero sequence, hence gives the zero fraction.

Choose pairwise distinct primes \(p_n\). Such an infinite sequence exists: if a finite list exhausted the primes, any prime divisor of one plus their product would not be on the list; a least divisor greater than one is prime. The sequence \((1/p_n)_n\) has no common integer denominator, since such a denominator would be divisible by every \(p_n\), whereas a fixed nonzero integer has only finitely many positive divisors. It is therefore outside the image.

The fraction exactness proof of Exercise 3 applies to arbitrary nonzero integer denominators as well, using their products for common denominators; thus \(\mathbb Q\) is flat. For finite free \(M=\mathbb Z^r\), both sides of the comparison are \(\mathbb Q^r\) and the map is the identity on basis values, including \(r=0\). The infinite example therefore isolates the missing finiteness condition, independently of the nonflat obstruction in Exercise 1.

Further reading

The Stacks Project gives ringed module pullback and pushforward, flat morphisms and the finite-presentation Hom comparison, and localization of ringed morphisms. The explicit algebraic factors, quotient exactness proof, generalized local verification, comparison maps and all four solutions used here are supplied above.