PERIODICITY FOR SUBQUOTIENTS OF THE MODULAR CATEGORY 𝒪

In this paper we study the category 𝒪 over the hyperalgebra of a reductive algebraic group in positive characteristics. For any locally closed subset K of weights, we define a subquotient 𝒪[K] of 𝒪. It has the property that its simple objects are parametrized by elements in K. We then show that 𝒪[K] is equivalent to 𝒪 [K +pl γ] for any dominant weight γ if l > 0 is an integer such that K ∩ (K – plη) = ∅ for all weights η > 0. Hence it is enough to understand the subquotients inside the dominant (or the antidominant) chamber.


Introduction
One of the cornerstones of the rational representation theory of a reductive algebraic group G over a field k of characteristic p > 0 is Steinberg's tensor product theorem.For a dominant weight µ we denote by L(µ) the irreducible representation of G with highest weight µ.For a dominant weight λ there is a p-adic extension λ = λ 0 + pλ 1 + p 2 λ 2 + • • •+ p l λ l with uniquely defined restricted weights λ 0 , . . ., λ l , and Steinberg's tensor product theorem states that where (•) [n] denotes the n-fold Frobenius twist.In order to prove the theorem, it is, by induction, sufficient to prove the following.If λ 0 is restricted and γ is dominant, then L(λ 0 ) ⊗ L(γ) [l] is a simple G-module (it must then be isomorphic to L(λ 0 + p l γ) for weight reasons).The theorem has numerous generalizations (for non-dominant weights, quantum groups, or for representations of the hyperalgebra of G, see, for example, [A]).
The above serves as a motivation for us to consider the functor (•) ⊗ L(γ) [l] on the category of modules of the hyperalgebra (or algebra of distributions) U associated with G.This algebra coincides with the universal enveloping algebra of the Lie algebra of G in the case that the ground field is of characteristic 0, but this is not the case in positive characteristics.The category of rational representations of G can be identified with the category of finite dimensional U -modules.As U admits a triangular decomposition one can, as in the classical characteristic 0 case, drop the finite dimensionality condition and instead consider the highest weight category O 1 .The category O for U shares many familiar properties with its characteristic 0 relative, which was studied intensely over multiple decades (for an overview on the most essential results, see [H2]).But in the modular case it has an additional property that does not occur in characteristic 0, and which is closely connected to Steinberg's theorem: it inhibits a periodicity structure on subquotients.
In this article we consider subquotient categories O [K] of O associated to locally closed subsets of the weight lattice X (a locally closed subset of X is a union of intervals [λ, µ] with respect to the usual partial order).We then prove the following.
Theorem.Suppose that K is a locally closed quasi-bounded subset of X. Suppose that l > 0 is such that K ∩ (K + p l η) = ∅ for all weights η > 0. Then the functor (•) ⊗ L(γ) [l] induces an equivalence O [K] ∼ − → O [K+p l γ] for all dominant weights γ ∈ X.
Apart from Steinberg's tensor product theorem also the periodicity of (p, ∆)-filtrations (cf.Corollary 4.3 in [A]) could find a conceptual foundation in the above theorem.
Note that if K is finite, then an integer l as required in the theorem always exists.Once such an l is fixed, we still can choose an arbitrary dominant weight γ for the statement to hold.This shows that we can transfer the subquotient O [K] deep inside the dominant chamber, or, by reversing the statement, deep inside the antidominant chamber.In both cases, the representation theory has some valuable additional features.By [A], projective covers exist in O for the simple objects L(λ) for antidominant weights λ, while the tilting modules T (λ) exist in O for all dominant λ.
We use this opportunity to also state and prove some basic results on the modular category O for future reference, such as Krull-Remak-Schmidt decompositions, the existence of projectives in truncated subcategories, BGGH-reciprocity, and some additional results on modules admitting a Verma flag.Most of these results are well-known in the characteristic 0 case and the proofs can be found in [H2].If the proof in the modular case does not need adjustment, we will simply refer to the appropriate result in [H2].We give a more detailed proof if some adjustments are required.

The modular category O
Let g be a semisimple complex Lie algebra with root system R.For α ∈ R we denote by α ∨ the associated coroot, and by X the weight lattice.We fix a basis Π ⊂ R and denote by R + ⊂ R the corresponding system of positive roots.Then we denote by ≤ the induced partial order on X, i.e. µ ≤ λ if and only if λ − µ can be written as a sum of elements of R + .We let U C = U (g) be the universal enveloping algebra of g.
with n > 0, α ∈ R + and β ∈ Π.We denote by U + Z the subring generated by the e α 's, and by U 0 Z the subring generated by the h β n 's.The following integral version of the PBW-theorem is one of the main results in [K].
(1) The algebra U + Z is free over Z, and the elements α∈R + e (nα) α with n α ≥ 0 form a basis.
(2) The algebra U − Z is free over Z, and the elements α∈R + f Z is free over Z, and the elements β∈Π h β n β with n β ≥ 0 form a basis.(4) The multiplication defines an isomorphism The products in parts (1) and (2) above should be taken with respect to a fixed, but arbitrary order on R + .The algebra U 0 Z is commutative, so no order is needed.Note that U Z can be considered as the algebra of distributions of the semisimple and simply connected Z-group scheme G Z associated with R (cf.Section II.1.12 in [J]).It inherits a Hopf algebra structure from U C .2.2.The modular category O. From now on we fix a field k of characteristic p > 0. We set U := U Z ⊗ Z k and define U + , U − , U 0 likewise.As before, we can consider U as the algebra of distributions of the group scheme G k .It inherits a Hopf algebra structure from U (g) via U Z .The category of U -modules hence obtains a tensor product structure (•) ⊗ (•).
To any λ ∈ X we can associate a character χ λ : U 0 → k that maps h β n to the image of λ,β ∨ n in k.For a U 0 -module M and λ ∈ X we define and call this the λ-weight space of M .We say that M is a weight module if M = λ∈X M λ , and we say that λ is a weight of M if M λ = 0. We say that a U + -module M is U + -locally finite if M is the union of its finite dimensional U + -submodules.
Definition 2.2.The category O is the full subcategory of the category of U -modules that contains all objects that are weight modules and U + -locally finite.
If M is an object in O, then every submodule and every quotient of M is also contained in O, and O is an abelian category.It is easy to check that O is stable under the tensor product on U -modules.
2.3.Verma modules and simple quotients.The algebra U 0 normalizes U + , so U ≥0 := U 0 U + ⊂ U is a subalgebra.For any λ ∈ X we denote by k λ the one-dimensional U ≥0 -module on which U 0 acts via the character χ λ , and U + acts via the augmentation U + → k (that sends all generators e (n) is called the Verma module with highest weight λ.We denote by v λ = 1 ⊗ 1 the obvious generator of ∆(λ).The following proposition collects the basic facts about Verma modules.The arguments for the proofs are standard and do not depend on the characteristic of k.The characteristic 0 version can be found in [H2].
(3) For each λ ∈ X there exists a unique simple quotient L(λ) of ∆(λ) in O. (4) The set {L(λ)} λ∈X is a full set of representatives for the simple isomorphism classes of O.
2.4.Finite dimensional simple modules.Denote by X + = {λ ∈ X | λ, α ∨ ≥ 0 for all α ∈ Π} the set of dominant weights.For a dominant weight λ we denote by V (λ) the Weyl module with highest weight λ.It is constructed as follows.We denote by L C (λ) the simple, finite dimensional U C -module with highest weight λ and choose a non-zero vector v ∈ L C (λ).The integral Weyl module is then defined by Proof.If λ is a dominant weight, then the Weyl module V (λ) is a finite dimensional U -module with highest weight λ and it contains L(λ) as a quotient.Hence L(λ) is finite dimensional.If λ is not dominant, then there is a simple root α such that λ, α ∨ < 0. We consider the α-string in L(λ) through the highest weight: n≥0 L(λ) λ−nα .This is a highest weight module for the subalgebra U α of U generated by the e and n ≥ 0. The algebra U α is the algebra of distributions associated to the Lie algebra sl 2 (C) and the field k.The structure of the highest weight modules can be worked out explicitely (for example, see [A]).The fact that λ, α ∨ < 0 implies that n≥0 L(λ) λ−nα must be infinite dimensional, and hence so is L(λ).
2.5.Characters.Suppose that M is an object in O that has the property that its set of weights is bounded, i.e. there exists some γ ∈ X such that M λ = 0 implies λ ≤ γ.Suppose furthermore that each weight space of M is finite dimensional.Then one can define the character of M as where Z[X] is a suitable completion of the group ring Z[X].This applies to any finitely generated object in O and, in particular, to the simple objects L(λ), so there is a well-defined character ch L(λ).Clearly, ch L(λ) ∈ e λ + µ<λ Z ≥0 e µ .Note that for any ν, γ ∈ X the interval [ν, γ] is finite.If M is as above, there are well defined numbers We write (M : L(µ)) = a µ and call this the multiplicity of L(µ) in M .

A duality on O.
Recall that there is an antiautomorphism σ on U C that maps e α to f α and h α to h α .It hence leaves U Z stable, so we obtain an antiautomorphism on U that we denote by the same symbol.For a U -module M we denote by M σ its twist by σ.Then (M σ ) λ = M λ as a vector space, so the σ-twist preserves weight modules.For a U -module M we denote by M * = λ∈X Hom k (M λ , k) the restricted linear dual.It acquires the structure of a U -module by setting (x.φ)(m) = φ(σ(x).m)for all m ∈ M , φ ∈ M * and x ∈ U .Lemma 2.5.Suppose that M is an object in O with finite dimensional weight spaces and such that its set of weights is bounded from above .Then M * is an object in O as well and we have a functorial identification (M * ) * = M .Proof.As observed above, M * is a weight module.As the set of weights of M * coincides with the set of weights of M , it is bounded from above, hence M * must be U + -locally finite.So it is contained in O as well.It again satisfies the assumptions of the above lemma, and it is immediate that (M * ) * = M canonically.
For the objects in the statement Clearly, L * is a simple object in O if L is.From the fact that the characters agree we deduce that L(λ) * ∼ = L(λ) for all λ ∈ X.
Lemma 2.6.We have (Here, Ext 1 O means the Yoneda-extension group.) Proof.For all λ, µ ∈ X we have by the universal property of Verma modules.This is a one-dimensional space if λ = µ.If λ ≤ µ, then ∇(µ) λ = 0 and hence there are no non-trivial homomorphisms from ∆(λ) to ∇(µ).From any non-zero homomorphism f : ∆(λ) → ∇(µ) we obtain by dualizing a non-zero homomorphism f * : ∆(µ) → ∇(λ).Hence the existence of a non-zero homomorphism implies λ ≤ µ and µ ≤ λ, hence λ = µ.We have proven the first statement.Let 0 → ∇(µ) → M → ∆(λ) → 0 be an exact sequence.If λ < µ, then λ is a maximal weight of M , so the universal property of ∆(λ) implies that this sequence splits.By dualizing we obtain another exact sequence 0 → ∇(λ) → M * → ∆(µ) → 0, which splits if µ < λ.But the conditions λ < µ and µ < λ cannot both hold.Hence the second statement is also true.2.7.The Frobenius twist.The group scheme G k is induced from an integral version G Z via base change.Hence there exists a Frobenius endomorphism G k → G k (an endomorphism of k-group schemes).This endomorphism induces an endomorphism on its k-algebra of distributions.We hence obtain an endomorphism Fr : U → U that is given by the following formulas: If M is a U -module, then we denote by M [1] the U -module that we obtain from M by pulling back the action homomorphism along Fr.If M is a weight module, then so is M [1] and we have M and we obtain a functor (•) [1] : O → O.This is called the Frobenius twist.We denote by (•) [l] the functor obtained by applying (•) [1] l-times.As Fr is surjective, L [1] is simple for all simple objects L in O.A quick check on the highest weights yields the following result.
2.8.Direct decompositions.The goal of this section is to show that we have a Krull-Remak-Schmidt decomposition for all finitely generated objects in O. Once the analogue of the Fitting lemma is proven, the arguments for this result are standard.
Lemma 2.8.Suppose that M is an object in O that is finitely generated as a U -module.Then any endomorphism f of M induces a Fitting-decomposition, i.e. there exists some n > 0 such that Proof.If M is finitely generated, then every weight space of M is finite dimensional and there exists a finite subset B of X such that M B := λ∈X M λ is finite dimensional and generates M .We denote by f B the vector space endomorphism on M B that is induced by f .Then there is some n ≫ 0 such that im generates im f l ⊂ M for all l > 0 and we deduce that the image of f stabilizes: im f n = im f n+1 = im f n+2 = . . . .Now let µ ∈ X and denote by f µ the endomorphism on M µ induced by f .Then im f n µ = im f n+1 µ = . . . .As M µ is finite dimensional we also deduce ker f n µ = ker f n+1 µ = . . . .We also obtain the Fittingdecomposition Lemma 2.9.Suppose that M is a finitely generated, indecomposable object in O. Then every endomorphism of M is either nilpotent or an automorphisms.In particular, End O (M ) is a local ring.
Proof.Let f be an endomorphism of M that is not an automorphism.The finite dimensionality of the weight spaces allows us to deduce that f is not injective.Hence ker f = {0} and we deduce M = ker f n for some n ≫ 0 from the indecomposability of M and Lemma 2.8.Hence f is nilpotent.So every endomorphism is either nilpotent or an automorphism.If f is nilpotent and g is an automorphism, then g − f is invertible, hence an automorphism.It follows that the sum of two nilpotent endomorphisms of M is again nilpotent.Moreover, g • f and f • g are not injective, hence nilpotent.So the nilpotent endomorphisms form an ideal in End O (M ) and each element in its complement is invertible.Hence End O (M ) is a local ring.
Proposition 2.10.Let M be a finitely generated object in O. Then M can be written as a finite direct sum of indecomposable objects in O.Moreover, such a decomposition is unique up to reordering and isomorphisms.
Proof.As in the proof of Lemma 2.8 we denote by B a subset of X such that M B = λ∈B M λ ⊂ M is finite dimensional and generates M .Then every direct summand of M must intersect M B nontrivially.In particular, the number of direct summands in a decomposition of M is bounded by the dimension of M B .So a decomposition of M into indecomposable objects exists.The uniqueness of this decomposition can be proven (with Lemma 2.9) using standard arguments.

Verma flags and projective objects in O
The modular category O does not possess enough projectives.In fact, only the L(λ) with antidominant λ admit a projective cover.But locally, i.e. for certain truncated subcategories O J , projective covers exist, as we will show in this section.This will be good enough for all our purposes.Note that this situation resembles the (related) case of representations of Kac-Moody algebras.We will also prove an appropriate version of the BGGH-reciprocity theorem that was first proven in the context of modular Lie algebras by Humphreys in [H1].Later, the characteristic 0 case was treated in [BGG].Many of the ideas used in the following originate in the article [RCW].
3.1.Verma flags.Let M be an object in O.
Definition 3.1.M is said to admit a Verma flag if there is a (finite) filtration A filtration like the one in the definition above is sometimes, for example in [H2], called a standard filtration.
In the situation of the definition above we set This number is independent of the chosen filtration and is called the multiplicity of ∆(µ) in M .Lemma 3.2.Suppose that M admits a Verma flag.Then there exists a filtration and λ 1 , . . ., λ n ∈ X such that M i /M i−1 ∼ = ∆(λ i ) for i = 1, . . ., n, and such that λ i > λ j implies i < j.
Proof.This follows, by induction, from the fact that a surjective homomorphism N → ∆(λ) splits if λ is maximal among the weights of N (by Proposition 2.3).Proof.If A and B admit Verma flags, then it is easily shown that M admits a Verma flag and that the claim about the multiplicities holds.It is hence enough to show that if M admits a Verma flag, then A and B do as well.This is proven using the same arguments as in the characteristic 0 case that can be found, for example, in Section 3.7 in [H2].

Some functors on O.
It is now useful to endow the set X with a topology.Definition 3.4.
(1) We say that a subset This indeed defines a topology.Note that arbitrary unions of closed subsets are closed again.Let M be an object in O, and let I ⊂ X be a closed subset with open complement J.We set (1) M admits a Verma flag.
(2) M J admits a Verma flag for any open set J.
(3) M I admits a Verma flag for any closed set I.

If either statement above is true, then we have
for all open subset J and all closed subsets I of X.
Proof.As M = M X = M X , either (2) or (3) imply (1).Suppose M admits a Verma flag.Let J be an open subset of X with closed complement I. Using Lemma 3.2 we deduce that there is a submodule M ′ of M , appearing in a Verma flag of M , such that M ′ and M/M ′ admit Verma flags and such that In particular, it follows that M ′ = M I and M/M ′ = M J .This shows that (1) implies ( 2) and ( 3), and that the statement about the multiplicities is true if (1) holds.

3.3.
Objects admitting a Verma flag as U ≤0 -modules.Set U ≤0 := U − U 0 .Again this is a subalgebra of U .We denote by U ≤0 -mod wt the full subcategory of the category of U ≤0 -modules that contains all objects that are weight modules, and we denote by Res the restriction functor from the category O to U ≤0 -mod wt .
Proof.Let f : M → Res ∆(λ) be a surjective homomorphism in U ≤0 -mod wt .Let g λ : Res ∆(λ) λ → M λ be a right inverse of f λ : M λ → Res ∆(λ) λ (in the category of vector spaces).Then g λ is a U 0 -module homomorphism.As ∆(λ) is free as a U − -module of rank 1 (by Kostant's version of the PBW-theorem) and as it is generated by any non-zero element in the one-dimensional space ∆(λ) λ , we have Hom U ≤0 (Res ∆(λ), M ) = Hom U 0 (Res ∆(λ) λ , M λ ).So g λ induces a morphism g : Res ∆(λ) → M with the property that its λ-component is right inverse to f λ .Hence f • g is an endomorphism of Res ∆(λ) that restricts to the identity on Res ∆(λ) λ .Hence f • g is the identity, so f splits.The claim follows.
(1) Let N be an object in O that admits a Verma flag.Then Res N is isomorphic to a direct sum of objects of the form Res ∆(λ) for various λ.
(2) Let M and N be objects in O and assume that N admits a Verma flag and let f : M → N be a surjective homomorphism.Then Res f : Res M → Res N splits.
Proof.Statement (1) follows from the fact that each Res ∆(λ) is projective in U ≤0 -mod wt .From (1) and Lemma 3.6 it then also follows that Res N is projective in U ≤0 -mod wt , which implies statement (2).
3.4.Tensor products.Let L be an object in O. Then the functor M → M ⊗ L preserves the category of weight modules and the category of U + -locally finite modules, hence it induces a functor from O to O.
Lemma 3.8.Suppose that L is finite dimensional.If M admits a Verma flag, then so does M ⊗ L and we have Proof.Note that the functor (•) ⊗ L is exact.It is hence enough to prove the statement for M = ∆(λ) for some λ ∈ X.In this case, the same arguments can be used as in the characteristic 0 case, which can be found, for example, in Section 3.6 of [H2].
3.5.Projective objects in truncated categories.Let J be an open subset of X.
Definition 3.9.We denote by O J the full subcategory of O that contains all objects M that have the property that M λ = 0 implies λ ∈ J.
Clearly, the functor M → M J is actually a functor from O to O J , and we have Definition 3.10.We say that an open subset J of X is quasi-bounded if for any λ ∈ X the set {µ ∈ J | λ ≤ µ} is finite.

Recall that in an abelian category A an epimorphism
Theorem 3.11.Suppose that J ⊂ X is open and quasi-bounded.For any λ ∈ J there exists a projective cover p λ : P J (λ) → L(λ) of L(λ) in O J , and the object P J (λ) admits a Verma flag.
Proof.Consider the U ≥0 -module C(λ) := U ≥0 ⊗ U 0 k λ .Theorem 2.1 implies that C(λ) is free of rank 1 as a U + -module.As an U 0 -module it is a weight module and its weights are contained in {µ | µ ≥ λ}.As before, define C(λ) J as the largest quotient U ≥0 -module with all weights contained in J, i.e.C(λ) J = C(λ)/ µ ∈J C(λ) µ .As J is supposed to be quasi-bounded, C(λ) J is a finite dimensional U ≥0 -module and a weight module supported in {µ | µ ∈ J, µ ≥ λ}.It hence has a U ≥0 -module filtration with one-dimensional subquotients isomorphic to k µ for various µ ∈ J (with µ ≥ λ).Moreover, k µ occurs with multiplicity dim k C(λ) J µ .Now consider Q J (λ) := U ⊗ U ≥0 C(λ) J .This is a weight module, and, as the functor U ⊗ U ≥0 (•) is exact (by Theorem 2.1), it has a finite filtration with subquotients isomorphic to ∆(µ) for various µ ∈ J (with µ ≥ λ).Hence it is an object in O J that admits a Verma flag and the multiplicities are given by (Q J (λ) : ∆(µ)) = dim k C(λ) J µ .Now let N be an object in O J .We then have a natural isomorphisms (that are functorial in N ): By the weight considerations above, ∆(λ) appears in a Verma flag of Q J (λ) with multiplicity 1 = dim k C(λ) λ and it is minimal among the highest weights of Verma subquotients.So there exists a surjection Q J (λ) → ∆(λ) in O J .As Q J (λ) is finitely generated (even cyclic) as a Umodule we can apply Proposition 2.10, so Q J (λ) splits into a finite direct sum of indecomposables.As a homomorphism M → ∆(λ) is surjective if and only if it is surjective on the λ-weight space, and as ∆(λ) λ is of dimension 1, there must be an indecomposable direct summand P J (λ) of Q J (λ) that admits a surjection onto ∆(λ).Then P J (λ) is projective and admits a Verma flag by Lemma 3.3.
As L(λ) is a quotient of ∆(λ), there exists a surjective homomorphism f : P J (λ) → L(λ).It remains to show that this is an essential homomorphism.So let g : M → P J (λ) be a morphism such that f • g : M → L(λ) is an epimorphism.The projectivity of P J (λ) allows us to find a morphism h : P J (λ) → M such that the diagram commutes.As f is an epimorphism, the composition g • h cannot be nilpotent, hence must be an automorphism of P J (λ) by Lemma 2.9.In particular, g is an epimorphism.So f is essential and indeed a projective cover.
3.6.Verma multiplicities and the BGGH-reciprocity.We are now going to prove an analogue of the reciprocity result that appeared in [H1] in the context of representations of modular Lie algebras.Proposition 3.12.Let M be an object in O that admits a Verma flag.Then for all µ ∈ X.
3.7.The exactness of (•) I and (•) J .Let us denote by V the full subcategory of O that contains all objects that admit a Verma flag.In general, it does not contain the subobjects and quotients of its objects, so it is not an abelian subcategory.However, it inherits an exact structure from the surrounding category O.We call a short sequence 0 → A → B → C → 0 in V exact if it is exact when considered as a sequence in O.
Let J be an open subset of X with closed complement I. Proposition 3.5 implies that (•) I and (•) J induce endofunctors on V that we denote by the same symbol.Then we have the following.
Proposition 3.14.Let 0 → A → B → C → 0 be a sequence in V. Then the following are equivalent.
(1) The sequence 0 1) is a special case of either one of the properties (2) or (3).It is hence enough to show that (1) implies ( 2) and (3).Let M be an object in O. Then λ∈I M λ is stable under the action of the subalgebra U ≥0 (as I is closed).Hence M I is generated by λ∈I M λ not only as a U -submodule, but already as a U − -submodule, and, in particular, as a U ≤0 -submodule.As the algebra U ≤0 detects the weight decomposition, we deduce that the functor (•) I factors through the restriction functor Res : O → U ≤0 -mod wt .But in the category U ≤0 -mod wt the short exact sequence appearing in (1) splits (cf.Lemma 3.7).As (•) I is additive, it produces an exact sequence when applied to a split sequence.Hence (1) implies (3).By the snake lemma, (1) and (3) imply (2).
3.8.Subcategories associated to locally closed subsets.Let K be a locally closed subset of X.
Then the following holds.
Theorem 3.17.Suppose that there exists some l > 0 such that the triple (J, J ′ , γ) satisfies the following.
Then the functors U and D induce mutually inverse equivalences of exact categories.

M
J := M/M I , i.e.M I ⊂ M is the minimal submodule of M that contains the weight spaces M λ with λ ∈ I, and M J the largest quotient of M with all weights contained in J. Clearly, these definitions yield functors (•) I , (•) J : O → O. Proposition 3.5.Let M be an object in O. Then the following are equivalent.
Now let γ ∈ X be a dominant weight.Then L = L(γ) is finite dimensional.Denote by Γ = Hom k (L, k) the dual U -module (without the σ-twist!).It is a finite dimensional module contained in O and the irreducible U -module of lowest weight −γ.Let J ′ ⊂ J be open subsets of X.Then J ′ = J ′ + γ ⊂ J = J + γ are open in X as well.We define the shift functorsU := ((•) ⊗ L) X\ J ′ : O → O, D := ((•) ⊗ Γ) J : O → O.Set K := J \ J ′ and K := J \ J ′ .Then K and K are locally closed, and the map (•) + γ yields a bijection K ∼ − → K.