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Toward a representation theory of the group scheme represented by the dual Steenrod algebra

Toward a representation theory of the group scheme represented by the dual Steenrod algebra. Atsushi Yamaguchi. Contents. §1. Motivation. §2. The Milnor coaction. §3. Topological graded rings and modules. §4. Suspensions. §5. Completion of topological modules.

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Toward a representation theory of the group scheme represented by the dual Steenrod algebra

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  1. Toward a representation theory ofthe group scheme represented bythe dual Steenrod algebra • Atsushi Yamaguchi

  2. Contents §1. Motivation §2. The Milnor coaction §3. Topological graded rings and modules §4. Suspensions §5. Completion of topological modules §6. Topologies on graded modules §7. Tensor products of topological modules §8. Spaces of homomorphisms §9. Adjointness §10. Completion of spaces of homomorphisms §11. Tensor products and spaces of homomorphisms §12. Reformulation of the Milnor coaction

  3. §13. Quasi-topological category §14. Topological affine scheme

  4. Let us denote by Athe dual Steenrod algebra. J. Milnor structure of H(X). We call this map the Milnor coaction. defined a map λ:H(X)→H(X)⊗A from the A-module in F. Then, H(X) has a structure of left A-module. Steenrod algebra over the prime field F. For a space X, sented by AonH(X). This motivates us to introduce various Let p be a prime number and let us denote by A the * * * * * we denote by H(X) the cohomology group with coefficients * * * ̂ * * p p * * §1. Motivation He showed that λ is coassociative and counital, that is, λ is a representation of the affine group scheme repre- methods in representation theory to study the category of modules over the Steenrod algebra.

  5. λ:M→M⊗A =⊕ΠM ⊗A Let A be a graded Hopf algebra over a field K and Ma A =⊕A of A. We review how Milnor defined the coaction Put Hom(A,K)=Aand consider the dual Hopf algebra graded left A-module with structure map α:A⊗M→M. n n * * M are finite dimensional for all n∈Z. n * * * * We assume that A and M are finite type, namely, A and * * i+j * * * ̂ -n n -i * * n∈Z j∈Z i∈Z §2. The Milnor coaction as follows.

  6. D :Hom(V,W)→Hom(Hom(W,K),Hom(V,K)) We note that D ,φ and χ are isomorphisms if V and φ :Hom(V,K)⊗Hom(W,K)→Hom(V⊗W,K) T is the switching map. χ :V→Hom(Hom(V,K),K) T :V⊗W→W⊗V D assigns a linear map to its dual map, φ is defined V,W V,W V,W V,W V V,W V,W V,W V V,W For vector spaces V and W, we consider the following maps. by φ (f⊗g)=f⊗g, χ is defined by χ (x)(f)=f(x) and V,W V,W V,W W are finite dimensional.

  7. Let β:A⊗Hom(M,K)→Hom(M,K) be the adjoint of the α:A⊗M→M of α:A⊗M→M. We put Let α:A→Hom(M,M) be the adjoint of a component D =(-1) D :Hom(M,M )→Hom(Hom(M,K),Hom(M,K)). ̃ α D i(i+j) j i+j i+j j A → Hom(M,M ) → Hom(Hom(M,K),Hom(M,K)) j i+j i+j j j,i+j i i i,j j i+j j i+j i i i+j j * ̃ * * j i+j j,i+j M,M i,j i,j i,j following composition.

  8. We put χ=(-1)χ :A→Hom(Hom(A,K),K)=Hom(A,K), Let γ :Hom(M⊗A,K)→Hom(M,K) be the following Since A and M are finite type, χandφ are isomorphisms. φ=(-1) φ :Hom(M,K)⊗Hom(A,K)→Hom(M⊗A,K) and T =(-1) T :Hom(M,K)⊗A→A⊗Hom(M,K). -1 β i(i+j) * * i i i i T i+j i i+j -1 (1⊗χ) φ Hom(M,K)⊗A→A⊗Hom(M,K)→Hom(M,K) i i+j j i+j i i,j i,j Hom(M ⊗A,K)→Hom(M,K)⊗Hom(A,K)→ j i(i+j) i+j i+j i i+j i,j i i i,j i+j i+j i i+j -i Hom(M ,K),A i,j A i i+j -i -i -i -i -i i,j M ,A i,j -i composition.

  9. Then, γ is regarded as an element of is the map whose component of degree j is λ. λ :M→M ⊗A Let λ:M→ΠM ⊗A be the map whose i-th component is λ . Finally, the Milnor coaction λ:M→M⊗A =⊕ΠM ⊗A of Hom(M,M⊗A) that maps to γ by D . Hom(Hom(M⊗A,K),Hom(M,K)). D :Hom(M,M⊗A)→Hom(Hom(M⊗A,K),Hom(M,K)) j j i+j i+j j i+j i+j j j i+j i+j j i+j * * j ̂ i,j j i+j -i i+j -i -i -i M,M ⊗A j i,j -i M,M ⊗A j -i -i i,j i,j -i * i∈Z -i j∈Z i∈Z Since is an isomorphism, there exists a unique element

  10. λ(m)= Σx ⊗m ∈ΠM⊗A. j+l -l j k k k∈Z l∈Z Milnor showed that the left A-module structure α:A⊗M→M * * * is recovered from the Milnor coaction as follows. Theorem 1.1 For x∈A and m∈M, α(x⊗m)= (-1) Σx(x)m if ij i j k k k∈Z

  11. (1) We say that a graded ring K is commutative if elements of ∪K. K is said to be homogeneous if it is generated by A submodule of M is said to be homogeneous if it is generated by elements of ∪M. Similarly, an ideal of From now on, “an ideal” of a graded ring always means a mn * * (2) Let K be a graded ring and M a graded K-module. * * * * m xy=(-1) yx for any m,n∈Z and x∈K, y∈K. n n n homogeneous ideal and “a submodule” of a graded module n∈Z n∈Z means a homogeneous submodule unless otherwise stated. §3. Topological graded rings and modules Definition 3.1

  12. If I is a fundamental system of neighborhoods of 0, set of open homogeneous two-sided ideals of A. (2) Let A and K be linearly topologized graded rings and (1) For a topological graded ring A, we denote by I the (A,η) (or A for short) is called a topological K-algebra. η:K→A a continuous homomorphism preserving degrees. * * mn A is said to be linearly topologized. * * * * * * m n * If η(x)y=(-1) yη(x) holds for any m,n∈Z and x∈K,y∈A, * A A * * Definition 3.2

  13. continuous homomorphism f:A→B preserving degrees topological K-algebras and homomorphisms of K-algebras. (4) For a commutative linearly topologized graded ring K, we denote by TopAlg the category of commutative * * * * * * * topological K-algebras. (3) Let (A,η) and (B,ι) be topological K-algebras. If a * K * * satisfies fη=ι, we call f a homomorphism of topological

  14. (i) β(L×M)⊂N for any l,m∈Z. (1) Let L, M and N be graded abelian groups. A map and z∈M for l,n∈Z. for any x,y∈L and z,w∈M. M, N are graded left K-modules. If β:L×M→N is β:L×M→N is said to be biadditive if β satisfies the * * * * * * * * * l (iii) β(rx,z)=rβ(x,z), β(x,rz)=(-1)rβ(x,z) if r∈K, x∈L * * ln n * * * m l+m l * Definition 3.3 following conditions (i) and (ii). (ii) β(x+y,z)=β(x,z)+β(y,z), β(x,z+w)=β(x,z)+β(x,w) (2) Suppose that K is a commutative graded ring and L, biadditive and satisfies the following condition (iii), we say that β is bilinear.

  15. of neighborhoods of M, we say that the topology of M is induced by K. {IM|I∈I } (resp. {MI|I∈I }) is a fundamental system V the set of homogeneous open submodules of M. the topology induced by K. topological graded left (resp.right) K-module. If we say that M is linearly topologized. If V is a fundamental system of neighborhoods of 0, * * * * * * * (right) K-module, then the topology of A is coarser than (2) Let K be a linearly topologized graded ring and M a * * * * * If A is a topological K-algebra and we regard A as a left * * M * M K * K * * Definition 3.4 * * (1) For a topological graded K-module M, let us denote by Remark 3.5

  16. groups. We say that a biadditive map β:L×M→N is Let L, M and N be linearly topologized graded abelian subgroup W of M such that β(V×M) and β(L×W) there exist an open subgroup V of L and an open strongly continuous if, for any open subgroup U of N, are contained in U. * * * * * * * * * * * * * * * * * Definition 3.6

  17. (1) The topology of M is coarser than the topology induced the topology of M is finer than the topology induced by linearly topologized graded left K-modules. induced by K, then f is continuous. by K if and only if the structure map α:K×M→M is * * * * * * * * * * * * K and the topology of N is coarser than the topology Let K be a linearly topologized graded ring and M, N * * * (2) Let f:M→N be a homomorphism of left K-modules. If * Proposition 3.7 strongly continuous.

  18. induced by K. the set of all morphisms in TopMod from M to N. (1) TopMod is complete and cocomplete. For objects M and N of TopMod,we denote by Hom (M,N) K-modules whose topology are coarser than the topology TopMod consisting of linearly topologized graded left TopMod the category of linearly topologized graded left * i * i * For a linearly topologized graded ring K, we denote by K-modules and continuous homomorphisms preserving * * (2) TopMod is complete and finitely cocomplete. * degrees. We denote by TopMod the full subcategory of * * * * K * * * * K K * K * K * K K K * Proposition 3.8

  19. Let τ :K→K be a homomorphism of graded rings given by For m∈Z and an object M of TopMod, define an object and ττ=id. We call τ the conjugation of K. ΣM of TopMod as follows. * n * n * m * * τ(r)=(-1)r if r∈K. Then, it is clear that τ is continuous K * K * K * K * K * K * K * K * K * §4. Suspensions

  20. Put (ΣM)={[m]}×M for i∈Z and give (ΣM) the ΣM by α (r,([m],x))=([m],α(τ(r),x)) for r∈K and x∈M, m i m m m i m i-m i-m m m m m m i m m * * V ={ΣU|U∈V }. If α:K×M→M is the K-module structure of M, we * m where τ :K→K is the m times composition of τ. m m * * * * * If U is an submodule of M, we can regard ΣU as a (ΣM)={[m]}×M onto the second component is an such that the set of open submodules of ΣM is given by define the K-module structure α :K×ΣM→ΣM of * * * * * * * submodule of ΣM. We give a linear topology on ΣM * * * * * * * * K K * * m M * ΣM * K * Definition 4.1 structure of an abelian group such that the projection isomorphism of abelian groups.

  21. If f:M→N is a morphism in TopMod , we denote by Σ:TopMod→TopMod. morphism in TopMod. Thus we have a functor m m m m m m i m m m i * * We call ΣM and Σf the m-fold suspension of M and f, * * ([m],f(x))∈(ΣN). It is easy to verify that Σf is a * Σf: ΣM→ΣN the map which maps ([m],x)∈(ΣM) to * * * * * * * K K K K respectively.

  22. D :V→TopMod given by D (U)=M/U. We denote by We say that an object M of TopMod is complete if M is for U∈V define a cone of D , there is a unique map Let M an object of TopMod . Regarding V as a category η :M→M satisfying πη =p for any U∈V . ̂ π n M the limit limD of D , namely, there is a limiting cone * ̂ * * * * * * * * * U * ̂ * * * (M →M/U) . Since the quotient maps p :M→M/U * * * * * M M * * M M * M ← * * * * K M M * U∈V K K M M U * * * * M U M * * * * U * M §5. Completion of topological modules Definition 5.1 complete for each n∈Z. whose morphisms are inclusion maps, consider a functor

  23. ̂ ̂ ̂ ̂ The image of η :M→M is dense and η is an open map (2) M is complete Hausdorff. (3) M is complete Hausdorff if and only if η :M→M is (1) M is Hausdorff if and only if η :M→M is injective. * * * * * * * * * M M * * M * M * Proposition 5.2 onto its image. Proposition 5.3 an isomorphism.

  24. the following diagram commute for any U∈V. The limit M of D :V→TopMod is called the completion of M. η fπ -1 ̂ * ̂ * ̂ * There exists a unique morphism f:M→N which makes we have a map f :M/f(U)→N/U induced by f. Then, * * * * * * -1 (M N/U) is a cone of D :V→TopMod. f (U ) * U * M * M * ̂ ̂ * * * * M * M → M * K * N U * * U∈V N * N * K * * * N ̂ ↓ ↓ f f η * N ̂ N → N * * Definition 5.4 Let f:M→N be a morphism in TopMod. For each U∈V , * * * N K * * N

  25. Let f:M→N be a morphism in TopMod such that N is of the inclusion functor TopMod→TopMod (resp. TopMod ) defined by C(M)=M and C(f)=f is a left adjoint ̂ i i i i i A functor C:TopMod→TopMod (resp. C:TopMod→ i Let us denote by TopMod (resp. TopMod) the full g:M→N such that gη =f. TopMod→TopMod). * * subcategory of TopMod(resp. TopMod ) consisting of ̂ ̂ * * * * * M cK cK * * cK * K * * K * K cK K * K cK * * cK * * * K * * K Proposition 5.5 complete Hausdorff. Then, there exists a unique morphism objects which are complete Hausdorff. Proposition 5.6

  26. that K has the cofinite topology if the set of all cofinite (1) A linearly topologized graded ring K is said to be finite an ideal I of K is cofinite if K/I is artinian. We say supercofinite). Hence K is subcofinite (resp. supercofinite) (2) For a linearly topologized graded ring K, we say that (3) If the topology of K is coarser (resp. finer) than the ideals of K is a fundamental system of the neighbor- cofinite topology, we say that K is subcofinite (resp. * * * * * * * * if K is discrete and artinian. * * §6. Topologies on graded modules Definition 6.1 hood of 0. if and only if every open ideal is cofinite (resp. every cofinite ideal is open).

  27. (3) If the topology of M is coarser (resp. finer) than the (1) An object M of TopMod is said to be finite if M is cofinite topology, we say that M is subcofinite (resp. N of M is cofinite if M/N is finite. We say that M has supercofinite). Hence M is subcofinite (resp. supercofinite) (2) For an object M of TopMod, we say that a submodule of M is a fundamental system of the neighborhood of 0. * * * * * * * * * * * * K * K * Definition 6.2 discrete and of finite length. the cofinite topology if the set of all cofinite submodules if and only if every open submodule is cofinite (resp. every cofinite submodule is open).

  28. {M[n]|n=0,1,2,...} is a fundamental system of the submodule of M generated by ∪ M. We say that an superskeletal). Hence M is subskeletal (resp. superskeletal) if and only if every open submodule contains M[n] for (6) If the topology of M is coarser (resp. finer) than the (5) For a non-negative integer n, let us denote by M[n] the M is profinite. (4) If M is complete Haudorff and subcofinite, we say that object M of TopMod has a skeletal topology if * * * skeletal topology, we say that M is subskeletal (resp. some n (resp. every submodule containing M[n] for some * * * * * * * i * K * |i|≧n neighborhood of 0. n is open).

  29. cofinite and N is subcofinite” or “M is superskeletal and K-modules if and only if M is subcofinite and Hausdorff. N is subskeletal”, then every linear map from M to N Let M and N be objects of TopMod. If “M is super- (1) If M is a subcofinite K-module, then each submodule (3) M is isomorphic to a submodule of product of finite and quotient module of M are subcofinite. ΠM is also subcofinite. * * * ̂ * (2) If (M) is a family of subcofinite K-modules, then * * * * * * (4) If M is subcofinite, the completion M is also subcofinite. * * * * * * * * * i i∈I i K * i∈I Proposition 6.3 Proposition 6.4 preserving degrees is continuous.

  30. We denote by M⊗ N the completion of M⊗ N and call For objects M, N of TopMod , we give a topology on this the completed tensor product of M and N. M⊗ N so that quotient maps for submodules U of M and V of N. {Ker(p ⊗q :M⊗ N→M/U⊗ N/V)|U∈V ,V∈V } Here, we denote by p :M→M/U and q :N→N/V the * * * * * * * * * * * * * * * * * * ̂ * * * * * * * * * * V * K * K * K * U * M * * N U * V * K * K K * * §7. Tensor products of topological modules is a fundamental system of the neighborhood of 0.

  31. (1) If M or N has a topology coarser than the topology A map β :M×N→M⊗ N defined by β (x,y)=x⊗y is morphism B:M⊗ N→L in TopMod satisfying Bβ =B. by K if and only if there is an isomorphism K⊗ M→M. induced by K, the topology on M⊗ N is coarser than continuous bilinear map B:M×N→L, there exists a unique (2) M has a topology coarser than the topology induced the topology induced by K. * * * * * * * * * * * ̃ * * ̃ * * * * * * * * M,N * M,N K K * * * * * K * * * K K M,N * * Proposition 7.1 a strongly continuous bilinear map and, for a strongly Proposition 7.2

  32. For objects M and N of TopMod, define a morphism m,n m,n i-m n j-n m,n m+n n(i-m) n m+n m,n m m m,n ̃ as follows. Define τ :Σ M×Σ N→Σ (M⊗ N) by Clearly, τ is a natural isomorphism. Let τ be the unique morphism satisfying m,n m,n ̃ τ β (x,y)=τ . τ :Σ M⊗ Σ N→Σ (M⊗ N) for (x,y)∈M ×N . Then, it is easy to verify that τ m,n ̃ ̃ τ (([m],x),([n],y))=([m+n],(-1) β (x,y)) * * * * * * * * * * ΣM,Σ N m n * * M,N M,N M,N M,N M,N * * * * * * * * * * M,N M,N K * K * K * M,N K * M,N * * * * * * * * is bilinear and strongly continuous.

  33. by K. the topology on M is coarser than the topology induced m m m m i-m We note that s is a natural isomorphism if and only if * * s :Σ M→(Σ K)⊗ M Then, s is a homomorphism of K-modules m m * by s ([m],x)= β (([m],1),x)for x∈M . * * * ΣK,M M M * * m M M * * * * K * For an object M of TopMod , define a morphism * K *

  34. by (r,f)→rf define a left K-module structure of The maps K×Hom(M,N)→Hom (M,N) for l,n∈Z given x∈M. Hom(M,N) of TopMod as follows. Put rf:Σ M→N in TopMod by (rf)([l+m],x)=rf([m],x) for For objects M and N of TopMod, we define an object Hom(M,N). K * l+m * l l n * m Hom(M,N)=(Hom(M,N))=Hom (ΣM,N). n n l+n n For r∈K and a morphism f:Σ M→N, define a morphism * * * * * * * * * * * * * * * * * * * * * * * K * K * K * §8. Spaces of homomorphisms Definition 8.1

  35. For morphisms f:M→N, g:N→L in TopMod, define maps p :N→N/U the quotient map. O(S,U)=Ker(i p :Hom(M,N)→Hom(S,N/U)). For a submodule S of M and a submodule U of N, we put f:Hom(N,L)→Hom(M,L) and g:Hom(M,N)→Hom(M,L) maps of K-modules. ψ∈Hom (M,N). It is easy to verify that f and g are by f(φ)=φΣf and g(ψ)=gψ for φ∈Hom(N,L) and n * * * * * Here we denote by i :S→M the inclusion map and by * * * * m * n * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * S U * * * S * U K * * * * *

  36. Let F be the set of finitely generated submodules of M. x∈N and s∈K. Then, δ is an isomorphism whose inverse Define a map δ :K→Hom(K,N) by (δ (x))([n],s)=sx for We denote by M the dual space Hom(M,K). E(f)=([-n],f([k],1)) for f∈Hom(K,N). Define a topology on Hom(M,N) such that {O(S,U)|S∈F,U∈V } is the evaluation map E:Hom(K,N)→K defined by * * k * * * * * * * * * * * * * * * * * * * * * * * N 1 M * N * 1 N * * M N * is a fundamental system of neighborhoods of 0. Proposition 8.2

  37. Let M, N, L be objects of TopMod and f:M⊗ N→L a K Φ=Φ : Hom (M⊗ N,L)→Hom (M,Hom(N,L)) morphism f :M→Hom(N,L) in TopMod. Define a map morphism in TopMod. For x∈M, define a map f:Σ N→L * a a a k k k n k n-k k k a k a by f(y)=f(x⊗y) for y∈(Σ N)=N . Then, f is an element k by Φ(f)=f. * (f )(x)=f and a family of linear maps ((f )) defines a Thus we have a map (f ):M→Hom (Σ N,L) given by * * * * * * * * * * * * * * * * * * * * * x x k∈Z x x M,N,L * * * K * K * K * K * K * K * K * §9. Adjointness of Hom(N,L)=Hom (ΣN,L). k k * * * * K *

  38. (ii) M⊗ N is superskeletal and L is subskeletal. Φ is injective and if one of the following conditions induced by K and the topology on L is coarser than the topology induced by K. (i) M⊗ N is supercofinite and L is subcofinite. is satisfied, Φ is an isomorphism. (iii) The topology on M⊗ N is finer than the topology * * * * * * * * * * * M,N,L * * * M,N,L * * * K * K * K * Proposition 9.1

  39. is an isomorphism in TopMod. If N is Hausdorff, so is Hom(M,N). If N is complete Hausdorff, η :Hom(M,N)→Hom(M,N) Suppose that N is complete Hausdorff. If there exists a finitely generated open submodule of M, Hom(M,N) is * ̂ * * * * * * * * * * * * * * * * K * M * §10. Completion of spaces of homomorphisms Proposition 10.1 Proposition 10.2 * Proposition 10.3 complete Hausdorff.

  40. (i) M is supercofinite and N is subcofinite. λ :Hom(M,N) →Hom(M,N) η (ii) M has a finitely generated open submodule. λ η * * * ̂ Hom(M,N) Hom(M,N) M,N * * * * * * * ̂ Hom(M,N) * ̂ N * * * * * * * * * * * M,N * * Hom(M,N) ̂ * * * Proposition 10.4 If one of the following conditions (i) or (ii) is satisfied, there exists a unique monomorphism that makes a diagram commute.

  41. (2) We say that a pair (M,N) of objects of TopMod is each (S,U)∈C. that i p :Hom(M,N)→Hom(S,N/U) is surjective for (1) We say that a pair (M,N) of objects of TopMod is such that i p :Hom(M,N)→Hom(S,N/U) is surjective and S is projective for each (S,U)∈C. op op nice if there exists a cofinal subset C of F×V such very nice if there exists a cofinal subset C of F×V * * * * * * * * * * * * * * * * * * * * * * * * * * * M * N M * * N * S * U S * U K * K * * * Definition 10.5

  42. (1) A pair (M,N) is nice if one of the following conditions (i) N is injective and there exists a cofinal subset S of F such that every element of S is projective. a field and M is supercofinite. (ii) M is projective and there exists a cofinal subset M (2) (M,N) is a very nice pair if the above (i) is satisfied. (iii) There exists a cofinal subset S of F such that every element of S is a direct summand of M and there element of M is a direct summand of N. op op op * of V such that N/U is injective for every U∈M. * * * * exists a cofinal subset M of V such that every (3) The above (iii) is satisfied for S= F and M=V if K is * * * * * * * * * M * N * M N * M * N * Remark 10.6 is satisfied.

  43. Suppose that (M,N) is nice. If one of the following λ :Hom(M,N) →Hom(M,N) (i) M is supercofinite and N is subcofinite. (ii) M has a finitely generated open submodule. * * * * * ̂ * * * * * ̂ * M,N * * Theorem 10.7 conditions (i) or (ii) is satisfied, then the morphism given in (10.4) is an isomorphism.

  44. for f∈Hom (M,N) and g∈Hom(M ,N). In other words, of graded K-modules by φ:Hom(M,N)⊗ Hom(M ,N )→Hom(M⊗ M ,N⊗ N ) Let M, N (s=1,2) be objects of TopMod. We define a map ˆ j-n -1 n(i-m) * m,n m+n m n m+n i-m if x∈M , y∈M . Then,φis a morphism in TopMod. φ (f⊗g)=(f⊗g)(τ ) φ (f⊗g)([m+n],x⊗y)=(-1) f([m],x)⊗g([n],y) * φ:Hom(M,N)⊗ Hom(M ,N )→Hom(M⊗ M ,N⊗ N ) * * * * * * * * * * * * * * * * * * * * * * * * * * ̂ * ̂ s s 1 1 1 2 2 2 2 1 2 1 M,M 2 1 1 * * 2 K * K * K * K * 1 1 2 2 1 2 1 K * 2 1 2 K * K * K * §11. Tensor products and spaces of homomorphisms We denote by the completion of φ.

  45. is a morphism in TopMod and it is an isomorphism if the φ :Hom(M,K)⊗ N→Hom(M,N) Let us define a map ι:M→M⊗ K by ι(x)=x⊗1. Then, ι of N induces an isomorphism α:K⊗ N→N by (7.2). Let Suppose that the topology on N is coarser than the topology induced by K. Then, the K-module structure map * 1⊗δ ̃ topology on M is coarser than the topology induced by K. M * * ι * * α * ̃ * * * * * * * * Hom(M⊗ K,K⊗ N) Hom(M,K⊗ N) Hom(M,N) * * * * * Hom(M,K)⊗ NHom(M,K)⊗ Hom(K,N) * * * * * 1 * φ * * * * * * * * * * * 1 1 * * 1 * * * * N * K * * * K * N * K * K * K * K * K * K * K * be the following composition of morphisms.

  46. topology induced by K. Let Let (M,K) be a very nice pair. (2) If M and N are objects of TopMod, then be the completion of φ :Hom(M,K)⊗ N→Hom(M,N). Suppose that the topology on N is coarser than the ˆ ˆ ˆ ˆ * If both (M ,N ) and (M ,N ) are very nice pairs, then * * φ :Hom(M,K)⊗ N→Hom(M,N) * * M * i * φ :Hom(M,K)⊗ N→Hom(M,N) is an isomorphism. * M * M * (1) φ:Hom(M,K)⊗ Hom(M,K)→Hom(M⊗ M,K) is an ̂ φ:Hom(M,N)⊗ Hom(M ,N )→Hom(M⊗ M ,N⊗ N ) * * ̂ * * * * * * * * * * * * * * * ̂ * * * * * ̂ * 1 1 * * * 2 2 * * * * * * ̂ * * * ̂ * * * * * * * * ̂ * ̂ N * N * N * K * K * 1 1 2 2 K * 1 2 1 K * 2 K * K * K * K * K * Theorem 11.1 is an isomorphism. Corollary 11.2 isomorphism. *

  47. λ :Hom(M⊗ M ,N⊗ N ) →Hom(M⊗ M ,N⊗ N ) Composing λ and an isomorphism are subcofinite” or “M⊗ M has a finitely generated open Suppose that “M⊗ M is supercofinite and both N and N φ:Hom(M,N)⊗ Hom(M ,N )→Hom(M⊗ M ,N⊗ N ) ̃ ˆ -1 (η ):Hom(M⊗ M ,N⊗ N )→Hom(M⊗ M ,N⊗ N ) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ̂ * ̂ ̂ * ̂ * ̂ ̂ ̂ * ̂ M⊗ M,N⊗ N 1 with φ, we have the following morphism. * * * * 1 2 2 1 2 1 1 2 2 1 1 2 1 M⊗ M,N⊗ N 1 1 2 2 1 2 2 1 2 1 1 1 2 2 * * * * 2 K * K * K * M⊗ M K K 2 K * 2 2 1 1 K * K * K * K * * * * * * * K K * K K * K * 2 2 1 1 * * K K 2 1 * K submodule”. By (10.7), there exists the following morphism.

  48. (ii) M⊗ M has a finitely generated open submodule. (i) M⊗ M is supercofinite and both N and N are φ:Hom(M,N)⊗ Hom(M ,N )→Hom(M⊗ M ,N⊗ N ) ̃ * Suppose that both (M ,N ) and (M ,N ) are very nice pairs. * * * * * * * * * * * * * * * * * * * ̂ * ̂ ̂ 1 1 1 2 2 2 1 2 1 2 1 1 2 2 1 2 1 K * 2 K * K * K * K * Combining (11.1) and (10.7), we have the following result. Corollary 11.3 If one of the following conditions (i) or (ii) is satisfied, then the morphism is an isomorphism. subcofinite.

  49. submodule”, then φ :Hom(M,K)⊗ N→Hom(M,N) is an and N are objects of TopMod . If “M is supercofinite and Suppose that (M,K) is a very nice pair and that both M “M has a finitely generated open submodule”. By (10.7), N is subcofinite” or “M has a finitely generated open there exists a morphism λ :Hom(M,N) →Hom(M,N). φ :Hom(M,K)⊗ N→Hom(M,N) ˆ * Suppose that “M is supercofinite and N is subcofinite” or * * * * * * * * * i M * M M M * ̂ ̂ ̂ ̃ ̃ * ̃ * * * * by φ =λ φ . The next result follows from (2) of (11.2) * * * * * * * * * * * * * ̂ * * ̂ ̂ * * N * N * N * M,N * * M,N K * N * K * K * * * We define a morphism and (10.7). Corollary 11.4 isomorphism.

  50. Let C be an object of TopMod. Suppose that morphisms γ:C→C⊗C and ε:C→K in TopModare given. We call a triple (C,γ,ε) a K-coalgebra if the following diagrams For the rest of this section, we assume that K is complete γ C * * * * C C⊗C * * ̂ * * ̂ * * * * * ι ι K * K * K * K * 2 1 γ 1⊗γ γ ε⊗1 1⊗ε C⊗K C⊗C K⊗C γ⊗1 ̂ ̂ ̂ * * * * * C⊗C C⊗C⊗C * ̂ ̂ * * * ̂ * * K K K * * * K K K * * * §12. Reformulation of the Milnor coaction Definition 12.1 commute. Hausdorff.

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