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

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

Struggle over how to understand the theory of unstable modules over the Steenrod algebra from a viewpoint of the representation theory Atsushi Yamaguchi

References Demazure, M., Gabriel, P., Introduction to Algebraic Geometry and Algebraic Groups, North-Holland MATHEMATICS STUDIES 39, North-Holland, 1980 Grothendieck, A., Le groupe fondamental : Generalites, Lecture Notes in Math., vol.224, Springer-Verlag, 1971, 105-144. Grothendieck, A., Categorie fibrees et Descente, Lecture Notes in Math., vol.224, Springer-Verlag, 1971, 145-194. Jantzen, J. C., Representations of Algebraic Groups Second Edition, Mathematical Surveys and Monographs, no.107, AMS, 2003 MacLane, S., Categories for the Working Mathematicians Second Edition, Graduate Texts in Math., vol.5, Springer-Verlag, 1997

Milnor, J. W., The Steenrod algebra and its dual, Ann. of Math., 67 (1958), 150-171. Schwartz, L., Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture, Chicago Lectures in Math. 1994. Yamaguchi, A., On excess filtration on the Steenrod algebra, Geometry & Topology Monographs Vol.10, (2007) 423-449. Yamaguchi, A., Representations of internal categories, Kyushu Journal of Math. Vol.62, No.1, (2008) 139-169. Yamaguchi, A., The structure of the Hopf algebroid associated with the elliptic homology theory, Osaka Journal of Math., vol.33, no.1, (1996), 57-68. Yamaguchi, A., Representations of the Steenrod group, http://www.las.osakafu-u.ac.jp/~yamaguti/archives/rsg.pdf

Contents of Representations of the Steenrod group 1 Topological graded rings and modules 1.1 Linear topology 1.2 Suspension 1.3 Completion of topological modules 1.4 Topologies on graded modules 2 Tensor products 2.1 Tensor product of topological modules 2.2 Change of rings 2.3 Completed tensor product 3 Spaces of homomorphisms 3.1 Topology on spaces of homomorphisms 3.2 Adjointness 3.3 Homomorphisms 3.4 Completion and spaces of homomorphisms

4 Relations between tensor products and spaces of linear maps 4.1 Completed tensor products of spaces of linear maps 4.2 Commutative diagrams 5 Algebras and coalgebras 5.1 Algebras, coalgebra and duality 5.2 Milnor coaction 6 Actions of group objects in a cartesian closed category 6.1 Group objects 6.2 Group objects in cartesian closed categories 6.3 Right induction 7 Study on fibered categories 7.1 Fibered categories 7.2 Bifibered category

7.3 Fibered category with products 7.4 Fibered category with exponents 7.5 Cartesian closed fibered category 8 Representations of group objects 8.1 Representations of group objects 8.2 Representations in fibered category with products 8.3 Representations in fibered category with exponents 8.4 Left induced representations 8.5 Right induced representations 9 Quasi-topological category 9.1 Quasi-topological category and continuous functor 9.2 Yoneda s lemma 9.3 Left adjoint of the Yoneda embedding 9.4 Colimit of representable functors 9.5 Exponential law 9.6 Kan extensions

10 Topological affine scheme 10.1 Definition and properties of topological affine schemes 10.2 Topological modules 11 Topological affine group schemes 11.1 Topological group functors 11.2 Hopf algebra and topological affine group scheme 11.3 Examples 11.4 General linear group 12 Fibered category of modules 12.1 Fibered category of affine modules 12.2 Fibered category of functorial modules 12.3 Cartesian closedness of the fibered category of functorial modules 12.4 Embedding of the fibered category of affine modules 12.5 Quasi-coherent modules

13 Representation of topological affine group schemes 13.1 Representation of topological group functors 13.2 Tensor product of representations 13.3 Regular representations 13.4 Fixed points 13.5 Examples 14 Unstable actions 15 Unstable coactions

Contents of this slide 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

13. Quasi-topological category 14. Topological affine scheme

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

2. The Milnor coaction Let A be a graded Hopf algebra over a field K and M a graded left A -module with structure map : A M M. We assume that A and M n are finite type, namely, A and n M are finite dimensional for all n Z. n Put Hom(A, K) = A -n and consider the dual Hopf algebra A = A of A. We review how Milnor defined the coaction n n Z in this section. i+j : M M A = M A j Z i Z -i

In Milnor s paper The Steenrod algebra and its dual, the original definition of the Milnor coaction is as follows. He also remarked the following in the next paragraph.

Namely, the target of the Milnor coaction should be the completed tensor product in general, hence we have to give suitable topologies on H and A. Since the above definition of the Milnor coaction is very brief, next we try to describe the Milnor coaction in detail.

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

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

i i i i A i -i We put = (-1) : A Hom(Hom(A, K), K) = Hom(A, K), i+j = (-1) i+j : Hom(M, K) Hom(A, K) Hom(M A, K) i,j i(i+j) M,A -i i+j -i -i and T = (-1) T i+j i: Hom(M, K) A A Hom(M, K). i,j i(i+j) Hom(M, K),A i+j i i i+j Since A and M are finite type, and are isomorphisms. i i,j Let : Hom(M A, K) Hom(M, K) be the following i,j composition. i+j -i j i+j Hom(M A, K) Hom(M, K) Hom(A, K) -i -1 i,j i+j -i -1 (1 i T i,j i+j i i i+j i,j Hom(M, K) A A Hom(M, K) Hom(M, K) j

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

Milnor showed that the left A-module structure : A M M is recovered from the Milnor coaction as follows. Theorem 2.1 For i x A and j m M, (x m) = (-1) a (x)m if ij k Z k k j+l (m) = m a M A. j k Z k k l Z -l

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

K We assume that Hom (C C, K ) is complete. K For morphisms : C C C and : C K in TopMod, we define morphisms : C C C and : K C to be the following compositions of morphisms, respectively. K K Hom (C, K ) Hom (C, K ) Hom (C, K ) Hom (C, K ) K K ˆ K Hom (C C, K ) Hom (C C, K ) Hom (C, K ) K Hom (K, K ) Hom (C, K ) ( ) K K -1 Proposition 12.2 (C,, ) is a K -algebra if and only if (C,, ) is a K -coalgebra.

We assume that Hom (A A, K ) is complete and that : ˆ Hom (A, K ) Hom (A, K ) Hom (A A, K ) is an isomorphism. ˆ K K -1 K Hom (A, K ) Hom (K, K ) K K following compositions of morphisms, respectively. Hom (A, K ) Hom (A A, K ) Hom (A, K ) Hom (A, K ) Proposition 12.3 (A,, ) is a K -coalgebra if and only if (A,, ) is a K -algebra. define morphisms : A A A and : A K K For morphisms : A A A and : K A in TopMod K -1 K K, we K to be the

(M, K ) is a very nice pair and that one of the following (i) M is supercofinite and N is subcofinite. (ii) M has a finitely generated open submodule. M N i K Let M and N be objects of TopMod. We assume that conditions is satisfied. Then, : Hom (M, K ) N Hom (M, N ) is defined and K it is an isomorphism by (11.4). Under these assumptions, let M, L, N K K K K = : Hom (M L, N ) Hom (L, N M ) be the following composition.

K K N K T L, M Hom (M L, N ) Hom (M L, N ) Hom (L M, N ) L, M, N K N T M, N K M -1 ( ) K K K Hom (L, Hom (M, N )) Hom (L, M N ) K Hom (L, N M ) We remark that is injective if N K K is Hausdorff and it is an isomorphism if N is profinite and L M K is supercofinite. Definition 12.4 Let (C,, ) be a K-coalgebra and M an object of TopMod. K A right C -comodule in TopMod K is a pair (M, ) of an that the following diagrams commute. M M C M C K 1 1 M C C K K K M M C K 1 M K 1 K

Theorem 12.5 satisfying the conditions (i), (ii), (iii) and (iv) or (v). (i) Hom (A A, K ) is complete. (iii) M (v) A : Hom (A M, M ) Hom (M, M A ) A, A, M K is an isomorphism. (iv) A is supercofinite and M is subcofinite. is Hausdorff. has a finitely generated open submodule. Then, a morphism : A M M in TopMod gives a left Let (A,, ) be a K -algebra and M A -module structure of M K K if and if the image of by K K K K gives a right A -comodule structure of M. an object of TopMod (ii) : ˆ Hom (A, K ) Hom (A, K ) Hom (A A, K ) is an K K K

Corollary 12.6 If the following conditions are satisfied, then maps the subset of Hom A -module structures of M structures of M. (i) Hom (A A, K ) is complete. K (ii) : ˆ Hom (A, K ) Hom (A, K ) Hom (A A, K ) is an K is an isomorphism. (iii) A is supercofinite or A has a finitely generated open (iv) M submodule. (v) A M Let (A,, ) be a K -algebra and M is profinite. K is supercofinite. K K bijectively onto the subset K of Hom (M, M A ) consisting of right A -comodule K an object of TopMod : Hom (A M, M ) Hom (M, M A ) A, A, M K K K K (A M, M ) consisting of left K K

13. Quasi-topological category We denote by Top the category of topological spaces and continuous maps. For x X, we denote by ev : Top(X, Y) Y the map defined x by ev (f) = f(x). For O Y, put W(x, O) = ev (O). x We give Top(X, Y) the pointwise convergent topology generated by {W(x, O) x X, O is an open set of Y}. In other words, the pointwise convergent topology on Top(X, Y) is the coarsest topology that ev is continuous x for every x X. -1 x

Proposition 13.1 Let X, Y and Z be topological spaces. (1) A map : Z Top(X, Y) is continuous if and only if ev : Z Y is continuous for any x X. x (2) For a continuous map f : X Y, the maps f : Top(Y, Z) Top(X, Z) and f : Top(Z, X) Top(Z, Y) induced by f are continuous.

Definition 13.2 A category T is called a quasi-topological category if the following conditions are satisfied. (1) For each R, S Ob T, T(R, S) is a topological space. (2) For any morphism f : R S in T and Z Ob T, the maps f : T(Z, R) T(Z, S) and f : T(S, Z) T(R, Z) are continuous. It follows from (2) of (13.1) that Top is a quasi-topological category.

Condition 13.3 Let T be a quasi-topological category and D : D T a functor. For an object X of T, define functors D : D Top X op X X X and D : D Top by D (i) = T(X, D(i)), D ( ) = D( ) We consider the following conditions. (T(C, X) T(D(i), X)) is a limiting cone of D. X X and D (i) = T(D(i), X), D ( ) = D( ) for i Ob D and Mor D. i (L) If (L D(i)) is a limiting cone of D, i Ob D (T(X, L) i T(X, D(i))) is a limiting cone of D. i i Ob D (C) If (D(i) C) is a colimiting cone of D, i i Ob D i Ob D Proposition 13.4 The conditions (L) and (C) of (13.3) are satisfied for any functor D : D Top and topological space X. X X

For categories C and D, we denote by Funct(C, D) the category of functors from C to D and natural trans- formations between them. Definition 13.5 Let C and T be quasi-topological categories. We say that a functor F : C T is continuous if F : C(R, S) T(F(R), F(S)) is continuous for any R,S Ob C. We denote by Funct (C, T) the full subcategory of c Funct(C, T) consiting of continuous functors.

Proposition 13.6 Let T be a quasi-topological category and R an object of T. Then, the functor h : T Top represented by R is R continuous. Proposition 13.7 Let C be a quasi-topological category and D : D Funct (C, Top) a functor. c (1) If (L i D(i)) is a limiting cone of D, L is a i Ob D continuous functor. i (2) If (D(i) C) is a colimiting cone of D, L is a i Ob D continuous functor.

Let us denote by Set the category of sets and maps and by : Top Set the forgetful functor. Corollary 13.8 For a quasi-topological category C, the composition of the inclusion functor and the functor : Funct c(c, Top) Funct(C, Set) Funct (C, Top) Funct(C, Top) c : Funct(C, Top) Funct(C, Set) induced by creates limits and colimits. Hence Funct (C, Top) is complete and cocomplete. c

Let C and T be categories. For R Ob C, define an evaluation functor E : Funct(C, T) T at R by E (F) = F(R) R R and E ( R ) = R. Let T be a quasi-topological category. For F, G Ob Funct(C, T), we give Funct(C, T)(F, G) the coarsest topology such that E : Funct(C, T)(F, G) T(F(R), G(R)) is R continuous for any object R of C. Proposition 13.9 Let C be a category and T a quasi-topological category. Then, Funct(C, T) is a quasi-topological category.

Proposition 13.10 Let T be a quasi-topological category and F : C T a functor. i (1) Suppose that (L D(i)) is a limiting cone of a (13.3) (T = Top, for example). Then, (Funct(C, T)(F, L) i Ob D functor D : D Funct(C, T) and that, for any R Ob C, E D : D T and F(R) Ob T satisfy the condition (L) of R Mor D. i Funct(C, T)(F, D(i))) i Ob D is a limiting cone of a functor D : D Top defined by F D (i) = Funct(C, T)(F, D(i)) and D ( ) = D( ) for i Ob D, F F

(2) Suppose that (D(i) i C) is a colimiting cone of a i Ob D functor D : D Funct(C, T) and that, for any R Ob C, E D : D T and F(R) Ob T satisfy the condition (C) of R (13.3) (T = Top, for example). Then, (Funct(C, T)(C, L) Mor D. i Funct(C, T)(D(i), F)) i Ob D is a limiting cone of a functor D : D Top defined by D F(i) = Funct(C, T)(D(i), F) and D F ( ) = D( ) for i Ob D, F op

Definition 13.11 If T = Set or Top, for a functor F : C T, we denote by C F the category of F-models, that is, C is given by F Ob C = {(R, x) R Ob C, x F(R)}, F C ((R, x), (Y, y)) = {f C(R, Y) F(f)(x) = y}. F Remark 13.12-1 R Since {E (W(x, O)) (R, x) Ob C, O is an open set of G(R)} is a subbasis of the topology on Funct(C, Top)(F, G), a map f : Z Funct(C, Top)(F, G) is continuous if and only if ev E f : Z G(R) is continuous for any (R, x) Ob C. x R F F

Lemma 13.13 Let C be a quasi-topological category and F : C Top a functor. For (R, x) Ob C and S Ob C, define a map F ( (F) ) : h (S) F(S) by ( (F) ) (f) = F(f)(x) for (R, x) S R (R, x) f C(R,S) = h (S). If F is continuous, ( (F) R Thus we have a morphism (F) (R, x) S : h F in Funct(C, Top). R (R, x) S ) is continuous. Let C be a category and R an object of C. Define a map R R R R R (G) : Funct(C, Top)(h, G) G(R) by (G)( ) = (id ).

Proposition 13.14 For an object R of C and a functor G : C Top, the following topologies O, O 1 and O 2 on Funct(C, Top)(h R, G) are the same. (i) O is the coarsest topology such that (ii) O (iii) O E : Funct(C, Top)(h, G) Top(h (S), G(S)) S is continuous for any S Ob C. 1 the coarsest topology such that (G) : Funct(C, Top)(h, G) G(R) is continuous. R R 2 is the coarsest topology such that R E : Funct(C, Top)(h, G) Top(h (R), G(R)) is continuous. R R R R

Corollary 13.15 (Yoneda embedding) A functor h : C Funct (C, Top) defined by h(r) = h and h(f) = h f op is continuous. Proposition 13.16 (Yoneda s lemma) c Let C be a quasi-topological category and F : C Top a continuous functor. Then, (F) : Funct(C, Top)(h, F) F(R) R R is a homeomorphism. Proposition 13.17 Let C be a quasi-topological category. The functor : Funct (C, Top) Funct(C, Set) c given in (13.8) has a left adjoint. R

If C is a quasi-topological category, we denote by op F D(F) : C Funct (C, Top) a functor given by D(F)(R, x) = h and D(F)(f) = h. Proposition 13.18 f c Let C be a quasi-topological category. If F : C Top is a colimit of representable functors, then (F) (R,x) (R, x) Ob C F op F (D (R,x) F) F is a colimiting cone of the functor D(F) : C Funct (C, Top). Hence F is in the image of a left adjoint of the functor given in (13.8). : Funct (C, Top) Funct(C, Set) c c R

For a quasi-topological category C and a functor op (13.15). Proposition 13.19 (1) If F : C Top is a colimit of representable functors and (h (i) F) is a cone of h such that c op D : D we denote by h : D Funct (C, Top) the composition of functors D D i op : D D op C and h : C Funct (C, Top) defined in i Ob D ( ) ( (h (i)) i (F)) is a colimiting cone of h. D i Then, (h (i) F) is a colimiting cone of h. D i Ob D (2) Suppose that F is a colimit of h : D Funct (C, Top) and that L is a limit of D. Then, L is a limit of the functor D(F) ˆ : C C defined by D(F)(R, ˆ x) = R and ˆD(F)(f) = f. op F i Ob D D D c op c D op D C,

Proposition 13.20 Let A be an object of a quasi-topological category C and F : C Top a functor. Suppose that a limit L(F) of the op F functor D(F) ˆ : C C defined in (13.19) exists. If F is a colimit of representable functors, there is a bijection : Funct(C, Top)(F, h ) C(A, L(F)) = C (L(F), A). F, A Moreover, if the condition (L) of (13.3) is satisfied for A and (L(F) (R,x) D(F)(R, x)), is a homeomorphism. (R, x) Ob C F, A F Proposition 13.21 For objects F and G of Funct(C, Top), suppose that limits of op F D(F) ˆ : C C and D(G) ˆ : C C exist and that F is a colimit L : Funct(C, Top)(F, G) op G of representable functors. If the condition (L) of (13.3) is (R,x) satisfied for L(G) and (L(F) D(F)(R, x)), then A C(L(G),L(F)) is continuous. op (R, x) Ob C F

For a quasi-topological category C, we denote by Funct (C, Top) the full subcategory of Funct(C, Top) r consisting of functors which are colimit of representable functors. Proposition 13.22 For F, G Ob Funct (C, Top), there is a product of F and G r in Funct (C, Top). r

Definition 13.23 Let A and B be objects of a quasi-topological category C. A topological coproduct of A and B is a coproduct 1 2 A A B B of A and B such that coproducts. 1 2 C(A,R) C(A B,R) C(B,R) is a product of C(A,R) and C(B,R) in Top for any R Ob C. If each pair of objects of C has a topological coproduct, we say that C is a category with finite topological Theorem 13.24 If C is quasi-topological category with finite topological coproducts, Funct (C, Top) is a cartesian closed category. r

14. Topological affine scheme For objects A and B of TopAlg, we give a topology on K the set of morphisms TopAlg B K (A, B ) by giving a uniform structure as follows. For p TopAlg (A, B ), S A and K J I, we put U(S, J) = {(f, g) TopAlg ( A, B ) TopAlg ( A, B K U(p;S, J) = {f TopAlg K ) f(x) - g(x) J if (A, B ) (f, p) U(S, J)}. K x S}, Recall that F denotes the set of finitely generated A K -submodules of A. We also put B = {U(S, J) S F, J I }, B = {U(p;S, J) S F, J I }. p A B A B

B is a basis of a uniform structure of TopAlg B p K (A, B ) and is a basis of the neighborhood of p with respect to the topology defined by the uniform structure of TopAlg (A, B ). K If C is a subcategory of TopAlg K, we give C(A,B ) the K topology induced by TopAlg (A, B ) for A, B Ob C. K We remark that TopAlg (A, B ) is a subspace of Hom (A, B ) if we regard A and B as objects of TopMod. K 0

Definition 14.1 Let Top be the category of topological spaces and continuous maps. For an object A of TopAlg and a subcategory K C of TopAlg, we denote by h : C Top the functor repre- K A A K sented by A, that is, h maps B Ob C to TopAlg (A, B ). We call h a topological affine K -scheme. Thus we have a A op functor h : C Funct(C, Top) given by h(a ) = h and h(f)=f. A Genarally, we call a functor from a subcategory of TopAlg K to Top a topological K -functor.

To be continued.