Hopf Pairings and (Co)induction Functors over Commutative Rings

Transcription

1 arxiv:math/ v2 [math.ra] 13 Jul 2004 Hopf Pairings and (Co)induction Functors over Commutative Rings Jawad Y. Abuhlail Mathematics Department Birzeit University P.O.Box 14, Birzeit - Palestine Abstract (Co)induction functors appear in several areas of Algebra in different forms. Interesting examples are the so called induction functors in the Theory of Affine Algebraic Groups. In this paper we investigate Hopf pairings (bialgebra pairings) and use them to study (co)induction functors for affine group schemes over arbitrary commutative ground rings. We present also a special type of Hopf pairings (bialgebra pairings) satisfying the so called α-condition. For those pairings the coinduction functor is studied and nice descriptions of it are obtained. Along the paper several interesting results are generalized from the case of base fields to the case of arbitrary commutative (Noetherian) ground rings. Introduction Hopf pairings (respectively bialgebra pairings) were presented by M. Takeuchi [35, Page 15] (respectively S. Majid [26, 1.4]). With the help of these, several authors studied affine group schemes and quantum groups over arbitrary commutative ground rings (e.g. [16], [33], [31]). In this paper we study the category of Hopf pairings P Hopf and the category of bialgebra pairings P Big over an arbitrary commutative base ring. In the case of Noetherian base rings we present the full subcategories P α Hopf P Hopf (respectively P α Big P Big) of Hopf pairings (respectively bialgebra pairings) satisfying the so called α-condition, see 1.4. For those a coinduction functor is presented and an interesting description of it is obtained. The paper is divided into seven sections. The first section includes some preliminaries about the so called measuring α-pairings, rational modules and dual coalgebras. In the MSC (2000): 16W30, 14L17, 16W35, 20G42 Keywords: Hopf Pairings, Bialgerba Pairings, Dual Bialgebras, Dual Hopf Algebras, Affine Group Schemes, Quantum Groups, Coinduction Functors, Cotensor Product. Current Address: Box # 281 King Fahd University of Petroleum & Minerals, Dhahran (Saudi Arabia), abuhlail@kfupm.edu.sa, Homepage: 1

2 second section we consider the cotensor functor and prove some properties of it that will be used in later sections. In the third section we introduce the coinduction functor in the category of measuring α-pairings and prove mainly that it can be obtained as a special case of a more general coinduction functor between categories of type σ[m]. Hopf pairings (bialgebra pairings) are presented in the fourth section, where an algebraically topological approach is used and several duality theorems are proved. In the fifth section we consider the category of Hopf α-pairings (bialgebra α-pairings) and generalize known results on admissible Hopf pairings over Dedekind domains to the case of quasi admissible bialgebra α-pairings and Hopf α-pairings over arbitrary commutative Noetherian ground rings. There the coinduction functor is also studied and different forms of it that appear in the literature are shown to be equivalent. The classical duality between groups and commutative Hopf algebras (e.g. [28], [30]) is the subject of the sixth section. In the seventh and last section we apply results obtained in the previous sections to affine group schemes over arbitrary commutative rings. Throughout this paper R denotes a commutative ring with 1 R 0 R. We consider R as a left (and a right) linear topological ring with the discrete topology. All R-modules are assumed to be unital and category of R-(bi)modules will be denoted by M R. With unadorned Hom(, ) and we mean Hom R (, ) and R respectively. For an R-module M we call an R-submodule K M pure (in the sense of Cohn), if the canonical mapping ι K R id N : K R N M R N is injective for every R-module N. For an R-module M and subsets X M (respectively Y M ) we set An(X) := {f M : f(x) = 0} (respectively Ke(Y ) := {m M : f(m) = 0 f Y }). Let A be an R-algebra (not necessarily with unity). A left A-module is said to be faithful, if An( A M) := {a A : am = 0} = (0 A ). We define a left A-module M to be A-faithful (respectively unital), if the canonical map ρ : M Hom R (A, M) is injective (respectively, if AM = M). With ÃM (respectively A M) we denote the category of A- faithful (respectively unital) left A-modules and left A-linear maps. The categories of A-faithful (respectively unital) right A-modules M A (respectively M A ) are analogously defined. If A has unity, then obviously every unital left (or right) A-module is A-faithful. For an R-algebra A and an A-module M, an A-submodule N M will be called R- cofinite, if M/N is finitely generated in M R. Unless otherwise explicitly mentioned, we assume that all R-algebras have unities respected by R-algebra morphisms and that all modules of R-algebras are unital. We assume the reader is familiar with the theory and notation of Hopf Algebras. For any needed definitions the reader may refer to any of the classical books on the subject (e.g. [1], [28], [30]) or to the recent monograph [12] for the theory of coalgebras over arbitrary base rings. For an R-coalgebra C, we call a right (respectively a left) C-comodule (M, M) counital if its structure map M is injective, compare [13, Lemma 1.1.]. For an R-coalgebra C we denote with M C (respectively C M) the category of counital right (respectively left) C-comodules. For an R-coalgebra (C, C, ε C ) and an R-algebra (A, µ A, η A ) we consider Hom R (C, A) as an R-algebra with multiplication the so called convolution product (f g)(c) := f(c 1 )g(c 2 ) and unity η A ε C. 2

3 1 Preliminaries In this section we present some definitions and lemmas to be referred to later in the paper Subgenerators. Let A be an R-algebra (not necessarily with unity) and K be a left A-module. We say a left A-module N is K-subgenerated, if N is isomorphic to a submodule of a K-generated left A-module (equivalently, if N is kernel of a morphism between K-generated left A-modules). The full subcategory of A M, whose objects are the K-subgenerated left A-modules is denoted by σ[ A K]. In fact σ[ A K] A M is the smallest Grothendieck full subcategory that contains K. If M is a left A-module, then Sp(σ[ A K], M) := {f(n) f Hom A (N, M), N σ[ A K]} is the largest A-submodule of M that belongs to σ[ A K]. The subcategory σ[ A K] A M can also be seen as the category of discrete left A-modules, where A is considered as a left linear topological R-algebra with the K-adic topology (e.g. [10]). The reader is referred to [38] and [37] for the well developed theory of categories of this type. An important result to which we will often refer is Lemma 1.2. ([38, 15.8], [12, 42.2]) Let A be a ring, K be a faithful left A-module and B A be a subring. Then σ[ B K] = σ[ A K] if and only if B A is K-dense. Remark 1.3. Let C be an R-algebra. Then C becomes two (left) linear topologies, the C- adic topology induced by C C and the finite topology induced by the embedding C R C. By [4, Lemma 2.2.4] the two topologies coincide The α-condition. With an R-pairing P = (V, W) we mean R-modules V, W with an R-linear map κ P : V W (equivalently χ P : W V ). For R-pairings (V, W) and (V, W ) a morphism (ξ, θ) : (V, W ) (V, W) consists of R-linear mappings ξ : V V and θ : W W, such that the induced R-bilinear map V W R, (v, w) < v, w >:= κ P (v)(w) = χ P (w)(v) has the property < ξ(v), w >=< v, θ(w ) > for all v V and w W. We say an R-pairing P = (V, W) satisfies the α-condition (or P is an α-pairing), if for every R-module M the following mapping is injective: α P M : M R W Hom R (V, M), mi w i [v m i < v, w i >]. (1) We say an R-module W satisfies the α-condition, if (W, W) satisfies the α-condition (equivalently, if R W is locally projective in the sense of Zimmermann-Huisgen [39, Theorem 2.1], [17, Theorem 3.2]). With P we denote the category of R-pairing with morphisms of pairings described above and with P α P the full subcategory of R-pairings satisfying the α-condition. 3

4 Remark 1.5. ([2, Remark 2.2]) Let P = (V, W) be an α-pairing. Then R W is flat and R-cogenerated. If moreover R W is finitely presented or R is perfect, then R W turns to be projective. Lemma 1.6. ([2, Lemma 2.3]) Let P = (V, W) P α. For every R-module M and every R-submodule N M we have for m i w i M R W : mi w i N R W m i < v, w i > N for all v V. (2) 1.7. Measuring R-pairings. For an R-coalgebra C and an R-algebra A (not necessarily with unity) we call an R-pairing P = (A, C) a measuring R-pairing, if the induced mapping κ P : A C is an R-algebra morphism. In this case C is an A-bimodule through the left and the right A-actions a c := c 1 < a, c 2 > and c a := < a, c 1 > c 2 for all a A, c C. (3) Let (A, C) and (B, D) be measuring R-pairings (A and B not necessarily with unities). Then we say an R-pairings morphism (ξ, θ) : (B, D) (A, C) is a morphism of measuring R-pairings, if ξ : A B is an R-algebra morphism and θ : D C is an R-coalgebra morphism. The category of measuring R-pairings and morphisms described above will be denoted by P m. If P = (A, C) is a measuring R-pairing, D C is a (pure) R-subcoalgebra and I A is an ideal with < I, D >= 0, then Q := (A/I, D) is a measuring R-pairing, (π I, ι D ) : (A/I, D) (A, C) is a morphism in P m and we call Q P a (pure) measuring R-subpairing. With P α m P m we denote the full subcategory of measuring R-pairings satisfying the α-condition. Obviously P α m P m is closed under pure measuring R-subpairings. Rational modules 1.8. Let P = (A, C) be a measuring α-pairing (A not necessarily with unity). Let M be an A-faithful left A-module and consider the injective canonical A-linear mapping ρ M : M Hom R (A, M). We put Rat C ( A M) := ρ 1 M (M R C). If Rat C ( A M) = M, then M is said to be C-rational and we define M := (α P M ) 1 ρ M : M M R C. For m Rat C ( A M) with M(m) = k m i c i we call {(m i, c i )} k i=1 M C a rational i=1 system for m. With Rat C ( A M) A M we denote the full subcategory of C-rational left A-modules. The full subcategory of C-rational right A-modules C Rat( M A ) M A is analogously defined (we will show in Theorem 1.14 that every C-rational left, respectively right, A-module is unital). As a preparation for the proof of the main results in this section (Theorems 1.14 and 1.15) and to make the paper more self-contained we begin with some technical lemmas. 4

5 Lemma 1.9. Let P = (A, C) be a measuring α-pairing (A not necessarily with unity). For every A-faithful left A-module M we have: 1. Rat C ( A M) M is an A-submodule. 2. For every A-submodule N M we have Rat C ( A N) = N Rat C ( A M). 3. Rat C (Rat C ( A M)) = Rat C ( A M). 4. For every L A M and f HomA (M, L) we have f(rat C ( A M)) Rat C ( A L). Proof. 1. Let b A and m Rat C ( A M) with rational system {(m i, c i )} k i=1 M C. Then we have for arbitrary a A : a(bm) = (ab)m = k m i < ab, c i >= i=1 k m i < a, bc i > i=1 and so bm Rat C ( A M) with rational system {(m i, bc i )} k i=1 M C. 2. Clearly Rat C ( A N) N Rat C ( A M). On the other hand take n N Rat C ( A M) with rational system {(m i, c i )} k i=1 M C. Then for arbitrary a A we have k m i < a, c i >= an N and so n Rat C ( A N) by Lemma 1.6. i=1 3. Follows from 1. and Let f : M L be a morphism of A-faithful left A-modules and take m Rat C ( A M) with rational system {(m i, c i )} k i=1 M C. Then for arbitrary a A we have k af(m) = f(am) = f( m i < a, c i >) = i=1 k f(m i ) < a, c i >, i=1 i.e. f(m) Rat C ( A L) with rational system {(f(m i ), c i )} k i=1 L C. Lemma Let P = (A, C) P m (A not necessarily with unity). 1. If (M, M) is a right C-comodule, then M is a left A-module through ρ M : M αp M M Hom R (A, M) (4) If M is counital and A has unity, then A M is unital (and A-faithful). 2. Let (M, M), (N, N) be right C-comodules and consider the induced left A-module structures (M, ρ M ), (N, ρ N ) as in (4). If f : M N is C-colinear, then f is A- linear. 5

6 3. Let N be a right C-comodule, K N be a right C-subcomodule and consider the induced left A-module structures (N, ρ N ), (K, ρ K ) as in (4). Then K N is an A-submodule. Proof. 1. Set P P := (A R A, C R C) and consider the following diagram with commutative trapezoids (where ζ l the isomorphism given by ζ l (δ)(a b) := δ(b)(a)): ρ M ρ M M Hom R (A, M) α P M M M M R C M M R C α P M Hom R (A, M) M RidC id M R M R C R C α P P M (µ,m) Hom R (A, Hom R (A, M)) Hom R (A R A, M) (A,ρ M ) ζ l (5) By assumption the internal rectangle is commutative and consequently the outer rectangle is commutative, i.e. (M, ρ M ) is a left A-module. If M is counital and A has unity, then for every m M : i.e. A M is unital (and A-faithful). 2. Consider the diagram 1 A m = ε C m = m <0> ε C (m <1> ) = m, M ρ M M Hom R (A, M) α P M M R C f (A,f) f id C N ρ N N Hom R (A, N) α P N N R C (6) The lower trapezoid is obviously commutative. Moreover both triangles are commutative by the definition of ρ M and ρ N (4). If f is C-colinear, then the outer rectangle is commutative and consequently the upper trapezoid is commutative, i.e. f is A-linear. 3. Trivial. Lemma Let P = (A, C) P α m (A not necessarily with unity). 6

7 1. If (M, ρ M ) A M is C-rational, then M is a counital right C-comodule through M : M (αp M ) 1 ρ M M R C (7) 2. Let (M, ρ M ), (N, ρ N ) A M be C-rational an consider the induced right C-comodule structures (M, M), (N, N) as in (7). Then Hom C (M, N) = Hom A (M, N). 3. Let (N, ρ N ) A M be C-rational and consider the induced right C-comodule structure (N, N) as in (7). If K N is an A-submodule, then K is a counital right C- subcomodule and moreover K = ( N) K. Proof. 1. If (M, ρ M ) is C-rational, then ρ M (M) α P M (M R C) (by definition). Moreover α P M is injective, hence M := (α P M ) 1 ρ M : M M R C is well defined and we have the commutative diagram Hom R (A, M) α P M ρ M M M C M The right trapezoid in diagram (5) is obviously commutative and by definition of M (7) all other trapezoids are commutative. By assumption M is a left A-module and so the outer rectangle is also commutative. By [2, Lemma 2.8] α P P M is injective and consequently the internal rectangle is commutative, i.e. (M, M) is a right C- comodule. Moreover, ρ M and α P M are by assumption injective and so M := α P M ρ M is injective, i.e. M is counital. 2. Let M, N Rat C ( A M) and f : M N be A-linear. The lower trapezoid in diagram (6) is obviously commutative and by definition of M, N all triangles are commutative. If f is A-linear then, by the injectivity of α P N, the upper trapezoid is commutative and consequently the outer triangle is commutative, i.e. f is C-colinear. So Hom A (M, N) Hom C (M, N) and the equality follows from Lemma 1.10 (2). 3. Let (N, ρ N ) be a C-rational left A-module. If K N is an A-submodule, then by Lemma 1.9 (2) Rat C (K) = K Rat C (M) = K, i.e. K is a C-rational left A-module. By (1) it follows that K is a counital right C-comodule through some R-linear map K : K K R C. Moreover K ι K N is by assumption A-linear and so C-colinear (by 2.), i.e. K N is a C-subcomodule. By remark 1.5 R C is flat and so K = ( N) K. Remark Let C be an R-coalgebra and (N, N) be an arbitrary right C-comodule. Let R (Λ) π N 0 be a free representation of N in M R. Then C (Λ) R (Λ) R C π id N R C 0 7

8 is an epimorphism in M C. Moreover the injective comodule structure map N : N N R C is C-colinear, i.e. N is a C-subcomodule of the C-generated C-comodule N R C and so C-subgenerated. Since N M C is arbitrary, we conclude that C is a subgenerator in M C For every R-coalgebra C we have an R-algebra isomorphism: Ψ : (C, ) (End C (C, C) op, ), f [c f(c 1 )c 2 ] with inverse Φ : g ε g. Analogously ( C End(C, C), ) (C, ) as R-algebras. If P = (A, C) P α m, then we have isomorphisms of R-algebras: and C C End(C) = End(C C ) = End(C End C (C) op) = End(C End( A C) op) := Biend( AC) C End C (C) op = End( C C) op = End(C End(C)C) op = End( End(CA )C) op := Biend(C A ), where Biend( A C) and Biend(C A ) are the biendomorphism rings of A C and C A, respectively (compare [38, 6.4]). We are now ready to prove the main result in this section: Theorem Let P = (A, C) P m, A not necessarily with unity. If R C is locally projective and κ P (A) C is dense with respect to the finite topology on C C C, then every right (respectively left) C-comodule is a unital left (respectively right) A-module and we have category isomorphisms and M C Rat C ( A M) = Rat C ( A M) = σ[ A C] Rat C ( C M) = Rat C ( C M) = σ[ C C] C M C Rat( M A ) = C Rat(M A ) = σ[c A ] C Rat( M C ) = C Rat(M C ) = σ[c C ]. (8) (9) Proof. We prove the category isomorphisms (8). The isomorphisms of categories (9) follow by symmetry. Step 1. M C Rat C ( A M). Since R C satisfies the α-condition and κ P (A) C is dense, it follows by [2, Proposition 2.4 (2)] that P P α m. For every counital (M, M) M C we conclude that ρ M := α P M M is injective, i.e. the induced left A-module is A-faithful. By Lemmas 1.10 and 1.11 we have covariant functors A( ) : M C Rat C ( A M), ( ) C : Rat C ( A M) M C, (M, M) (M, α P M M), (M, ρ M ) (M, (α P M ) 1 ρ M ), (10) 8

9 acting as the identity on morphisms. Obviously ( ) C A ( ) id M C and A ( ) ( ) C id Rat C ( A M), i.e. M C Rat C ( A M). Step 2. Rat C ( A M) = Rat C ( A M). Let (N, ρ N ) Rat C ( A M) and n N with N(n) = k i=1 n i c i. By assumption κ P (A) C is dense and so there exists some a A, so that κ P (a)(c i ) = ε(c i ) for i = 1,..., k. Hence n = k n i ε(c i ) = i=1 k n i < a, c i >= an N (i.e. A N is unital). i=1 Step 3. M C = σ[ A C]. By Remark 1.5 R C is flat and so M C is a Grothendieck category (e.g. [12, 3.13]). Moreover by the previous lemmas and Remark 1.12 M C σ[ A C] is a full subcategory of AM. The equality follows now by the fact that σ[ A C] A M is the smallest Grothendieck full subcategory of A M that contains C. Step 4. C has unity ε C and by assumption (C, C) Pm α, so the proof above can be repeated to get M C Rat C ( C M) = Rat C ( C M) = σ[ C C]. Theorem For a measuring R-pairing P = (A,C) (A not necessarily with unity), the following are equivalent: 1. P satisfies the α-condition; 2. M C σ[ A C] = σ[ C C]; 3. R C is locally projective and κ P (A) C is dense; 4. C M σ[c A ] = σ[c C ]. Proof Follows from Theorem By assumption we have σ[ A C] = σ[ C C] = σ[ Biend(A C)C] and the density of κ P (A) C follows by Lemma 1.2. The proof of M C = σ[ C C] R C locally projective follows from [36, 3.5] (which appeared also as [12, 4.3]) Follows from general theory of dual pairings over rings (e.g. [2, Proposition 2.4 (2)]) Follows by symmetry. Example An interesting example for a measuring pairing for which Theorem 1.15 applies is P := (C, C), where C is a locally projective R-coalgebra and C := Rat C ( C C). 9

10 Dual coalgebras Definition Let A be an R-algebra and consider the class of R-cofinite A-ideals K A. For every class F of R-cofinite A-ideals we define the set A F := {f A f(i) = 0 for some I F}. (11) 1. A filter F = {I λ } Λ consisting of R-cofinite A-ideals will be called an α-filter, if the R-pairing (A, A F ) satisfies the α-condition; cofinitary, if for every I λ F there exists I κ I λ for some κ Λ, such that A/I κ is finitely generated and projective in M R ; cofinitely R-cogenerated, if A/I is R-cogenerated for every I F. 2. We call A : an α-algebra, if K A is an α-filter; cofinitary, if K A is a cofinitary filter; cofinitely R-cogenerated, if A/I is R-cogenerated for every I K A. Notation. With Cog R (respectively Big R, Hopf R ) denote the category of R-coalgebras (respectively R-bialgebras, Hopf R-algebras) and with CAlg R (respectively CCog R ) the category of commutative R-algebras (respectively cocommutative R-coalgebras). With CBig R (respectively CCBig R ) we denote the category of commutative (respectively cocommutative) R-bialgebras and with CHopf R (respectively CCHopf R ) the category of commutative (respectively cocommutative) Hopf R-algebras. For two R-coalgebras C, D we denote with Cog R (C, D) the set of all R-coalgebra morphisms from C to D. For two R-algebras (respectively R-bialgebras, Hopf R-algebras) H, K we denote with Alg R (H, K) (respectively Big R (H, K), Hopf R (H, K)) the set of all R- algebra morphisms (respectively R-bialgebra morphisms, Hopf R-morphisms) from H to K. Remark We make the convention that an R-bialgebra (respectively a Hopf R-algebra) is an α-bialgebra (respectively a Hopf α-algebra), is cofinitary or is cofinitely R-cogenerated, if it is so as an R-algebra. With Big α R Big R (respectively Hopf α R Hopf R ) we denote the full subcategory of α-bialgebras (respectively Hopf α-algebras). Lemma ([6, Proposition 2.6]) Let R be Noetherian and A be an R-algebra. Then A := {f A f(i) = 0 for some R-cofinite ideal I A}; = {f A f(i) = 0 for some R-cofinite left (right) A-ideal}; = {f A Af (fa) is f.g. in M R } = {f A AfA is f.g. in M R }. 10

11 Theorem ([3, Theorem 3.3.]) Let R be Noetherian, A be an R-algebra and consider A A as an A-bimodule under the left and the right regular A-actions (af)(ã) = f(ãa) and (fa)(ã) = f(aã) for all a, ã A and f A. (12) For an A-subbimodule C A and P := (A, C) the following are equivalent: 1. R C is locally projective and κ P (A) C is dense; 2. R C satisfies the α-condition and κ P (A) C is dense; 3. (A, C) is an α-pairing; 4. C R A is pure; 5. C is an R-coalgebra and (A, C) P m α. If R is a QF Ring, then these are moreover equivalent to 6. R C is projective. Corollary ([3, Corollary 3.16]) Let A be an R-algebra and F be a filter consisting of R-cofinite A-ideals. If R is Noetherian and F is an α-filter, or if F is cofinitary then we have isomorphisms of categories M A F Rat A F (A M) = σ[ A A F ] Rat A F (A M) = σ[ F A A F F ] & A F M A F Rat(MA ) = σ[a FA ] A F Rat(MA ) = σ[a F FA ]. F 2 The cotensor functor Dual to the tensor product of modules, J. Milnor and J. Moore introduced in [27] the cotensor product of comodules. For a closer look on the properties of the cotensor product over arbitrary (commutative) base rings the interested reader may refer to [20] (and [7]) Let C be an R-coalgebra, (M, M) M C, (N, N) C M and consider the R-linear mapping M,N := M id N id M N : M R N M R C R N. The cotensor product of M and N (denoted with M C N) is defined through the exactness of the following sequence in M R : 0 M C N M R N M,N M R C R N. For M, M M C and N, N C M, the cotensor product of f Hom C (M, M ) and g C Hom(N, N ) is defined as the R-linear mapping f C g : M C N M C N, 11

12 that completes the following diagram commutatively 0 M C N M R N M,N M R C R N f C g f g f id C g 0 M C N M R N M,N M R C R N (13) In this way we get the cotensor functor M C : C M M R (respectively C N : M C M R ), which is left exact if R C and M R (respectively R C and R N) are flat. Definition 2.2. Let C be a flat R-coalgebra (hence M C and C M are abelian categories). A right (respectively a left) C-comodule M is called coflat, if the functor M C : C M M R (respectively C M : M C M R ) is exact. Lemma 2.3. (Compare [29, Page 127], [12, 10.6]) Let C be an R-coalgebra and M M C, N C M. If W R is flat, then there are isomorphisms of R-modules W R (M C N) (W R M) C N and (M C N) R W M C (N R W). (14) The following result can easily be derived with the help of Lemma 2.3: Corollary 2.4. Let C, D be R-coalgebras and (M, CM, DM ) C M D. 1. Assume C R to be flat. For every N D M, M D N is a left C-comodule through CM D id N : M D N (C R M) D N C R (M D N). 2. Assume R D to be flat. For every L M C, L C M is a right D-comodule through id L C DM : L CM L C (M R D) (L C M) R D. Remark 2.5. ([7, Lemma II.2.5, Folgerung II.2.6]) Let C be a flat R-coalgebra. For every M M C, the mapping M : M M C C is an isomorphism in M C with inverse λ M : m c mε(c) and moreover we have If M is coflat in M C, then M R is flat. M R M C (C R ) : M R M Z. The Associativity of the cotensor products is not valid in general (see [18]). However we have it in special cases, e.g. : 12

13 Lemma 2.6. ([7, Folgerung II.3.4.]) Let C, D be flat R-coalgebras, N D M, M C M D and L M C. If L M C (or N D M) is coflat, then we have an isomorphism of R-modules (L C M) D N L C (M D N). (15) Notation. For an R-algebra A we denote with A e := A R A op the enveloping R-algebra of A. Lemma 2.7. Let A be an R-algebra, M, N A M and consider A, Hom R (N, M) with the canonical left A e -module structures. Then we have a functorial isomorphism Proof. The isomorphism is given by Hom A e (A, Hom R (N, M)) Hom A (N, M). Φ N,M : Hom A e (A, Hom R (N, M)) Hom A (N, M), f f(1 A ) with inverse Ψ N,M : g [a ag( )]]. One can easily show that Φ N,M and Ψ N,M are functorial in M and N. In the case of a base field, the cotensor functor is equivalent to a suitable Hom-functor (e.g. [9, Proposition 3.1]). Over arbitrary ground rings we have Proposition 2.8. Let P = (A, C) P m, (M, M) M C, (N, N) C M and consider A, M R N with the canonical left A e -module structures. 1. If α P M R N is injective, then we have for m i n i M R N : mi n i M C N am i n i = m i n i a for all a A. 2. If P Pm α, then we have a functorial isomorphism Proof. M C N Hom A e (A, M R N). 1. Let α P M R N be injective and set ψ := αp M R N τ (23). Then mi n i M C N m i<0> m i<1> n i = m i n i< 1> n i<0>, ψ( m i<0> m i<1> n i )(a) = ψ( m i n i< 1> n i<0> )(a), a A m i<0> < a, m i<1> > n i = m i < a, n i< 1> > n i<0>, a A am i n i = m i n i a, a A. 2. The isomorphism is given through γ M,N : M C N Hom A e (A, M R N), m n [a am n (= m na)] with inverse β M,N : f f(1 A ). It is easy to see that γ M,N and β M,N are functorial in M and N. 13

14 Lemma 2.9. ([37, 15.7], [11, II, 4.2, Proposition 2]) Let A be an R-algebra, K, K be left A-modules, L be an R-module and consider the R-linear mapping υ : Hom A (K, K ) R L Hom A (K, K R L), h l h( ) l. (16) 1. If R L is flat and A K is finitely generated (respectively finitely presented), then υ is injective (respectively bijective). 2. If A K be K -projective and A K is finitely generated, then υ is bijective. 3. If A K be K -projective and R L is finitely presented, then υ is bijective. 4. If R L is projective (respectively finitely generated projective), then υ is injective (respectively bijective) ([12, 3.11]) Let R C be a flat R-coalgebra. Let M be a left C-comodule and consider the R-linear mapping γ : M Hom R (M, C), f [m f(m < 1> )m <0> ]. (17) If R M is finitely presented, then Hom R (M, C) M R C (see Lemma 2.9) and M is a right C-comodule through M : M γ Hom R (M, C) M R C. (18) If M is a right C-comodule and M R is finitely presented, then M becomes analogously a left C-comodule. With the help of Lemmas 2.7 and 2.9, the following result can be derived directly from Proposition 2.8: Corollary Let P = (A, C) P α m. 1. Let M, N M C. If M R is flat and R N is finitely presented, or N R is finitely generated projective, then we have functorial isomorphisms M C N Hom A e (A, M R N ) Hom A e (A, Hom R (N, M)) Hom A (N, M) = Hom C (N, M). 2. Let M M C, N be a C-bicomodule and consider N with the induced left A e -module structure. Then we have isomorphisms of R-modules M C N Hom A e (A, M R N) M R Hom A e (A, N), if any one of the following conditions is satisfied: (a) M R is flat and A ea is finitely presented (e.g. A is an affine R-algebra [37, 23.6]); 14

15 (b) A ea is N-projective and finitely generated; (c) A ea is N-projective and M R is finitely presented; (d) M R is finitely generated projective. 3. Let N C M, M be a C-bicomodule and consider M with the induced left A e -module structure. Then we have an isomorphism of R-modules M C N Hom A e (A, M R N) Hom A e (A, M) R N, if any one of the following conditions is satisfied: (a) R N is flat and A ea is finitely presented (e.g. A is an affine R-algebra [37, 23.6]); (b) A ea is M-projective and finitely generated; (c) A ea is M-projective and R N is finitely presented; (d) R N is finitely generated projective. Injective comodules For P = (A, C) Pm α we get from [38, 16.3] the following characterizations of the injective objects in M C Rat C ( A M) = σ[ A C] : Lemma Let P = (A, C) P α m. For every U RatC ( A M) the following are equivalent: 1. U is injective in Rat C ( A M); 2. Hom C (, U) Hom A (, U) : Rat C ( A M) M R is exact; 3. U is C-injective in Rat C ( A M); 4. U is K-injective for every (finitely generated, cyclic) left A-submodule K C; 5. every exact sequence 0 U L N 0 in Rat C ( A M) splits. 6. every exact sequence 0 U L N 0 in Rat C ( A M), in which N is a factor module of C (or A) splits. The following Lemma plays an important role in the study of injective objects in the category Rat C ( A M), where (A, C) P α m : Lemma Let (A, C) P α m. If R is a QF ring then a C-rational left A-module M, with R M flat, is injective in Rat C ( A M) if and only if M is coflat in M C. 15

16 Proof. By Theorem 1.15 we have the isomorphism of categories and we get the result by [12, 10.12]. σ[ A C] = Rat C ( A M) M C Lemma If P = (A, C) P α m, then R C : M R Rat C ( A M) respects injective objects. Proof. By Theorem 1.15 M C Rat C ( A M) = σ[ A C], i.e. Rat C ( A M) A M is a closed subcategory. The exact forgetful functor : M C M R is left adjoint to R C : M R M C and the result follows then by [38, 45.6]. Proposition Let (A, C) P α m and M RatC ( A M). 1. M is an A-submodule of an injective C-rational left A-module. 2. Every injective object in Rat C ( A M) is C-generated. 3. M is injective in Rat C ( A M) if and only if there exists an injective R-module X for which A M is a direct summand of X R C. 4. Let M be injective in M R. Then M is injective in Rat C ( A M) if and only if M : M M R C splits in A M. 5. Let R be Noetherian. Then M is injective in Rat C ( A M) if and only if M (Λ) is injective in Rat C ( A M) for every index set Λ. Moreover, direct limits of injectives in Rat C ( A M) are injective. 6. Let A be separable (i.e. A ea is projective). Then M M C is coflat if and only if M R is flat. Proof. 1. Let M Rat C ( A M) and denote with E(M) the injective hull of M in M R. By Lemma 2.14 E(M) R C is injective in Rat C ( A M). Obviously (ι M R id C ) M : M E(M) R C is A-linear and the result follows. 2. Let (M, M) Rat C ( A M) be injective. By Lemma 2.12, there exists an epimorphism of left A-modules β : M R C M, such that β M = id M. If R (Λ) π M 0 is a free representation of M in M R, then we get the following exact sequence in Rat C ( A M) : C (Λ) R (Λ) R C β (π id C) M Let X be an injective R-module, such that A M is a direct summand of X R C. By Lemma 2.14 X R C is injective in Rat C ( A M) and consequently M is injective in Rat C ( A M). On the other hand, let M be injective in Rat C ( A M) and denote with E(M) the injective hull of M in M R. Then we get an exact sequence in Rat C ( A M) 0 M (ι id C) M E(M) R C. (19) Now (19) splits in Rat C ( A M) by Lemma 2.12 and the result follows. 16

17 4. Follows from Lemmata 2.12 and By [4, Folgerung ] A C is locally Noetherian. The result follows then from the isomorphism of categories Rat C ( A M) σ[ A C] and [38, 27.3]. 6. If M M C is coflat, then M R is flat (by Remark 2.5). Assume now that A ea is projective. If M R is flat, then by Proposition 2.8 is exact, i.e. M is coflat. M C Hom A e (A, ) (M R ) Corollary Let (A, C) P α m. If R is semisimple (e.g. a field), then: 1. M Rat C ( A M) is injective if and only if A M is a direct summand of A C (Λ) for some index set Λ. 2. If A is separable, then C is right semisimple (i.e. every right C-comodules is injective). 3 Coinduction Functors in P α m By his study of the induced representations of quantum groups, Z. Lin ( [24, 3.2], [23]) considered induction functors for admissible Hopf R-pairings over Dedekind rings. His aspect was inspired by the induction functors in the theory of affine algebraic groups and quantum groups. We generalize his results to the coinduction functor for the category of measuring α-pairings P α m P m and show that it is isomorphic to a coinduction functor between categories of Type σ[m]. Moreover we get as nice description of it as a composition of a suitable Hom-functor and a Trace-functor Let A, B be R-algebras and ξ : A B be an R-algebra morphism. Then every left B-module becomes a left A-module in a canonical way and we get the so called restriction functor ( ) ξ : B M A M. Considering B with the canonical A-bimodule structure, we have the functor Hom A (B, ) : A M B M. Moreover (( ) ξ, Hom A (B, )) is an adjoint pair of covariant functors through the functorial canonical isomorphisms Hom A (M ξ, N) Hom A (B B M, N) Hom B (M, Hom A (B, N)). If we consider the induction functor B A : AM B M, then (B A, ( ) ξ ) is an adjoint pair of covariant functors through the functorial canonical isomorphisms Hom B (B A N, M) Hom A (N, Hom B (B, M)) Hom A (N, M ξ ). 17

18 3.2. The general coinduction functor. Let A, B be R-Algebras and ξ : A B be an R-algebra morphism. If L is a left B-module, then we get the covariant functor HOM A (B, ) := Sp(σ[ B L],Hom A (B, )) : A M σ[ B L]. (20) For every left A-module K (20) restricts to the covariant coinduction functor Coind L K( ) := Sp(σ[ B L],Hom A (B, )) : σ[ A K] σ[ B L], (21) i.e. Coind L K( ) is defined through the commutativity of the following diagram: AM Hom A (B, ) BM HOM A (B, ) Sp(σ[ B L], ) σ[ A K] Coind L K ( ) σ[ B L] If (L) ξ is K-subgenerated as a left A-module, then ( ) ξ : B M A M restricts to ( ) ξ : σ[ B L] σ[ A K] and (( ) ξ, Coind L K( )) turns to be an adjoint pair of covariant functors The ad-corestriction functor. Let C, D be R-coalgebras and θ : D C be an R-coalgebra morphism. Then we get the covariant corestriction functor ( ) θ : M D M C, (M, M) (M, (id M θ) M). (22) On the other hand, consider D as a left C-comodule through CD : D D D R D θ id C R D. If R D is flat, then for every M M C the cotensor product M C D becomes a right D-comodule through M C D id C D and we get the ad-corestriction functor M C (D R D) (M C D) R D C D : M C M D, M M C D Let Q = (B, D) Pm α. For every R-algebra A with R-algebra morphism ξ : A B we have the covariant functor HOM A (B, ) := Rat D ( ) Hom A (B, ) : AM Rat D ( B M). If P = (A, C) P α m, then HOM A (B, ) restricts to the coinduction functor from P to Q : Coind Q P ( ) : RatC ( A M) Rat D ( B M), M Rat D (Hom A (B, M)), 18

19 i.e. Coind Q P ( ) is defined through the commutativity of the following diagram: AM Hom A (B, ) HOM A (B, ) BM Rat D ( ) Rat C ( A M) Coind Q P ( ) Rat D ( B M) Proposition 3.5. Let P = (A, C), Q = (B, D) P m and (ξ, θ) : (B, D) (A, C) be a morphism in P m. 1. If R D is flat, then (( ) θ, C D) is an adjoint pair of covariant functors. 2. If P, Q P α m and B is commutative, then we have for every N A M : HOM A (B, N) = HOM A (B, Rat C ( A N)). Proof. 1. One can show easily that the mapping Φ N,L : Hom D (N, L C D) Hom C (N θ, L), f (id L C θ) f is an isomorphism with inverse g (g C id D ) N and moreover that it is functorial in N M D and L M C. 2. If g HOM A (B, N), then we have for all a A and b B : a(g(b)) = g(a b) = g(ξ(a)b) = g(bξ(a)) = (ξ(a)g)(b) = g <0> (b) < ξ(a), g <1> > = g <0> (b) < a, θ(g <1> ) >. Consequently g(b) Rat C ( A N) and the result follows Let P = (A, C), Q = (B, D) Pm α, (ξ, θ) : (B, D) (A, C) be a morphism in Pm α and denote the restriction of ( ) ξ : B M A M on Rat D ( B M) = σ[ B D] also with ( ) ξ. Through the isomorphism of categories M C Rat C ( A M) = σ[ A C] and M D Rat D ( B M) = σ[ B D] (compare Theorem 1.15) we get an equivalence of functors ( ) θ ( ) ξ. Considering the covariant functors (10) we get a commutative diagram of pairwise 19

20 adjoint covariant functors M D ( ) D B( ) Rat D ( B M) σ[ B D] Sp(σ[ B D], ) ι D BM ( ) θ C D ( ) ξ Coind Q P ( ) ( ) ξ Coind D C ( ) ( ) ξ Hom A (B, ) M C A( ) ( ) C Rat C ( A M) σ[ A C] ι C AM Sp(σ[ A C], ) (23) Theorem 3.7. Let P = (A, C), Q = (B, D) P α m (so that in particular RC and R D are flat) and (ξ, θ) : (B, D) (A, C) be a morphism in P α m. Through the isomorphisms of categories M C Rat C ( A M) = σ[ A C] and M D Rat D ( B M) = σ[ B D] (compare Theorem 1.14) the following functors are equivalent C D : M C M D, Coind Q P ( ) : RatC ( A M) Rat D ( B M), Hom A e (A, R D) : Rat C ( A M) Rat D ( B M), Coind D C ( ) : σ[ AC] σ[ B D]. Proof. Consider for every N M C the injective R-linear mapping γ N := (α Q N ) N C D : N C D Hom R (B, N), ni d i [b n i < b, d i >]. Then we have for all a A and b B : γ N ( n i d i )(a b) = n i < a b, d i > = n i < b, d i a > = γ N ( n i d i a)(b) = γ N ( an i d i )(b) (compare Lemma 2.8 (1)) = an i < b, d i > = a(γ N (n i d i )(b)), i.e. γ N (N C D) Hom A (B, N). Moreover we have for arbitrary n i d i N C D and b, b B : γ N ( b( n i d i ))(b) = γ N ( n i b d i )(b) = n i < b, b d i > = n i < b b, d i > = γ N ( n i d i )(b b) = ( bγ N ( n i d i ))(b), 20

21 i.e. γ N is B-linear. But R D is flat, so N C D M D by Corollary 2.4 and it follows by Lemma 1.9 that γ N (N C D) HOM A (B, N). Now we show that the following R-linear mapping is well defined β N : HOM A (B, N) N C D, f f <0> (1 B ) f <1>. For all f HOM A (B, N), a A and b B we have γ N ( a(f <0> (1 B )) f <1> )(b) = a(f <0> (1 B )) < b, f <1> > = f <0> (a 1 B )) < b, f <1> > = f <0> (ξ(a)) < b, f <1> > = (ξ(a)f <0> )(1 B ) < b, f <1> > = f <0><0> (1 B ) < ξ(a), f <0><1> >< b, f <1> > = f <0> (1 B ) < ξ(a), f <1>1 >< b, f <1>2 > = f <0> (1 B ) < ξ(a)b, f <1> > = f <0> (1 B ) < a b, f <1> > = f <0> (1 B ) < b, f <1> a > = γ N ( f <0> (1 B ) f <1> a)(b), i.e. a(f<0> (1 B )) f <1> = f <0> (1 B ) f <1> a (since γ N is injective). It follows then by Proposition 2.8 (1) that f <0> (1 B ) f <1> N C D, i.e. β N is well defined. Moreover, we have for all f HOM A (B, N) and b B : (γ N β N )(f)(b) = γ N ( f <0> (1 B ) f <1> )(b) = f <0> (1 B ) < b, f <1> > = (bf)(1 B ) = f(b), hence γ N β N = id. Obviously β N γ N = id. Consequently γ N and β N are isomorphisms. It is easy to show that γ N and β N are functorial in N, hence C D Coind Q P ( ). The equivalences Coind Q P ( ) CoindD C ( ) and CD Hom A e (A, R D) follow now by Theorem 1.15 and Proposition 2.8 (2), respectively Let Q = (B, D) Pm α and consider the trivial R-pairing P = (R, R) Pα m with the morphism of measuring R-pairings (η B, ε D ) : (B, D) (R, R). Then we have for every M M R M R Coind Q P ( ) := HOM R(B, ) R D. Notice that ( ) ε : M D M R, where is the forgetful functor, hence (, Coind Q P ( )) is an adjoint pair of covariant functors Universal Property. Let P = (A, C), Q = (B, D) Pm α and (ξ, θ) : (B, D) (A, C) be a morphism in Pm. α Then Coind Q P ( ) has the following universal property: if N M D, M M C and φ Hom C (N θ, M), then there exists a unique φ Hom D (N, Coind Q P (M)), such that φ(n) = φ(n)(1 B ) for every n N. In what follows we list some properties of the coinduction functor: 21

22 3.10. Let P = (A, C), Q = (B, D) P α m and (ξ, θ) : (B, D) (A, C) be a morphism in P α m. 1. Coind Q P ( ) respects direct limits: if {N λ} Λ is a directed system in Rat C ( A M), then Coind Q P (lim N λ) lim N λ C D lim (N λ C D) = lim Coind Q P (N λ). 2. Rat D ( ) & Hom A (B, ) are left-exact, hence Coind Q P ( ) := RatD ( ) Hom A (B, ) is left-exact. If moreover A B is projective (hence Hom A (B, ) is exact) and Rat D ( ) is exact, then Coind Q P ( ) is exact. 3. Coind Q P ( ) CD is exact if and only if D is coflat in C M. If R is a QF ring, then Coind Q P ( ) is exact if and only if D is injective in C Rat(M A ). 4. By Lemma 3.5 (1) (( ) θ, C D) is an adjoint pair of covariant functors, hence Coind Q P ( ) CD respects inverse Limit, i.e. direct products, kernels and injective objects (since ( ) θ : M D M C is exact). In particular, if C is injective in Rat C ( A M), then D C C D Coind Q P (C) is injective in RatD ( B M). 5. Let A be separable. Then C D Coind Q P ( ) Hom A e (A, R D) is exact, i.e. D is coflat in C M. If moreover R is a QF ring, then D is injective in C M. A version of the following result was obtained by Y. Doi [14, Proposition 5] in the case of a base field: Proposition Let P = (A, C), Q = (B, D) Pm α and (ξ, θ) : (B, D) (A, C) be a morphism in Pm. α If R is a QF ring, then the following are equivalent: 1. The functor Coind Q P ( ) : RatC ( A M) Rat D ( B M) is exact; 2. D is coflat in C M; 3. D is injective in C Rat(M A ); 4. If M is an injective left D-comodule that is flat in M R, then M is injective in C Rat(M A ). Proof. (1) (2) Follows from the isomorphism of functors Coind Q P ( ) CD : M C M D. (2) (3) By Remark 1.5 R D is flat, so the equivalence follows from Lemma (2) (4) Let M be a left D-comodule and assume that M is injective in D Rat(M B ) and flat in M R. Then M is coflat in D M (by Lemma 2.13) and we have (by Lemma 2.6) an isomorphism of functors C M ( C D) D M : M C M R 22

23 By assumption C D : M C M D and D M : M D M R are exact and so C M is exact. By Lemma 2.13 M is injective in C Rat(M A ). (4) (3) Since R is injective, D is injective in D Rat(M B ) D M. It follows then from the assumption that D is injective in C Rat(M A ). As a consequence of Theorem 1.15 and [3, Proposition 3.23] we get: Corollary Let R be Noetherian, A, B be R-algebras and ξ : A B be an R-algebra morphism. 1. Let A, B be cofinitely R-cogenerated α-algebras, P := (A, A ), Q := (B, B ) and consider the morphism of measuring α-pairings (ξ, ξ ) : (B, B ) (A, A ). Then we have for every right A -comodule N : Coind Q P (N) = {f Hom A (B, N) Bf is finitely generated in M R }. 2. Let F A, F B be cofinitely R-cogenerated α-filters of R-cofinite A-ideals, B-ideals respectively and consider A, B as a left linear topological R-algebra with the induced left linear topologies T(F A ), T(F B ) respectively. If ξ : A B be an R-algebra morphism that is continuous with respect to (T(F A ), T(F B )), P := (A, A F A ) and Q := (B, B F B ), then we have for every N M A F A : Coind Q P (N) = {f Hom A (B, N) (0 : f) Ĩ for some Ĩ F B}. 4 Hopf R-pairings Definition 4.1. Let H be an R-bialgebra. An H-ideal, which is also an H-coideal, is called a bi-ideal. If H is a Hopf R-algebra with antipode S H and J H is an H-bi-ideal with S H (J) J, then J is called a Hopf ideal The category P Big. A bialgebra R-pairing is an R-pairing P = (H, K), where H, K are R-bialgebras and κ P : H K, χ P : K H are R-algebra morphisms. For bialgebra R-pairings (H, K), (Y, K) a morphism of R-pairings (ξ, θ) : (Y, Z) (H, K) is said to be a morphism of bialgebra R-pairings, if ξ : H Y and θ : Z K are R-bialgebra morphisms. With P Big P m we denote the subcategory of bialgebra R-pairings and with P α Big P Big the full subcategory, whose objects satisfy the α-condition. If P = (H, K) P Big, Z K is a (pure) R-subbialgebra and J H is an H-bi-ideal with < J, Z >= 0, then Q = (H/J, Z) is a bialgebra R-pairing, (π J, ι Z ) : (H/J, Z) (H, K) is a morphism in P Big and Q P is called a (pure) bialgebra R-subpairing. Obviously P α Big P Big is closed under pure bialgebra R-subpairings The category P Hopf. A Hopf R-pairing P = (H, K) is a bialgebra R-pairing with H, K Hopf R-algebras. With P Hopf P Big we denote the full subcategory of Hopf 23

24 R-pairings and with P α Hopf P Hopf the full subcategory, whose objects satisfy the α- condition. If P = (H, K) is a Hopf R-pairing, Z K a (pure) Hopf R-subalgebra and J H a Hopf ideal with < J, Z >= 0, then Q := (H/J, Z) is a Hopf R-pairing, (π J, ι Z ) : (H/J, Z) (H, K) is a morphism in P Hopf and Q P is called a (pure) Hopf R-subpairing. Obviously P α Hopf P Hopf is closed under pure Hopf R-subpairings. Example 4.4. Let R be Noetherian and H be an α-bialgebra (respectively a Hopf α- algebra). Then H is by ([6, Theorem 2.8]) an R-bialgebra (respectively a Hopf R-algebra). Moreover it is easy to see that (H, H ) is a bialgebra α-pairing (respectively a Hopf α- pairing). Remarks (Compare [33]) If P = (H, K) is a Hopf R-pairing, then < S H (h), k >=< h, S K (k) > for all h H and k K. 2. Let R be Noetherian. If P = (H, K) is a bialgebra R-pairing (respectively a Hopf R-pairing), then κ P (H) K and χ P (K) H. If (H, K) P Big and H Big α R (respectively K Big α R ), then χ P : K H (respectively κ P : H K ) is an R-bialgebra morphism. Quasi-Admissible filters. By the study of induced representations of quantum groups, Z. Lin [24] and M. Takeuchi [32] studied the so called admissible filters of ideals of a Hopf R-algebra over arbitrary (Dedekind) rings. In what follows we introduce what we call the quasi-admissible filters and generalize some of their results to the class of (not necessarily cofinitary) quasi-admissible α-filters Let A, B be R-algebras and F A, F B be filters consisting of R-cofinite A-ideals, B-ideals respectively. Then the filter basis F A F B := {Im(ι I id B ) + Im(id A ι J ) I F A, J F B } (24) induces on A R B a topology T(F A F B ), such that (A R B, T(F A F B )) is a linear topological R-algebra and F A F B is a neighbourhood basis of 0 A R B Let H be an R-bialgebra (that is not a Hopf R-algebra), F K H be a filter and consider the induced linear topological R-algebras (H, T(F)) and (H R H, T(F F)). We call F quasi-admissible, if H : H H R H and ε H : H R are continuous, i.e. if F satisfies the following axioms: (A1) I, J F there exists L F, such that H (L) Im(ι I id H ) + Im(id H ι J ) (25) and (A2) I F, such that Ker(ε H ) I. (26) 24

25 If H is a Hopf R-algebra, then we call a filter F K H quasi-admissible, if it satisfies (A1), (A2) as well as (A3) for every I F there exists J F, such that S H (J) I (27) (i.e. if H, ε H and S H are continuous). In [24] and [32], a cofinitary quasi-admissible filter of R-cofinite H-ideals (for a Hopf R-algebra H) is called admissible. Definition 4.8. We call an R-bialgebra (respectively Hopf R-algebra) H a quasi-admissible R-bialgebra (respectively a quasi-admissible Hopf R-algebra), if the class of R-cofinite H- ideals K H is a quasi-admissible filter. Lemma 4.9. If the ground ring R is Noetherian, then every R-bialgebra (Hopf R-algebra) is quasi-admissible. Proof. Let H be an R-bialgebra. Since R is Noetherian, K H is a filter. Moreover, H R Ker(ε H ), hence Ker(ε H ) K H. Let I, J K H and set L := Im(ι I id H )+Im(id H ι J ). Notice that (H R H)/L H/I R H/J (e.g. [11, II-3.6, III-4.2]), hence L K H R H. By definition : H H R H is an R-algebra morphism and it follows, by the assumption R is Noetherian, that 1 (L) H is an R-cofinite ideal. Consequently H is a quasi-admissible R-bialgebra. If H is moreover a Hopf R-algebra, then S H : H H is an R-algebra antimorphism and it follows, from the assumption R is Noetherian, that for every R-cofinite ideal I H the H-ideal S 1 H (I) H is R-cofinite. Consequently H is a quasi-admissible Hopf R-algebra. Definition ([34]) An R-coalgebra C is called infinitesimal flat, if C = lim C λ for a directed system of finitely generated projective R-subcoalgebras {C λ } Λ. Proposition Let H be an R-bialgebra (respectively a Hopf R-algebra) and F K H be a quasi-admissible filter. 1. If R is Noetherian and F is an α-filter, then HF is an R-bialgebra (respectively a Hopf R-algebra) and (H, HF ) is a bialgebra α-pairing (respectively a Hopf α-pairing). 2. If F is moreover cofinitary, then HF is an infinitesimal flat R-bialgebra (Hopf R- algebra) and (H, HF ) is a bialgebra α-pairing (a Hopf α-pairing). Proof. 1. Let H be an R-bialgebra. Obviously HF H is an H-subbimodule under the regular left and the right H-actions (12) and so an R-coalgebra by Theorem If f(i) = 0 and g(j) = 0 for I, J F, then there exists by (25) some L F, such that (L) Im(ι I id H )+Im(id H ι J ). Consequently (f g)(l) = (f g)( (L)) = 0, i.e. f g An(L) HF. By (26) ε H HF and so H F H is an R-subalgebra. It is easy to see that : HF R HF H F and ε : R HF are coalgebra morphisms, i.e. HF is an R-bialgebra. If H is a Hopf R-algebra with Antipode S, then it follows from (27) that S (HF ) H F, hence H F is a Hopf R-algebra with antipode S. 25

26 2. See [32]. As a consequence of Lemma 4.9 and Proposition 4.11 we get Corollary Let R be Noetherian. If H is an α-bialgebra (respectively a Hopf α- algebra), then H is an R-bialgebra (respectively a Hopf R-algebra). If H is cofinitary, then H is an infinitesimal flat R-bialgebra (respectively Hopf R-algebra). Proposition Let H be an R-bialgebra, F be a quasi-admissible filter of R-cofinite H-ideals and consider H as a left linear topological R-algebra with the induced left linear topology T(F). If R is an injective cogenerator, then the following are equivalent: 1. T(F) is Hausdorff; 2. the canonical R-linear mapping λ : H H F 3. H F H is dense; 4. σ[ H F H] = σ[ H H]. is injective; Proof. By assumption H/I is R-cogenerated for every I F (hence I = KeAn(I) by [38, 28.1]) and so 0 A := I F I = I F KeAn(I) = Ke( I F An(I)) = Ke(H F) = Ker(λ). Since R is an injective cogenerator, the equivalence (2) (3) follows from [2, Theorem 1.8 (2)]. By assumption F is quasi-admissible, hence H F H is an R-subalgebra and the equivalence (3) (4) follows by Lemma ]: The proof of the following Proposition is along the lines of the proof of [5, Theorem Proposition Let H, K be R-bialgebras (Hopf R-algebras) with quasi-admissible filters F H, F K and consider the canonical R-linear mapping δ : H R K (H R K) and the filter F of R-cofinite H R K-ideals generated by F H F K. 1. If F H and F K are moreover cofinitary (i.e. admissible filters), then (H R K) F is an R-bialgebra (respectively a Hopf R-algebra). If R is Noetherian, then δ induces an isomorphism of R-bialgebras (respectively Hopf R-algebras) HF H R KF K (H R K) F. 2. Let R be Noetherian. If F K is an α-filter and F H is cofinitary, then (H R K) F is an R-bialgebra (a Hopf R-algebra) and δ induces an isomorphism of R-bialgebras (Hopf R-algebras) HF H R KF K (H R K) F. Definition The ring R is called hereditary, if every ideal I R is projective. 26