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Let basis for B. if there exists an (A, B#RA}-bimodule map (j>: Horn (B, A) ^ B* A -> B#RA such that 2) B^RA/A is split if and only if there exists an (A, B^RA)-bimodule map B#RA -» Horn (B, A) such that 3) B#RA/A is Frobenius if and only if B* ® A and B#RA are isomorphic as (A,B#RA)-bimodules. >: B#RA^Kom(B,A), $(b#a)(d) = K(bdR)aR are inverses of each other. The same method can be applied to the extension B#RA/B. 7 (left and right separable (resp. Frobenius) extension coincide). Another possibility is to use "op" -arguments.

To appear. [4] T. Brzeziriski, Coalgebra-Galois extensions from the extension point of view, in "Hopf algebras and quantum groups", S. Caenepeel and F. ), Lee. Notes Pure Appl. Math. 209, Marcel Dekker, New York, 2000. [5] T. Brzezinski, The structure of corings. RA/0002105. [6] T. Brzeziriski and P. M. Hajac, Coalgebra extensions and algebra coextensions of Galois type, Comm. Algebra 27 (1999), 1347-1367. [7] T. Brzeziriski and S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), 467-492.

N} a n supp(a fc ) ^ 0 V/c = 1, . . , s} The following result allows the effective computation of the dimension of any monoideal, and hence, of any stable subset. 3], can be found in [5, Section 4]. 10 Let E be a proper monoideal o/N™. (1) dim(£) = n - min{card(cr) a 6 V(E)} (2) If m is the maximum of the entries of all vectors in a finite set of generators for E, then there is an unique polynomial h(x] G Q[z] such that HF£(s) = h(s) for every s ^ mn. 1]. The fact that HFB(S) coincides with a polynomial for s big enough was proved in [19, Lemma 16, p.

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