By Michel Broué
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version 20 Dec 2001
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Additional resources for Symmetric algebras
Ii) εM,N is a split epimorphism in A ModA . (iii) ηN,M is a split monomorphism in A ModA . (iv) The trace map G B TrF GF (A) : (N ⊗A M ⊗B N ⊗A M ) → ZA is onto. (v) Every A–module is M –split. If the preceding conditions are satisﬁed, we say that the algebra A is B–split (through (M, N )). Example : Induction–Restriction with R. Choose B := R, M :=A AR , N :=R AA and a, a := t(aa ). 1. The following conditions are equivalent. (i) A is strongly symmetric. (ii) R is A–split. 2. The following conditions are equivalent.
The M -Stable Category Generalities. Let A and B be two categories. We denote by HomM A (X, X ) the image of M TrN (X, X ) in HomA (X, X ) and call these morphisms the “M -split morphisms”. , such that all endomorphisms are M –split). 55), we have the following property. 60. Lemma. A morphism X → X in A is M -split if and only if it factorizes through an M –split object of A. 61. Definition. Let A be an abelian category. The category Stab(A), is deﬁned as follows: 1. the objects of Stab(A) are the objects of A, 2.
We know that both, (M, N ) and (ΩA , Ω−1 ), are biadjoint pairs. Thus the functor A −1 −1 ΩA M is left adjoint to the functor N ΩA . 68, N ΩA is naturally equivalent to the functor Ω−1 B N. D4. Stable Equivalences of Morita Type Let (M, N ) be a selfdual exact pair of bimodules for A and B. Since the functors M ⊗B and N ⊗A factorize through the functors StA : A Mod −→ A Stab and StB : B Mod −→ B Stab , the bimodules M and N induce two functors M ⊗B : B Stab −→ A Stab and N ⊗A : A Stab −→ B Stab .