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A construction for quasi-hereditary algebras. (English) Zbl 0677.16007
Let C, D be rings, \({}_ CS_ D\), \({}_ DT_ C\) bimodules, \(\gamma\) : \({}_ CS_ D\otimes_ DT_ C\to_ CC_ C\) a bimodule homomorphism, \({\mathcal T}(S,T)\) the tensor algebra of the \(C\times D-D\times C\)-bimodule \(S\oplus T\) and \(A(\gamma)={\mathcal T}(S,T)/R(\gamma)\) where R(\(\gamma)\) is the ideal of \({\mathcal T}(S,T)\) generated by all elements of the form \(s\otimes t-\gamma (s\otimes t)\), with \(s\in S\), \(t\in T\). The rings A(\(\gamma)\) have been introduced by R. Mirollo and K. Villonen [Ann. Sci. Ec. Norm. Supér., IV. Sér. 20, 311-323 (1987; Zbl 0651.14010)] to study the categories of perverse sheaves over suitable spaces. The aim of this paper is to exhibit the precise relationship between the rings A(\(\gamma)\) and quasi-hereditary rings introduced by E. Cline, B. Parshall and L. Scott [J. Reine Angew. Math. 391, 85-99 (1988; Zbl 0657.18005)] to study highest weight categories. Recall that an ideal J of a semiprimary ring A is called a heredity ideal if \(J^ 2=J\), \(JN(A)J=0\) and \(J_ A\) is projective. Then A is called quasi-hereditary if there exists a (heredity) chain \(0=J_ 0\subset J_ 1\subset...\subset J_ m=A\) of ideals of A such that, for any \(1\leq t\leq m\), \(J_ t/J_{t-1}\) is a heredity ideal of \(A/J_{t-1}\). Given an A-module \(X_ A\) the induced filtration \(0=XJ_ 0\subset XJ_ 1\subset..\subset XJ_ m=X\) of \(X_ A\) is called good provided \(XJ_ i/XJ_{i-1}\) is a projective \(A/J_{i-1}\)-module, for \(0\leq i\leq m.\)
In the paper the following results are proved: (1) Assume that, in the above notation, C and D are quasi-hereditary and \({}_ CS\) and \(T_ C\) have good filtrations with respect to some heredity chain of C. Then A(\(\gamma)\) is quasi-hereditary. (2) Let A be a non-zero quasi-hereditary finite dimensional algebra over a field k with a heredity chain \((J_ i)\), \(0\leq i\leq m\), and such that \(D=A/J_{m-1}\) is a separable k- algebra. Then there exists a quasi-hereditary k-algebra C with a heredity chain \((I_ i)\), \(0\leq i\leq m-1\), less number of nonisomorphic simple modules than A, bimodules \({}_ CS_ D\), \({}_ DT_ C\) such that the induced filtrations of \({}_ CS\) and \(T_ C\) are good, and a bimodule homomorphism \(\gamma\) : \({}_ CS_ D\otimes_ DT_ C\to_ CC_ C\) such that A is isomorphic to A(\(\gamma)\).

MSC:
16P10 Finite rings and finite-dimensional associative algebras
16Gxx Representation theory of associative rings and algebras
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References:
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