# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Bass, H. , Algebraic K-Theory , Benjamin, New York (1968). · Zbl 0174.30302 [2] Beilinson, A. , Bernstein, J. and Deligne, P. , Faisceaux pervers , Asterisque 100 (1983). · Zbl 0536.14011 [3] Bernstein, J. , Gelfand, I. and Gelfand, S. , Category of g-modules , Funct. Anal. Appl. 10 (1976), 87-92. · Zbl 0353.18013 · doi:10.1007/BF01077933 [4] Cline, E. , Parshall, B. and Scott, L. , Finite dimensional algebras and highest weight categories (To appear), J. Reine Angew. Math. · Zbl 0657.18005 · crelle:GDZPPN002205971 · eudml:153071 [5] Dlab, V. and Ringel, C.M. , Quasi-hereditary algebras (To appear), Ill. J. Math. · Zbl 0666.16014 [6] Macpherson, R. and Vilonen, K. , Elementary construction of perverse sheaves , Inv. Math. 84 (1986), 403-435. · Zbl 0597.18005 · doi:10.1007/BF01388812 · eudml:143344 [7] Mebkhout, Z. , Une équivalence des catégories. Une autre équivalence des catégories , Comp. Math. 51 (1984), 51-88. · Zbl 0566.32021 · numdam:CM_1984__51_1_51_0 · eudml:89634 [8] Mirollo, R. and Vilonen, K. , Bernstein-Gelfand-Gelfand reciprocity on perverse sheaves . Ann. Scient. Éc. Norm. Sup. 4e série 20 (1987), 311-324. · Zbl 0651.14010 · doi:10.24033/asens.1536 · numdam:ASENS_1987_4_20_3_311_0 · eudml:82205 [9] Parshall, B.J. , Finite dimensional algebras and algebraic groups (To appear). · Zbl 0682.20029 [10] Parshall, B.J. and Scott, L.L. , Derived categories, quasi-hereditary algebras, and algebraic groups (To appear). · Zbl 0711.18002 [11] Scott, L.L. , Simulating algebraic geometry with algebra I: Derived categories and Morita theory , Proc. Symp. Pure Math., Amer. Math. Soc., Providence 47 (1987), part 1, 271-282. · Zbl 0659.20038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.