Integral and graded quasi-hereditary algebras. I
The paper extends the notion of a quasi-hereditary algebra from finite dimensional algebras to finitely generated algebras over a commutative Noetherian ring. Moreover, the authors begin the study of graded quasi- hereditary algebras. Let k be a commutative Noetherian ring and let A be a k-algebra, which is a finitely generated projective k-module. An ideal J in A is called a heredity ideal if \(J^ 2=J\), J is a projective right R-module and the k-algebra \(End_ A(J_ A)\) is semisimple. The algebra A is called quasi-hereditary if there exists a sequence \(0=J_ 0\subseteq J_ 1\subseteq...\subseteq J_ t=A\) of ideals in A such that \(J_ i/J_{i-1}\) is a heredity ideal in \(A/J_{i-1}\). Several results for quasi-hereditary Noetherian algebras are established in analogy with the finite dimensional case. In particular it is shown that gl.dim \(A\leq 2t-2+gl.\dim k\). Moreover, if \(J_ i=Ae_ iA\) where \(e_ i\) is an idempotent in A such that \(gl.\dim (e_ iAe_ i)\) is finite then there is a full recollement diagram \[ D^ b(mod A/J_ i)\quad D^ b(mod A)\quad D^ b(mod e_ iAe_ i). \] It is shown in the paper that the q-Schur algebras \(S_ q(n,r,k)\) are quasi-hereditary if \(q^{1/2}\in k\). A positively Z-graded k-algebra A is called a graded quasi-hereditary algebra if A is a quasi-hereditary k-algebra in the sense above. In this case each of the idempotent ideals \(J_ i\) in the defining sequence is in fact homogeneous. Elementary properties of graded quasi-hereditary algebras are established and several examples are presented. In connection with recent work of Beilingson and Ginsberg the concept of a formal quasi hereditary algebra is briefly discussed.
scientific article; zbMATH DE number 4146190
scientific article