Green, Edward L.; Huang, Rosa Q. Projective resolutions of straightening closed algebras generated by minors. (English) Zbl 0824.13012 Adv. Math. 110, No. 2, 314-333 (1995). The authors define a class of algebras called straightening closed algebras, generalizing the algebras generated by minors in a generic matrix. Gröbner bases for the defining ideals of these algebras (viewed as factor rings of free associative \(k\)-algebras, \(k\) a field) are calculated. These are used to get projective resolutions of \(k\), using a construction of Anick. The authors note that, if the Gröbner basis consists of elements of degree two, then the algebra is a Koszul algebra, i.e. \((\text{Tor}_i(k,k))_j=0\) if \(i\neq j\). In fact the setting is more general, factor rings of so called path algebras instead of free associative algebras are allowed. Reviewer: R.Fröberg (Stockholm) Cited in 1 ReviewCited in 19 Documents MSC: 13F50 Rings with straightening laws, Hodge algebras 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 13C40 Linkage, complete intersections and determinantal ideals Keywords:straightening closed algebras; projective resolutions; Gröbner basis; Koszul algebra PDFBibTeX XMLCite \textit{E. L. Green} and \textit{R. Q. Huang}, Adv. Math. 110, No. 2, 314--333 (1995; Zbl 0824.13012) Full Text: DOI