Projective resolutions of straightening closed algebras generated by minors
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.
scientific article; zbMATH DE number 745286