New algebra and module varieties
3rd November 2021, 2:30 pm – 3:30 pm
Fry Building, Room G.11 and Zoom
In this talk we introduce two new kinds of affine algebraic varieties: namely varieties of algebras and varieties of modules where each point corresponds to a quotient of a path algebra of a quiver or to a module over such an algebra. Both constructions are based on non-commutative Gröbner basis theory and the main idea of these new varieties is the following: In the case of algebra varieties, each variety contains a monomial algebra, that is a quotient of a path algebra of a quiver where the ideal of relations is generated by paths in the quiver - in terms of polynomial algebras this would be an algebra of the form K[x]/x^i as opposed to an algebra K[x]/(f(x) for some general polynomial f(x). Monomial algebras are much better understood than a general quotient of a path algebra of a quiver and we show that the homological properties of the monomial algebra in a variety govern the homological properties of all the algebras in the variety. In the case of modules a similar phenomenon occurs and leads to the introduction of a new type of module, which we name monomial module.