On Noncommutative Varieties

Abstract

We have shown that the noncommutative equivalent of the set of closed points in the projective variety associated with an algebra A is in bijection with the set F of shift-equivalence classes of 1-critical graded modules such that the Gel'fand-Kirillov dimension d(A/Ann(M)) = 1, and with the set C of twist-equivalence classes of non-trivial finite dimensional simple A-modules. This means that we can use either of these sets to describe a candidate for noncommutative projective varieties. We then outlined how multilinearisation of an algebra can be used to parametrise its point modules, which are objects in C.