Abstract
In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same variety have the same dimension. The cases of finite dimensional algebras as well as that of graded algebras arise as subvarieties of the varieties we define. As an application we show that for algebras of global dimension two over the complex numbers, any algebra in the variety continuously deforms to a monomial algebra.
Highlights
The interplay between commutative algebra and algebraic geometry plays a fundamental role in these areas, see for example [8]
The use of algebraic geometry in other areas of mathematics has led to important results
The Cartan determinant conjecture states that the determinant of the Cartan matrix of a finite dimensional algebra of finite global dimension is equal to 1
Summary
The interplay between commutative algebra and algebraic geometry plays a fundamental role in these areas, see for example [8]. Commutative finitely generated algebras are quotients of polynomial rings by ideals of polynomials, where the zero loci of these polynomials give rise to the associated affine geometry. We use non-commutative Grobner basis theory to construct new affine algebraic varieties whose points are in one-to-one correspondence with quotients of path algebras. The Cartan determinant conjecture states that the determinant of the Cartan matrix of a finite dimensional algebra of finite global dimension is equal to 1 This conjecture has been shown to hold for monomial algebras [23]. A consequence of the above theorem is that if the Cartan determinant of a finite dimensional algebra of finite global dimension is equal to 1, the Cartan determinant conjecture holds for all algebras in that variety. To prove the Cartan determinant conjecture one need only study algebras of finite global dimension whose associated monomial algebra is of infinite global dimension for any order
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