Algebras and varieties

In this paper we introduce new affine algebraic varieties whose pointscorrespond to associative algebras. We show that the algebras within a variety sharemany important homological properties. In particular, any two algebras in the samevariety have the same dimension. The case of finite dimensional algebras as wellas that of graded algebras arise as subvarieties of the varieties we define. As anapplication we show that for algebras of global dimension two over the complexnumbers, any algebra in the variety continuously deforms to a monomial algebra.

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