Deformations of finite-dimensional algebras and their idempotents

Schaps, M.

doi:10.1090/S0002-9947-1988-0940231-4

Deformations of finite-dimensional algebras and their idempotents
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Trans. Amer. Math. Soc. 307 (1988), 843-856 Request permission

Abstract:

Let $B$ be a finite dimensional algebra over an algebraically closed field $K$. If we represent primitive idempotents by points and basis vectors in ${e_i}B{e_j}$ by "arrows" from ${e_j}$ to ${e_i}$, then any specialization of the algebra acts on this directed graph by coalescing points. This implies that the number of irreducible components in the scheme parametrizing $n$-dimensional algebras is no less than the number of loopless directed graphs with a total of $n$ vertices and arrows. We also show that the condition of having a distributive ideal lattice is open.

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