transactions of the american mathematical society Volume 307, Number 2, June 1988 DEFORMATIONS OF FINITE DIMENSIONAL ALGEBRAS AND THEIR IDEMPOTENTS M. SCHAPS 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 eiBej by "arrows" from ej to e,-, 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. 1. Introduction. In [5], P. Gabriel leaves as an open challenge the determi- nation of the number of irreducible components in the structure constant scheme Algn, which parametrizes multiplication structures of n-dimensional algebras. The number of quivers in which each vertex is either a source or a sink provides a very weak lower bound, since the radical-squared zero algebras generated by such quiv- ers are known to be rigid. This was converted into a numerical bound by Mazzola in [11]. However, as an examination of the list of irreducible components for di- mension 5 will show, most of the quivers of rigid radical-squared zero algebras have vertices which are neither sources nor sinks. Let the directed graph described in the abstract above be called the basis graph of an algebra, and conversely, given a directed graph, let the unique radical-squared zero algebra with that configuration of idempotents and radical elements be called the basis graph algebra generated by that graph. The long range purpose of the present work is to develop machinery for determining and classifying the irreducible components of Alg„ by studying the effect of deformation on the basis graph, but one of its immediate applications is to prove that every irreducible component contains no more than one loopless basis graph algebra. For example, in dimension 5 there are nine loopless basis graph algebras, contained in nine of the ten different irreducible components. Thus in low dimensions the number of loopless basis graphs gives a fairly reasonable lower bound. A different strand in the analysis of Alg„ began with the work of M. Gerstenhaber on formal deformations of algebras [6] and culminated in the work of Flanigan [3, 4]. In the second section we broaden Flanigan's "straightening-out" theorems, [3], so that they will be valid for deformations over an arbitrary base space. In the third section we stratify Alg„ by associating to each algebra a weighted basis graph, demonstrate that this is indeed an algebraic stratification, and show that the only Received by the editors May 22, 1986 and, in revised form, May 7, 1987. Presented at Rep- resentations of Finite Dimensional Algebras, Mathematisches Forschungsinstitut, Oberwolfach, March 26, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A46, 16A58. ©1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page 843 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use