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I nt r oduc t i onReflection functors were introduced into the representation theory of quivers by Bernstein, Gelfand and Ponomarev in their work on the 4-subspace problem and on

A method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself. The method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal Ž of relations that is either homogeneous or admissible with some additional . finiteness restrictions in the latter case . In particular, it applies to any finitedimensional algebra over an algebraically closed field. The method is illustrated by a number of examples, viz. truncated algebras, monomial algebras, and Koszul algebras, with the aim of unifying existing treatments of these in the literature.ᮊ 1999 Academic Press ⌳ ym y . Any two such resolutions are homotopic, but, when ⌳ admits a ⌳ minimal resolution, then this resolution is unique up to isomorphism and should give the most natural and efficient method for making the computations already mentioned. Of course, the minimal resolution should also 323

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Let = ZG be the integer group ring of a group, G, of prime order. A main result of this note is that every -module with a free underlying abelian group decomposes into a direct sum of copies of the well-known indecomposable -lattices of finite rank. The first part of the proof reduces the problem to one about countably generated modules, and works in a wider context of suitably restricted modules over orders of finite lattice type of a quite general type. However, for countably generated modules, use is seemingly needed of the classical theory of -lattices.