Minimal Resolutions of Algebras

✍ Scribed by M.C.R. Butler; A.D. King

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 230 KB
• Volume: 212
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7599

✦ Synopsis

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