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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Minimal projective resolutions

Authors: E. L. Green, Ø. Solberg and D. Zacharia
Journal: Trans. Amer. Math. Soc. 353 (2001), 2915-2939
MSC (2000): Primary 16E05, 18G10; Secondary 16P10
Published electronically: March 8, 2001
MathSciNet review: 1828479
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In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the $\operatorname{Ext}$-algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the ``no loop'' conjecture.

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Additional Information

E. L. Green
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123

Ø. Solberg
Affiliation: Institutt for matematiske fag, NTNU, Lade, N–7491 Trondheim, Norway

D. Zacharia
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244

Keywords: Projective resolutions, finite dimensional and graded algebras
Received by editor(s): September 21, 1998
Received by editor(s) in revised form: January 3, 2000
Published electronically: March 8, 2001
Additional Notes: Partially supported by a grant from the NSA
Partially supported by NRF, the Norwegian Research Council
Dedicated: Dedicated to Helmut Lenzing for his 60th birthday
Article copyright: © Copyright 2001 American Mathematical Society