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Transactions of the American Mathematical Society

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

 
 

 

Classification of the Tor-algebras of codimension four almost complete intersections


Author: Andrew R. Kustin
Journal: Trans. Amer. Math. Soc. 339 (1993), 61-85
MSC: Primary 13D03; Secondary 13C05, 13C40
DOI: https://doi.org/10.1090/S0002-9947-1993-1132435-7
MathSciNet review: 1132435
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Abstract: Let $(R,m,k)$ be a local ring in which $2$ is a unit. Assume that every element of $k$ has a square root in $k$. We classify the algebras $\operatorname {Tor}_ \bullet ^R(R/J,k)$ as $J$ varies over all grade four almost complete intersection ideals in $R$ . The analogous classification has already been found when $J$ varies over all grade four Gorenstein ideals [21], and when $J$ varies over all ideals of grade at most three [5, 30]. The present paper makes use of the classification, in [21], of the Tor-algebras of codimension four Gorenstein rings, as well as the (usually nonminimal) ${\text {DG}}$-algebra resolution of a codimension four almost complete intersection which is produced in [25 and 26].


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Keywords: Almost complete intersection, Betti numbers, <!– MATH ${\text {DG}}$ –> <IMG WIDTH="37" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${\text {DG}}$">-algebra, Gorenstein ideal, linkage, perfect ideal, Poincar&#233; series, Tor-algebra
Article copyright: © Copyright 1993 American Mathematical Society