Mathematical Research Letters

Volume 9, Issue 1, January 2002 pp. 117-128.

Cohen--Macaulay quotients of normal semigroup rings via irreducible resolutions

Authors: Ezra Miller
Author institution: Massachusetts Institute of Technology

Summary: For a radical monomial ideal $I$\/ in a normal semigroup ring~$k[Q]$, there is a unique minimal {\it irreducible resolution} $0 \to k[Q]/I \to \WW^0 \to \WW^1 \to \cdots$\ by modules $\WW^i$ of the form $\bigoabstract {\hskip -5 pt}_j k[F_{ij}]$, where the $F_{ij}$ are (not necessarily distinct) faces of~$Q$. That is, $\WW^i$ is a direct sum of quotients of $k[Q]$ by prime ideals. This paper characterizes Cohen--Macaulay quotients $k[Q]/I$ as those whose minimal irreducible resolutions are {\it linear}, meaning that $\WW^i$ is pure of dimension $\dim(k[Q]/I) - i$ for $i \geq 0$. The proof exploits a graded ring-theoretic analogue of the Zeeman spectral sequence~\cite{zeeIII}, thereby also providing a combinatorial topological version involving no commutative algebra. The characterization via linear irreducible resolutions reduces to the Eagon--Reiner theorem \cite{ER} by Alexander duality \mbox{when $Q = \bignabstract^d$.}

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