Cohen–macaulay Quotients of Normal Semigroup Rings via Irreducible Resolutions

For a radical monomial ideal I in a normal semigroup ring k[Q], there is a unique minimal irreducible resolution 0 → k[Q]/I → W 0 → W 1 → · · · by modules W i of the form ⊕ j k[Fij ], where the Fij are (not necessarily distinct) faces of Q. That is, W 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 linear, meaning that W i is pure of dimension dim(k[Q]/I) − i for i ≥ 0. The proof exploits a graded ring-theoretic analogue of the Zeeman spectral sequence [Zee63], thereby also providing a combinatorial topological version involving no commutative algebra. The characterization via linear irreducible resolutions reduces to the Eagon–Reiner theorem [ER98] by Alexander duality when Q = N. 2000 AMS Classification: 13C14, 14M05, 13D02, 55Txx (primary) 14M25, 13F55 (secondary)