Cohen-Macaulay quotients of normal semioroup rings via irreducible resolutions

For a radical monomial ideal I in a normal semigroup ring κ[Q], there is a unique minimal irreducible resolution 0 → κ[Q]/I → W̄⁰ → W̄¹ ... by modules W̄ⁱ of the form ⊕jk[Fij], where the Fij are (not necessarily distinct) faces of Q. That is, W̄ⁱ is a direct sum of quotients of κ[Q] by prime ideals. This paper characterizes Cohen-Macaulay quotients κ[Q]/I as those whose rainimal irreducible resolutions are linear, meaning that W̄ⁱ is pure of dimension dim(κ[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^d.

Duke Scholars

Author Ezra Miller Mathematics