We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the mo...

Let R be a local Cohen-Macaulay ring, I an R-ideal and G the associated graded ring of I. We give an estimate for the depth of G when G fails to be Cohen-Macaulay. We assume that I has small reduction number, sufficiently good residual intersection properties, and satisfies local conditions on the depth of some powers. The main theorem unifies and generalizes several known results. We also give...

Let S be an unramified regular local ring of mixed characteristic p≥3 and Sp the subring obtained by lifting to image Frobenius map on S/pS. R integral closure in a biradical extension degree p2 its quotient field adjoining p-th roots sufficiently general square free elements f,g∈Sp. We show that admits birational maximal Cohen-Macaulay module. It is noted not automatically Cohen-Macaulay.

If V is a faithful module for a nite group G over a eld of characteristic p > 0, then the ring of invariants need not be Cohen-Macaulay if p divides the order of G. In this article the cohomology of G is used to study the question of Cohen-Macaulayness of the invariant ring. Let R = S(V) be the polynomial ring on which G acts. Then the main result can be stated as follow: If H r (G; R) 6 = 0 fo...

We study the singularities of integral models Shimura varieties and moduli stacks shtukas with parahoric level structure. More generally, our results apply to Pappas–Zhu Levin mixed characteristic local models, their equal analogues. For any such model we prove under minimal assumptions that entire is normal reduced special fiber and, if $p > 2$, it also Cohen–Macaulay. This proves a conjecture...

For several important classes of manifolds acted on by the torus, the information about the action can be encoded combinatorially by a regular n-valent graph with vector labels on its edges, which we refer to as the torus graph. By analogy with the GKM-graphs, we introduce the notion of equivariant cohomology of a torus graph, and show that it is isomorphic to the face ring of the associated si...

The absolute order on the hyperoctahedral group Bn is investigated. Using a poset fiber theorem, it is proved that the order ideal of this poset generated by the Coxeter elements is homotopy Cohen–Macaulay. This method results in a new proof of Cohen–Macaulayness of the absolute order on the symmetric group. Moreover, it is shown that every closed interval in the absolute order on Bn is shellab...

We prove that the generic quantized coordinate ringOq(G) is Auslander-regular, Cohen-Macaulay, and catenary for every connected semisimple Lie group G. This answers questions raised by Brown, Lenagan, and the first author. We also prove that under certain hypotheses concerning the existence of normal elements, a noetherian Hopf algebra is Auslander-Gorenstein and Cohen-Macaulay. This provides a...

We describe some recent work concerning Gorenstein liaison of codimension two subschemes of a projective variety. Applications make use of the algebraic theory of maximal Cohen–Macaulay modules, which we review in an Appendix.

Let S be a finite alphabet. An injective word over S is a word over S such that each letter in S appears at most once in the word. We study Boolean cell complexes of injective words over S and their commutation classes. This generalizes work by Farmer and by Björner and Wachs on the complex of all injective words. Specifically, for an abstract simplicial complex ∆, we consider the Boolean cell ...