In this paper we consider bi-Cohen-Macaulay graphs, and give a complete classification of such graphs in the case they are bipartite or chordal. General biCohen-Macaulay graphs are classified up to separation. The inseparable bi-CohenMacaulay graphs are determined. We establish a bijection between the set of all trees and the set of inseparable bi-Cohen-Macaulay graphs.

In this paper we study topological properties of the poset of injective words and the lattice of classical non-crossing partitions. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This extends the well-known result that those posets are shellable. Both results rely on a new poset fiber theorem, for doubly homo...

Let S = K[x1, . . . ,xn] be a polynomial ring and R = S/I where I ⊂ S is a graded ideal. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan states that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S as well as bounded below by a function of the minimal shifts if R is Cohen–Macaulay. In this paper we study t...

We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of indecomposable maximal Cohen-Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a certain family of filtrations. Within th...

Let (R,m) denote an n-dimensional Gorenstein ring. For an ideal I ⊂ R of height c we are interested in the endomorphism ring B = HomR(H c I (R),H c I (R)). It turns out that B is a commutative ring. In the case of (R, m) a regular local ring containing a field B is a Cohen-Macaulay ring. Its properties are related to the highest Lyubeznik number l = dimk Ext d R(k, H c I (R)). In particular R ≃...

Given a reduced local algebra T over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections R → T which induce isomorphisms Ω R/k ⊗ T → Ω T /k. Such maps, called evolutions, arise naturally in the study of Hecke algebras, as they implicitly do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the existence of non-trivial evolutions ...

In this paper, we study non-Cohen–Macaulay Buchsbaum Stanley– Reisner rings with linear free resolution. In particular, for given integers c, d, q with c ≥ 1, 2 ≤ q ≤ d, we give an upper bound hc,d,q on the dimension of the unique non-vanishing homology H̃q−2(∆; k) of a d-dimensional Buchsbaum ring k[∆] with q-linear resolution and codimension c. Also, we discuss about existence for such Buchsba...

Let S be an unramified regular local ring of mixed characteristic two and R the integral closure in a biquadratic extension its quotient field obtained by adjoining roots sufficiently general square free elements f,g?S. S2 denote subring lifting to image Frobenius map on S/2S. When at least one f,g?S2, we characterize Cohen-Macaulayness show that admits birational small Cohen-Macaulay module. I...

For a skew version of graded (A ∞ ) hypersurface singularity A, we study the stable category maximal Cohen-Macaulay modules over A. As consequence, see that A has countably infinite Cohen–Macaulay representation type and is not noncommutative isolated singularity.