Let I be a monomial ideal of the polynomial ring S = K[x1, . . . , x4] over a field K. Then S/I is sequentially Cohen-Macaulay if and only if S/I is pretty clean. In particular, if S/I is sequentially Cohen-Macaulay then I is a Stanley ideal.

in this paper, we introduce a subclass of chordal graphs which contains $d$-trees and show that their complement are vertex decomposable and so is shellable and sequentially cohen-macaulay.

Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially CohenMacaulay graph. We focus on modifications given by adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex in G. We give various sufficient conditions and...

Let k be an algebraically closed field of characteristic zero, S = k[X0, X1, X2, X3, X4] and P = Proj(S). By a curve we always mean a closed one-dimensional subscheme of P which is locally Cohen-Macaulay and equidimensional. The main purpose of this paper is to show that arithmetically Cohen-Macaulay curves C ⊂ P lying on a “general” arithmetically Cohen-Macaulay surface X ⊂ P with degree matri...

1.1 Statement (Matijevic–Roberts type theorem (MRTT)). Let C be a class of noetherian local rings. Let R be a noetherian Z-graded ring, and P its prime ideal. Let P ∗ be the prime ideal generated by the all homogeneous elements of P . If RP ∗ ∈ C, then RP ∈ C. Clearly, the truth of the statement depends on the choice of C. Nagata conjectured the Matijevic–Roberts type theorem for the case that ...

We prove that sequentially Cohen-Macaulay rings in positive characteristic, as well as sequentially Cohen-Macaulay Stanley-Reisner rings in any characteristic, have trivial Lyubeznik table. Some other configurations of Lyubeznik tables are also provided depending on the deficiency modules of the ring.

We use a Mayer–Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality properties of spaces and groups. In the process, we consider cohomology of local systems with a general, Cohen– Macaulay-type condition. As a result, we recover kn...

We investigate the transfer of the Cohen-Macaulay property from a commutative ring to a subring of invariants under the action of a finite group. Our point of view is ring theoretic and not a priori tailored to a particular type of group action. As an illustration, we briefly discuss the special case of multiplicative actions, that is, actions on group algebras k[Z n ] via an action on Z n .

For the Spencer δ-cohomologies of a symbolic system we construct a spectral sequence associated with a subspace. We calculate the sequence for the systems of Cohen-Macaulay type and obtain a reduction theorem, which facilitates computation of δ-cohomologies by reducing dimension of the system. Using this algebraic result we prove an efficient compatibility criterion for a system of two scalar n...