We study reflexive modules over one dimensional Cohen-Macaulay rings. Our key technique exploits the concept of I I -Ulrich modules.

In a previous article (J. Algebra 367 (2012), 142-165) we established axiomatic parametrised Cohen-Macaulay approximation which in particular was applied to pairs consisting of finite type flat family rings and modules. this sequel study the induced maps deformation functors deduce properties like smoothness injectivity under general, mainly cohomological conditions on module.

Let H ⊆ Z be a positive semigroup generated by A ⊆ H, and let K[H] be the associated semigroup ring over a field K. We investigate heredity of the Cohen–Macaulay property from K[H] to both its A -Newton graded ring and to its face rings. We show by example that neither one inherits in general the Cohen–Macaulay property. On the positive side we show that for every H there exist generating sets ...

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 irreducib...

Let G be a simple undirected graph on n vertices, and let I(G) ⊆ R = k[x 1 ,. .. , x n ] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and im...

In this paper, we characterize the shellable complete $t$-partite graphs. We also show for these types of graphs the concepts vertex decomposable, shellable and sequentially Cohen-Macaulay are equivalent. Furthermore, we give a combinatorial condition for the Cohen-Macaulay complete $t$-partite graphs.