We give several characterizations for the linearity property for a maximal Cohen-Macaulay module over a local or graded ring, as well as proofs of existence in some new cases. In particular, we prove that the existence of such modules is preserved when taking Segre products, as well as when passing to Veronese subrings in low dimensions. The former result even yields new results on the existenc...

In the winter of 1999 I gave a series of lectures at Queen’s university about some recent results concerning the Cohen-Macaulay property of invariants of Hopf algebras. Tony Geramita asked me to write up my notes for the Queen’s Papers, and I happily took up his suggestion. Although this article focuses on the proof of one main theorem (Theorem 2.11 on page 12), it has some of the character of ...

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...

We prove that certain class of Stanley–Reisner rings having sufficiently large multiplicities are Cohen–Macaulay using Alexander duality.

We settle the negativity conjecture of Vasconcelos for the Chern number of an ideal in certain unmixed quotients of regular local rings by explicit calculation of the Hilbert polynomials of all ideals generated by systems of parameters. Introduction Let I be an m-primary ideal in a local ring (R,m) of dimension d. Let H(I, n) = λ(R/I) denote the Hilbert function of I where λ(M) denotes the leng...

We study upper bounds of the colength trace canonical module in one-dimensional Cohen–Macaulay rings. answer two questions posed by Herzog–Hibi–Stamate and Kobayashi.

The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjecture is also established for the face rings of all doubly Cohen-Macaulay complexes whose 1-skeleton’s connectivi...