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On the structure of sequentially Cohen-Macaulay bigraded modules

Leila Parsaei Majd, Ahad Rahimi (2015)

Czechoslovak Mathematical Journal

Let K be a field and S = K [ x 1 , ... , x m , y 1 , ... , y n ] be the standard bigraded polynomial ring over K . In this paper, we explicitly describe the structure of finitely generated bigraded “sequentially Cohen-Macaulay” S -modules with respect to Q = ( y 1 , ... , y n ) . Next, we give a characterization of sequentially Cohen-Macaulay modules with respect to Q in terms of local cohomology modules. Cohen-Macaulay modules that are sequentially Cohen-Macaulay with respect to Q are considered.

On the symmetric algebra of certain first syzygy modules

Gaetana Restuccia, Zhongming Tang, Rosanna Utano (2022)

Czechoslovak Mathematical Journal

Let ( R , 𝔪 ) be a standard graded K -algebra over a field K . Then R can be written as S / I , where I ( x 1 , ... , x n ) 2 is a graded ideal of a polynomial ring S = K [ x 1 , ... , x n ] . Assume that n 3 and I is a strongly stable monomial ideal. We study the symmetric algebra Sym R ( Syz 1 ( 𝔪 ) ) of the first syzygy module Syz 1 ( 𝔪 ) of 𝔪 . When the minimal generators of I are all of degree 2, the dimension of Sym R ( Syz 1 ( 𝔪 ) ) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.

Orderings of monomial ideals

Matthias Aschenbrenner, Wai Yan Pong (2004)

Fundamenta Mathematicae

We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the Hilbert-Samuel polynomial, and we compute bounds on the maximal order type.

Poincaré duality and commutative differential graded algebras

Pascal Lambrechts, Don Stanley (2008)

Annales scientifiques de l'École Normale Supérieure

We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincaré duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincaré duality in the same dimension. This has applications in rational homotopy, giving Poincaré duality at the cochain level, which is of interest in particular in the study of configuration spaces and in string topology.

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