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On the free character of the first Koszul homology module.

Antonio García Rodicio (1991)

Extracta Mathematicae

Let (A,M,K) denote a local noetherian ring A with maximal ideal M and residue field K. Let I be an ideal of A and E the Koszul complex generated over A by a system of generators of I.The condition: H1(E) is a free A/I-module, appears in several important results of Commutative Algebra. For instance:- (Gulliksen [3, Proposition 1.4.9]): The ideal I is generated by a regular sequence if and only if I has finite projective dimension and H1(E) is a free A/I-module.- (André [2]): Assume that A is a complete...

On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence

Kosar Abolfath Beigi, Kamran Divaani-Aazar, Massoud Tousi (2022)

Czechoslovak Mathematical Journal

Let R be a local ring and C a semidualizing module of R . We investigate the behavior of certain classes of generalized Cohen-Macaulay R -modules under the Foxby equivalence between the Auslander and Bass classes with respect to C . In particular, we show that generalized Cohen-Macaulay R -modules are invariant under this equivalence and if M is a finitely generated R -module in the Auslander class with respect to C such that C ⊗ R M is surjective Buchsbaum, then M is also surjective Buchsbaum.

On the Jacobian ideal of the binary discriminant.

Carlos D'Andrea, Jaydeep Chipalkatti (2007)

Collectanea Mathematica

Let Δ denote the discriminant of the generic binary d-ic. We show that for d ≥ 3, the Jacobian ideal of Δ is perfect of height 2. Moreover we describe its SL2-equivariant minimal resolution and the associated differential equations satisfied by Δ. A similar result is proved for the resultant of two forms of orders d, e whenever d ≥ e-1. If Φn denotes the locus of binary forms with total root multiplicity ≥ d-n, then we show that the ideal of Φn is also perfect, and we construct a covariant which...

On the minimaxness and coatomicness of local cohomology modules

Marzieh Hatamkhani, Hajar Roshan-Shekalgourabi (2022)

Czechoslovak Mathematical Journal

Let R be a commutative Noetherian ring, I an ideal of R and M an R -module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and 𝒞 -minimaxness of local cohomology modules. We show that if M is a minimax R -module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if n is a nonnegative integer such that ( H I i ( M ) ) 𝔪 is a minimax R 𝔪 -module for all 𝔪 ∈ Max ( R ) and for all i < n , then the set Ass R ( H I n ( M ) ) is finite. Also, if H I i ( M ) is minimax for...

On the regularity and defect sequence of monomial and binomial ideals

Keivan Borna, Abolfazl Mohajer (2019)

Czechoslovak Mathematical Journal

When S is a polynomial ring or more generally a standard graded algebra over a field K , with homogeneous maximal ideal 𝔪 , it is known that for an ideal I of S , the regularity of powers of I becomes eventually a linear function, i.e., reg ( I m ) = d m + e for m ≫ 0 and some integers d , e . This motivates writing reg ( I m ) = d m + e m for every m ≥ 0 . The sequence e m , called the defect sequence of the ideal I , is the subject of much research and its nature is still widely unexplored. We know that e m is eventually constant. In this article, after...

Currently displaying 341 – 360 of 559