The rational homology groups of packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre–Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability,...

On considère l’espace de modules $M(c_1,c_2)$ des fibrés stables de rang $2$ sur ${\mathbb{P}}_k^3$, de classes de Chern $c_1,c_2$, $k$ étant un corps algébriquement clos de caractéristique quelconque. Si ($c_1=0,c_2\>0$) ou ($c_1=-1,c_2=2p\ge 6$), on sait ([7], [9]) que $M(c_1,c_2)$ a une composante irréductible dont le point générique $ℱ(c_1,c_2)$ a la cohomologie naturelle. Nous avons calculé ([16]) la résolution minimale de $ℱ(0,c_2)$. Dans cet article, nous voulons déterminer celle de $ℱ(-1,c_2)$ si ${\displaystyle c_2\>\frac{(v+2)(2v^2+3v-1)}{6v+7},}$ où $v$ est le plus petit entier tel que $h^0\left(ℱ\left(v\right)\right)\>0$. Par un procédé standard rappelé dans [16], on se ramène à des...

We define and study restricted projective, injective and flat dimensions over local homomorphisms. Some known results are generalized. As applications, we show that (almost) Cohen-Macaulay rings can be characterized by restricted homological dimensions over local homomorphisms.

This paper deals with the rings which satisfy $DC{C}_d$ condition. This notion has been introduced recently by R. Dastanpour and A. Ghorbani (2017) as a generalization of Artnian rings. It is of interest to investigate more deeply this class of rings. This study focuses on commutative case. In this vein, we present this work in which we examine the transfer of these rings to the trivial, amalgamation and polynomial ring extensions. We also investigate the relationship between this class of rings and the...

We consider Stanley-Reisner rings $k[x_1,...,x_n]/I\left(\mathscr{H}\right)$ where $I\left(\mathscr{H}\right)$ is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize some known results about chordal graphs and study a weak form of shellability.

In this paper, we prove that any pure submodule of a strict Mittag-Leffler module is a locally split submodule. As applications, we discuss some relations between locally split monomorphisms and locally split epimorphisms and give a partial answer to the open problem whether Gorenstein projective modules are Ding projective.

The aim of this paper is to discuss the flat covers of injective modules over a Noetherian ring. Let R be a commutative Noetherian ring and let E be an injective R-module. We prove that the flat cover of E is isomorphic to ${\prod }_{p\in At{t}_R\left(E\right)}{T}_p$. As a consequence, we give an answer to Xu’s question [10, 4.4.9]: for a prime ideal p, when does $T_p$ appear in the flat cover of E(R/m̲)?

In this note we study some algebraic properties of Borel Ideals in the ring of polynomials over an effective field of characteristic zero by using a suitable partial order relation defined on the set of terms of each degree. In particular, in the three variable case, we characterize all the 0-dimensional Borel ideals corresponding to an admissible $h$-vector and their minimal free resolutions.

Propagation of regularity is considered for solutions of rectangular systems of infinite order partial differential equations (resp. convolution equations) in spaces of hyperfunctions (resp. C∞ functions and distributions). Known resulys of this kind are recovered as particular cases, when finite order partial differential equations are considered.

The most fundamental complexes of free modules over a commutative ring are the Koszul complex, which is constructed from a vector (i.e., a 1-tensor), and the Eagon-Northcott and Buchsbaum-Rim complexes, which are constructed from a matrix (i.e., a 2-tensor). The subject of this paper is a multilinear analogue of these complexes, which we construct from an arbitrary higher tensor. Our construction provides detailed new examples of minimal free resolutions, as well as a unifying view on a wide variety...

Using syzygies computed via Gröbner bases techniques, we present algorithms for testing some homological properties for submodules of the free module $A^m$, where A = R[x₁,...,xₙ] and R is a Noetherian commutative ring. We will test if a given submodule M of $A^m$ is flat. We will also check if M is locally free of constant dimension. Moreover, we present an algorithm that computes the rank of a flat submodule M of $A^m$ and also an algorithm that computes the projective dimension of an arbitrary submodule...

Let $A=K[x_1,\cdots ,x_m]$ be a polynomial ring in $m$ variables and let $\mathbf{d}=(d_0\<\cdots \<d_m)$ be a strictly increasing sequence of $m+1$ integers. Boij and Söderberg conjectured the existence of graded $A$-modules $M$ of finite length having pure free resolution of type $\mathbf{d}$ in the sense that for $i=0,\cdots ,m$ the $i$-th syzygy module of $M$ has generators only in degree $d_i$.This paper provides a construction, in characteristic zero, of modules with this property that are also $GL\left(m\right)$-equivariant. Moreover, the construction works over rings of the form $A{\otimes }_{K}B$ where $A$ is a polynomial...

We consider the Hilbert scheme $H(d,g)$ of space curves $C$ with homogeneous ideal $I\left(C\right):=H_{*}^0\left({ℐ}_C\right)$ and Rao module $M:=H_{*}^1\left({ℐ}_C\right)$. By taking suitable generizations (deformations to a more general curve) $C^{\prime }$ of $C$, we simplify the minimal free resolution of $I\left(C\right)$ by e.g making consecutive free summands (ghost-terms) disappear in a free resolution of $I\left(C^{\prime }\right)$. Using this for Buchsbaum curves of diameter one ($M_v\ne 0$ for only one $v$), we establish a one-to-one correspondence between the set $𝒮$ of irreducible components of $H(d,g)$ that contain $\left(C\right)$ and a set of minimal...