Given any diagonal cyclic subgroup $Λ\subset GL(n+1,k)$ of order $d$, let $I_d\subset k[x_0,\ldots, x_n]$ be the ideal generated by all monomials $\{m_{1},\ldots, m_{r}\}$ of degree $d$ which are invariants of $Λ$. $I_d$ is a monomial Togliatti system, provided $r \leq \binom{d+n-1}{n-1}$, and in this case the projective toric variety $X_d$ parameterized by $(m_{1},\ldots, m_{r})$ is called a $GT$-variety with group $Λ$. We prove that all these $GT$-varieties are arithmetically Cohen-Macaulay and we give a combinatorial expression of their Hilbert functions. In the case $n=2$, we compute explicitly the Hilbert function, polynomial and series of $X_d$. We determine a minimal free resolution of its homogeneous ideal and we show that it is a binomial prime ideal generated by quadrics and cubics. We also provide the exact number of both types of generators. Finally, we pose the problem of determining whether a surface parameterized by a Togliatti system is aCM. We construct examples that are aCM and examples that are not.
In this paper we give necessary and sufficient conditions for the Cohen-Macaulayness of the tangent cone of a monomial curve in the 4-dimensional affine space. We study particularly the case where $C$ is a Gorenstein non-complete intersection monomial curve.
In this note, we study Togliatti systems generated by invariants of the dihedral group $D_{2d}$ acting on $k[x_{0},x_{1},x_{2}]$. This leads to the first family of non monomial Togliatti systems, which we call $GT-$systems with group $D_{2d}$. We study their associated varieties $S_{D_{2d}}$, called $GT-$surfaces with group $D_{2d}$. We prove that they are arithmetically Cohen-Macaulay surfaces whose homogeneous ideal, $I(S_{D_{2d}})$, is minimally generated by quadrics and we find a minimal free resolution of $I(S_{D_{2d}})$.
In this paper we study the finitely generated bigraded modules over a standard bigraded polynomial ring which are relative Cohen-Macaulay or relative unmixed with respect to one of the irrelevant bigraded ideals. A generalization of Reisner's criterion for Cohen-Macaulay simplicial complexes is considered.
We give the Castelnuovo-Mumford regularity of arrangements of (n-2)-planes in P^n whose incidence graph is a sufficiently large complete bipartite graph, and determine when such arrangements are arithmetically Cohen-Macaulay.
Let $C({\bf n})$ be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve $C({\bf n}+w{\bf v})$, where $w>0$ is an integer and ${\bf v} \in \mathbb{N}^{4}$. Also we investigate the Cohen-Macaulayness of the tangent cone of $C({\bf n}+w{\bf v})$.
Given two determinantal rings over a field k. We consider the Rees algebra of the diagonal ideal, the kernel of the multiplication map. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety. When the Rees algebra and the Symmetric algebra coincide, we show that the Rees algebra is Cohen-Macaulay.
We prove that, modulo any power of a prime $p$, the absolute integral closure of an excellent noetherian domain is Cohen-Macaulay. A graded analog is also established, yielding variants of Kodaira vanishing "up to finite covers" in mixed characteristic. Our main tools are (log) prismatic cohomology (which yields a Frobenius action in mixed characteristic) and the $p$-adic Riemann-Hilbert functor for constructible étale $\mathbf{F}_p$-sheaves on varieties over a $p$-adic field (which almost controls perfectified prismatic cohomology).
We present criteria for the Cohen-Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen-Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen-Macaulayness of the second symbolic power or of all symbolic powers of a Stanley-Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen-Macaulay. In particular, all symbolic powers a…
Let S be a standard N^r-graded algebra over a local ring A, and let M be a finitely generated Z^r-graded S-module. We characterize the Cohen-Macaulayness of M in terms of the vanishing of certain sheaf cohomology modules. As a consequence, we apply our result to study the Cohen-Macaulayness of multi-Rees modules (also called Rees modification). Our work extends previous studies on the Cohen-Macaulayness of multi-Rees algebras.
We show the Cohen-Macaulayness and describe the canonical module of residual intersections $J=\mathfrak{a}\colon_R I$ in a Cohen-Macaulay local ring $R$, under sliding depth type hypotheses. For this purpose, we construct and study, using a recent article of Hassanzadeh and the second named author, a family of complexes that contains important informations on a residual intersection and its canonical module. We also determine several invariants of residual intersections as the graded canonical module, the Hilbert series, the Castelnuovo-Mumford regularity and the type. Finally, whenever $I$ is…
We study the singularities of integral models of Shimura varieties and moduli stacks of shtukas with parahoric level structure. More generally our results apply to the Pappas-Zhu and Levin mixed characteristic parahoric local models, and to their equal characteristic analogues. For any such local model we prove under minimal assumptions that the entire local model is normal with reduced special fiber and, if $p>2$, it is also Cohen-Macaulay. This proves a conjecture of Pappas and Zhu, and shows that the integral models of Shimura varieties constructed by Kisin and Pappas are Cohen-Macaulay as …
Let Z_K be the moment angle complex associated to a simplicial complex K, with the canonical torus T-action. In this paper, we prove that, for any possibly disconnected subgroup G of T, G-equivariant cohomology of Z_K over the integer Z is isomophic to the Tor-module Tor_{H(BR;Z)}(Z[K],Z) as graded modules, where Z_[K] is the Stanley-Reisner ring of K. Based on this, we prove that the surjectivity of the natural map H_T(Z_K;Z) to H_G(Z_K;Z) is equivalent to the vanishing of Tor^{H(BR;Z)}_1(Z[K],Z). Since the integral cohomology of various toric orbifolds can be identified with H_G(Z_K;Z), we s…
In this paper we solve a problem, originally raised by Grothendieck, on the transfer of Cohen-Macaulayness to tensor products of algebras over a field. As a prelude to this, we investigate the grade for some specific types of ideals that play a primordial role within the ideal structure of such constructions.
We introduce the use of liaison addition to the study of hyperplane arrangements. For an arrangement, $\mathcal A$, of hyperplanes in $\mathbb P^n$, $\mathcal A$ is free if $R/J$ is Cohen-Macaulay, where $J$ is the Jacobian ideal of $\mathcal A$. Terao's conjecture says that freeness of $\mathcal A$ is determined by the combinatorics of the intersection lattice of $\mathcal A$. We study the Cohen-Macaulayness of three other ideals, all unmixed, that are closely related to $\mathcal A$. Let $\overline J = \mathfrak q_1 \cap \dots \cap \mathfrak q_s$ be the intersection of height two primary com…
In this article we study the structure of residual intersections via constructing a finite complex which is acyclic under some sliding depth conditions on the cycles of the Koszul complex. This complex provides information on an ideal which coincides with the residual intersection in the case of geometric residual intersection; and is closely related to it in general. A new success obtained through studying such a complex is to prove the Cohen-Macaulayness of residual intersections of a wide class of ideals. For example we show that, in a Cohen-Macaulay local ring, any geometric residual inter…
Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions derived from simplicial complexes. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen-Macaulay property implies shellability for both the Scarf complex and the Stanley-Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen-Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case the Cohen-Macaulay type is greater than or equal to the numbe…
This paper addresses problems related to the existence of arithmetic Macaulayfications of projective schemes. Let Y be the blow-up of a projective scheme X = Proj R along the ideal sheaf of I \subset R. It is known that there are embeddings Y \cong Proj k[(I^e)_c] for c \ge d(I)e + 1, where d(I) denotes the maximal generating degree of I, and that there exists a Cohen-Macaulay ring of the form k[(I^e)_c] if and only if H^0(Y,O_Y) = k, H^i(Y,O_Y) = 0 for i = 1,...,dim Y-1, Y is equidimensional and Cohen-Macaulay. Cutkosky and Herzog asked when there is a linear bound on c and e ensuring that k[…
