We give a two step method to study certain questions regarding associated graded module of a Cohen-Macaulay (CM) module $M$ w.r.t an $mathfrak{m}$-primary ideal $mathfrak{a}$ in a complete Noetherian local ring $(A,mathfrak{m})$. The first step, we call it complete intersection approximation, enables us to reduce to the case when both $A$, $ G_mathfrak{a}(A) = bigoplus_{n geq 0} mathfrak{a}^n/mathfrak{a}^{n+1} $ are complete intersections and $M$ is a maximal CM $A$-module. The second step consists of analyzing the classical filtration ${Hom_A(M,mathfrak{a}^n) }_{mathbb{Z}}$ of the dual $Hom_A(M,A)$. We give many applications of this point of view. For instance let $(A,mathfrak{m})$ be equicharacteristic and CM. Let $a(G_mathfrak{a}(A))$ be the $a$-invariant of $G_mathfrak{a}(A)$. We prove: 1. $a(G_mathfrak{a}(A)) = -dim A$ iff $mathfrak{a}$ is generated by a regular sequence. 2. If $mathfrak{a}$ is integrally closed and $a(G_mathfrak{a}(A)) = -dim A + 1$ then $mathfrak{a}$ has minimal multiplicity. We extend to modules a result of Ooishi relating symmetry of $h$-vectors. As another application we prove a conjecture of Itoh, if $A$ is a CM local ring and $mathfrak{a}$ is a normal ideal with $e_3^mathfrak{a}(A) = 0$ then $G_mathfrak{a}(A)$ is CM.
Let $R$ be a commutative $k-$algebra over a field $k$. Assume $R$ is a noetherian, infinite, integral domain. The group of $k-$automorphisms of $R$,i.e.$Aut_k(R)$ acts in a natural way on $(R-k)$.In the first part of this article, we study the structure of $R$ when the orbit space $(R-k)/Aut_k(R)$ is finite.We note that most of the results, not particularly relevent to fields, in [1,S 2] hold in this case as well. Moreover, we prove that $R$ is a field. In the second part, we study a special case of the Conjecture 2.1 in [1] : If $K/k$ is a non trivial field extension where $k$ is algebraically closed and $mid (K-k)/Aut_k(K) mid = 1$ then $K$ is algebraically closed. In the end, we give an elementary proof of [1,Theorem 1.1] in case $K$ is finitely generated over its prime subfield.
Our goal is to settle a fading problem, the Jacobian Conjecture $(JC_n)$~: If $f_1, cdots, f_n$ are elements in a polynomial ring $k[X_1, cdots, X_n]$ over a field $k$ of characteristic zero such that $ det(partial f_i/ partial X_j) $ is a nonzero constant, then $k[f_1, cdots, f_n] = k[X_1, cdots, X_n]$. Practically, what we deal with is the generalized one, oindent The Generalized Jacobian Conjecture$(GJC)$ :{it Let $S hookrightarrow T$ be an unramified homomorphism of Noetherian domains. Assume that $S$ is a simply connected UFD ({sl i.e.,} ${rm Spec}(S)$ is simply connected and $S$ is a unique factorization domain) and that $T^times cap S = S^times$. Then $T = S$.} In addition, for consistency of the discussion, we raise some serious (or idiot) questions and some comments about the examples appeared in the papers published by the certain excellent mathematicians (though we are not willing to deal with them). However, the existence of such examples would be against our Main Result above, so that we have to dispute in Appendix B their arguments about the existence of their respective (so called) counter-examples. Our conclusion is that they are not perfect counter-examples which is shown explicitly.
Denote by p_k the k-th power sum symmetric polynomial n variables. The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular sequence. We consider then the following problem: describe the subsets n powersums forming a regular sequence. A necessary condition is that n! divides the product of the degrees of the elements. To find an easily verifiable sufficient condition turns out to be surprisingly difficult already in 3 variables. Given positive integers a<b<c with GCD(a,b,c)=1, we conjecture that p_a, p_b, p_c is a regular sequence for n=3 if and only if 6 divides abc. We provide evidence for the conjecture by proving it in several special instances.
The profile of a relational structure $R$ is the function $varphi_R$ which counts for every integer $n$ the number, possibly infinite, $varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $A(R)$, introduced by P.~J.~Cameron. In a previous paper, we studied the relationship between the properties of a relational structure and those of their algebra, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. In this paper, we investigate how far the well know algebraic properties of those rings extend to age algebras. The main result is a combinatorial characterization of when the age algebra is finitely generated. In the special case of tournaments, we show that the age algebra is finitely generated if and only if the profile is bounded. We explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra.
We define specific multiplicities on the braid arrangement by using edge-bicolored graphs. To consider their freeness, we introduce the notion of bicolor-eliminable graphs as a generalization of Stanleys classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of the deformation of the braid arrangement in terms of directed graphs.
Let $M$ be an $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = cap lbrace I colon I text{is an ideal of} R text{and} x in IM rbrace $. $M$ is said to be a content $R$-module if $x in c(x)M $, for all $x in M$. $B$ is called a content $R$-algebra, if it is a faithfully flat and content $R$-module and it satisfies the Dedekind-Mertens content formula. In this article, we prove some new results for content modules and algebras by using ideal theoretic methods.
To any toric ideal $I_A$, encoded by an integer matrix $A$, we associate a matroid structure called {em the bouquet graph} of $A$ and introduce another toric ideal called {em the bouquet ideal} of $A$. We show how these objects capture the essential combinatorial and algebraic information about $I_A$. Passing from the toric ideal to its bouquet ideal reveals a structure that allows us to classify several cases. For example, on the one end of the spectrum, there are ideals that we call {em stable}, for which bouquets capture the complexity of various generating sets as well as the minimal free resolution. On the other end of the spectrum lie toric ideals whose various bases (e.g., minimal generating sets, Grobner, Graver bases) coincide. Apart from allowing for classification-type results, bouquets provide a new way to construct families of examples of toric ideals with various interesting properties, such as robustness, genericity, and unimodularity. The new bouquet framework can be used to provide a characterization of toric ideals whose Graver basis, the universal Grobner basis, any reduced Grobner basis and any minimal generating set coincide.
In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform numerical experiments suggesting a very promising efficiency. On the way of describing the method, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal.
We introduce and study a class of objects that encompasses Christensen and Foxbys semidualizing modules and complexes and Kubiks quasi-dualizing modules: the class of $mathfrak{a}$-adic semidualizing modules and complexes. We give examples and equivalent characterizations of these objects, including a characterization in terms of the more familiar semidualizing property. As an application, we give a proof of the existence of dualizing complexes over complete local rings that does not use the Cohen Structure Theorem.
