Let R be a Noetherian commutative ring of dimension n >2 and let A=R[T,T^{-1}]. Assume that the height of the Jacobson radical of R is atleast 2. Let P be a projective A-module of rank n=dim A - 1 with trivial determinant. We define an abelian group called the "Euler class group of A", denoted by E(A). Let χbe an isomorphism from A to det(P). To the pair (P,χ), we associate an element of E(A), called the Euler class of P, denoted by e(P,χ). Then we prove that a necessary and sufficient condition for P to have a unimodular element is the vanishing of e(P,χ) in E(A).
Earlier, Bhatwadekar and Raja Sridharan have defined the Euler class group of R, denoted by E(R), and have proved similar results for projective R-module of rank n. Later, Mrinal K. Das defined the Euler class group of the polynomial ring R[T], denoted by E(R[T]), and proved similar results for projective R[T]-modules of rank n with trivial determinant.
If $R$ is an integral domain and $A$ is an $R$-algebra, then $A$ has the {\it Laurent cancellation property over $R$} if $A^{[\pm n]}\cong_RB^{[\pm n]}$ implies $A\cong_RB$ ($n\ge 0$ and $B$ an $R$-algebra). Here, $A^{[\pm n]}$ denotes the ring of Laurent polynomials in $n$ variables over $A$. Our main result (Thm. 4.3) is that, if the transcendence degree of $A$ over $R$ is one, then $A$ has the Laurent cancellation property. The proof uses the characterization of Laurent polynomial rings given in Thm. 3.2.
We prove constructively that for any finite-dimensional commu- tative ring R, every stably free module over R[X;X^{1}] of rank > dim R is free, i.e., R[X;X^{-1}] is (dimR)-Hermite.
In this paper, we present several algorithms for dealing with graded components of Laurent polynomial rings. To be more precise, let $S$ be the Laurent polynomial ring $k[x_1,...,x_{r},x_{r+1}^{\pm 1},..., x_n^{\pm 1}]$, $k$ algebraicaly closed field of characteristic 0. We define the multigrading of $S$ by an arbitrary finitely generated abelian group $A$. We construct a set of fans compatible with the multigrading and use this fans to compute the graded components of $S$ using polytopes. We give an algorithm to check whether the graded components of $S$ are finite dimensional. Regardless of …
We study the question if projective modules over formal Laurent series rings are extended. We relate this question to the Bass-Quillen conjecture for commutative regular local rings and to the Hermite ring conjecture for all commutative local rings. Using our result about projective modules over formal Laurent series rings, we prove a reduction step for the Hermite ring conjecture. We show that the Hermite ring conjecture holds for all commutative local rings if and only if it holds for complete intersection rings which are also unique factorization domains.
In this paper we proved that if $R$ is right $ω$-Euclidean domain, then skew Laurent formal series ring is right $ω$-Euclidean domain. We also showed that if $R$ is a right $ω$-Euclidean domain with multiplicative norm, then skew Laurent formal series ring is a right principal ideal domain. In addition, we proved that if $R$ is a noncommutative $ω$-Euclidean domain with a multiplicative norm, then $R$ and skew Laurent formal series ring is a ring with elementary reduction of matrices.
Duistermaat and van der Kallen show that there is no nontrivial complex Laurent polynomial all of whose powers have a zero constant term. Inspired by this, Sturmfels posed two questions: Do the constant terms of a generic Laurent polynomial form a regular sequence? If so, then what is the degree of the associated zero-dimensional ideal? In this note, we prove that the Eulerian numbers provide the answer to the second question. The proof involves reinterpreting the problem in terms of toric geometry.
Dolgachev proved that, for any field k, the ring naturally associated to a generic Laurent polynomial in d variables, $d \geq 4$, is factorial. We prove a sufficient condition for the ring associated to a very general complex Laurent polynomial in d=3 variables to be Q-factorial.
A finitely generated module over the ring L=Z[t, t^{-1}] of integer Laurent polynomials that has no Z-torsion is determined by a pair of sub-lattices of L^d. Their indices are the absolute values of the leading and trailing coefficients of the order of the module. This description has applications in knot theory.
We first present a filtration on the ring L of Laurent polynomials such that the direct sum decomposition of its associated graded ring gr L agrees with the direct sum decomposition of gr L, as a module over the complex general linear Lie algebra gl(n), into its simple submodules. Next, generalizing the simple modules occurring in gr L, we give some explicit constructions of weight multiplicity-free irreducible representations of gl(n).
We study chains of lattice ideals that are invariant under a symmetric group action. In our setting, the ambient rings for these ideals are polynomial rings which are increasing in (Krull) dimension. Thus, these chains will fail to stabilize in the traditional commutative algebra sense. However, we prove a theorem which says that "up to the action of the group", these chains locally stabilize. We also give an algorithm, which we have implemented in software, for explicitly constructing these stabilization generators for a family of Laurent toric ideals involved in applications to algebraic sta…
If $i:A\subset B$ is a commutative ring extension, we show that the group $\mathcal I(A,B)$ of invertible $A$-submodules of $B$ is contracted in the sense of Bass, with $L\mathcal I(A,B)=H^0_{et}(A,i_*\mathbb Z/\mathbb Z)$. This gives a canonical decomposition for $\mathcal I(A[t,\frac1t],B[t,\frac1t])$.
Let $K$ be a non-empty set of ideals of the commutative ring $R$, closed under taking smaller ideals. A subset $X$ of the group ring $R[\mathbb{Z}^s]$ is called a $K$-set if the ideal generated by the coefficients of the elements of $X$ is in $K$. For $X$ not a $K$-set we investigate the set of those homomorphisms $p \colon \mathbb{Z}^s \to \mathbb{Z}^t$ such that $p_*(X)$ is a $K$-set. We also consider corresponding notions of rank of matrices and Betti numbers of chain complexes; this includes an analysis of the case of McCoy rank. Our setup also recovers results on jump loci obtained by Koh…
Let $R=\bigoplus_{n\ges0}R_n$ be a graded commutative ring generated over a field $K=R_0$ by homogeneous elements $x_1,\dots,x_e$ of positive degrees $d_1,\dots,d_e$. The Hilbert-Serre Theorem shows that for each finite graded $R$--module $M=\bigoplus_{n\in\BZ}M_n$ the {\it Hilbert series\/} $\sum_{n\in\BZ}(\rank_K M_n)t^n$ is the Laurent expansion around $0$ of a rational function $$ H_M(t)=\frac{q_M(t)}{\prod_{i=1}^e(1-t^{d_i})} $$ with $q_M(t)\in\BZ[t,\ti]$. We demonstrate that Laurent expansions $\left[M\right]_z$ of $H_M(t)$ around other points $z$ of the extended complex plane $\overline…
If every subring of an integral domain is atomic, then we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectivel…
We will study the Euler class group of a commutative Noetherian ring $A$ of dimension $d$, developed by S.M. Bhatwadekar and Raja Sridharan. This is my M.Phil thesis from 2001.
Let $A\subseteq B$ be a commutative ring extension. Let $\mathcal I(A, B)$ be the multiplicative group of invertible $A$-submodules of $B$. In this article, we extend a result of Sadhu and Singh by finding a necessary and sufficient condition on an integral birational extension $A\subseteq B$ of integral domains with $\dim A\leq 1$, so that the natural map $\mathcal I(A,B) \rightarrow \mathcal I (A [X, X^{-1}],B [X, X^{-1}])$ is an isomorphism. In the same situation, we show that if $\dim A\geq 2$ then the condition is necessary but not sufficient. We also discuss some properties of the cokern…
Let R be a commutative Noetherian ring of dimension d and B=R[X_1,\ldots,X_m,Y_1^{\pm 1},\ldots,Y_n^{\pm 1}] a Laurent polynomial ring over R. If A=B[Y,f^{-1}] for some f\in R[Y], then we prove the following results: (i) If f is a monic polynomial, then Serre dimension of A is \leq d. In case n=0, this result is due to Bhatwadekar, without the condition that f is a monic polynomial. (ii) The p-th Euler class group E^p(A) of A, defined by Bhatwadekar and Raja Sridharan, is trivial for p\geq max \{d+1, \dim A -p+3\}. In case m=n=0, this result is due to Mandal-Parker.
Let $K$ be the field of Laurent series with complex coefficients, let $\mathcal{R}$ be the inverse limit of the standard-graded polynomial rings $K[x_1, \ldots, x_n]$, and let $\mathcal{R}^{\flat}$ be the subring of $\mathcal{R}$ consisting of elements with bounded denominators. In previous joint work with Erman and Sam, we showed that $\mathcal{R}$ and $\mathcal{R}^{\flat}$ (and many similarly defined rings) are abstractly polynomial rings, and used this to give new proofs of Stillman's conjecture. In this paper, we prove the complementary result that $\mathcal{R}$ is a polynomial algebra ove…
The set of the first Hilbert coefficients of parameter ideals relative to a module--its Chern coefficients--over a local Noetherian ring codes for considerable information about its structure--noteworthy properties such as that of Cohen-Macaulayness, Buchsbaumness, and of having finitely generated local cohomology. The authors have previously studied the ring case. By developing a robust setting to treat these coefficients for unmixed rings and modules, the case of modules is analyzed in a more transparent manner. Another series of integers arise from partial Euler characteristics and are show…
Let $(R,\mm,K)$ be a regular local ring containing a field $k$ such that either char $k=0$ or char $k=p$ and tr-deg $K/\BF_p\geq 1$. Let $g_1,\ldots,g_t$ be regular parameters of $R$ which are linearly independent modulo $\mm^2$. Let $A=R_{g_1\cdots g_t} [Y_1,\ldots,Y_m,f_1(l_1)^{-1},\ldots, f_n(l_n)^{-1}]$, where $f_i(T)\in k[T]$ and $l_i=a_{i1}Y_1+\ldots+a_{im}Y_m$ with $(a_{i1},\ldots,a_{im})\in k^m-(0)$. Then every projective $A$-module of rank $\geq t$ is free. Laurent polynomial case $f_i(l_i)=Y_i$ of this result is due to Popescu.
We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon from math.RT/0104151, we show that, under an assumption of "acyclicity", a cluster algebra coincides with its "upper" counterpart, and is finitely generated. In this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple comple…
A cluster is a finite set of generators of a cluster algebra. The Laurent Phenomenon of Fomin and Zelevinsky says that any element of a cluster algebra can be written as a Laurent polynomial in terms of any cluster. The upper cluster algebra of a cluster algebra is the ring of rational functions that can be written as a Laurent polynomial in every cluster of the cluster algebra. By the Laurent phenomenon a cluster algebra is always contained in its upper cluster algebra, but they are not always equal. In 2014 it was conjectured that the equality of the cluster algebra and upper cluster algebra…
The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1, ..., x_d]]$, where $O$ is the ring of integers of a finite extension of the field of p-adic integers $Q_p$. The specialization method is a technique that recovers the information on the characteristic ideal $char_R(M)$ from $char_{R/I}(M/IM)$, where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen-Macaulay normal domains by …
