Regular Local Ring Research Articles

Analytic spread of filtrations and symbolic algebras

In this paper we define and explore the analytic spread ℓ ( I ) $\ell (\mathcal {I})$ of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non-Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal I $I$ , we have the classical bounds ht ( I ) ⩽ ℓ ( I ) ⩽ dim R $\mbox{ht}(I)\leqslant \ell (I)\leqslant \dim R$ . The upper bound ℓ ( I ) ⩽ dim R $\ell (\mathcal {I})\leqslant \dim R$ is true for filtrations I $\mathcal {I}$ , but the lower bound is not true for all filtrations. We show that for the filtration I $\mathcal {I}$ of symbolic powers of a height two prime ideal p $\mathfrak {p}$ in a regular local ring of dimension three (a space curve singularity), so that ht ( I ) = 2 $\mbox{ht}(\mathcal {I}) =2$ and dim R = 3 $\dim R=3$ , we have that 0 ⩽ ℓ ( I ) ⩽ 2 $0\leqslant \ell (\mathcal {I})\leqslant 2$ and all values of 0, 1 and 2 can occur. In the cases of analytic spread 0 and 1 the symbolic algebra is necessarily non-Noetherian. The symbolic algebra is non-Noetherian if and only if ℓ ( p ( n ) ) = 3 $\ell (\mathfrak {p}^{(n)})=3$ for all symbolic powers of p $\mathfrak {p}$ and if and only if ℓ ( I a ) = 3 $\ell (\mathcal {I}_a)=3$ for all truncations I a $\mathcal {I}_a$ of I $\mathcal {I}$ .

Torsors on semistable curves and degenerations

In this paper, we answer two long-standing questions on the classification of G-torsors on curves for an almost simple, simply connected algebraic group G over the field of complex numbers. The first is the construction of a flat degeneration of the moduli of G-torsors on smooth projective curves when the smooth curve degenerates to an irreducible nodal curve and the second one is to give an intrinsic definition of (semi)stability for a G-torsor on an irreducible projective nodal curve. A generalization of the classical Bruhat–Tits group schemes to two-dimensional regular local rings and an application of the geometric formulation of the McKay correspondence provide the key tools.

Products of ideals and Golod rings

In this paper, we study conditions guaranteeing that a product of ideals defines a Golod ring. We show that for a 3 3 -dimensional regular local ring (or 3 3 -variable polynomial ring) ( R , m ) (R, \mathfrak {m}) , the ideal I m I \mathfrak {m} always defines a Golod ring for any proper ideal I ⊂ R I \subset R . We also show that non-Golod products of ideals are ubiquitous; more precisely, we prove that for any proper ideal with grade ⩾ 4 \geqslant 4 , there exists an ideal J ⊆ I J \subseteq I such that I J IJ is not Golod. We conclude by showing that if I I is any proper ideal in a 3 3 -dimensional regular local ring and a ⊆ I \mathfrak {a} \subseteq I a complete intersection, then a I \mathfrak {a} I is Golod.

On a theorem of Keune

In this paper we first prove that for regular local ring R containing an infinite field, the natural map S L r ( R [ X ] , ( X 2 − X ) / E r ( R [ X ] , ( X 2 − X ) ) → S K 1 ( R [ X ] , ( X 2 − X ) ) is an isomorphism for r ≥ 3 . Then we prove that for a smooth affine domain R over an infinite field of dimension d , a unimodular row of length d + 1 over R is completable, if U m d + 1 ( Q ( R ) [ X 1 , X 2 , … , X d ] , ( X 1 2 − X 1 ) ( X 2 2 − X 2 ) … ( X d 2 − X d ) ) = e 1 S L d + 1 ( Q ( R ) [ X 1 , X 2 , … , X d ] , ( X 1 2 − X 1 ) ( X 2 2 − X 2 ) … ( X d 2 − X d ) ) where Q ( R ) is the quotient field of R .

Exact categories, big Cohen-Macaulay modules and finite representation type

Abstract One of the first remarkable results in the representation theory of artin algebras , due to Auslander and Ringel-Tachikawa, is the characterisation of when an artin algebra is representation-finite. In this paper, we investigate aspects of representation-finiteness in the general context of exact categories in the sense of Quillen. In this framework, we introduce “big objects” and prove an Auslander-type “splitting-big-objects” theorem. Our approach generalises and unifies the known results from the literature. As a further application of our methods, we extend the theorems of Auslander and Ringel-Tachikawa to arbitrary dimension, i.e. we characterise when a Cohen-Macaulay order over a complete regular local ring is of finite representation type.

Regular local rings of dimension four and Gorenstein syzygetic prime ideals

Let R be a Noetherian local ring. We prove that R is regular of dimension at most 4 if, and only if, every prime ideal, defining a Gorenstein quotient ring, is syzygetic. We deduce a characterization of these rings in terms of the André-Quillen homology.

Remark on Rees valuations of finitely supported complete monomial ideals in a regular local ring

Let be an equicharacteristic regular local ring of dimension and and consider the sequence of d-dimensional local monomial quadratic transformations where is a local monomial quadratic transformation of Ri for We prove that where and is the special -simple complete ideal associated with Let I and J be finitely supported complete -primary monomial ideals in R. We show that if the set of base points of the ideal IJ is linearly ordered, then

Motivic and ell -adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

We study the motivic and ell -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the ell -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the ell -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.

An Extension of a Depth Inequality of Auslander

In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question of whether such a depth inequality can be extended for $n$-Tor-rigid modules, and obtain an affirmative answer for $2$-Tor-rigid modules that are generically free. Furthermore, in the appendix, we use Dao's eta function and determine new classes of Tor-rigid modules over hypersurfaces that are quotient of unramified regular local rings.

Quasi-projective dimension

In this paper, we introduce a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. We discuss various properties of quasi-projective dimension. Among other things, we prove the following. (1) Over a quotient of a regular local ring by a regular sequence, every finitely generated module has finite quasi-projective dimension. (2) The Auslander--Buchsbaum formula and the depth formula for modules of finite projective dimension remain valid for modules of finite quasi-projective dimension. (3) Several results on vanishing of Tor and Ext hold for modules of finite quasi-projective dimension.

On the Grothendieck–Serre conjecture on principal bundles in mixed characteristic

Let $R$ be a regular local ring. Let $\mathbf {G}$ be a reductive $R$-group scheme. A conjecture of Grothendieck and Serre predicts that a principal ${\mathbf {G}}$-bundle over $R$ is trivial if it is trivial over the quotient field of $R$. The conjecture is known when $R$ contains a field. We prove the conjecture for a large class of regular local rings

Completion of skew completable unimodular rows

In this paper, we prove that if R is a local ring of dimension \(d\ge 3\), d odd and \(\frac{1}{(d-1)!}\in R\) then any skew completable unimodular row \(v\in Um_{d}(R[X])\) is completable. It is also proved that skew completable unimodular rows of size \(d\ge 3\) over a regular local ring of dimension d are first row of a 2- stably elementary matrix.

On the existence of birational maximal Cohen-Macaulay modules over biradical extensions in mixed characteristic

Let $S$ be an unramified regular local ring of mixed characteristic $p\geq 3$ and $S^p$ the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/pS$. Let $R$ be the integral closure of $S$ in a biradical extension of degree $p^2$ of its quotient field obtained by adjoining $p$-th roots of sufficiently general square free elements $f,g\in S^p$. We show that $R$ admits a birational maximal Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay.

Existence of birational small Cohen-Macaulay modules over biquadratic extensions in mixed characteristic

Let S be an unramified regular local ring of mixed characteristic two and R the integral closure of S in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements f,g∈S. Let S2 denote the subring of S obtained by lifting to S the image of the Frobenius map on S/2S. When at least one of f,g∈S2, we characterize the Cohen-Macaulayness of R and show that R admits a birational small Cohen-Macaulay module. It is noted that R is not automatically Cohen-Macaulay in case f,g∈S2 or if f,g∉S2.

Local cohomology of module of differentials of integral extensions

The main focus of this paper is on determining the highest non-vanishing local cohomology modules of ΩB/R,ΩB/V(ΩB/k) where R is either a complete regular local ring or a complete local normal domain with coefficient ring V (field k) and B is its integral closure in an algebraic extension of Q(R). Similar problem is also studied over a normal domain R containing a field k of characteristic 0. In this connection new observations on the direct summand property for integral extensions are also presented.

Loose edges and factorization theorems

Let $ R $ be a regular local ring with maximal ideal $ \mathfrak{m} $. We consider elements $ f \in R $ such that their Newton polyhedron has a loose edge. We show that if the symbolic restriction of $f$ to such an edge is a product of two coprime polynomials, then $f$ factorizes in the $ \mathfrak{m} $-adic completion.

Enlarging localized polynomial rings while preserving their prime ideal structure

Let n be an integer greater than 1 and let x1,…,xn be indeterminates over a countable field k. In this paper, we employ techniques of Heitmann and Nagata to show there exists an uncountable regular local ring S between the localized polynomial ring k[x1,…,xn](x1,…,xn) and the power series ring k[[x1,…,xn]] such that the prime ideal spectrum of S is homeomorphic to the prime ideal spectrum of k[x1,…,xn](x1,…,xn) as topological spaces with the Zariski topology (Theorem 3.17). Thus S is a local n-dimensional Noetherian domain and the cardinality of the set of prime ideals of S is strictly less than the cardinality of S. We also show that every Noetherian ring A with infinitely many prime ideals has a Noetherian subring B such that the prime ideal spectrum of B is homeomorphic to the prime ideal spectrum of A and the cardinality of the set of prime ideals of B equals the cardinality of B.

Regular Parameter Elements and Regular Local Hyperrings

Inspired by the concept of regular local rings in classical algebra, in this article we initiate the study of the regular parameter elements in a commutative local Noetherian hyperring. These elements provide a deep connection between the dimension of the hyperring and its primary hyperideals. Then, our study focusses on the concept of regular local hyperring R, with maximal hyperideal M, having the property that the dimension of R is equal to the dimension of the vectorial hyperspace MM2 over the hyperfield RM. Finally, using the regular local hyperrings, we determine the dimension of the hyperrings of fractions.

Criteria for prescribed bound on projective dimension

We establish a prescribed upper bound for the projective dimension of a finitely generated module over a Cohen-Macaulay local ring (with canonical module) satisfying certain cohomological conditions. The central hypothesis is the vanishing of finitely many suitable Ext modules. We then derive corollaries which provide freeness criteria and a characterization of regular local rings. Finally, we raise two main questions related to our results; one generalizes a problem due to D. Jorgensen, and the other retrieves the long-standing Auslander-Reiten conjecture in the Gorenstein case.

Covers of rational double points in mixed characteristic

We further the classification of rational surface singularities. Suppose $(S, \mathfrak{n}, \mathcal{k})$ is a strictly Henselian regular local ring of mixed characteristic $(0, p > 5)$. We classify functions $f$ for which $S/(f)$ has an isolated rational singularity at the maximal ideal $\mathfrak{n}$. The classification of such functions are used to show that if $(R, \mathfrak{m}, \mathcal{k})$ is an excellent, strictly Henselian, Gorenstein rational singularity of dimension $2$ and mixed characteristic $(0, p > 5)$, then there exists a split finite cover of $\mbox{Spec}(R)$ by a regular scheme. We give an application of our result to the study of $2$-dimensional BCM-regular singularities in mixed characteristic.