Regular Local Ring Research Articles

Derived functors and Hilbert Polynomials over regular local rings

Abstract Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$ , I an $\mathfrak{m}$ -primary ideal. Let N be a nonzero finitely generated A-module. Consider the functions \begin{equation*} t^I(N, n) = \sum_{i = 0}^{d}\ell(\text{Tor}^A_i(N, A/I^n)) \ \text{and}\ e^I(N, n) = \sum_{i = 0}^{d}\ell(\text{Ext}_A^i(N, A/I^n)) \end{equation*} of polynomial type and let their degrees be $t^I(N) $ and $e^I(N)$ . We prove that $t^I(N) = e^I(N) = \max\{\dim N, d -1 \}$ . A crucial ingredient in the proof is that $D^b(A)_f$ , the bounded derived category of A with finite length cohomology, has no proper thick subcategories.

Quasi-pure resolutions and some lower bounds of Hilbert coefficients of Cohen-Macaulay modules

Let (A,m) be a Noetherian local ring and let M be a finitely generated Cohen Macaulay A module. Let G(A)=⨁n≥0mn/mn+1 be the associated graded ring of A and G(M)=⨁n≥0mnM/mn+1M be the associated graded module of M. If A is regular and if G(M) has a quasi-pure resolution then we show that G(M) is Cohen-Macaulay. If A is Gorenstein and G(A) is Cohen-Macaulay and if M has finite projective dimension then we give lower bounds on e0(M) and e1(M). Finally let A=Q/(f1,…,fc) be a strict complete intersection with ord(fi)=s for all i, where Q is a regular local ring. Let M be an Cohen-Macaulay module with complexity cxA(M)=r<c. We give lower bounds on e0(M) and e1(M).

The integral closure of a primary ideal is not always primary

In 1936, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic p=2. We provide counterexamples to Krull's question regarding polynomial rings over any fields. We also find that the Jacobian ideal J of the polynomial f=x6+y6+x4zt+z3 given by Briançon and Speder (1975) is a counterexample to Krull's question.

Sequence-regular commutative DG-rings

We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings (A,m¯) such that the maximal ideal m¯⊆H0(A) can be generated by an A-regular sequence. We call these DG-rings sequence-regular DG-rings, and make a detailed study of them. Using methods of Cohen-Macaulay differential graded algebra, we prove that the Auslander-Buchsbaum-Serre theorem about localization generalizes to this setting. This allows us to define global sequence-regular DG-rings, and to introduce this regularity condition to derived algebraic geometry. It is shown that these DG-rings share many properties of classical regular local rings, and in particular we are able to construct canonical residue DG-fields in this context. Finally, we show that sequence-regular DG-rings are ubiquitous, and in particular, any eventually coconnective derived algebraic variety over a perfect field is generically sequence-regular.

Indecomposable integrally closed modules of rank 3 over two-dimensional regular local rings

We characterise ideals in two-dimensional regular local rings that arise as ideals of maximal minors of indecomposable integrally closed modules of rank three.

FORMALLY REGULAR RINGS AND DESCENT OF REGULARITY

Abstract Valuation rings and perfectoid rings are examples of (usually non-Noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-Noetherian rings so that it includes valuation and perfectoid rings and it is related to Grothendieck’s definition of formal smoothness as in the Noetherian case. For that, we have to take into account the topologies. We prove a descent theorem for regularity along flat homomorphisms (in fact for homomorphisms of finite flat dimension), extending some known results from the Noetherian to the non-Noetherian case, as well as generalizing some recent results in the non-Noetherian case, such as the descent of regularity from perfectoid rings by B. Bhatt, S. Iyengar and L. Ma.

Pfister forms and a conjecture due to Colliot-Thélène in the mixed characteristic case

Let $R$ be a regular local ring of mixed characteristic $(0,p)$, where $p\neq 2$ is a prime number. Suppose that the quotient ring $R/pR$ is also regular. We fix a non-degenerate Pfister form $Q(T_{1},\ldots,T_{2^{m}})$ over $R$ and an invertible element $c$ in $R$. Then the equation $Q(T_{1},\ldots,T_{2^{m}})=c$ has a solution over $R$ if and only if it has a solution over the fraction field $K$.

Формы Пфистера и гипотеза Кольe-Телена для случая смешанной характеристики

Let $R$ be a regular local ring of a mixed characteristic $(0,p)$ where $p\neq 2$ is a prime number. Suppose that the quotient ring $R/pR$ is also regular. Fix a non-degenerate Pfister form $Q(T_{1},\ldots,T_{2^{m}})$ over $R$ and an invertible element $c$ in $R$. Then the equation $Q(T_{1},\ldots,T_{2^{m}})=c$ has a solution over $R$ if and only if it has a solution over the fraction field $K$.

Numbers of generators of perfect ideals

This article is concerned with bounds on the number of generators of perfect ideals J in regular local rings (R,m). If J is sufficiently large modulo mn, a bound is established depending only on n and the projective dimension of R/J. More ambitious questions are also introduced with some partial results.

On a generalization of a result of Peskine and Szpiro

Let (R,\mathfrak{m}) be a regular local ring containing a field K . Let I be a Cohen–Macaulay ideal of height g . If \operatorname{char} K = p &gt; 0 then by a result of Peskine and Szpiro the local cohomology modules H^i_I(R) vanish for i &gt; g . This result is not true if \operatorname{char} K = 0 . However, we prove that the Bass numbers of the local cohomology module H^g_I(R) completely determine whether H^i_I(R) vanish for i &gt; g . The result of this paper has been proved more generally for Gorenstein local rings by Hellus and Schenzel (2008) (Theorem 3.2). However, our result is elementary to prove. In particular, we do not use spectral sequences in our proof.

On associated graded modules of maximal Cohen–Macaulay modules over hypersurface rings

Let [Formula: see text] where [Formula: see text] be a complete regular local ring of dimension [Formula: see text], [Formula: see text] for some [Formula: see text] and [Formula: see text] an MCM [Formula: see text]-module with [Formula: see text] then we prove that [Formula: see text]. If [Formula: see text] is a complete hypersurface ring of dimension [Formula: see text] with infinite residue field and [Formula: see text], let [Formula: see text] be an MCM [Formula: see text]-module with [Formula: see text] or [Formula: see text] then we prove that [Formula: see text]. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurface rings.

Branched covers and matrix factorizations

AbstractLet be a regular local ring and a non‐zero element of . A theorem due to Knörrer states that there are finitely many isomorphism classes of maximal Cohen–Macaulay (CM) ‐modules if and only if the same is true for the double branched cover of , that is, the hypersurface ring which is defined by in . We consider an analogue of this statement in the case of the hypersurface ring defined instead by for . In particular, we show that this hypersurface, which we refer to as the ‐fold branched cover of , has finite CM representation type if and only if, up to isomorphism, there are only finitely many indecomposable matrix factorizations of with factors. As a result, we give a complete list of polynomials with this property in characteristic zero. Furthermore, we show that reduced ‐fold matrix factorizations of correspond to Ulrich modules over the ‐fold branched cover of .

Surjectivity of some local cohomology map and the second vanishing theorem

The second vanishing theorem has a long history in the theory of local cohomology modules, which connects the vanishing of a complete regular local ring with a topological property of the punctured spectrum of the ring under some conditions. However, the case of complete ramified regular local rings is unresolved. In this paper, we give a partial answer to the second vanishing theorem in the ramified case. Our proof is inspired by the theory of surjective elements in the theory of local cohomology.

On accumulation points of F-pure thresholds on regular local rings

Blickle, Mustaţă and Smith proposed two conjectures on the limits of F-pure thresholds. One conjecture asks whether or not the limit of a sequence of F-pure thresholds of principal ideals on regular local rings of fixed dimension can be written as an F-pure thresshold in lower dimension. Another conjecture predicts that any F-pure threshold of a formal power series can be written as the F-pure threshold of a polynomial. In this paper, we prove that the first conjecture has a counterexample but a weaker statement still holds. We also give a partial affirmative answer to the second conjecture.

The asymptotic Samuel function and invariants of singularities

The asymptotic Samuel function generalizes to arbitrary rings the usual order function of a regular local ring. In this paper, we use this function to introduce the notion of the Samuel slope of a Noetherian local ring, and we study some of its properties. In particular, we focus on the case of a local ring at singular point of a variety, and, among other results, we prove that the Samuel slope of these rings is related to some invariants used in algorithmic resolution of singularities.

Unimodular rows over monoid extensions of overrings of polynomial rings

Let R be a commutative Noetherian ring of dimension d and M a commutative cancellative torsion-free seminormal monoid. Then: (1) Let A be a ring of type R[d,m,n] and P be a projective A[M]-module of rank r≥max {2,d+1}. Then the action of E(A[M]⊕P) on (A[M]⊕P) is transitive and (2) Assume (R,m,K) is a regular local ring containing a field k such that either char k=0 or char k=p and tr-deg K∕𝔽p≥1. Let A be a ring of type R[d,m,n]∗ and f∈R be a regular parameter. Then all finitely generated projective modules over A[M], A[M]f and A[M]⊗RR(T) are free. When M is free both results are due to Keshari and Lokhande (2014).

Vanishing of Avramov obstructions for products of sequentially transverse ideals

Two ideals I and J are called transverse if I ∩ J = I J . We show that the obstructions defined by Avramov for products of (sequentially) transverse ideals in regular local rings are always 0. In particular, we compute an explicit free resolution and Koszul homology for all such ideals. Moreover, we construct an explicit trivial Massey operation on the associated Koszul complex and hence (by Golod's construction) a minimal free resolution of the residue field over the quotient defined by the product of transverse ideals. We conclude with questions about the existence of associative multiplicative structures on the minimal free resolution of the quotient defined by products of transverse ideals.

Local Cohomology of Module of Differentials of integral extensions II

In this note (R,m) denotes a complete regular local ring and B mostly denotes its absolute integral closure. The four objectives of this paper are the following: i) to determine the highest non-vanishing local cohomology of ΩB/R in equicharacteristic 0, ii) to establish a connection between each of ΩB/R and ΩB/V and pull-back of ΩA/V via a short exact sequence together with new observations on corresponding local cohomologies in mixed characteristic where V is the coefficient ring of R and A is its absolute integral closure, iii) to demonstrate that ΩB/R can be mapped onto a cohomologically Cohen-Macaulay module and iv) to study torsion-free property for ΩC/V and ΩC/k along with their respective completions where C is an integral domain and a module finite extension of R. In this connection an extension of Suzuki's theorem on normality of complete intersections to the formal set-up in all characteristics is accomplished.

DG structure on length 3 trimming complexes and applications to Tor algebras

In this paper, we consider the iterated trimming complex associated to data yielding a complex of length 3. We compute an explicit algebra structure in this complex in terms of the algebra structures of the associated input data. Moreover, it is shown that many of these products become trivial after descending to homology. We apply these results to the problem of realizability for Tor-algebras of grade 3 perfect ideals, and show that under mild hypotheses, the process of “trimming” an ideal preserves Tor-algebra class. In particular, we construct new classes of ideals in arbitrary regular local rings defining rings realizing Tor-algebra classes G ( r ) and H ( p , q ) for a prescribed set of homological data.

Indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring

In this paper, we construct indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring. The modules are quite explicitly constructed from a given complete monomial ideal. We also give structural and numerical results on integrally closed modules. These are used in the proof of indecomposability of the modules. As a consequence, we have a large class of indecomposable integrally closed modules of arbitrary rank whose ideal is not necessarily simple. This extends the original result on the existence of indecomposable integrally closed modules and strengthens the non-triviality of the theory developed by Kodiyalam.