Conjecture For Rings Research Articles

The Telescope Conjecture for von Neumann regular rings

In this note, we show that any epimorphism originating at a von Neumann regular ring (not necessary commutative) is a universal localization. As an application, we prove that the Telescope Conjecture holds for the unbounded derived categories of von Neumann regular rings (not necessary commutative).

The multivariate Serre conjecture ring

It is well-known that for any integral domain R, the Serre conjecture ring R〈X〉, i.e., the localization of the univariate polynomial ring R[X] at monic polynomials, is a Bézout domain of Krull dimension ≤1 if R is a Bézout domain of Krull dimension ≤1. Consequently, defining by induction R〈X1,…,Xn〉:=(R〈X1,…,Xn−1〉)〈Xn〉, the ring R〈X1,…,Xn〉 is a Bézout domain of Krull dimension ≤1 if so is R. The fact that R〈X1,…,Xn〉 is a Bézout domain when R is a valuation domain of Krull dimension ≤1 was the cornerstone of Brewer and Costa's theorem stating that if R is a one-dimensional arithmetical ring then finitely generated projective R[X1,…,Xn]-modules are extended. It is also the key of the proof of the Gröbner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain R of Krull dimension ≤1, any n∈N>0, and any finitely generated ideal I of R[X1,…,Xn], the ideal LT(I) generated by the leading terms of the elements of I with respect to the lexicographic monomial order is finitely generated. Since the ring R〈X1,…,Xn〉 can also be defined directly as the localization of the multivariate polynomial ring R[X1,…,Xn] at polynomials whose leading coefficients according to the lexicographic monomial order with X1<X2<⋯<Xn is 1, we propose to generalize the fact that R〈X1,…,Xn〉 is a Bézout domain of Krull dimension ≤1 when R is a Bézout domain of Krull dimension ≤1 to any rational monomial order, bolstering the evidence for the Gröbner Ring Conjecture in the rational case. We give an example showing that this result is no more true in the irrational case.

The Auslander-Reiten conjecture for certain non-Gorenstein Cohen-Macaulay rings

The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In a Cohen-Macaulay complete local ring R with a parameter ideal Q, the Auslander-Reiten conjecture holds for R if and only if it holds for the residue ring R/Q. In the former part of this paper, we study the Auslander-Reiten conjecture for the ring R/Qℓ in connection with that for R, and prove the equivalence of them for the case where R is Gorenstein and ℓ≤dim⁡R. In the latter part, we generalize the result of the minimal multiplicity by J. Sally. Due to these two of our results, we see that the Auslander-Reiten conjecture holds if there exists an Ulrich ideal whose residue ring is a complete intersection. We also explore the Auslander-Reiten conjecture for determinantal rings.

Polynomial structure of Gromov–Witten potential of quintic 3-folds

We prove two structure theorems for the Gromov-Witten theory of the quintic threefolds, which together give an effective algorithm for the all genus Gromov-Witten potential functions of quintics. By using these structure theorems, we prove Yamaguchi-Yau's Polynomial Ring Conjecture in this paper and prove Bershadsky-Cecotti-Ooguri-Vafa's Feynman rule conjecture in the subsequent paper.

A counterexample to the Gröbner ring conjecture

We propose a counterexample to the Gröbner ring conjecture in the irrational case. More precisely, we construct a valuation domain V of Krull dimension 1 and a finitely generated ideal J of V[X,Y] equipped with some irrational monomial order <, such that the ideal LT<(J) generated by the leading terms of the elements of J is not finitely generated.

Algebraic cycles on a very special EPW sextic

Motivated by the Beauville–Voisin conjecture about Chow rings of powers of K3 surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We prove part of this conjecture for the very special EPW sextic studied by Donten-Bury et al. We also prove some other results concerning the Chow groups of this very special EPW sextic, and of certain related hyperkähler fourfolds.

Algebraic cycles on a very special EPW sextic

Motivated by the Beauville–Voisin conjecture about Chow rings of powers of $K3$ surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We prove part of this conjecture for the very special EPW sextic studied by Donten-Bury et al. We also prove some other results concerning the Chow groups of this very special EPW sextic, and of certain related hyperkähler fourfolds.

Huneke–Wiegand Conjecture for complete intersection numerical semigroup rings

We give a positive answer to the Huneke–Wiegand Conjecture for monomial ideals over free numerical semigroup rings, and for two-generated monomial ideals over complete intersection numerical semigroup rings.

The Gröbner ring conjecture in the lexicographic order case

We prove that a valuation domain \(\mathbf{V}\) has Krull dimension \(\le \)1 if and only if, for any \(n\), fixing the lexicographic order as monomial order in \(\mathbf{V}[X_1,\ldots ,X_n]\), for every finitely generated ideal \(I\) of \(\mathbf{V}[X_1,\ldots ,X_n]\), the ideal generated by the leading terms of the elements of \(I\) is also finitely generated. This proves the Grobner ring conjecture in the lexicographic order case. The proof we give is both simple and constructive. The same result is valid for Prufer domains. As a “scoop”, contrary to the common idea that Grobner bases can be computed exclusively on Noetherian ground, we prove that computing Grobner bases over \(\mathbf{R}[X_1,\ldots , X_n]\), where \(\mathbf{R}\) is a Prufer domain, has nothing to do with Noetherianity, it is only related to the fact that the Krull dimension of \(\mathbf{R}\) is \(\le \)1.

The hoop conjecture for black rings

A precise formulation of the hoop conjecture for four-dimensional spacetimes proposes that the Birkhoff invariant β for an apparent horizon in a spacetime with mass M should satisfy β⩽4πM. The invariant β is the least maximal length of any sweepout of the 2-sphere apparent horizon by circles. An analogous conjecture in five spacetime dimensions was recently formulated, asserting that the Birkhoff invariant β for S1×S1 sweepouts of the apparent horizon should satisfy β⩽163πM. Although this hoop inequality was formulated for conventional five-dimensional black holes with 3-sphere horizons, we show here that it is also obeyed by a wide variety of black rings, where the horizon instead has S2×S1 topology.

Approximate roots of a valuation and the Pierce-Birkhoff conjecture

This paper is a step in our program for proving the Piece-Birkhoff Conjecture for regular rings of any dimension (this would contain, in particular, the classical Pierce-Birkhoff conjecture which deals with polynomial rings over a real closed field). We first recall the Connectedness and the Definable Connectedness conjectures, both of which imply the Pierce - Birkhoff conjecture. Then we introduce the notion of a system of approximate roots of a valuation v on a ring A (that is, a collection Q of elements of A such that every v-ideal is generated by products of elements of Q). We use approximate roots to give explicit formulae for sets in the real spectrum of A which we strongly believe to satisfy the conclusion of the Definable Connectedness conjecture. We prove this claim in the special case of dimension 2. This proves the Pierce-Birkhoff conjecture for arbitrary regular 2-dimensional rings.

Irreducible values of polynomials

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem à la Hilbert.

The Gröbner ring conjecture in one variable

We prove that a valuation domain V has Krull dimension ≤ 1 if and only if for every finitely generated ideal I of V[X] the ideal generated by the leading terms of elements of I is also finitely generated. This proves the Grobner ring conjecture in one variable. The proof we give is both simple and constructive. The same result is valid for semihereditary rings.

On sums and products in ℂ[x

We prove that there exists an absolute constant c>0 such that if A is a set of n monic polynomials, and if the product set A.A has at most n 1+c elements, then |A+A|≫n2. This can be thought of as step towards proving the Erdős–Szemeredi sum-product conjecture for polynomial rings. We also show that under a suitable generalization of Fermat’s Last Theorem, the same result holds for the integers. The methods we use to prove are a mixture of algebraic (e.g. Mason’s theorem) and combinatorial (e.g. the Ruzsa–Plunnecke inequality) techniques.

The Hermite ring conjecture in dimension one

We prove constructively that for any ring R of Krull dimension ⩽1 and n ⩾ 3 , the group E n ( R [ X ] ) acts transitively on Um n ( R [ X ] ) . In particular, we obtain that for any ring R with Krull dimension ⩽1, all finitely generated stably free modules over R [ X ] are free. This settles the long-standing Hermite ring conjecture for rings of Krull dimension ⩽1.

Quadratic form theory over preordered von Neumann-regular rings

To each pair, 〈 R , T 〉 , consisting of a unitary commutative von Neumann-regular ring, R, where 2 is a unit and T is a preorder on R, we associate a reduced special group, G T ( R ) , which faithfully reflects quadratic form theory, modulo T, over free R-modules and then show, using the representation of R as the ring of global sections of its affine scheme, together with results from [M. Dickmann, F. Miraglia, On quadratic forms whose total signature is zero mod 2 n . Solution to a problem of M. Marshall, Invent. Math. 133 (1998) 243–278; M. Dickmann, F. Miraglia, Lam's Conjecture, Algebra Colloq. 10 (2003) 149–176; M. Dickmann, F. Miraglia, Algebraic K-theory of special groups, J. Pure Appl. Algebra 204 (2006) 195–234], that G T ( R ) satisfies a powerful K-theoretic property, the [SMC]-property. From this we conclude that quadratic form theory modulo T over free R-modules verifies Marshall's signature conjecture, Lam's conjecture, as well as a reduced version of Milnor's Witt ring conjecture.

Annihilators of local cohomology in characteristic zero

This paper discusses the problem of whether it is possible to annihilate elements of local cohomology modules by elements of arbitrarily small order under a fixed valuation. We first discuss the general problem and its relationship to the Direct Summand Conjecture, and next present two concrete examples where annihilators with small order are shown to exist. We then prove a more general theorem, where the existence of such annihilators is established in some cases using results on abelian varieties and the Albanese map. 1. Almost vanishing of local cohomology The concept of almost vanishing that we use here comes out of recent work on Almost Ring Theory by Gabber and Ramero [4]. This theory was developed to give a firm foundation to the results of Faltings on Almost etale extensions [3], and these ideas have their origins in a classic work of Tate on p-divisible groups [21]. The use of the general theory, for our purposes, is comparatively straightforward, but it illustrates the main questions in looking at certain homological conjectures, as discussed later in the section. The approach is heavily influenced by Heitmann’s proof of the Direct Summand Conjecture for rings of dimension three [8]. Let A be an integral domain, and let v be a valuation on A with values in the abelian group of rational numbers; more precisely, v is a function from A to Q ∪ {∞} such that (1) v(a) =∞ if and only if a = 0, (2) v(ab) = v(a) + v(b) for all a, b ∈ A, and (3) v(a+ b) > min{v(a), v(b)} for all a, b ∈ A. Received November 6, 2006; received in final form April 4, 2007. 2000 Mathematics Subject Classification. Primary 13D22. Secondary 13D45, 14K05. P.R. and A.K.S. were supported in part by grants from the National Science Foundation. V.S. was supported by a Swarnajayanthi Fellowship of the DST. c ©2007 University of Illinois

CORRIGENDA: MACKEY FUNCTORS AND CONTROL OF FUSION

In the introduction to [3] it is mistakenly claimed that Mislin's proof uses Carlsson's proof of Segal's Burnside ring conjecture. In fact, Mislin uses instead the work of Carlsson, Miller and Lannes on Sullivan's fixed-point conjecture, and work of Dwyer and Zabrodski [1]. There is, however, a proof by Snaith [2] that depends on a version of the Segal conjecture. 2000 Mathematics Subject Classification 20J06 (primary), 20C05 (secondary).

On the codimension of modules over skew power series rings with applications to Iwasawa algebras

The first purpose of this paper is to set up a general notion of skew power series rings S over a coefficient ring R, which are then studied by filtered ring techniques. The second goal is the investigation of the class of S-modules which are finitely generated as R-modules. In the case that S and R are Auslander regular we show in particular that the codimension of M as S-module is one higher than the codimension of M as R-module. Furthermore its class in the Grothendieck group of S-modules of codimension at most one less vanishes, which is in the spirit of the Gersten conjecture for commutative regular local rings. Finally we apply these results to Iwasawa algebras of p-adic Lie groups.

The Woods–Erdös conjecture for polynomial rings

The elementary theories of polynomial rings over finite fields with the coprimeness predicate and two kinds of “successor” functions are studied. It is proved that equality is definable in these languages. This gives an affirmative answer to the polynomial analogue of the Woods–Erdös conjecture. It is also proved that these theories are undecidable.