Trace Ideals Research Articles

Partial trace ideals, torsion and canonical module

For any finitely generated module M with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant h(M) was introduced and studied in [25]. We establish a bound on it which helps capture information about the torsion submodule of M when M has rank one and generalizes the discussion in [25]. We further study bounds and properties of h(M) in the case when M is the canonical module ωR. This in turn helps in answering a question of S. Greco and then provides classifications in the Gorenstein, almost Gorenstein and far-flung Gorenstein setups.

Reflexive modules over the endomorphism algebras of reflexive trace ideals

In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong ([20, Theorem 5.1]) and T. Kobayashi ([30, Theorem 1.3]) concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified and has only finitely many Ulrich ideals. As a consequence, Arf local rings contain only finitely many Ulrich ideals once the normalization is a local ring.

When are Trace Ideals Finite?

When are Trace Ideals Finite?

Compatible ideals in ℚ-Gorenstein rings

Suppose R R is a F F -finite and F F -pure Q \mathbb {Q} -Gorenstein local ring of prime characteristic p > 0 p>0 . We show that an ideal I ⊆ R I\subseteq R is uniformly compatible ideal (with all p − e p^{-e} -linear maps) if and only if exists a module finite ring map R → S R\to S such that the ideal I I is the sum of images of all R R -linear maps S → R S\to R . In other words, the set of uniformly compatible ideals is exactly the set of trace ideals of finite ring maps.

Formal matrix rings: Automorphisms

Automorphism groups of formal matrix algebras with zero trace ideals are studied. Such an algebra is represented as a splitting extension of some ring by some nilpotent ideal. Using this extension, the study of the structure of the automorphism group of an algebra in a certain sense is reduced to the study of the structure of some its subgroups and quotient groups. Then the structure of these subgroups and factor groups is found. The case of formal triangular matrix rings is specially considered. At the end of the paper we study automorphisms of ordinary matrix rings. In this case, a well-known homomorphism from the group of outer automorphisms to the Picard group of a certain ring is used.

The trace property in preenveloping classes

We develop the theory of trace modules up to isomorphism and explore the relationship between preenveloping classes of modules and the property of being a trace module, guided by the question of whether a given module is trace in a given preenvelope. As a consequence we identify new examples of trace ideals and trace modules, and characterize several classes of rings with a focus on the Gorenstein and regular properties.

Stable trace ideals and applications

We study stable trace ideals in one dimensional local Cohen–Macaulay rings and give numerous applications.

The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied.

Trace ideals of semidualizing modules and two generalizations of nearly Gorenstein rings

In this article, we study the trace ideal of semidualizing modules. Also, we introduce two new classes of rings, namely quasi nearly Gorenstein and weakly nearly Gorenstein, which are generalizations of nearly Gorenstein rings. Furthermore, we show that the trace ideal of a semidualizing module is annihilators of certain Ext modules. Moreover, we give new versions of Tachikawa conjecture using quasi and weakly nearly Gorenstein rings.

On the nonrigidity of trace modules

We establish a link between trace modules and rigidity in modules over Noetherian rings. We identify classes of modules which must have self-extensions and use the theory of trace ideals to verify the Auslander–Reiten conjecture for syzygies of ideals over Artinian Gorenstein rings.

The reduction number of canonical ideals

In this article, we introduce an invariant of Cohen-Macaulay local rings in terms of the reduction number of canonical ideals. The invariant can be defined in arbitrary Cohen-Macaulay rings and it measures how close to being Gorenstein. First, we clarify the relation between almost Gorenstein rings and nearly Gorenstein rings by using the invariant in dimension one. We next characterize the idealization of trace ideals over Gorenstein rings in terms of the invariant. It provides better prospects for a result on the almost Gorenstein property of idealization.

On the set of trace ideals of a Noetherian ring

We consider trace ideals in Noetherian rings and focus our attention to one-dimensional analytically irreducible local rings. For such rings we classify those Gorenstein rings which admit only a finite number of trace ideals.

Partial trace ideals and Berger's conjecture

For any finitely generated module M with non-zero rank over a commutative one-dimensional Noetherian local domain, we study a numerical invariant h(M) based on a partial trace ideal of M. We study its properties and explore relations between this invariant and the colength of the conductor. Finally we apply this to the universally finite module of differentials ΩR/k, where R is a complete k-algebra with k any perfect field, to study a long-standing conjecture due to R. W. Berger.

Microscopic conservation laws for the derivative Nonlinear Schrödinger equation

Compared with macroscopic conservation law for the solution of the derivative nonlinear Schrödinger equation (DNLS) with small mass in Klaus and Schippa (A priori estimates for the derivative nonlinear Schrödinger equation. Accepted by Funkcial. Ekvac), we show the corresponding microscopic conservation laws for the Schwartz solutions of DNLS with small mass. The new ingredient is to make use of the logarithmic perturbation determinant \(A(\kappa )\) introduced in Rybkin (in: Topics in Operator Theory, Birkhäuser Verlag, Basel, 2010), Simon (in: Trace ideals and their applications, American Mathematical Society, Providence, 2005) to show one-parameter family of microscopic conservation laws of the \(A(\kappa )\) flow and the DNLS flow, which is motivated by Harrop-Griffiths et al. (Sharp well-poseness for the cubic NLS and MKdV in \(H^s\). arXiv:2003.05011), Killip and Visan (Ann Math 190:249–305, 2019), Killip et al. (Geom Funct Anal 28:1062–1090, 2018).

Some Results on Pure Ideals and Trace Ideals of Projective Modules

Let R be a commutative ring with the unit element. It is shown that an ideal I in R is pure if and only if Ann(f) + I = R for all f ∈ I. If J is the trace of a projective R-module M, we prove that J is generated by the “coordinates” of M and JM = M. These lead to a few new results and alternative proofs for some known results.

Optimal constants in non-commutative Hölder inequality for quasi-norms

We resolve an open problem due to B. Simon [Trace ideals and their applications, Cambridge University Press, Cambridge–New York, 1979; Trace ideals and their applications, American Mathematical Society, Providence, RI, 2005] concerning optimal constants in Hölder inequality for certain quasi-normed principal ideals of importance in Mathematical Physics and Noncommutative Geometry.

Trace ideals of canonical modules, annihilators of Ext modules, and classes of rings close to being Gorenstein

In this note we study trace ideals of canonical modules. Characterizations of the trace ideals in terms of annihilators of certain Ext modules are given. We apply our results to study many classes of rings close to being Gorenstein that appear in recent literature. We discuss the behavior of near Gorensteinness, introduced by Herzog, Hibi and Stamate, under reductions by regular sequences. A nearly Gorenstein version of the Tachikawa conjecture is posed, and an affirmative answer is given for type 2 rings. We also introduce the class of weakly almost Gorenstein rings using the canonical module, and show that they are almost Gorenstein rings in dimension one, and are closely related to Teter rings in the artinian case. We explain their basic properties and give some examples.

V-Proximal Trustworthy Banach Spaces

In a recent work (2016), the first author proved the fuzzy sum rule for the V-proximal subdifferential under some natural assumptions on an equivalent norm of the Banach spaces. In the present paper, we are going to prove that the class of Banach spaces satisfying the fuzzy sum rule is very large and contains all Lp spaces 1<p<∞ as well as the sequence spaces lp1<p<∞, the Sobolev spaces Wp,n1<p<∞, and the Schatten trace ideals Cp1<p<∞.

Correction to: C*-algebras, H*-algebras and trace ideals of pseudo-differential operators on locally compact, Hausdorff and abelian groups

Correction to: C*-algebras, H*-algebras and trace ideals of pseudo-differential operators on locally compact, Hausdorff and abelian groups

Trace ideals, normalization chains, and endomorphism rings

In this paper we consider reduced (non-normal) commutative noetherian rings R.With the help of conductor ideals and trace ideals of certain R-modules we deduce a criterion for a reflexive R-module to be closed under multiplication with scalars in an integral extension of R. Using results of Greuel and Knorrer this yields a characterization of plane curves of finite Cohen–Macaulay type in terms of trace ideals. Further, we study one-dimensional local rings (R,m) such that that their normalization is isomorphic to the endomorphism ring EndR(m): we give a criterion for this property in terms of the conductor ideal, and show that these rings are nearly Gorenstein. Moreover, using Grauert–Remmert normalization chains, we show the existence of noncommutative resolutions of singularities of low global dimensions for curve singularities.