arXiv subjects

Milen T. Yakimov

Publications and source records attributed to Milen T. Yakimov.

7 recordsLinked to original sources

arXiv2026

The homological spectrum for monoidal triangulated categories

The authors develop a notion of homological prime spectrum for an arbitrary monoidal triangulated category, ${\mathbf C}$. Unlike the symmetric case due to Balmer, the homological primes of ${\mathbf C}$ are not defined as the maximal Serre ideals of the small module category ${\sf mod}\text{-}{\mathbf C}$, or via a noncommutative ring theory inspired version of this construction. Instead, the authors work with an extended comparison map from the Serre prime spectrum $\operatorname{Spc} ({\sf mod}\text{-}{\mathbf C})$ to the Balmer spectrum $\operatorname{Spc}{\mathbf C}$, and select the maximal elements of each fiber to define the homological spectrum $\operatorname{Spc}^{\text{h}}{\mathbf C}$. A surjective continuous homological comparison map $\operatorname{Spc}^{\text{h}}{\mathbf C} \to \operatorname{Spc}{\mathbf C}$ is constructed and used to formulate an extended Nerves of Steel Condition, stating that this map is a homeomorphism. It is believed that this condition holds for many naturally arising monoidal triangulated categories. Nerves of Steel is shown to hold under stratification and uniformity conditions on ${\mathbf C}$. The proof is based on a general theorem giving an explicit description of the Balmer--Favi support of the pure-injectives associated to all Serre primes of ${\sf mod}\text{-}{\mathbf C}$. It is shown that the validity of Nerves of Steel carries over from ${\mathbf C}$ to a semidirect product ${\mathbf C} \rtimes G$ with an arbitrary group $G$, and when $G$ is infinite, this provides examples of monoidal triangulated categories satisfying Nerves of Steel and whose thick ideals are not centrally generated. Important cases in which the condition is verified include the stable module categories of the coordinate rings of all finite group schemes and the Benson--Witherspoon Hopf algebras.

math.CT↗

arXiv2023

A Chinese remainder theorem and Carlson's theorem for monoidal triangulated categories

In this paper the authors prove fundamental decomposition theorems pertaining to the internal structure of monoidal triangulated categories (M$Δ$Cs). The tensor structure of an M$Δ$C enables one to view these categories like (noncommutative) rings and to attempt to extend the key results for the latter to the categorical setting. The main theorem is an analogue of the Chinese Remainder Theorem involving the Verdier quotients for coprime thick ideals. This result is used to obtain orthogonal decompositions of the extended endomorphism rings of idempotent algebra objects of M$Δ$Cs. The authors also provide topological characterizations on when an M$Δ$C contains a pair of coprime proper thick ideals, and additionally, when the latter are complementary in the sense that their intersection is contained in the prime radical of the category. As an application of the aforementioned results, the authors establish for arbitrary M$Δ$Cs a general version of Carlson's theorem on the connnectedness of supports for indecomposable objects. Examples of our results are given at the end of the paper for the derived category of schemes and for the stable module categories for finite group schemes.

math.CT↗

arXiv2023

On the spectrum and support theory of a finite tensor category

Finite tensor categories (FTCs) $\bf T$ are important generalizations of the categories of finite dimensional modules of finite dimensional Hopf algebras, which play a key role in many areas of mathematics and mathematical physics. There are two fundamentally different support theories for them: a cohomological one and a universal one based on the noncommutative Balmer spectra of their stable (triangulated) categories $\underline{\bf T}$. In this paper we introduce the key notion of the categorical center $C^\bullet_{\underline{\bf T}}$ of the cohomology ring $R^\bullet_{\underline{\bf T}}$ of an FTC, $\bf T$. This enables us to put forward a complete and detailed program for determining the exact relationship between the two support theories, based on $C^\bullet_{\underline{\bf T}}$ of the cohomology ring $R^\bullet_{\underline{\bf T}}$ of an FTC, $\bf T$. More specifically, we construct a continuous map from the noncommutative Balmer spectrum of an FTC, $\bf T$, to the $\text{Proj}$ of the categorical center $C^\bullet_{\underline{\bf T}}$, and prove that this map is surjective under a weaker finite generation assumption for $\bf T$ than the one conjectured by Etingof-Ostrik. Under stronger assumptions, we prove that (i) the map is homeomorphism and (ii) the two-sided thick ideals of $\underline{\bf T}$ are classified by the specialization closed subsets of $\text{Proj} C^\bullet_{\underline{\bf T}}$. We conjecture that both results hold for all FTCs. Many examples are presented that demonstrate how in important cases $C^\bullet_{\underline{\bf T}}$ arises as a fixed point subring of $R^\bullet_{\underline{\bf T}}$ and how the two-sided thick ideals of $\underline{\bf T}$ are determined in a uniform fashion. The majority of our results are proved in the greater generality of monoidal triangulated categories.

math.CT↗

arXiv2021

Noncommutative tensor triangular geometry and the tensor product property for support maps

The problem of whether the cohomological support map of a finite dimensional Hopf algebra has the tensor product property has attracted a lot of attention following the earlier developments on representations of finite group schemes. Many authors have focussed on concrete situations where positive and negative results have been obtained by direct arguments. In this paper we demonstrate that it is natural to study questions involving the tensor product property in the broader setting of a monoidal triangulated category. We give an intrinsic characterization by proving that the tensor product property for the universal support datum is equivalent to complete primeness of the categorical spectrum. From these results one obtains information for other support data, including the cohomological one. Two theorems are proved giving compete primeness and non-complete primeness in certain general settings. As an illustration of the methods, we give a proof of a recent conjecture of Negron and Pevtsova on the tensor product property for the cohomological support maps for the small quantum Borel algebras for all complex simple Lie algebras.

math.CT↗

arXiv2021

Noncommutative tensor triangular geometry

We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$Δ$Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M$Δ$C, ${\bf K}$, and then to associate to ${\bf K}$ a topological space--the Balmer spectrum $\operatorname{Spc} {\bf K}$. We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that $\operatorname{Spc} {\bf K}$ is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M$Δ$C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of ${\bf K}$, which in turn can be applied to classify the thick two-sided ideals and $\operatorname{Spc} {\bf K}$. As a special case, our approach can be applied to the stable module categories of arbitrary finite dimensional Hopf algebras that are not necessarily cocommutative (or quasitriangular). We illustrate the general theorems with classifications of the Balmer spectra and thick two-sided/right ideals for the stable module categories of all small quantum groups for Borel subalgebras, and classifications of the Balmer spectra and thick two-sided ideals of Hopf algebras studied by Benson and Witherspoon.

math.CT↗

arXiv2018

Integral operators, bispectrality and growth of Fourier algebras

In the mid 80's it was conjectured that every bispectral meromorphic function $ψ(x,y)$ gives rise to an integral operator $K_ψ(x,y)$ which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions $ψ(x,y)$ where the commuting differential operator is of order $\leq 6$. We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second and first order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel $K_ψ(x,y)$ and a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.

math.CA↗

arXiv2017

Azumaya loci and discriminant ideals of PI algebras

We prove that, under mild assumptions, for all positive integers $\ell$, the zero set of the discriminant ideal $D_{\ell}(R/Z(R); tr)$ of a prime polynomial identity (PI) algebra $R$ coincides with the zero set of the modified discriminant ideal $MD_{\ell}(R/Z(R); tr)$ of $R$. Furthermore, we prove that, when $\ell$ is the square of the PI-degree of $R$, this zero set is precisely the complement of the Azumaya locus of $R$. This description is used to classify the Azumaya loci of the mutiparameter quantized Weyl algebras at roots of unity. As another application, we prove that the zero set of the top discriminant ideal of a prime PI algebra $R$ coincides with the singular locus of the center of $R$, provided that the discriminant ideal has height at least 2, $R$ has finite global dimension and $R$ is a Cohen-Macaulay module over its center.

math.RA↗