arXiv subjects

Changchang Xi

Publications and source records attributed to Changchang Xi.

At least 37 records · Page 2Linked to original sources

arXiv2020

Structure of centralizer algebras

Given an $n\times n$ matrix $c$ over a unitary ring $R$, the centralizer of $c$ in the full $n\times n$ matrix ring $M_n(R)$ is called a principal centralizer matrix ring, denoted by $S_n(c,R)$. We investigate its structure and prove: $(1)$ If $c$ is an invertible matrix with a $c$-free point, or if $R$ has no zero-divisors and $c$ is a Jordan-similar matrix with all eigenvalues in the center of $R$, then $M_n(R)$ is a separable Frobenius extension of $S_{n}(c,R)$ in the sense of Kasch. $(2)$ If $R$ is an integral domain and $c$ is a Jordan-similar matrix, then $S_n(c,R)$ is a cellular $R$-algebra in the sense of Graham and Lehrer. In particular, if $R$ is an algebraically closed field and $c$ is an arbitrary matrix in $M_n(R)$, then $S_n(c,R)$ is always a cellular algebra, and the extension $S_n(c,R)\subseteq M_n(R)$ is always a separable Frobenius extension.

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arXiv2020

Representation dimensions linked by Frobenius bimodules with applications to group algebras

We establish relations between representation dimensions of two algebras connected by a Frobenius bimodule or extension. Consequently, upper bounds and equality formulas for representation dimensions of group algebras, symmetric separably equivalent algebras and crossed products are obtained. Particularly, for any subgroup H of a finite group G, if [G:H] is invertible in a field, then the representation dimensions of the group algebras of G and H over the field are the same.

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arXiv2019

Cellularity of centrosymmetric matrix algebras and Frobenius extensions

Centrosymmetric matrices of order $n$ over an arbitrary algebra $R$ form a subalgebra of the full $n\times n$ matrix algebra over $R$. It is called the centrosymmetric matrix algebra of order $n$ over $R$ and denoted by $S_n(R)$. We prove (1) $S_n(R)$ is Morita equivalent to $S_2(R)$ if $n$ is even, and to $S_3(R)$ if $n\ge 3$ is odd; (2) the full $n\times n$ matrix algebra over $R$ is a separable Frobenius extension of $S_n(R)$; and (3) if $R$ is a commutative ring, then $S_n(R)$ is a cellular $R$-algebra in the sense of Graham-Lehrer for all $n\ge 1$.

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arXiv2019

Frobenius bimodules and flat-dominant dimensions

We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture on stable equivalences. We show that the Frobenius parts of Frobenius extensions are again Frobenius extensions. Further, let $A$ and $B$ be finite-dimensional algebras over a field $k$, and let $\dm(_AX)$ stand for the dominant dimension of an $A$-module $X$. If $_BM_A$ is a Frobenius bimodule, then $\dm(A)\le \dm(_BM)$ and $\dm(B)\le \dm(_A\Hom_B(M, B))$. In particular, if $B\subseteq A$ is a left-split (or right-split) Frobenius extension, then $\dm(A)=\dm(B)$. These results are applied to calculate flat-dominant dimensions of a number of algebras: shew group algebras, stably equivalent algebras, trivial extensions and Markov extensions. Finally, we prove that the universal (quantised) enveloping algebras of semisimple Lie algebras are $QF$-$3$ rings in the sense of Morita.

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arXiv2018

Derived decompositions of abelian categories I

Derived decompositions of abelian categories are introduced in internal terms of abelian subcategories to construct semi-orthogonal decompositions (or Bousfield localizations, or hereditary torsion pairs) in various derived categories of abelian categories. We give a sufficient condition for arbitrary abelian categories to have such derived decompositions and show that it is also necessary for abelian categories with enough projectives and injectives. For bounded derived categories, we describe which semi-orthogonal decompositions are determined by derived decompositions. The necessary and sufficient condition is then applied to the module categories of rings: localizing subcategories, homological ring epimorphisms, commutative noetherian rings and nonsingular rings. Moreover, for a commutative noetherian ring of Krull dimension at most $1$, a derived stratification of its module category is established.

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arXiv2017

An application of Nakayama functor in representation stability theory

Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI$_G$ and VI$_q$, and answer positively an open question of Nagpal on representation stability theory.

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arXiv2017

Milnor squares of algebras, I: derived equivalences

Derived equivalences for Artin algebras (and almost $ν$-stable derived equivalences for finite-dimensional algebras) are constructed from Milnor squares of algebras. Particularly, three operations of gluing vertices, unifying arrows and identifying socle elements on derived equivalent algebras are presented to produce new derived equivalences of the resulting algebras from the given ones. As a byproduct, we construct a series of derived equivalences, showing that derived equivalences may change Frobenius type of algebras in general, though both tilting procedure and almost $ν$-stable derived equivalences do preserve Frobenius type of algebras.

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arXiv2015

Finitistic dimension conjecture and radical-power extensions

The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field should have finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this paper, we investigate those extensions of Artin algebras in which some radical-power of smaller algebras is a one-sided ideal in bigger algebras. Our results, however, are formulated more generally for an arbitrary ideal: Let $B\subseteq A$ be an extension of Artin algebras and $I$ an ideal of $B$ such that the full subcategory of $B/I$-modules is $B$-syzygy-finite. Then: (1) If the extension is right-bounded (for example, proj.dim$(A_B)$ is finite), $I A\, rad(B)\subseteq B$ and fin.dim$(A)$ is finite, then fin.dim$(B)$ is finite. (2) If $I\, rad(B)$ is a left ideal of $A$ and $A$ is torsionless-finite, then fin.dim$(B)$ is finite. Particularly, if $I$ is specified to a power of the radical of $B$, then our results not only generalize some ones in the literature (see Corollaries 1.3 and 1.4), but also provide some completely new ways to detect algebras of finite finitistic dimensions.

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arXiv2015

Dominant dimensions, derived equivalences and tilting modules

The Nakayama conjecture states that an algebra of infinite dominant dimension should be self-injective. Motivated by understanding this conjecture in the context of derived categories, we study dominant dimensions of algebras under derived equivalences induced by tilting modules, specifically, the infinity of dominant dimensions under tilting procedure. We first give a new method to produce derived equivalences from relatively exact sequences, and then establish relationships and lower bounds of dominant dimensions for derived equivalences induced by tilting modules. Particularly, we show that under a sufficient condition the infinity of dominant dimensions can be preserved by tilting, and get not only a class of derived equivalences between two algebras such that one of them is a Morita algebra in the sense of Kerner-Yamagata and the other is not, but also the first counterexample to a question whether generalized symmetric algebras are closed under derived equivalences.

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arXiv2014

Derived equivalences and stable equivalences of Morita type, II

Motivated by understanding the Broué's abelian defect group conjecture from algebraic point of view, we consider the question of how to lift a stable equivalence of Morita type between arbitrary finite dimensional algebras to a derived equivalence. In this paper, we present a machinery to solve this question for a class of stable equivalences of Morita type. In particular, we show that every stable equivalence of Morita type between Frobenius-finite algebras over an algebraically closed field can be lifted to a derived equivalence. %Thus Frobenius-finite algebras share many common %invariants of both derived equivalences and stable equivalences. Especially, Auslander-Reiten conjecrure is true for stable equivalences of Morita type between Frobenius-finite algebras without semisimple direct summands. Examples of such a class of algebras are abundant, including Auslander algebras, cluster-tilted algebras and certain Frobenius extensions. As a byproduct of our methods, we further show that, for a Nakayama-stable idempotent element $e$ in an algebra $A$ over an arbitrary field, each tilting complex over $eAe$ can be extended to a tilting complex over $A$ that induces an almost $ν$-stable derived equivalence studied in the first paper of this series. Moreover, we demonstrate that our techniques are applicable to verify the Broué's abelian defect group conjecture for several cases mentioned by Okuyama.

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arXiv2014

Recollements of derived categories II: Algebraic K-theory

For a recollement of derived module categories of rings, we provide sufficient conditions to guarantee the additivity formula of higher algebraic K-groups of the rings involved, and establish a long Mayer-Vietoris exact sequence of higher algebraic K-groups for homological exact contexts introduced in the first paper of this series. Our results are then applied to recollements induced from homological ring epimorphisms and noncommutative localizations. Consequently, we get an infinitely long Mayer-Vietoris exact sequence of K-theory for Milnor squares, re-obtain a result of Karoubi on localizations and a result on generalized free products pioneered by Waldhausen and developed by Neeman and Ranicki. In particular, we describe algebraic $K$-groups of the free product of two groups over a regular coherent ring as the ones of the noncommutative tensor product of an exact context. This yields a new description of algebraic $K$-theory of infinite dihedral group.

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arXiv2014

Recollements of derived categories III: finitistic dimensions

In this paper, we study homological dimensions of algebras linked by recollements of derived module categories, and establish a series of new upper bounds and relationships among their finitistic or global dimensions. This is closely related to a longstanding conjecture, the finitistic dimension conjecture, in representation theory and homological algebra. Further, we apply our results to a series of situations of particular interest: exact contexts, ring extensions, trivial extensions, pullbacks of rings, and algebras induced from Auslander-Reiten sequences. In particular, we not only extend and amplify Happel's reduction techniques for finitistic dimenson conjecture to more general contexts, but also generalise some recent results in the literature.

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arXiv2014

Recollements of derived categories I: Exact contexts

Recollements were introduced originally by Beilinson, Bernstein and Deligne to study the derived categories of perverse sheaves, and nowadays become very powerful in understanding relationship among three algebraic, geometric or topological objects. The purpose of this series of papers is to study recollements in terms of derived module categories and homological ring epimorphisms, and then to apply our results to both representation theory and algebraic K-theory. In this paper we present a new and systematic method to construct recollements of derived module categories. For this aim, we introduce a new ring structure, called the noncommutative tensor product, and give necessary and sufficient conditions for noncommutative localizations which appears often in representation theory, topology and K-theory, to be homological. The input of our machinery is an exact context which can be easily obtained from a rigid morphism that exists in very general circumstances. The output is a recollement of derived module categories of rings in which the noncommutative tensor product of an exact context plays a crucial role. Thus we obtain a large variety of new recollements from commutative and noncommutative localizations, ring epimorphisms and extensions.

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arXiv2012

Algebraic K-theory of endomorphism rings

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let $Y\ra X$ be a covariant morphism of objects in ${\mathcal C}$. Then $K_n\big(_{\mathcal C}(X\oplus Y)\big)\simeq K_n\big(_{{\mathcal C},Y}(X)\big)\oplus K_n\big(_{\mathcal C}(Y)\big)$ for all $1\le n\in \mathbb{N}$, where $_{{\mathcal C},Y}(X)$ is the quotient ring of the endomorphism ring $_{\mathcal C}(X)$ of $X$ modulo the ideal generated by all those endomorphisms of $X$ which factorize through $Y$. Moreover, let $R$ be a ring with identity, and let $e$ be an idempotent element in $R$. If $J:=ReR$ is homological and $_RJ$ has a finite projective resolution by finitely generated projective $R$-modules, then $K_n(R)\simeq K_n(R/J)\oplus K_n(eRe)$ for all $n\in \mathbb{N}$. This reduces calculations of the higher algebraic $K$-groups of $R$ to those of the quotient ring $R/J$ and the corner ring $eRe$, and can be applied to a large variety of rings: Standardly stratified rings, hereditary orders, affine cellular algebras and extended affine Hecke algebras of type $\tilde{A}$.

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arXiv2012

Ringel modules and homological subcategories

Given a good $n$-tilting module $T$ over a ring $A$, let $B$ be the endomorphism ring of $T$, it is an open question whether the kernel of the left-derived functor $T\otimes^L_B-$ between the derived module categories of $B$ and $A$ could be realized as the derived module category of a ring $C$ via a ring epimorphism $B\rightarrow C$ for $n\ge 2$. In this paper, we first provide a uniform way to deal with the above question both for tilting and cotilting modules by considering a new class of modules called Ringel modules, and then give criterions for the kernel of $T\otimes^L_B-$ to be equivalent to the derived module category of a ring $C$ with a ring epimorphism $B\rightarrow C$. Using these characterizations, we display both a positive example of $n$-tilting modules from noncommutative algebra, and a counterexample of $n$-tilting modules from commutative algebra to show that, in general, the open question may have a negative answer. As another application of our methods, we consider the dual question for cotilting modules, and get corresponding criterions and counterexamples. The case of cotilting modules, however, is much more complicated than the case of tilting modules.

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arXiv2011

Higher algebraic $K$-groups and $\mathcal D$-split sequences

In this paper, we use $\mathcal D$-split sequences and derived equivalences to provide formulas for calculation of higher algebraic $K$-groups (or mod-$p$ $K$-groups) of certain matrix subrings which cover tiled orders, rings related to chains of Glaz-Vasconcelos ideals, and some other classes of rings. In our results, we do not assume any homological requirements on rings and ideals under investigation, and therefore extend sharply many existing results of this type in the algebraic $K$-theory literature to a more general context.

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arXiv2011

Stratifications of derived categories from tilting modules over tame hereditary algebras

In this paper, we consider the endomorphism algebras of infinitely generated tilting modules of the form $R_{\mathcal U}\oplus R_{\mathcal U}/R$ over tame hereditary $k$-algebras $R$ with $k$ an arbitrary field, where $R_{\mathcal{U}}$ is the universal localization of $R$ at an arbitrary set $\mathcal{U}$ of simple regular $R$-modules, and show that the derived module category of $\End_R(R_{\mathcal U}\oplus R_{\mathcal U}/R)$ is a recollement of the derived module category $\D{R}$ of $R$ and the derived module category $\D{{\mathbb A}_{\mathcal{U}}}$ of the adèle ring ${\mathbb A}_{\mathcal{U}}$ associated with $\mathcal{U}$. When $k$ is an algebraically closed field, the ring ${\mathbb A}_{\mathcal{U}}$ can be precisely described in terms of Laurent power series ring $k((x))$ over $k$. Moreover, if $\mathcal U$ is a union of finitely many cliques, we give two different stratifications of the derived category of $\End_R(R_{\mathcal U}\oplus R_{\mathcal U}/R)$ by derived categories of rings, such that the two stratifications are of different finite lengths.

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arXiv2011

Derived equivalences from cohomological approximations, and mutations of $Φ$-Yoneda algebras

In this article, a new construction of derived equivalences is given. It relates different endomorphism rings and more generally cohomological endomorphism rings - including higher extensions - of objects in triangulated categories. These objects need to be connected by certain universal maps that are cohomological approximations and that exist in very general circumstances. The construction turns out to be applicable in a wide variety of situations, covering finite dimensional algebras as well as certain infinite dimensional algebras, Frobenius categories and $n$-Calabi-Yau categories.

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