公开问题
  本人对数学很感兴趣，特别是对代数学兴趣浓厚。 热忱欢迎与志同道合的各位学者进行有意义的数学讨论和交流。真诚欢迎各位前辈、 老师、同行和学生批评指导。这里介绍一些在学习和研究的过程中遇到的还没有解决的问题, 愿与朋友们共同探讨。
(Beijing, December 2023)
    Given a square matrix c, describe all square matrices d (not necessarily have the same size) such that the centralizer algebras of these d in the full matrix algebras are derived equivalent to the one of the given c.

    More generally, given a commutative subalgebra C of the n by n full matrix algebra over a field R, and a commutative subalgebra D of the m by m full matrix algebra over R, find conditions on C and D such that the centralizers of C and D in their matrix algebras are derived equivalent.

    For a background of the question, click here

(Beijing, November 2023)
    Given two square matrices (not necessarily have the same size), when are their centralizers in the full matrix algebras stably equivalent?

    For a background of the question, click here

(Beijing, November 2022)
    Recall that an abelian category is complete (respectively, cocomplete) if it has products (respectively, coproducts) indexed over sets; and bicomplete if it is both complete and cocomplete. Let n be a nonnegative integer, and let A be a bicomplete abelian category. An additive full subcategory B of A is said to be n-symmetric if (i) B is closed under extensions, products and coproducts, and (ii) for any exact sequence of n+3 terms in the category A , if the middle n+1 terms lie in B, then the reminder two terms are in B.

    Given a right A-module T with finite projective dimension n, define E to be the full subcategory of A-Mod consisting of all left A-module X such that the tensor product of T with X over A as well as all higher torsion groups vanish.
    (1) Is E an n-symmetric subcategory of A-Mod?
    (2) Is every n-symmetric subcategory of A-Mod obtained as in (1)?

    For a background of the questions, click here

(Beijing, July 2022)
    If there is a derived equivalence between the bounded derived module categories of noetherian rings A and B, then the delooping level of A is finite if and only if so is the one of B, that is, derived equivalences preserve finiteness of delooping levels of noetherian rings.

    For a background of the question, click here

(Beijing, March 2022)
    This is to introduce a conjecture by L'vov and Kaplansky, proposed in about 1993, which states the following:

    Let p be a multilinear polynomial. Then the set of values of p on the d by d matrix algebra over an infinite field k is a vector space.

    For a survey of this conecture we refer to [Alexei Kanel-Belov; Sergey Malev; Louis Rowen; Roman Yavich, Evaluations of noncommutative polynomials on algebras: methods and problems, and the L'vov-Kaplansky conjecture. SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 071, 61 pp ].

(Beijing, February 2022)
    Let B be a subalgebra of an algebra A, we consider them as an extension of algebras. Suppose this extension is semisimple, that is, every A-module X is a direct summand of the tensor product (over B) of A with the B-module X. Is the global dimension of A less than or equal to the one of B ?

    Note (April 26, 2022): The question should read as follows: What is the relation between global dimensions of A and B?
    If we take a universal localization A of an algebra B of global dimension n, then we may get an injective ring epimorphism from B to A. This is a semisimple extension, but A may have arbitrary global dimension. So one cannot get positive answer to the question !
(Beijing, March 2021)
    Given a partition p of a natural number n, we can define a partition polynomial fp(x) in Z[x] and say that two partitions p and q of n are equivalent if their partition polynomials are equal.

  1. For which partition p does its equivalence class just contain one element ?

  2. For which partitions p are the partition polynomials fp(x) irreducible over the field of rational numbers?

    For a background of the questions, click here

(Beijing, November 2020)
  1. Given a permutation p in the symmetric group of degree n, there is associated to p a permutation matrix P of degree n, which is a 0-1 matrix. For any ring R with identity, we may consider the centralizer of P in the full n by n matrix ring, denoted by S(n,p,R). Let p and q be two permutations of degree n. When are the centralizer rings S(n,p,R) and S(n,q,R) Morita (or derived) equivalent?

    Note (May 8, 2023): This question has some progresses recently. See [LX]

  2. Is the full n by n matrix ring over a ring R always a Frobenius extension of the centralizer subring of any n by n matrix c in the matrix ring?

    For backgrounds of the questions, click here , see also xy-laa

    Note (March 28, 2021): Partial answer to the question 1 is given in [XZ-2]

    Note (April 29, 2021): Answer to question 2 is YES for R a field. See [XZ-2]

    Note (August 12, 2023): Answer to question 2 is No for arbitrary ring R. See [Zhu]

(Beijing, February 2020)
  1. Given a finite-dimensional algebra A with a nilpotent ideal I and that A is a split extension of I by the quotient A/I, if A/I is derived simple, then we conjecture that A is derived simple.

  2. Given a finite-dimensional algebra A which is derived simple, is the dual extension of A again derived simple?

    Recall that an algebra is called derived simple if its bounded derived module category has no non-trivial recollements of derived module categories.

(Beijing, December 2019)
  1. Let A be the n by n matrix algebra over a field k. Find all subalgebras B of A such that A is a Frobenius extension of B.

    Recall that a subring B of a ring A with the same identity is called a Frobenius extension if A is a finitely generated projective B-module such that the A-B-bimodule A is isomorphic to Hom_B(A,B) as a bimodule.

    For a background of the question, please click here , see also SCM2021

(Beijing, December 2017)
  1. For a given algebra A, we consider the class C of algebras which are derived equivalent to A. Are the domiant dimensions of algebras in C upper (or lower) bounded by a function depending on the number of simple moduless and the length of tilting complexes over A?

 (Beijing, December 2016)
  1. Given two symmetric algebras A and B which are presented by quivers with relations. Let x and y be elements in the socles of A and B, respectively. Then we may define a new algebra C in which x and y are identified (see Subsection 4.3 and Proposition 5.4 for details). Suppose that A and B are symmetric algebras. Is C also symmetric ? We do know that C is self-injective.

    For a background of the above question, please click here

(Beijing, September 2015)
  1. Let B be a subalgebra of A such that the cubes of the radicals of B and A are equal. If gl.dim(A) is at most three, can you prove that the finitistic dimension of B is finite? Or more generally, Let B be a subalgebra of A such that rad^s(A)=rad^s(B) and gl.dim(A) is at most s for a positive integer s. Can we show that fin.dim(B) is finite?

    For a background of the above question, please click here

(Beijing, October 2014)
  1. Given a ring R with two distinct ideals I and J, under which coditions are the two quotient rings R/I and R/J derived equivalent?

    This is motivated by the following consideration: Given an AR-sequence Z -> Y -> X, we know that End(Z+Y) is derived equivalent to End(Y+X). If we factorize the ideals generated by the projection from Z+Y to Z and the projection from Y+X to X, respectively, then we get two quotient algebras End(Y)/(Y,Z,Y) and End(Y)/(Y,X,Y), where (Y,Z,Y) meas the set of all endomorphism of Y which factorise through the module Z. We would like to kow when they are derived equivalent.

  2. Let A and B be finite-dimensional algebras over a field. Suppose that A and B are derived equivalent. Is it true that the dominant dimension of A is finite if and only if so is the dominant dimnsion of B?

    For a background of the above questions, please click here , see also a question posed in 2012

 (Beijing, June 2013, the following open questions were mentioned in a talk at the mini-workshop: Localizing and tilting in abelain and triangulated categories)
  1. Parametrize or determine all homological subcategories of the derived module category of a ring R.

  2. Find a 2-tilting R-module over an algebra R, which is not built from a finitely generated tilting module, such that the kernel of the derived tensor functor induecd from T is homological.

    January, 2017 : There is a method to get such tilting modules (see CX ).

  3. A ring R is called absolutely derived simple if the derived module category of R has only trivial homological subcategories. Which rings are absolutely derived simple?

    For a background of the above questions, please click here .



(Beijing, December 2012)
  1. Let f be a homological ring epimorphism from R to S such that the left R-module S has an infinite projective resolution by finitely generated projective R-modules.Is the n-th algebraic K-groups of the ring S always a direct summand of the n-th algebraic K-group of the ring R?

    For a background of this question, please click here.

  2. Let R be a ring with identity, and let I be an ideal of R such that the square of I is zero. Let S be the ring of 2 by 2 matrix ring with main diagonal being R and other entries are I. Is there any formula for the algebraic K-groups of S in terms of the algebraic K-groups of R and rings produced from I (for example, R/I)?

    For a background of this question, please click here.

  3. Let T be an n-tilting R-module, and let S be the endomorphism ring of T. Find more conditions for T so that the kernel of the total left-derived functor of the tensor functor defined by the left R, right S-module T is equivalent to a derived module category of a ring.

    For a background of this question, please click here.

  4. Let A and B are Artin algebras. Suppose that A and B are derived equivalent. What is the relationship between the dominant dimensions of A and B?

    For finitistic dimensions, there is a bound, see [PX].

    January 18, 2018: Recently, there are some discussions on the question, click here , see also papers by Rene Marczinzik, and others.

  5. Let A be a finite dimensional algebra, and let e be an idempotent element in A. We conjecture that the representation dimension of eAe is less than or equal to the representation dimension of A.

    For a background of this question, please click here to see the last section of an unpublished paper.

(Beijing, October 2011)
  1. Suppose that A and B are two indecomposable finite dimensional algebras over a field. If they are stably equivalent of Morita type, is it true that the n-th algebraic K-groups of A is isomorphisc with the n-th algebraic K-group of B for all n>0 ? Here by K-theory we mean the algebraic K-theory in the sense of Quillen.
(Beijing, July 2011)
  1. Let R be a ring with identity, and let I, J be two ideals in R. We denote by K_n(R) the n-th algebraic K-group of R. Let S be the (m by m) marix ring with the entries R in the principal diagonal, I above the principal diagonal, and J below the principal diagonal. Is it true that K_n(S) is isomorphic to a direct sum of K_n(R) and m-1 copies of K_n(R/(IJ+JI)) ?

    Note: For some backgrounds of this problem, please click here.

  2. Let C be a triangulated category and F an auto-equivalence of C. Suppose X is an admissible subset of the integers. Then we may define an X-orbit category of C with respect to F, denoted by C(F,X). Is C(F,X) a triangulated category ? If not, when is it a triangulated category ?

    Note: For some backgrounds of this problem, please click here. and here (Section 2.2).

 (Beijing, November 2010)
  1. Is there a ring R with identity such that its derived module category has two stratifications by derived module categories of rings, so that one of them is of finite length, and the other is of infinite length ?

    Note: For some backgrounds of this problem, please click here and here
(Beijing, January 2010)
  1. Let C be the bounded derived category of an artin algebra, and X a complex in C. Suppose that S is an admissible subset of the set of natural numbers. When is the S-Auslander-Yoneda algebra of X self-injective ?

    Note: For some backgrounds of this problem, please click here.
(Beijing, September 2009)
  1. Left-right question for the strong no loop conjecture: Suppose A is an Artin algebra and S is a simple A-module. The strong no loop conjecture says that if S has non-trivial self extension, then the projective dimension of S should be infinite. Suppose the strong no loop conjecture is true for each simple A-moule. Is it possible to show that the strong no loop conjecture is also true for each simple module over the opposite algebra of A ?

(Beijing, February 2009)
  1. Suppose an Artin algebra A is an extension of an Artin algebra B such that rad(B) is a left ideal in A. We have proved that if the relative global dimension of A related to B is zero, then the validity of the finitistic dimension conjecture for A implies the validity of the conjecture for B. Our question is: what could we say about this statement if the relative global dimension of A related to B is at mot 1 ?

    Note: The affirmative answer to this question would imply that the finitistic dimension conjecture is true. For some backgrounds, please click here.
(Beijing, September 2008)
  1. Given two indecomposable modules X and Y over an Artin algebra A, suppose Y is a direct summand of the middle term of the AR-sequence starting at X. If X is isomorphic with TrD(X), is it true that Y is isomorphic to TrD(Y) ? (or equivalently, is an AR-component containing a homogeneous module homogeneous ?)

    Note: For some backgrounds of this problem, please see Proposition 3.12(2) of the paper XH-2. When we write TrD(Y), we always mean that the module TrD(Y) exists, that is, TrD(Y) is non-zero.

    Note (March 18, 2009): Piotr Malicki points out that the algebra k[x]/(x^n) is a counterexample to the question: is an AR-component containing a homogeneous module homogeneous ? This example shows also that the first question is not equivalent to the second question. So the second question should be modified: Is an infinite AR-component containing a homogeneous module homogeneous ? or: Is a stable AR-component containg a homogeneous module homogeneous ?


    Note (March 31, 2009): Piotr Malicki points out that the whole problem was in fact solved completely by M.Hoshino in the paper: DTr-invariant modules. Tsukuba J. Math. 7 (1983), no. 2, 205--214.
(Beijing, June 2008)
  1. Given an Artin algebra A. Could you find some methods to construct another Artin algebra B such that there is an almost N-stable derived equivalence between A and B ? where N is the Nakayama functor. (A general question is: find all such algebras B for a given A.)

    Note: For some backgrounds of this problem, please click here.
(Beijing, April 2008)
  1. Suppose A and B are representation-finite self-injective Artin algebras. If the Auslander algebras of A and B are derived-equivalent, are the algebras A and B themselves already derived-equivalent ?

    Note: For some backgrounds of this problem, please click here.

    Note (December 22, 2012): Recently, M. Fang has some progress on this problem.
(Beijing, January 2008)
  1. Given an algebra A, how to find all those algebras B that are stably equivalent of Morita type to A ?
(Beijing, September 2007)
  1. If B is a subalgebra of an Artin algebra A with the same identity, we call A an extension of B, and denote by P(A,B) the full subcategory of all finitely generated relatively projective A-modules with respect to B. Under which conditions is P(A,B) closed under taking A-syzygies ?

    Note: For a background of this question, please click here.

  2. Suppose A is an extension of B. Find a criterion for the extension to have relative global dimension at most one.
(Beijing, June 2006)
  1. Suppose two artin algebras A and B are stably equivalent of Morita type. Are the determinants of Cartan matrcies of A and B equal ?

    Note: For a background of this question, please click here.
(Beijing, December 2005)
  1. Are there two finite-dimensional algebras A and B with the following properties: (1) they are stably equivalent of Morita type; (2) If M and N define a stable equivelence of Morita type between them, then one (or both) of the two natural pairs of tensor functors defined by M and N are not adjoint pairs ?

    Note ( September 2006): This problem is solved by Dugas and Martinez-Villa in a current paper.
(Beijing, June 2004)
The following question was presented in the 4-th China-Japan-Korea International Syposium on Ring Theory (24-28 Jue 2004)
  1. Let A,B,C,D be algebras such that D,C and B are subalgebras of C,B and A, respectively. Suppose that the radicals of D, C and B are left ideals in C,B and A, respectively. If A is representation-finite, is the finititsic dimension of D finite ? (More generally, consider the case of more than 4 algebras.)
(Beijing, May 2004)
The following questions were presented at the Workshop on Representations and Structures of Algebras (17-21 May 2004):
  1. Suppose that A and B are representation-finite. If the Auslander algebras of A and B are stably equivalent of Morita type, are A and B stably equivalent of Morita type, too ?

    Note (October 2006): This problem is solved currently. The answer is YES. For a proof in details, click here

  2. Is there a series of infinitely many algebras such that they have the same dimension and are stably equivalent of Morita type to each other, but they are pairwise non-Morita equivalent ?

    Note (December 2010): This problem is completely solved for derived equivalences in the paper [Chen, Pan and Xi].

  3. [J.Rickard]: Suppose that A,B,C and D are indecomposable algebras. If A and B are stably equivalent of Morita type and C and D are stably equivalent of Morita type, are the tensor products of A and C, and B and D stably equivalent of Morita type ?
  4. [M.Auslander]: If A and B are stably equivalent of Morita type, are the numbers of non-projective simple modules over A and B equal ?

 (Beijing, March 2004)
  1. If A and B are stably equivalent of Morita type, are the n-th Hochschild cohomology groups of A and B isomorphic for all positive number n ?

    Note: This is true for self-injective algebras proved by Pogorza`ly , see also a paper of Liu and Xi. For Hochschild homology groups this was proved to be true for general algebras by Liu and Xi.
    Note (December 2005): For non-self-injective algebras, a partial answer to this question is found recently by Xi.
    Note (September 2006): Using the result in Xi. and a current result of Dugas and Martinez-Villa, this question is completely solved. The answer is YES.
(Beijing, Jan. 2003)
  1. Let C and B be two representation-finite algebras over a field. Does the trivially twisted extension of C and B at S has the representation dimension of at most 3 ?
  2. Let A be an artin algebra and J an ideal in A such that the cube of J vanishes. If A/J is representation-finite, is the finitistic dimension conjecture true for A ?
  3. Let A be an artin algebra and J an ideal in A such that the square of J vanishes. If A/J is representation-finite, does the algebra A has the representation dimension at most 3 ?
  4. Let A be an artin algebra and let e be an idempotent element in A. We conjecture that the representation dimension of eAe is less than or equal to that of A.

    Note: For some backgrounds of the first three problems, please see a paper of Xi.
Last modified: May 11, 2024
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