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, 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) ?

    Note: For some backgrounds of this problem, please see Proposition 3.13(2) of the paper XH-2.
(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 amost 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.
(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 recent 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 recently. The answer is YES.

  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 ?
  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 recent 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: October 5, 2008
Copyright © Changchang Xi