公开问题

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本人对数学很感兴趣，特别是对代数学兴趣很强。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。
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(Beijing, June 2006)

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)

Are there two 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)

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):

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.


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 ?

[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 ?
[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)

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)
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 ?
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 ?
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 ?
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.
这是他的主页http://math.bnu.edu.cn/~ccxi/Problems.php
有兴趣的网友不妨做一做这些题目,有几个题目我比较感兴趣.

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