Open problems

(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.

(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?

(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?

(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.

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?

(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?

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)?

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.

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].

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)) ?

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 ?

(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 ?

(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 ?

(Beijing, September 2009)
1. Left-right question for 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 A. Is it possible to show that the strong no loop conjecture is also true for 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 ?

(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 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.)

(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 ?

(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 ?

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 ?
(Beijing, December 2005)
1. 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 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 not Morita equivalent ?

Note (December 2010): This problem is completely solved 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 currently 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.