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Results tagged with homological-algebra
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user 468
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
32
votes
3
answers
4k
views
Replacing triangulated categories with something better
Gelfand and Manin in their 1988 book on homological algebra write that the non-functoriality of cones means that "something is going wrong in the axioms of a triangulated category. Unfortunately at th …
- 6,302
6
votes
Accepted
Why is the representation dimension of an Artin algebra never equal to 1?
First of all, you have to assume that $A$ is non-semi-simple. For a semi-simple Artin algebra, the representation dimension is defined to be 1.
For a non-semi-simple algebra, the representation di …
- 6,302
4
votes
History of an open problem on partial tilting modules
There is relevant information here, including a statement of the conjecture (as Conjecture 5.1).
http://www.math.uni-bonn.de/people/schroer/fd-problems-files/FD-RigidModulesConj.pdf
That preprint …
- 6,302
4
votes
Accepted
Finding exceptional regular representations of $\tilde{D}_4$ efficiently
The AR quiver of the regular representations of an affine quiver consists of infinitely many "tubes". A tube of rank $r$ has $r$ modules on what you call the border. Let me number them $B_1, B_2, \dot …
- 6,302
2
votes
Bijection on tilting modules
I am going to give a negative answer for the first question, under a stronger notion of canonicity.
The approach I want to take is to consider the poset of tilting modules. They are ordered by in …
- 6,302
1
vote
Accepted
Representation dimension of a special algebra
I don't think (*) is correctly copied from the paper. The corresponding claim in the paper is that every morphism from an indecomposable summand of $M$ except for the identity morphism from $T$ to $T …
- 6,302