Timeline for Auslander-Reiten theory for Gorenstein algebras
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 29, 2017 at 16:26 | vote | accept | Mare | ||
| Jun 29, 2017 at 16:03 | comment | added | Matthew Pressland | @Mare Further to my previous comment, I now do have an example of a finite dimensional algebra for which the category of Gorenstein projectives is not the mesh category of its AR-quiver: see Example 9.2 in arxiv.org/pdf/1702.05352.pdf. (The AR-quiver is drawn in Figure 2 on page 51, and the caption explicitly points out this property!) | |
| Aug 25, 2016 at 23:30 | comment | added | Alex Dugas | Nice answer. In the Kleinian singularity examples, I believe the extra mesh relation at R comes from the "fundamental sequence", which here takes the form $0\rightarrow R \rightarrow E \stackrel{p}{\rightarrow} R$. Moreover $p$ is a right almost split map with image equal to the maximal ideal of R. (See Ch. 11 in Yoshino's book on CM modules for more info.) | |
| Jun 15, 2016 at 9:41 | comment | added | Matthew Pressland | OK - in that case I don't know an example of this failing. (Although the problem in the Kleinian singularity case isn't really to do with finiteness, because the algebras are finitely generated over a Krull dimension $1$ ring - the problem might be to do with this Krull dimension being non-zero, but if it isn't then I would expect similar examples to exist in the case of finite dimensional algebras over fields.) | |
| Jun 15, 2016 at 9:13 | comment | added | Mare | Thanks for the edit, but Im mainly interested in algebras with finite dimension. | |
| Jun 15, 2016 at 9:07 | history | edited | Matthew Pressland | CC BY-SA 3.0 |
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| Jun 15, 2016 at 8:49 | comment | added | Matthew Pressland | Ah, great - thanks for clarifying. It is in fact not true that the "GP"-Auslander algebra is given by the AR-quiver of the category of GP modules modulo meshes from the AR sequences: this doesn't hold in the case of Kleinian singularities. I will add some details to the answer. | |
| Jun 14, 2016 at 6:47 | comment | added | Mare | Thanks for the answer. To Q2: I actually meant quiver and relations. In the case of Auslander algebras, one can obtain the relations immediatly as mesh relations in most cases. Is something similar true in case of Gorenstein projective modules? | |
| Jun 13, 2016 at 18:39 | history | answered | Matthew Pressland | CC BY-SA 3.0 |