Timeline for Cohomology of commutative monoid acting on module
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 5, 2021 at 15:22 | comment | added | xir | okay, thanks for your help! :) | |
| May 5, 2021 at 7:31 | comment | added | Dmitry Vaintrob | Right, this is true on the nose in the cancellative case only. In the non-cancellative case this is true in a derived sense, if you define the localization in an infinity-categorical sense. (See e.g. ncatlab.org/nlab/show/group+completion or the Dwyer-Kan paper referenced there.) | |
| May 5, 2021 at 4:08 | comment | added | xir | this is true even in the non-cancellative case, right? it's basically the discrete version of the group completion theorem. I was slightly confused about this point because of Benjamin's comment below about classifying space and derived functor giving the same cohomology, which maybe maybe also only applies in the non-cancellative case, since it seems to me like group completion can change the derived functor construction but not the classifying space one. | |
| May 5, 2021 at 0:31 | comment | added | Dmitry Vaintrob | I see. That will actually always be equivalent to the classifying space for the group completion, essentially since the classifying space construction doesn't distinguish "forwards" and "backwards" arrows | |
| May 5, 2021 at 0:08 | comment | added | xir | i meant the cohomology of the space BM (the classifying space construction, which applies to a general category, including the one-object category M). | |
| May 5, 2021 at 0:08 | comment | added | xir | oh that's really dumb of me haha, thank you! for some unfathomable reason i was convinced A also had to be supported there. | |
| May 4, 2021 at 22:53 | comment | added | Dmitry Vaintrob | The cohomology Ext_M(Z,A) most certainly is computable by pullback to the group completion! The module Z is supported in the open Z[M^{gp}], and so any Ext can be computed after localizing to this open. By "same as the categorical cohomology" I mean it is the same as the cohomology of the trivial representation of the corresponding category (not sure if this is what you mean by "classifying space as a category") | |
| May 4, 2021 at 22:42 | comment | added | xir | I'm not sure how to use it to compute an actual specific group of this sort though. also, by "the same as the categorical cohomology," what do you mean? | |
| May 4, 2021 at 22:41 | comment | added | xir | I mostly asked because I was just curious, but the original use case I had was figuring out something about Ext_M(Z, A) for commutative cancellative M and a certain non-trivial module A which definitely can't be computed via pullback to the group completion. and yeah that geometric picture seems very interesting and promising to me. | |
| May 4, 2021 at 22:39 | vote | accept | xir | ||
| May 4, 2021 at 22:09 | comment | added | Dmitry Vaintrob | Maybe more generally, a very good first thing to do when working with a commutative monoid M is to pass to the corresponding "toric" geometry object X = Spec Z[M]. The category of M-representations is equivalent to the category of quasicoherent sheaves on X, and invariants computed in this category have geometric meaning. If you want to encode the Hopf algebra of M and not just its underlying ring, this is equivalent to considering its spectrum X as a semigroup object. Most interesting monoid invariants should have meaning as invariants of commutative geometric semigroups. | |
| May 4, 2021 at 21:56 | comment | added | Dmitry Vaintrob | What is the use case that you have in mind? The most straightforward generalization of group cohomology would be Ext_M(Z, Z) which is the same as the categorical cohomology. The problem is that this is a boring invariant in the commutative cancellative case since it is invariant under group completion. Indeed, transposing to geometry, this is the self-Ext of the skyscraper sheaf at 1 in Spec(Z[M]), which can be computed inside the open chart Spec(Z[M^gp]) corresponding to the group completion. | |
| May 4, 2021 at 18:35 | answer | added | Benjamin Steinberg | timeline score: 2 | |
| May 4, 2021 at 16:41 | history | asked | xir | CC BY-SA 4.0 |