Timeline for When does grading pass to (co)-homology?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 20, 2012 at 3:44 | vote | accept | Hugo Chapdelaine | ||
| Feb 20, 2012 at 1:00 | history | edited | Hugo Chapdelaine | CC BY-SA 3.0 |
edited title
|
| Feb 19, 2012 at 22:51 | answer | added | Ralph | timeline score: 2 | |
| Feb 19, 2012 at 22:44 | comment | added | Mariano Suárez-Álvarez | (By the way: your title mentions cohomology but you then refer to Tor... If you wanted Ext instead, then things are a bit more delicate because hom(M,N) is not G-graded in the obvious way unless M is finitely generated.) | |
| Feb 19, 2012 at 22:42 | comment | added | Mariano Suárez-Álvarez | It follows from that that there are enough projectives in the category of graded modules and homogeneous maps, so you can compute Tors there. The homology groups will be themselves in the category (because your $R$ is commutative) and in particular will be graded. The usual well-definedness of Tor then gives you a naturality result for the grading. | |
| Feb 19, 2012 at 22:29 | comment | added | Mariano Suárez-Álvarez | For Q1: it is enough to show that every graded module $M$ is an epimorphic image of a graded free one by an homogeneous map: just pick an homogeneous set of generators of $M$ and proceed as in the ungraded case. | |
| Feb 19, 2012 at 22:13 | history | asked | Hugo Chapdelaine | CC BY-SA 3.0 |