Timeline for Primitive Cohomology Useful?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 11, 2022 at 19:55 | comment | added | Z. M | A technical question about Thm 1.5 in Deligne's paper: does a choice of $u\colon X\to X[2]$ correspond naturally to a splitting $X\simeq\bigoplus_i\pi_i(X)[i]$? It might be possible to prove this without reference to spectral sequences, I guess. | |
| Jan 12, 2013 at 18:34 | comment | added | ACL | @LMN: Right, there is a hard Lefschetz theorem in étale cohomology, proved by Deligne in La conjecture de Weil, II (Section 4.1). | |
| Jan 12, 2013 at 7:27 | comment | added | LMN | I guess here $\omega$ is $c_1()$ of very ample line bundle defining a $K$-rational map to projective space. | |
| Jan 12, 2013 at 7:23 | comment | added | LMN | Aakumadula, are you saying that if we pick a Kahler form, and our variety is defined over a number field $K$, then the representation of the galois group $Gal(\bar{K}/K)$ on etale, and hence singular cohom, has each of the primitive cohomology groups as invariant subspaces? This is interesting. | |
| Jan 12, 2013 at 7:21 | comment | added | Dan Petersen | What I was referring to is the very first proposition in Deligne's paper. The Leray SS comes from the equality $\mathrm R(g\circ f)_\ast = \mathrm R g_\ast \circ \mathrm R f_\ast$ in the derived category, where $g \colon S \to \mathrm{pt}$ is the projection to a point. The decomposition of $\mathrm Rf $ in the derived category implies the degeneration not only of this particular spectral sequence, but also of the ones obtained by replacing $g_\ast$ by any left exact functor, or more generally replacing $Rg_\ast$ by any cohomological functor. | |
| Jan 12, 2013 at 7:19 | comment | added | Venkataramana | the point is that all the cohomology (of a smooth projective variety) is obtained from the primitive one by wedging with a Lefschetz class. Thus primitive cohomology is to be viewed as the subspace of highest weight vectors for the (Lefschetz) $SL_24 action. So, if you know for example, the Galois representations in the primitive cohomology, then you know the reps for all of the variety | |
| Jan 12, 2013 at 7:08 | comment | added | LMN | Dan, this is very nice, Thanks! Could you say a few words about why we should think of the derived category result as better? | |
| Jan 12, 2013 at 6:45 | history | answered | Dan Petersen | CC BY-SA 3.0 |