Timeline for Describing the crystalline extension of $\mathbb{Q}_p$ by $\mathbb{Q}_p$
Current License: CC BY-SA 3.0
13 events
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| Jan 16, 2016 at 10:56 | history | edited | Sam Derbyshire | CC BY-SA 3.0 |
Changed K to K_0 throughout.
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| Jan 13, 2016 at 14:11 | comment | added | Sam Derbyshire | Yes, but in that case there are different weights; the weight filtration yields an extension of something of weight $1$ by something of weight $0$, in accordance with the sequence $0 \to W_0 \to W_1 \to W_1/W_0 \to 0$. I am arguing that such an extension shouldn't occur in geometry for the OP's situation because everything is in weight $0$. | |
| Jan 13, 2016 at 13:56 | history | edited | Sam Derbyshire | CC BY-SA 3.0 |
Made more precise the claims about realizing E geometrically.
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| Jan 13, 2016 at 13:49 | comment | added | David Loeffler | Semisimplicity is also false for proper but non-smooth schemes (take an elliptic curve and quotient out by the equivalence relation that identifies a non-torsion $\mathbf{Q}_p$-point with the identity). You really need both smoothness and properness here. | |
| Jan 13, 2016 at 13:16 | comment | added | Sam Derbyshire | However I don't know of any specific conjectures that address this, except for the conjectured semisimplicity of Frobenius acting on the log-crystalline cohomology of a proper semistable scheme over $\mathcal{O}_K$, which is not exactly relevant (non-smooth case vs non-proper case). I thought that some yoga of comparison theorems would lead us to expect the behaviour on the p-adic side to mirror what happens with mixed Hodge structures. | |
| Jan 13, 2016 at 13:11 | comment | added | Sam Derbyshire | Sure, I was deliberately a bit vague in the last paragraph, and I should've been more specific. I didn't mean to consider the p-adic étale cohomology, but maybe instead a version of crystalline cohomology obtained through some simplicial resolution by smooth proper schemes. My impression is that there still won't be extensions of this form (although of course as you say other nontrivial extensions are possible); for instance there are no nontrivial extensions in the category of mixed Hodge structures of the trivial Hodge structure by itself. | |
| Jan 13, 2016 at 12:38 | comment | added | David Loeffler | Hang on. A more serious point: in the last paragraph it looks like you're trying to consider the p-adic etale cohomology of a mod p variety, which is generally rather pathological, and won't compare well to the generic fibre. Moreover, it definitely isn't true that the Galois action on the cohomology of the generic fibre of a smooth $\mathbf{Z}_p$-scheme is semisimple; you need the scheme also to be proper. For non-proper schemes there are lots of interesting extensions -- this is the whole point of the theory of mixed motives. | |
| Jan 12, 2016 at 21:42 | history | edited | Sam Derbyshire | CC BY-SA 3.0 |
Corrected the statement about subrings of A_cris.
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| Jan 12, 2016 at 19:53 | comment | added | David Loeffler | Two minor nitpicks: $\mathcal{O}_{\mathbb{C}_p}$ is not a subring of $\mathbf{A}_{\mathrm{cris}}$ -- if this were so, then every finite-image representation would be crystalline, which is not true. Moreover, $W(\overline{\mathbf{F}}_q)$ is bigger than $\mathcal{O}_{K^{\mathrm{nr}}}$ (the former is the completion of the latter). | |
| Jan 12, 2016 at 17:01 | comment | added | user84144 | Perfect! Exactly the sort of answer I was hoping for. | |
| Jan 12, 2016 at 16:56 | vote | accept | user84144 | ||
| Jan 12, 2016 at 16:24 | history | edited | Sam Derbyshire | CC BY-SA 3.0 |
added 104 characters in body
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| Jan 12, 2016 at 15:09 | history | answered | Sam Derbyshire | CC BY-SA 3.0 |