Timeline for Why was it so difficult to define the relative de Rham-Witt complex?
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| Dec 6, 2023 at 22:11 | comment | added | Z. M | A late remark: there seems no obvious reason that the description #2 generalizes to the case that the base is imperfect. This is problematic even when $n=1$ and both schemes are affine: the Cartier isomorphism tells us that de Rham cohomology groups are isomorphic to Frobenius twisted differential forms, so a priori you will not define differentials on it so that it recovers the de Rham complex. | |
| Jan 10, 2019 at 3:34 | comment | added | LSpice | @LeoAlonso's reference, clickable: Cuntz and Deninger - An alternative to Witt vectors. | |
| Dec 17, 2013 at 1:38 | comment | added | Piotr Achinger | @BenjaminDickman Thanks, I saw that. But again, their generalization is about replacing the base $\mathbb{F}_p$ by $\mathbb{Z}_{(p)}$ rather than an arbitrary $\mathbb{F}_p$-scheme $S$. I've been told once that the topologists actually like the fact that the de Rham-Witt complex is absolute (because algebraic K-theory is absolute?). | |
| Dec 16, 2013 at 10:59 | comment | added | Benjamin Dickman | Perhaps of interest: Hesselholt, L., & Madsen, I. (2004, February). On the De Rham–Witt complex in mixed characteristic. (See, e.g., p.4 just after Theorem C, and the start of p.6. The version I reference is pay-walled at: sciencedirect.com/science/article/pii/S0012959304000096) | |
| Dec 16, 2013 at 9:37 | comment | added | Piotr Achinger | @LeoAlonso no, I haven't seen that one. Very interesting, though not (immediately) applicable to the question. They treat the case of a perfect ring $R$ over $\mathbb{F}_p$, but in this case $W_n(R)/p = R$, i.e. $W_n(R)$ is the (unique!) lift of $R$ over $\mathbb{Z}/p^n$. | |
| Dec 16, 2013 at 9:28 | comment | added | Leo Alonso | It seems that leaving the "base field" case adds a lots of complexity, as in general Witt vectors vs. $p$-typical Witt vectors. Are you aware of arXiv:1311.2774? I guess, these kind of ideas would lead to a simpler construction of DRW over a field. | |
| Dec 16, 2013 at 9:28 | history | edited | Piotr Achinger | CC BY-SA 3.0 |
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| Dec 16, 2013 at 9:24 | comment | added | Piotr Achinger | @LeoAlonso I'm only interested in the situation when the base has characteristic $p$. Langer and Zink indeed treat the case over $\mathbb{Z}_{(p)}$ rather than $\mathbb{F}_p$, so maybe that is where the difficulty comes from? | |
| Dec 16, 2013 at 9:20 | history | edited | Piotr Achinger | CC BY-SA 3.0 |
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| Dec 16, 2013 at 9:15 | history | edited | Piotr Achinger | CC BY-SA 3.0 |
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| Dec 16, 2013 at 9:14 | comment | added | Leo Alonso | I guess this is connected to Grothendieck's problem of finding a good theory of crystals "over $\mathbb{Z}$". And this seems to have revealed as a very complicated task. | |
| Dec 16, 2013 at 9:02 | history | asked | Piotr Achinger | CC BY-SA 3.0 |