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- W2904592945 abstract "Let $k$ be a field of characteristic $p>0$. Denote by $W_r(k)$ the ring of truntacted Witt vectors of length $r geq 2$, built out of $k$. In this text, we consider the following question, depending on a given profinite group $G$. $Q(G)$: Does every (continuous) representation $Glongrightarrow GL_d(k)$ lift to a representation $Glongrightarrow GL_d(W_r(k))$? We work in the class of cyclotomic pairs (Definition 4.3), first introduced in [DCF] under the name profinite groups. Using Grothendieck-Hilbert' theorem 90, we show that the algebraic fundamental groups of the following schemes are cyclotomic: spectra of semilocal rings over $mathbb{Z}[frac{1}{p}]$, smooth curves over algebraically closed fields, and affine schemes over $mathbb{F}_p$. In particular, absolute Galois groups of fields fit into this class. We then give a positive partial answer to $Q(G)$, for a cyclotomic profinite group $G$: the answer is positive, when $d=2$ and $r=2$. When $d=2$ and $r=infty$, we show that any $2$-dimensional representation of $G$ stably lifts to a representation over $W(k)$: see Theorem 6.1. When $p=2$ and $k=mathbb{F}_2$, we prove the same results, up to dimension $d=4$. We then give a concrete application to algebraic geometry: we prove that local systems of low dimension lift Zariski-locally (Corollary 6.3)." @default.
- W2904592945 created "2018-12-22" @default.
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- W2904592945 date "2018-12-19" @default.
- W2904592945 modified "2023-09-25" @default.
- W2904592945 title "Lifting low-dimensional local systems" @default.
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