Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lifting vector bundles to Witt vector bundles   In this paper, we lay the fundations of the theory of Witt-Frobenius modules
and of Witt vector bundles. It is the main geometric tool in a series of papers
by the first two authors in which they prove the mod $p^2$ liftability of mod
$p$ Galois representations of all fields and of all dimensions, as well as
provide a brand new proof of the Norm Residue Isomorphism Theorem. Let $p$ be a
prime, and let $S$ be a scheme of characteristic $p$. Let $r \geq 2$ be an
integer. Denote by $W_r(S)$ the scheme of Witt vectors of length $r$, built out
of $S$. The main objective of this paper concerns the question of extending
(=lifting) vector bundles on $S$, to vector bundles on $W_r(S)$. We first study
two elementary but significant cases, in which the answer is positive: line
bundles, and the tautological vector bundle of a projective bundle over an
affine base. We give several applications, showcasing the systematic use of the
Teichm\"uller lift of line bundles. First, we offer a simple formulation of
classical results in deformation theory of smooth varieties over a field $k$ of
characteristic $p$ and we extend them to reduced $k$-schemes. These results
were recently recovered, in another form, by Stefan Schr\"oer. Then, we show
that the tautological vector bundle of the Grassmannian
$Gr_{\mathbb{F}_p}(m,n)$ does not extend to $W_2(Gr_{\mathbb{F}_p}(m,n))$, if
$2 \leq m \leq n-2$. This permits to recover, and generalize, a recent result
of Maciej Zdanowicz about the non-liftability of some projective bundles. In
the Appendix, we give algebraic details on our new approach to Witt vectors,
using divided powers and polynomial laws. In particular, we produce a natural
surjection $W_2 \circ W_r \longrightarrow W_{r+1}$, of ring schemes over
$\mathbb{F}_p$.

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The categories of this paper are: econ
4
The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex