Let X be a scheme. Let r ≥ 2 be an integer. Denote by Wr(X) the scheme of Witt vectors of length r, built out of X. We are concerned with the question of extending (=lifting) vector bundles on X, to vector bundles on Wr(X)—promoting a systematic use of Witt modules and Witt vector bundles. To begin with, we investigate two elementary but significant cases, in which the answer to this question is positive: line bundles, and the tautological vector bundle of a projective bundle over an affine base. We then offer a simple (re)formulation of classical results in deformation theory of smooth varieties over a field k of characteristic p > 0, and extend them to reduced k-schemes. Some of these results were recently recovered, in another form, by Stefan Schröer. As an application, we prove that the tautological vector bundle of the Grassmannian <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2775_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$\text{Gr}_{\mathbb{F}_{p}}(m,n)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mtext>Gr</mtext> <mrow> <msub> <mrow> <mi mathvariant="double-struck">F</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> does not extend to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2775_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbf{W}_{2}(\text{Gr}_{\mathbb{F}_{p}}(m,n))$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mi mathvariant="bold">W</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mtext>Gr</mtext> <mrow> <msub> <mrow> <mi mathvariant="double-struck">F</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>, if 2 ≤ m ≤ n − 2. To conclude, we establish a connection to the work of Zdanowicz, on non-liftability of some projective bundles.

Lifting vector bundles to Witt vector bundles

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发表日期：

2025年6月29日

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Charles De Clercq,
Mathieu Florence,
Giancarlo Lucchini-Arteche

摘要

Let X be a scheme. Let r ≥ 2 be an integer. Denote by W_r(X) the scheme of Witt vectors of length r, built out of X. We are concerned with the question of extending (=lifting) vector bundles on X, to vector bundles on W_r(X)—promoting a systematic use of Witt modules and Witt vector bundles. To begin with, we investigate two elementary but significant cases, in which the answer to this question is positive: line bundles, and the tautological vector bundle of a projective bundle over an affine base. We then offer a simple (re)formulation of classical results in deformation theory of smooth varieties over a field k of characteristic p > 0, and extend them to reduced k-schemes. Some of these results were recently recovered, in another form, by Stefan Schröer. As an application, we prove that the tautological vector bundle of the Grassmannian $\text{Gr}_{\mathbb{F}_{p}}(m,n)$ ${Gr}_{F_{p}} (m, n)$ does not extend to $\mathbf{W}_{2}(\text{Gr}_{\mathbb{F}_{p}}(m,n))$ $W_{2} ({Gr}_{F_{p}} (m, n))$ , if 2 ≤ m ≤ n − 2. To conclude, we establish a connection to the work of Zdanowicz, on non-liftability of some projective bundles.