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<p>We study relative Wach modules, generalising our previous works on this subject. Our main result shows a categorical equivalence between relative Wach modules and lattices inside relative crystalline representations. Using this result, we deduce a purity statement for relative crystalline representations and provide a criteria for checking the crystallinity of relative <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p\text {-adic}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mtext>-adic</mtext> </mrow> </math></EquationSource> </InlineEquation> representations. Furthermore, we interpret relative Wach modules as modules with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(q\text {-connections}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mtext>-connections</mtext> </mrow> </math></EquationSource> </InlineEquation> and show that for a crystalline representation, its associated Wach module together with the Nygaard filtration is the canonical <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(q\text {-deformation}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mtext>-deformation</mtext> </mrow> </math></EquationSource> </InlineEquation> (after inverting <i>p</i>) of the filtered <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((\varphi ,\partial )\text {-module}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>∂</mi> <mo stretchy="false">)</mo> <mtext>-module</mtext> </mrow> </math></EquationSource> </InlineEquation> associated to the representation.</p>

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Crystalline representations and Wach modules in the relative case II

  • 期刊论文

  • 发表日期：

    2026年5月15日

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  • Abhinandan

摘要

We study relative Wach modules, generalising our previous works on this subject. Our main result shows a categorical equivalence between relative Wach modules and lattices inside relative crystalline representations. Using this result, we deduce a purity statement for relative crystalline representations and provide a criteria for checking the crystallinity of relative \(p\text {-adic}\) p -adic representations. Furthermore, we interpret relative Wach modules as modules with \(q\text {-connections}\) q -connections and show that for a crystalline representation, its associated Wach module together with the Nygaard filtration is the canonical \(q\text {-deformation}\) q -deformation (after inverting p) of the filtered \((\varphi ,\partial )\text {-module}\) ( φ , ∂ ) -module associated to the representation.