Abstract

We study the motivic and ell -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the ell -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the ell -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n.

Highlights

  • The connection between categories of singularities and vanishing cycles is well known, thanks to works of Dyckerhoff [23], Preygel [45], Efimov [24] and many others

  • The main purpose of the above mentioned paper is to identify a classical object of singularity theory, namely the -adic sheaf of vanishing cycles, with the -adic cohomology of a non-commutative space, the dg category of singularities of the special fiber

  • We want to compute RS,∨(Sing(X0)), where X0 is the fiber of sX : X → V(LS) along the zero section S → V(LS). One can view the former generalization as a particular case of the latter thanks to a theorem of Orlov [42] and Burke and Walker [11], which tells us that the dg category of singularities of (X, π ◦ p) is equivalent to the dg category of singularities of (PnX−1, (π1 ◦ p) · T1 + · · · + · Tn ∈ O(1)(PnX−1))

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Summary

Introduction

The connection between categories of singularities and vanishing cycles is well known, thanks to works of Dyckerhoff [23], Preygel [45], Efimov [24] and many others. The main result of Blanc et al ([9, Theorem 4.39]) reads as follows: Let p : X → S be a proper, flat, regular scheme over an excellent strictly henselian trait S. We want to compute RS,∨(Sing(X0)), where X0 is the fiber of sX : X → V(LS) along the zero section S → V(LS) One can view the former generalization as a particular case of the latter thanks to a theorem of Orlov [42] and Burke and Walker [11], which tells us that the dg category of singularities of (X , π ◦ p) is equivalent to the dg category of singularities of (PnX−1, (π1 ◦ p) · T1 + · · · + (πn ◦ p) · Tn ∈ O(1)(PnX−1)). The first thing we can think of are vanishing cycles over general

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Some notation and convention
Reminders on dg categories
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Derived algebraic geometry
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The stable homotopy category of schemes
The stable homotopy category of non-commutative spaces
The bridge between motives and non-commutative motives
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The category of twisted LG models
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The dg category of singularities of a twisted LG model
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The formalism of vanishing cycles
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Monodromy-invariant vanishing cycles
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The comparison theorem
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The main theorem
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Reduction of codimension
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Tame vanishing cycles over A1S
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Some remarks on the regularity hypothesis
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Full Text
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