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On singular supports in mixed characteristic

Takeshi Saito

Abstract

We fix an excellent regular noetherian scheme $S$ over ${\mathbf Z}_{(p)}$ satisfying a certain finiteness condition. For a constructible étale sheaf ${\cal F}$ on a regular scheme $X$ of finite type over $S$, we introduce a variant of the singular support relatively to $S$ and prove the existence of a saturated relative variant of the singular support by adopting the method of Beilinson using the Radon transform. We may deduce the existence of the singular support itself, if we admit an expected property on the micro support of tensor product and if the scheme $X$ is sufficiently ramified over the base $S$.

On singular supports in mixed characteristic

Abstract

We fix an excellent regular noetherian scheme over satisfying a certain finiteness condition. For a constructible étale sheaf on a regular scheme of finite type over , we introduce a variant of the singular support relatively to and prove the existence of a saturated relative variant of the singular support by adopting the method of Beilinson using the Radon transform. We may deduce the existence of the singular support itself, if we admit an expected property on the micro support of tensor product and if the scheme is sufficiently ramified over the base .
Paper Structure (12 sections, 79 theorems, 40 equations)

This paper contains 12 sections, 79 theorems, 40 equations.

Table of Contents

  1. Micro support
  2. ${\cal F}$-transversality
  3. ${\cal F}$-transversality and local acyclicity
  4. $C$-transversality
  5. Micro support
  6. Singular support
  7. $S$-acyclicity
  8. $S$-micro support
  9. $C$-acyclicity over $S$
  10. ${\cal F}$-acyclicity over $S$
  11. $S$-micro support
  12. $S$-singular support

Key Result

Lemma 1.1.1

The following diagram is commutative: \begin{CD} h^*{\cal F}\otimes h^!{\cal G} @>{c_{h,{\cal F},{\cal G}}}>> h^!({\cal F}\otimes{\cal G}) \\ @A{1_{h^*{\cal F}}\otimes c_{h,{\cal G}}}AA @AA{c_{h,{\cal F}\otimes{\cal G}}}A \\ h^*{\cal F}\otimes h^*{\cal G}\otimes h^!\Lambda @>{{\rm can}\otimes 1}>>

Theorems & Definitions (95)

  • Lemma 1.1.1
  • Lemma 1.1.2
  • Lemma 1.1.3
  • Definition 1.1.4
  • Lemma 1.1.5
  • Definition 1.1.6
  • Lemma 1.1.7
  • Lemma 1.1.8
  • Lemma 1.1.9
  • Lemma 1.1.10
  • ...and 85 more