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Ramification theory from homotopical point of view, I

Tomoyuki Abe

Abstract

We prove the compatibility of pushforward along a proper morphism of an étale constructible sheaf and the pushforward of its characteristic cycle up to $p$-torsion. This was conjectured by Takeshi Saito. For this, we revisit the construction of the characteristic cycle, due to Saito and Beilinson, from more homotopical point of view. In particular, the language of $\infty$-categories is indispensable to carry this out.

Ramification theory from homotopical point of view, I

Abstract

We prove the compatibility of pushforward along a proper morphism of an étale constructible sheaf and the pushforward of its characteristic cycle up to -torsion. This was conjectured by Takeshi Saito. For this, we revisit the construction of the characteristic cycle, due to Saito and Beilinson, from more homotopical point of view. In particular, the language of -categories is indispensable to carry this out.
Paper Structure (15 sections, 52 theorems, 97 equations)

This paper contains 15 sections, 52 theorems, 97 equations.

Table of Contents

  1. Abstract formalism
  2. Proof of Theorem \ref{['mainconscc']}
  3. Variant
  4. Construction of reversible coefficient
  5. Microlocalization
  6. Microlocalization
  7. Functorialities
  8. Preliminary --- Nearby cycles and flat étale systems
  9. Construction of deformation system
  10. Summary
  11. Comparison with Saito's characteristic cycles
  12. Functoriality of $\infty$-topoi
  13. Family of morphism objects
  14. Proof of Proposition \ref{['concdescM']}
  15. Enhanced cohomological operations and motivic cohomology

Key Result

Theorem 1

Let $k$ be a perfect field of characteristic $p>0$, and let $\Lambda$ be a finite local ring whose residue field is of characteristic different from that of $k$. Now, let $h\colon X\rightarrow Y$ be a morphism between separatedThe theory of characteristic cycles works without separatedness assumptio in $\mathrm{CH}_{\dim(Y)}(h_{\circ}\mathrm{SS}(\mathcal{F}))[1/p]$.

Theorems & Definitions (137)

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  • Example
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