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Cohomological Methods in Intersection Theory

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Book cover Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2292))

Abstract

The goal of these notes is to see how motives may be used to enhance cohomological methods, giving natural ways to prove independence of ℓ results and constructions of characteristic classes (as 0-cycles). This leads to the Grothendieck-Lefschetz formula, of which we give a new motivic proof. There are also a few additions to what have been told in the lectures:

  • A proof of Grothendieck-Verdier duality of étale motives on schemes of finite type over a regular quasi-excellent scheme (which slightly improves the level of generality in the existing literature).

  • A proof that Q-linear motivic sheaves are virtually integral (Theorem 3.3.2.12).

  • A proof of the motivic generic base change formula.

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Notes

  1. 1.

    As shown by D. Rydh [41], this topology can be extended to all schemes, at the price of adding compatiblities with the constructible topology.

  2. 2.

    We refer to [31, 32] in general. However, most of the literature on motives is written using the theory of Quillen model structures. The precise way to translate this language to the one of ∞-categories is discussed in Chapter 7 of [11].

  3. 3.

    Also known as Lichtenbaum cohomology.

  4. 4.

    It is standard terminology to call such ℓ-adic sheaves ‘lisses’. This comes from Deligne’s work, which is written in French. I prefer to translate into ‘smooth’ because this is what we do for morphisms of schemes. The reason is that this terminology comes from the fact that there are transersality conditions one can define between (motivic or ℓ-adic) sheaves and morphisms of schemes, and that a basic intuition about smoothness is that a smooth object is transverse to anything: indeed, a smooth sheaf is transverse to any morphism, while any sheaf is transverse to a smooth morphism. This why I think it is better to use the same word to express the smoothness of both sheaves and morphisms of schemes.

  5. 5.

    This is where ∞-category theory appears seriously: proving that the construction f↦f ! actually defines a presheaf is a highly non-trivial homotopy coherence problem. Such construction is explained in [38, Chapter 10], using the general results of [29, 30].

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Acknowledgements

These notes are an account of a series of lectures I gave at the LMS-CMI Research School ‘Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects’, in July 2018, at the Imperial College London. I am grateful to Shachar Carmeli for having allowed me to use the notes he typed from my lectures, and to Kévin François for finding a gap in the proof of the motivic generic base formula. While preparing these lectures and writing these notes, I was partially supported by the SFB 1085 “Higher Invariants” funded by the Deutsche Forschungsgemeinschaft (DFG).

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  1. Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany

    Denis-Charles Cisinski

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  1. School of Computing and Mathematical Sciences, University of Leicester, Leicester, UK

    Frank Neumann

  2. Department of Mathematics, Imperial College London, London, UK

    Ambrus Pál

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Cisinski, DC. (2021). Cohomological Methods in Intersection Theory. In: Neumann, F., Pál, A. (eds) Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects. Lecture Notes in Mathematics, vol 2292. Springer, Cham. https://doi.org/10.1007/978-3-030-78977-0_3

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