arXiv subjects

Denis-Charles Cisinski

Publications and source records attributed to Denis-Charles Cisinski.

At least 19 recordsLinked to original sources

arXiv2025

Completion of motivic sheaves

We study the process of $\ell$-adic completion of motivic sheaves. We observe that, in equal characteristic, when restricted to constructible objets, it is compatible with the six operations. This implies that one can reconstruct $\ell$-adic sheaves of geometric origin over a scheme of finite type over a field from $\ell$-adic cohomology of smooth schemes. In the case of finite fields, this includes perverse $\ell$-adic sheaves of geometric orgin. However, the analogous behaviour fails systematically in mixed characteristic: the reason is that it would imply strong independence of $\ell$ results that can be proven to be too optimistic.

math.AG↗

arXiv2024

Uniform approximation of Betti numbers

We prove that Lefschetz's principle of approximating the cohomology of a possibly singular affine scheme of finite type over a field by the cohomology of a suitable (thickening of a) hyperplane section can be made uniform: in the affine case, we can choose the hyperplane section independently of the cohomology. Using Jouanolou's trick, this gives a new way to bound the Betti numbers of quasi-projective schemes over a field, independently of the cohomology. This is achieved through a motivic version of Deligne's generic base change formula and an axiomatic presentation of the theory of perverse sheaves. These methods produce generating families of Voevodsky motivic sheaves that are realized in perverse sheaves over any base of equal characteristic.

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arXiv2024

Controlled objects in left-exact $\infty$-categories and the Novikov conjecture

We associate to every $G$-bornological coarse space $X$ and every left-exact $\infty$-category with $G$-action a left-exact infinity-category of equivariant $X$-controlled objects. Postcomposing with algebraic K-theory leads to {new} equivariant coarse homology theories. This allows us to apply the injectivity results for assembly maps by Bunke, Engel, Kasprowski and Winges to the algebraic K-theory of left-exact $\infty$-categories.

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arXiv2023

Homotopy theory of schemes and $R$-equivalence

We prove that, for any smooth and projective scheme $X$ over a field $k$ of char. $0$, the set of maps from Spec $k$ to $X$ in the $\mathbf{A}^1$-homotopy category of schemes $\mathcal{H}_{\mathbf{A}^1}(k)$ is in bijection with the quotient of $X(k)$ by $R$-equivalence, and is a birational invariant of $X$. This is achieved by establishing a precise relation between the localization of the category of smooth $k$-schemes by birational maps and the category $\mathcal{H}_{\mathbf{A}^1}(k)$, and by applying results of the second named author and R. Sujatha on birational invariants. This gives a new proof of results obtained by A. Asok and F. Morel.

math.AG↗

arXiv2022

The universal coCartesian fibration

We give a new proof of the straightening/unstraightening correspondence by proving a generalization of the univalence property of the universal coCartesian fibration.

math.CT↗

arXiv2022

Étale tame vanishing cycles over $[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$

We develop a theory of tame vanishing cycles for schemes over $[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$ in the context of étale sheaves. We show some desired properties of this formalism, among which: a compatibility with tame vanishing cycles over a (strctly) henselian trait, a compatibility with the theory of tame vanishing cycles over $\mathbb{A}^1_{S}$, a compatibility with tensor product and with duality. In the last section, we prove that monodromy-invariant vanishing cycles, introduced by the second named author, are the homotopy fixed points with respect to a canonical continuous action of $μ_{\infty}$ of tame vanishing cycles over $[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$.

math.AG↗

arXiv2020

A^1-homotopy invariance in spectral algebraic geometry

We study two different flavours of A^1-homotopy theory in the setting of spectral algebraic geometry, and compare them to classical A^1-homotopy theory. As an application we show that the spectral analogue of Weibel's homotopy invariant K-theory collapses to the classical theory. Along the way we give a new construction of nonconnective algebraic K-theory of stable infinity-categories via a generalization of the Bass-Thomason-Trobaugh construction.

math.AT↗

arXiv2020

Cohomological Methods in Intersection Theory

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. The goal of these notes is to see how motives may be used to enhance cohomological methods, giving natural ways to prove independence of $\ell$ results for traces and zeta-functions, 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 $\mathbf{Q}$-linear motivic sheaves are virtually integral; a proof of the motivic generic base change formula.

math.AG↗

arXiv2019

Triangulated categories of mixed motives

This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular, it is shown that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following and extending insights and constructions of Deligne, Beilinson, Bloch, Thomason, Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, Röndigs, Østvær, and others. In particular, the relation of motives with $K$-theory is addressed in full, and we prove the absolute purity theorem with rational coefficients, using Quillen's localization theorem in algebraic $K$-theory together with a variation on the Grothendieck-Riemann-Roch theorem. Using resolution of singularities via alterations of de Jong-Gabber, this leads to a version of Grothendieck-Verdier duality for constructible motivic sheaves with rational coefficients over rather general base schemes. We also study versions with integral coefficients, constructed via sheaves with transfers, for which we obtain partial results. Finally, we associate to any mixed Weil cohomology a system of categories of coefficients and well behaved realization functors, establishing a correspondence between mixed Weil cohomologies and suitable systems of coefficients. The results of this book have already served as ground reference in many subsequent works on motivic sheaves and their realizations, and pointers to the most recent developments of the theory are given in the introduction.

math.AG↗

arXiv2018

The dendroidal category is a test category

We prove that the category of trees $Ω$ is a test category in the sense of Grothendieck. This implies that the category of dendroidal sets is endowed with the structure of a model category Quillen-equivalent to spaces. We show that this model category structure, up to a change of cofibrations, can be obtained as an explicit left Bousfield localisation of the operadic model category structure.

math.AT↗

arXiv2017

A universal coarse K-theory

In this paper, we construct an equivariant coarse homology theory with values in the category of non-commutative motives of Blumberg, Gepner and Tabuada, with coefficients in any small additive category. Equivariant coarse K-theory is obtained from the latter by passing to global sections. The present construction extends joint work of the first named author with Engel, Kasprowski and Winges by promoting codomain of the equivariant coarse K-homology functor to non-commutative motives.

math.KT↗

arXiv2015

Integral mixed motives in equal characteristic

For noetherian schemes of finite dimension over a field of characteristic exponent $p$, we study the triangulated categories of $\mathbf{Z}[1/p]$-linear mixed motives obtained from cdh-sheaves with transfers. We prove that these have many of the expected properties. In particular, the formalism of the six operations holds in this context. When we restrict ourselves to regular schemes, we also prove that these categories of motives are equivalent to the more classical triangulated categories of mixed motives constructed in terms of Nisnevich sheaves with transfers. Such a program is achieved by comparing these various triangulated categories of motives with modules over motivic Eilenberg-MacLane spectra.

math.AG↗

arXiv2014

Étale motives

We define a theory of etale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of these categories coincides with the triangulated categories of Beilinson motives (and is thus strongly related to algebraic $K$-theory). We extend the rigity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion etale motives essentially coincide with the usual complexes of torsion etale sheaves (at least if we restrict ourselves to torsion prime to the residue characteristics). As a consequence, we obtain the expected results of absolute purity, of finiteness, and of Grothendieck duality for etale motives with integral coefficients, by putting together their counterparts for Beilinson motives and for torsion etale sheaves. Following Thomason's insights, this also provides a conceptual and convenient construction of the $\ell$-adic realization of motives, as the homotopy $\ell$-completion functor.

math.AG↗

arXiv2014

Univalent universes for elegant models of homotopy types

We construct a univalent universe in the sense of Voevodsky in some suitable model categories for homotopy types (obtained from Grothendieck's theory of test categories). In practice, this means for instance that, appart from the homotopy theory of simplicial sets, intensional type theory with the univalent axiom can be interpreted in the homotopy theory of cubical sets (with connections or not), or of Joyal's cellular sets.

math.AT↗

arXiv2014

Note on the tensor product of dendroidal sets

In our paper "Dendroidal sets as models for homotopy operads" (J. Topol. 4 (2011), no. 2, 257-299, and arXiv:0902.1954), we made the wrong claim about the behaviour of the tensor product with respect to cofibrations of dendroidal sets. We added an erratum at the end of the arXiv version of loc. cit. This short note contains the proof of a technical lemma used in the erratum.

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arXiv2014

Dendroidal sets as models for homotopy operads

The homotopy theory of infinity-operads is defined by extending Joyal's homotopy theory of infinity-categories to the category of dendroidal sets. We prove that the category of dendroidal sets is endowed with a model category structure whose fibrant objects are the infinity-operads (i.e. dendroidal inner Kan complexes). This extends the theory of infinity-categories in the sense that the Joyal model category structure on simplicial sets whose fibrant objects are the infinity-categories is recovered from the model category structure on dendroidal sets by simply slicing over the point.

math.CT↗

arXiv2013

Multitensor lifting and strictly unital higher category theory

In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result -- the lifting theorem for multitensors -- enables us to see the Gray tensor product of 2-categories and the Crans tensor product of Gray categories as part of this framework. We define weak n-categories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak n-categories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)-categories with strict units.

math.CT↗

arXiv2013

Dendroidal Segal spaces and infinity-operads

We introduce the dendroidal analogs of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the property of being fibrant. We prove that these two model categories are Quillen equivalent to each other, and to the monoidal model category for infinity-operads which we constructed in an earlier paper. By slicing over the monoidal unit objects in these model categories, we derive as immediate corollaries the known comparison results between Joyal's quasi-categories, Rezk's complete Segal spaces, and Segal categories.

math.CT↗