We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we construct its punctual weight zero part $ω^0_X(M)$ as the universal Artin motive mapping to M. We use this to define a motive E_X over X which is an invariant of the singularities of X. The first half of the paper is devoted to the study of the functors $ω^0_X$ and the computation of the motives E_X.
In the second half of the paper, we develop the application to locally symmetric varieties. Specifically, let Y be a locally symmetric variety and denote by p:W-->Z the projection of its reductive Borel-Serre compactification W onto its Baily-Borel Satake compactification Z. We show that $Rp_*(\Q_W)$ is naturally isomorphic to the Betti realization of the motive E_Z, where Z is viewed as a scheme. In particular, the direct image of E_Z along the projection of Z to Spec(C) gives a motive whose Betti realization is naturally isomorphic to the cohomology of W.
This paper studies Artin-Tate motives over number rings. As a subcategory of geometric motives, the triangulated category of Artin-Tate motives DATM(S) is generated by motives of schemes that are finite over the base S. After establishing stability of these subcategories under pullback and pushforward along open and closed immersions, a motivic t-structure is constructed. Exactness properties of these functors familiar from perverse sheaves are shown to hold in this context. The cohomological dimension of mixed Artin-Tate motives is two, and there is an equivalence of the triangulated category…
We first study the weight structure on the triangulated category of Artin-Tate motives over a perfect base field k, building on results of Bondarko's. We then study the t-structure on the triangulated category of Artin-Tate motives, when k is algebraic over the rationals, generalizing a result of Levine's. We finally study the interaction of the weight structure and the t-structure. When k is a number field, this will give a useful criterion identifying the weight structure via realizations.
In this paper, we present a noncommutative scheme theory for the semi-graded rings generated in degree one defined by Lezama and Latorre \cite{LezamaLatorre2017} following the ideas about schematicness introduced by Van Oystaeyen and Willaert \cite{VanOystaeyenWillaert1995} for $\mathbb{N}$-graded algebras. With this theory, we prove the Serre-Artin-Zhang-Verevkin theorem \cite{ArtinZhang1994, EGAII1961, Hartshorne1977, Serre1955, Verevkin1992a, Verevkin1992} for several families of non-$\mathbb{N}$-graded algebras and finitely non-$\mathbb{N}$-graded algebras appearing in ring theory and nonc…
We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full subcategory of NC mixed Artin motives. Making use of Hochschild homology, we then apply Ayoub's weak Tannakian formalism to these motivic categories. In the case of NC mixed motives, we obtain a motivic Hopf dg algebra, which we describe explicitly in terms of Hochschild homology and complexes of exact cubes. In the case of NC mixed Artin motives, we compute th…
In this article we introduce the categories of noncommutative (mixed) Artin motives. In the pure world, we start by proving that the classical category AM(k) of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives we then show that the image of this full embedding, which we call the category NAM(k) of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the sym…
We develop a general theory which enables the computation of the Picard group of a symmetric monoidal triangulated category, equipped with a weight structure, in terms of the Picard group of the associated heart. As an application, we compute the Picard group of several categories of motivic nature - mixed Artin motives, mixed Artin-Tate motives, motivic spectra, noncommutative mixed Artin motives, noncommutative mixed motives of central simple algebras, noncommutative mixed motives of separable algebras - as well as the Picard group of the derived categories of symmetric ring spectra.
Deligne has conjectured that certain mixed Hodge theoretic invariants of complex algebraic invariants are motivic. This conjecture specializes to an algebraic construction of the Jacobian for smooth projective curves, which was done by A. Weil. The conjecture (and one-motives) are motivated by means of Jacobians, generalized Jacobians of Rosenlicht, and Serre's generalized Albanese varieties. We discuss the connections with the Hodge and the generalized Hodge conjecture. We end with some applications to number theory by providing partial answers to questions of Serre, Katz and Jannsen.
Let $G$ be a split connected semisimple group over a field. We give a conjectural formula for the motive of the stack of $G$-bundles over a curve $C$, in terms of special values of the motivic zeta function of $C$. The formula is true if $C=\pp^1$ or $G=\sln$. If $k=\cc$, upon applying the Poincaré or Serre characteristic, the formula reduces to results of Teleman and Atiyah-Bott on the gauge group. If $k=\ffq$, upon applying the counting measure, it reduces to the fact that the Tamagawa number of $G$ over the function field of $C$ is $|π_1(G)|$.
According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti-Tate group. We extend this result to $1$-motives.
The goal of this paper is to give an explicit description of the triangulated categories of Tate and Artin-Tate motives with finite coefficients Z/m over a field K containing a primitive m-root of unity as the derived categories of exact categories of filtered modules over the absolute Galois group of K with certain restrictions on the successive quotients. This description is conditional upon (and its validity is equivalent to) certain Koszulity hypotheses about the Milnor K-theory/Galois cohomology of K. This paper also purports to explain what it means for an arbitrary nonnegatively graded …
We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing the construction in terms of fiber functors from the motivic category of absolute Hodge cycles into a suitable category of Hodge structures of odd weight. This extends the case of abelian varietes, which we treated in a previous paper. That description was used by Fite--Kedlaya--Rotger--Sutherland to classify Sato-Tate groups of abelian surfaces; the present description is used by Fite--Kedlaya--Sutherland to make a similar classification for certain motives of weight 3. We also give conditions under…
We construct a notion of p-adic measure on Artin n-stacks which are strongly of finite type over the ring of p-adic integers. We also prove the rationality of of the Poincare series and the Serre series associated with such stacks. Finally, we use motivic integration to show that the rationality of the Serre series is uniform with respect to p.
We formulate a notion of "punctual gluing" of $t$-structures and weight structures. As our main application we show that the relative version of Ayoub's $1$-motivic $t$-structure restricts to compact motives. We also demonstrate the utility of punctual gluing by recovering several constructions in literature. In particular we construct the weight structure on the category of motivic sheaves over any base $X$ and we also construct the relative Artin motive and the relative Picard motive of any variety $Y/X$.
In this note, we prove a criteria for supersingularity when the variety has a large automorphism group and a perfect bilinear pairing. This criteria unifies and extends many known results on the supersingularity of curves and varieties and in particular, applies to a large family of Artin-Schreier curves.
We propose a construction of a tensor exact category F_X^m of Artin-Tate motivic sheaves with finite coefficients Z/m over an algebraic variety X (over a field K of characteristic prime to m) in terms of etale sheaves of Z/m-modules over X. Among the objects of F_X^m, in addition to the Tate motives Z/m(j), there are the cohomological relative motives with compact support M_cc^m(Y/X) of varieties Y quasi-finite over X. Exact functors of inverse image with respect to morphisms of algebraic varieties and direct image with compact supports with respect to quasi-finite morphisms of varieties Y\to …
We prove that the functor associating to a rigid analytic variety the singular complex of the underlying Berkovich topological space is motivic, and defines the maximal Artin quotient of a motive. We use this to generalize Berkovich's results on the weight-zero part of the étale cohomology of a variety defined over a non-archimedean valued field.
We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of one-motives (considered up to isogeny in positive characteristic). The algebraic definition of these invariants presented here proves a conjecture of Deligne. Applications include some cases of conjectures of Serre, Katz and Jannsen on the independence of $\ell$ of parts of the étale cohomology of arbitrary varieties over number fields and finite fields.
This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This paper extends the previous three from (derived) schemes to (derived) Artin stacks. We prove four main results: (a) If $(X,ω)$ is a $k$-shifted symplectic derived Artin stack for $k<0$ in the sense of arXiv:1111.3209, then near each $x\in X$ we can find a 'minimal' smooth atlas $φ:U\to X$ with $U$ an affine derived scheme, such that $(U,φ^*(ω))$ may be written explicitly in coordinat…
Let X be a proper scheme over a field k which satisfies Serre's condition S2 and G a reductive group over k. We prove that the functor of principal G-bundles defined away from a non-fixed closed subset in X of codimension at least 3, is an algebraic stack in the sense of Artin.
We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety $X$ over a complete discretely valued field $K$ with perfect residue field $k$. If $K$ has characteristic zero, we extend the definition to arbitrary $K$-varieties using Bittner's presentation of the Grothendieck ring and a process of Néron smoothening of pairs of varieties. The motivic Serre invariant can be considered as a measure for the set of unramified points on $X$. Under certain tameness conditions, it admits a cohomological interpretation by means of a trace formula. In the curve case, we use T. Sait…
This is a revised and slightly expanded version. We point out that in the previous summary, "without cohomology" should really read "almost without cohomology" because of the proof of Lemma 2, that the idea to consider effective motives divisible by the Lefschetz motive was anticipated by Serre in letters to Gilles Lachaud and Marc Perret, and finally that the birational invariance of the number of points modulo q in fact follows from a 1983 Comptes Rendus note of Torsten Ekedahl (I am grateful to Antoine Chambert-Loir for making this observation and indicating this reference).
In this note we continue our development of tannakizations of symmetric monoidal infinity-categories, begun in our previous paper. The issue treated in this paper is the calculation of tannakizations of examples of symmetric monoidal stable infinity-categories with fiber functors. We consider the case of symmetric monoidal infinity-categories of perfect complexes on perfect derived stacks. The first main result especially says that our tannakization includes the bar construction for an augmented commutative ring spectrum and its equivariant version as a special case. We apply it to the study o…
We give a method for constructing maps from a non-commutative scheme to a commutative projective curve. With the aid of Artin-Zhang's abstract Hilbert schemes, this is used to construct analogues of the extremal contraction of a $K$-negative curve with self-intersection zero on a smooth projective surface. This result will hopefully be useful in studying Artin's conjecture on the birational classification of non-commutative surfaces. As a non-trivial example of the theory developed, we look at non-commutative ruled surfaces and, more generally, at non-commutative $\PP^1$-bundles. We show in pa…
Let $U$ be a smooth scheme over an algebraically closed field $\mathbb K$ of characteristic zero and $f:U\to{\mathbb A}^1$ a regular function, and write $X=$Crit$(f)$, as a closed subscheme of $U$. The motivic vanishing cycle $MF_{U,f}^ϕ$ is an element of the $\hatμ$-equivariant motivic Grothendieck ring ${\mathcal M}^{\hatμ}_X$ defined by Denef and Loeser math.AG/0006050 and Looijenga math.AG/0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) $MF_{U,f}^ϕ$ depends only on the third-order thickenings $…
Let X be a smooth proper variety over a perfect field k of arbitrary characteristic. Let D be an effective divisor on X with multiplicity. We introduce an Albanese variety Alb(X, D) of X of modulus D as a higher dimensional analogon of the generalized Jacobian of Rosenlicht-Serre with modulus for smooth proper curves. Basing on duality of 1-motives with unipotent part (which are introduced here), we obtain explicit and functorial descriptions of these generalized Albanese varieties and their dual functors. We define a relative Chow group of zero cycles w.r.t. the modulus D and show that Alb(X,…
