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We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including
$\overline{\mathbb{Q}}$
and
$\overline{\mathbb{F}_{p}}$
, we arrive at a complete description of the tensor triangular spectrum and a classification of the thick tensor ideals.
We produce an isomorphism
$E_{\infty }^{m,-m-1}\cong \text{Nrd}_{1}(A^{\otimes m})$
between terms of the
$\text{K}$
-theory coniveau spectral sequence of a Severi–Brauer variety
$X$
associated with a central simple algebra
$A$
and a reduced norm group, assuming
$A$
has equal index and exponent over all finite extensions of its center and that
$\text{SK}_{1}(A^{\otimes i})=1$
for all
$i>0$
.
In this paper, we construct surfaces in
$\mathbf{P}^{3}$
with large higher Chow groups defined over a Laurent power series field. Explicit elements in higher Chow group are constructed using configurations of lines contained in the surfaces. To prove the independentness, we compute the extension class in the Galois cohomologies by comparing them with the classical monodromies. It is reduced to the computation of linear algebra using monodromy weight spectral sequences.
In this paper we construct a
$\mathbb{Q}$
-linear tannakian category
$\mathsf{MEM}_{1}$
of universal mixed elliptic motives over the moduli space
${\mathcal{M}}_{1,1}$
of elliptic curves. It contains
$\mathsf{MTM}$
, the category of mixed Tate motives unramified over the integers. Each object of
$\mathsf{MEM}_{1}$
is an object of
$\mathsf{MTM}$
endowed with an action of
$\text{SL}_{2}(\mathbb{Z})$
that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over
${\mathcal{M}}_{1,1}$
whose fiber over the tangential base point
$\unicode[STIX]{x2202}/\unicode[STIX]{x2202}q$
at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of
$\mathsf{MEM}$
is determined and the lowest order terms of a set (conjecturally, a minimal generating set) of relations are deduced from computations of Brown. This set of relations includes the arithmetic relations, which describe the ‘infinitesimal Galois action’. We use the presentation to give a new and more conceptual proof of the Ihara–Takao congruences.
Given a smooth variety
$X$
and an effective Cartier divisor
$D\subset X$
, we show that the cohomological Chow group of 0-cycles on the double of
$X$
along
$D$
has a canonical decomposition in terms of the Chow group of 0-cycles
$\text{CH}_{0}(X)$
and the Chow group of 0-cycles with modulus
$\text{CH}_{0}(X|D)$
on
$X$
. When
$X$
is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of
$\text{CH}_{0}(X|D)$
. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that
$\text{CH}_{0}(X|D)$
is torsion-free and there is an injective cycle class map
$\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$
if
$X$
is affine. For a smooth affine surface
$X$
, this is strengthened to show that
$K_{0}(X,D)$
is an extension of
$\text{CH}_{1}(X|D)$
by
$\text{CH}_{0}(X|D)$
.
Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy
$K$
-theory groups of a Noetherian scheme
$X$
of Krull dimension
$d$
vanish below
$-d$
. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy
$K$
-theory groups vanish below
$-1$
. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy
$K$
-theory group.
We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky–MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity homology theory and a companion perversity cohomology theory based on generalized cocycle spaces. These theories lead to conditions on pairs of cycles which can be intersected and a suitable equivalence relation on cocycles/cycles enabling pairings on equivalence classes. We establish suspension and splitting theorems, as well as a localization property. Some examples of intersections on singular varieties are computed.
We start developing a notion of reciprocity sheaves, generalizing Voevodsky’s homotopy invariant presheaves with transfers which were used in the construction of his triangulated categories of motives. We hope that reciprocity sheaves will eventually lead to the definition of larger triangulated categories of motivic nature, encompassing non-homotopy invariant phenomena.
For a prime number
$p$
, we show that differentials
$d_{n}$
in the motivic cohomology spectral sequence with
$p$
-local coefficients vanish unless
$p-1$
divides
$n-1$
. We obtain an explicit formula for the first non-trivial differential
$d_{p}$
, expressing it in terms of motivic Steenrod
$p$
-power operations and Bockstein maps. To this end, we compute the algebra of operations of weight
$p-1$
with
$p$
-local coefficients. Finally, we construct examples of varieties having non-trivial differentials
$d_{p}$
in their motivic cohomology spectral sequences.
In this paper we demonstrate that non-commutative localizations of arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category
$\text{}\underline{C}$
by a set
$S$
of morphisms in the heart
$\text{}\underline{Hw}$
of a weight structure
$w$
on it one obtains a triangulated category endowed with a weight structure
$w^{\prime }$
. The heart of
$w^{\prime }$
is a certain version of the Karoubi envelope of the non-commutative localization
$\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$
(of
$\text{}\underline{Hw}$
by
$S$
). The functor
$\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$
is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of
$S$
invertible. For any additive category
$\text{}\underline{A}$
, taking
$\text{}\underline{C}=K^{b}(\text{}\underline{A})$
we obtain a very efficient tool for computing
$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$
; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that
$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$
coincides with the ‘abstract’ localization
$\text{}\underline{A}[S^{-1}]$
(as constructed by Gabriel and Zisman) if
$S$
contains all identity morphisms of
$\text{}\underline{A}$
and is closed with respect to direct sums. We apply our results to certain categories of birational motives
$DM_{gm}^{o}(U)$
(generalizing those defined by Kahn and Sujatha). We define
$DM_{gm}^{o}(U)$
for an arbitrary
$U$
as a certain localization of
$K^{b}(Cor(U))$
and obtain a weight structure for it. When
$U$
is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general
$U$
the result is completely new. The existence of the corresponding adjacent
$t$
-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over
$U$
.
In this article we construct symmetric operations for all primes (previously known only for
$p=2$
). These unstable operations are more subtle than the Landweber–Novikov operations, and encode all
$p$
-primary divisibilities of characteristic numbers. Thus, taken together (for all primes) they plug the gap left by the Hurewitz map
$\mathbb{L}{\hookrightarrow}\mathbb{Z}[b_{1},b_{2},\ldots ]$
, providing an important structure on algebraic cobordism. Applications include questions of rationality of Chow group elements, and the structure of the algebraic cobordism. We also construct Steenrod operations of tom Dieck style in algebraic cobordism. These unstable multiplicative operations are more canonical and subtle than Quillen-style operations, and complement the latter.
We show that the torsion in the group of indecomposable
$(2,1)$
-cycles on a smooth projective variety over an algebraically closed field is isomorphic to a twist of its Brauer group, away from the characteristic. In particular, this group is infinite as soon as
$b_{2}-{\it\rho}>0$
. We derive a new insight into Roǐtman’s theorem on torsion
$0$
-cycles over a surface.
It is shown that the Grayson tower for
$K$
-theory of smooth algebraic varieties is isomorphic to the slice tower of
$S^{1}$
-spectra. We also extend the Grayson tower to bispectra, and show that the Grayson motivic spectral sequence is isomorphic to the motivic spectral sequence produced by the Voevodsky slice tower for the motivic
$K$
-theory spectrum
$\mathit{KGL}$
. This solves Suslin’s problem about these two spectral sequences in the affirmative.
We define a theory of étale 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 rigidity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion étale motives essentially coincide with the usual complexes of torsion étale 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 étale motives with integral coefficients, by putting together their counterparts for Beilinson motives and for torsion étale 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.
The goal of this paper is to prove that if certain ‘standard’ conjectures on motives over algebraically closed fields hold, then over any ‘reasonable’ scheme
$S$
there exists a motivic
$t$
-structure for the category
$\text{DM}_{c}(S)$
of relative Voevodsky’s motives (to be more precise, for the Beilinson motives described by Cisinski and Deglise). If
$S$
is of finite type over a field, then the heart of this
$t$
-structure (the category of mixed motivic sheaves over
$S$
) is endowed with a weight filtration with semisimple factors. We also prove a certain ‘motivic decomposition theorem’ (assuming the conjectures mentioned) and characterize semisimple motivic sheaves over
$S$
in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic
$t$
-structure is transversal to the Chow weight structure for
$\text{DM}_{c}(S)$
(that was introduced previously by Hébert and the author). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow weight spectral sequences for ‘perverse étale homology’ (which we prove unconditionally); this statement also yields the existence of the Chow weight filtration for such (co)homology that is strictly restricted by (‘motivic’) morphisms.
We consider étale motivic or Lichtenbaum cohomology and its relation to algebraic cycles. We give an geometric interpretation of Lichtenbaum cohomology and use it to show that the usual integral cycle maps extend to maps on integral Lichtenbaum cohomology. We also show that Lichtenbaum cohomology, in contrast to the usual motivic cohomology, compares well with integral cohomology theories. For example, we formulate integral étale versions of the Hodge and the Tate conjecture, and show that these are equivalent to the usual rational conjectures.
We construct new indecomposable elements in the higher Chow group
$CH^2(A,1)$
of a principally polarized Abelian surface over a
$p$
-adic local field, which generalize an element constructed by Collino [Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6 (1997), 393–415]. These elements are constructed using a generalization, due to Birkenhake and Wilhelm [Humbert surfaces and the Kummer plane, Trans. Amer. Math. Soc. 355 (2003), 1819–1841 (electronic)], of a classical construction of Humbert. They can be used to prove a non-Archimedean analogue of the Hodge-
${\mathcal{D}}$
-conjecture – namely, the surjectivity of the boundary map in the localization sequence – in the case where the Abelian surface has good and ordinary reduction.
Let
$k$
be a base commutative ring,
$R$
a commutative ring of coefficients,
$X$
a quasi-compact quasi-separated
$k$
-scheme, and
$A$
a sheaf of Azumaya algebras over
$X$
of rank
$r$
. Under the assumption that
$1/r\in R$
, we prove that the noncommutative motives with
$R$
-coefficients of
$X$
and
$A$
are isomorphic. As an application, we conclude that a similar
isomorphism holds for every
$R$
-linear additive invariant. This leads to several computations.
Along the way we show that, in the case of finite-dimensional algebras of finite
global dimension, all additive invariants are nilinvariant.
For a perfect field k, we use the techniques of Bondal-Kapranov and Hanamura to construct a tensor triangulated category of mixed motives over the truncated polynomial ring k[t]/(tm+1). The extension groups in this category are given by Bloch's higher Chow groups and the additive higher Chow groups. The main new ingredient is the moving lemma for additive higher Chow groups by the authors and its refinements.