A cycle class map from Chow groups with modulus to relative KK-theory

Let $\bar{X}$ be a smooth quasi-projective $d$-dimensional variety over a field $k$ and let $D$ be an effective Cartier divisor on it. In this note, we construct cycle class maps from (a variant of) the higher Chow group with modulus of the pair $(\bar{X},D)$ in the range $(d+n, n)$ to the relative $K$-groups $K_n(\bar{X}, D)$ for every $n\geq 0$.Comment: 24 pages. Final version to appear in Documenta Mat

Torsion zero cycles with modulus on affine varieties

In this note we show that given a smooth affine variety $X$ over an algebraically closed field $k$ and an effective (possibly non reduced) Cartier divisor $D$ on it, the Kerz-Saito Chow group of zero cycles with modulus ${\rm CH}_0(X|D)$ is torsion free, except possibly for $p$-torsion if the characteristic of $k$ is $p>0$. This generalizes to the relative setting classical theorems of Rojtman (for $X$ smooth) and of Levine (for $X$ singular). A stronger version of this result, that encompasses $p$-torsion as well, was proven with a different and more sophisticated method by A. Krishna and the author in another paper.Comment: Final version. 12 pages, exposition improved. Several gaps in the proofs fixe

Relative cycles with moduli and regulator maps

Let X be a separated scheme of finite type over a field k and D a non-reduced effective Cartier divisor on it. We attach to the pair (X, D) a cycle complex with modulus, whose homotopy groups - called higher Chow groups with modulus - generalize additive higher Chow groups of Bloch-Esnault, R\"ulling, Park and Krishna-Levine, and that sheafified on $X_{Zar}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight 1. When X is smooth over k and D is such that $D_{red}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El-Zein's explicit construction. This is used to define a natural regulator map from the relative motivic complex of (X,D) to the relative de Rham complex. When X is defined over $\mathbb{C}$, the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch's regulator from higher Chow groups. Finally, when X is moreover connected and proper over $\mathbb{C}$, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J^r_{X|D}$ of the pair (X,D). For r= dim X, we show that $J^r_{X|D}$ is the universal regular quotient of the Chow group of 0-cycles with modulus.Comment: 46 pages. Final version: Section 9 added and material rearranged. To appear in Journal of the Inst. of Math. Jussie

Zero cycles with modulus and zero cycles on singular varieties

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 ${\rm CH}_0(X)$ and the Chow group of 0-cycles with modulus ${\rm CH}_0(X|D)$ on $X$. When $X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of ${\rm CH}_0(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that ${\rm CH}_0(X|D)$ is torsion-free and there is an injective cycle class map ${\rm CH}_0(X|D) \hookrightarrow K_0(X,D)$ if $X$ is affine. For a smooth affine surface $X$, this is strengthened to show that $K_0(X,D)$ is an extension of ${\rm CH}_1(X|D)$ by ${\rm CH}_0(X|D)$.Comment: 62 pages. Final version to appear in Compositio Mat

Torsion and divisibility for reciprocity sheaves and 0-cycles with modulus

The notion of modulus is a striking feature of Rosenlicht-Serre's theory of generalized Jacobian varieties of curves. It was carried over to algebraic cycles on general varieties by Bloch-Esnault, Park, R\"ulling, Krishna-Levine. Recently, Kerz-Saito introduced a notion of Chow group of $0$-cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields. We study the non-homotopy invariant part of the Chow group of $0$-cycles with modulus and show their torsion and divisibility properties. Modulus is being brought to sheaf theory by Kahn-Saito-Yamazaki in their attempt to construct a generalization of Voevodsky-Suslin-Friedlander's theory of homotopy invariant presheaves with transfers. We prove parallel results about torsion and divisibility properties for them.Comment: 15 pages, exposition improve

Connectivity and Purity for logarithmic motives

The goal of this paper is to extend the work of Voevodsky and Morel on the homotopy $t$-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel's connectivity theorem and show a purity statement for $(\mathbf{P}^1, \infty)$-local complexes of sheaves with log transfers. The homotopy $t$-structure on $\mathbf{logDM}^{\textrm{eff}}(k)$ is proved to be compatible with Voevodsky's $t$-structure i.e. we show that the comparison functor $R^{\overline{\square}}\omega^*\colon \mathbf{DM}^{\textrm{eff}}(k)\to \mathbf{logDM}^{\textrm{eff}}(k)$ is $t$-exact. The heart of the homotopy $t$-structure on $\mathbf{logDM}^{\textrm{eff}}(k)$ is the Grothendieck abelian category of strictly cube-invariant sheaves with log transfers: we use it to build a new version of the category of reciprocity sheaves in the style of Kahn--Saito--Yamazaki and R\"ulling.Comment: 43 pages, final version, to appear in J. Inst. Math. Jussie

GAGA problems for the Brauer group via derived geometry

In this paper we prove that the Brauer group of any (derived) scheme $X$, proper over the spectrum of a quasi-excellent Henselian ring, injects into the Brauer group of the Henselization of $X$ along the base, generalizing a classical result of Grothendieck. We offer two proofs of this fact, one based on a formal GAGA-type theorem for smooth and proper stable $\infty$-categories enriched over the $\infty$-category $\mathrm{QCoh}(X)$ of quasi-coherent $\mathscr{O}_X$-modules, and a second one based on a GAGA-type theorem for perfect complexes on $\mathbb{G}_m$-gerbes.Comment: 33 page

Connectivity and Purity for logarithmic motives

The goal of this article is to extend the work of Voevodsky and Morel on the homotopy t-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel’s connectivity theorem and show a purity statement for (P1,∞)-local complexes of sheaves with log transfers. The homotopy t-structure on logDMeff(k) is proved to be compatible with Voevodsky’s t-structure; that is, we show that the comparison functor R□¯¯¯¯ω∗:DMeff(k)→logDMeff(k) is t-exact. The heart of the homotopy t-structure on logDMeff(k) is the Grothendieck abelian category of strictly cube-invariant sheaves with log transfers: we use it to build a new version of the category of reciprocity sheaves in the style of Kahn-Saito-Yamazaki and Rülling