Related papers: Zero-cycles with modulus and relative $K$-theory

Let $k$ be a field of characteristic zero, and let $X$ be a projective variety embedded into a projective space over $k$. For two natural numbers $r$ and $d$ let $C_{r,d}(X)$ be the Chow scheme parametrizing effective cycles of dimension…

We show the existence of a regular universal quotient as a smooth commutative algebraic group of the Chow group of 0-cycles on a projective reduced variety, and give over the field of complex numbers an analytic description of it. This…

The paper has two parts. First we prove that the specialization maps on R-equivalence and on the Chow group of zero cycles are isomorphisms for families over a local, Henselian, Dedekind ring when the special fiber is smooth and separably…

We study a local to global principle for certain higher zero-cycles over global fields. We thereby verify a conjecture of Colliot-Th\'el\`ene for these cycles. Our main tool are the Kato conjectures proved by Jannsen, Kerz and Saito. Our…

The classical cycle class map for a smooth complex variety sends cycles in the Chow ring to cycles in the singular cohomology ring. We study two cycle class maps for smooth real varieties: the map from the I-cohomology ring to singular…

We show that for an associative algebra A and its ideal I such that the I-adic topology on A coincides with the p-adic topology, the relative continuous K-theory pro-spectrum "lim"K(A_i, IA_i), where A_i :=A/p^i A, is naturally isogenous to…

Let k be an algebraically closed field and X a smooth projective k-variety. A famous theorem of A. A. Roitman states that the canonical map from the degree zero part of the Chow group of zero cycles on X to the group of k-points of its…

Bloch and Esnault defined additive higher Chow groups with modulus (m+1) on the level of zero cycles over a field k, denoted by TH^n(k,n;m), n,m >0. They prove that TH^n(k,n;1) is isomorphic to the group of absolute Kaehler differentials of…

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…

Let $C$ be a smooth and projective curve over the truncated polynomial ring $k_m:=k[t]/(t^m), $ where $k$ is a field of characteristic 0. Using a candidate for the motivic cohomology group ${\rm H}^{3}_{\pazocal{M}}(C,\mathbb{Q}(3))$ based…

Let k be a characteristic zero field, C a k-algebra and M a square zero two sided ideal of C. We obtain a new mixed complex, simpler that the canonical one, giving the Hochschild and cyclic homologies of C relative to M. This complex…

The Chow group of zero cycles in the moduli space of stable pointed curves of genus zero is isomorphic to the integer additive group. Let $M$ be monomial in this Chow group. If no two factors of $M$ fulfill a particular quadratic relation,…

We show that the characteristic homomorphism of Colliot-Th{\'e}l{\`e}ne and Sansuc from the Chow group of 0-cycles of degree~0 is compatible with restriction and corestriction maps.

We prove Bloch's formula for 0-cycles on affine schemes over algebraically closed fields. We prove this formula also for projective schemes over algebraically closed fields which are regular in codimension one. Several applications,…

We extend the definition of relative Gromov--Witten invariants with negative contact orders to all genera. Then we show that relative Gromov--Witten theory forms a partial CohFT. Some cycle relations on the moduli space of stable maps are…

We prove Bloch's formula for the Chow group of 0-cycles with modulus on smooth projective varieties over finite fields. The proof relies on two new results in global ramification theory.

We generalize a recent result of Pavic--Schreieder regarding the surjectivity of the obstruction morphism defined in [PS23]. As a consequence of this result, we show that geometrically (retract) rational varieties over a Laurent field of…

In this article we describe the $G\times G$-equivariant $K$-ring of $X$, where $X$ is a regular compactification of a connected complex reductive algebraic group $G$. Furthermore, in the case when $G$ is a semisimple group of adjoint type,…

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let $Y\ra X$ be a covariant morphism…

We study the deformations of the Chow group of zero-cycles of the special fibre of a smooth scheme over a henselian discrete valuation ring. Our main tools are Bloch's formula and differential forms. As a corollary we get an algebraization…