Related papers: A decomposition theorem for 0-cycles and applicati…

Let $R$ be a regular semi-local ring, essentially of finite type over an infinite perfect field of characteristic $p \ge 3$. We show that the cycle class map with modulus from an earlier work of the authors induces a pro-isomorphism between…

We determine the Chow group of zero-cycles on a rational surface X defined over a finite extension K of the field of p-adic numbers (p a prime) when X is split by an unramified extension of K.

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 -…

The Chow groups of codimension-p algebraic cycles modulo rational equivalence on a smooth algebraic variety X have steadfastly resisted the efforts of algebraic geometers to fathom their structure. This book explores a "linearization"…

In this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension $\le 1$. In this process, we establish the decomposition of Chow groups for the cases of Cayley's trick and…

Given a smooth surface $X$ over a field and an effective Cartier divisor $D$, we provide an exact sequence connecting $CH_0(X,D)$ and the relative $K$-group $K_0(X,D)$. We use this exact sequence to answer a question of Kerz and Saito…

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…

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…

We present a relation between the classical Chow group of relative $0$-cycles on a regular scheme $\mathcal{X}$, projective and flat over an excellent Henselian discrete valuation ring, and the Levine-Weibel Chow group of 0-cycles on the…

We prove that the Chow-Witt group of zero-cycles is a birational invariant of smooth proper schemes over a base field.

We consider proper, algebraic semismall maps f from a complex algebraic manifold X. We show that the topological Decomposition Theorem implies a "motivic" decomposition theorem for the rational algebraic cycles of X and, in the case X is…

Using recent developments in the theory of mixed motives, we prove that the log Bloch conjecture holds for an open smooth complex surface if the Bloch conjecture holds for its compactification. This verifies the log Bloch conjecture for all…

In this paper, we study $\mathbb{A}^1$-equivalence classes of zero cycles on open complex algebraic surfaces. We prove the logarithmic version of Mumford's theorem on zero cycles and prove that log Bloch's conjecture holds for…

We show that the Levine-Weibel Chow group of 0-cycles $\CH^d(A)$ of a reduced affine algebra $A$ of dimension $d \ge 2$ over an algebraically closed field is torsion-free. Among several applications, it implies an affirmative solution to an…

We prove that Chow groups of certain non-commutative Hilbert schemes have a basis consisting of monomials in Chern classes of the universal bundle. Furthermore, we realize the cohomology of non-commutative Hilbert schemes as a module over…

We prove a blow-up formula for cyclic homology which we use to show that infinitesimal $K$-theory satisfies $cdh$-descent. Combining that result with some computations of the $cdh$-cohomology of the sheaf of regular functions, we verify a…

We prove that all points of a toroidal compactification lying over 0-dimensional cusps are rationally equivalent in the integral Chow group for most classical modular varieties (Siegel, Hilbert, orthogonal, Hermitian, quaternionic). This…

Let $k$ be a perfect field of characteristic $p > 0$, $W_n = W_n(k)$. For separated $k$-schemes of finite type, we explain how rigid cohomology with compact supports can be computed as the cohomology of certain de Rham-Witt complexes with…

Let U be a smooth quasi-projective variety over a field k that is finite, the algebraic closure of a finite field or algebraically closed of characteristic 0. Let X be a suitable projective compactification of U, and D an effective divisor…

The decomposition theorem for smooth projective morphisms $\pi:\mathcal{X}\rightarrow B$ says that $R\pi_*\mathbb{Q}$ decomposes as $\oplus R^i\pi_*\mathbb{Q}[-i]$. We describe simple examples where it is not possible to have such a…