We prove a moving lemma which implies the contravariance of Bloch-Esnault's additive higher Chow group in smooth affine varieties and Binda-Saito's higher Chow group (taken in the Nisnevich topology) in smooth varieties equipped with effective Cartier divisors. The new ingredients in the moving method are parallel translation with modulus in the affine space that involves a new integer parameter, and Noether's normalization lemma over a Dedekind base.

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, Rülling, 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.

In this paper, we construct Chern classes from the relative $K$ -theory of modulus pairs to the relative motivic cohomology defined by Binda–Saito. An application to relative motivic cohomology of henselian dvr is given.

The moving lemma of Suslin states that a cycle on X × A n meeting all faces properly can be moved so that it becomes equidimensional over A n . This leads to an isomorphism of motivic Borel-Moore homology and higher Chow groups.In this short paper we formulate and prove a variant of this. It leads to an isomorphism of Suslin homology with modulus and higher Chow groups with modulus, in an appropriate pro setting. Recently the context has been extended to cycles with modulus by Binda-Kerz-Saito [KS, BS] and Kahn-Saito-Yamazaki [KSY]. The reader finds the definitions below. There is an obvious injection (r ≥ 0)for each pair (X, Y ) consisting of a finite type k-scheme X and an effective Cartier divisor Y on it. We usually write X := X \ Y .

Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated Green-Tao theorem on arithmetic progressions of rational primes and Tao's theorem on constellations of Gaussian primes. Furthermore, we prove a constellation theorem on prime representations of binary quadratic forms with integer coefficients. More precisely, for a non-degenerate primitive binary quadratic form F which is not negative definite, there exist arbitrarily shaped constellations consisting of pairs of integers (x, y) for which F (x, y) is a rational prime. The latter theorem is obtained by extending the framework from the ring of integers to the pair of an order and its invertible fractional ideal.