Suslin homology via cycles with modulus and applications

Abstract

We show that for a smooth projective variety X X over a field k k and a reduced effective Cartier divisor D ⊂ X D \subset X , the Chow group of 0-cycles with modulus C H 0 ( X | D ) CH_0(X|D) coincides with the Suslin homology H 0 S ( X ∖ D ) H^S_0(X \setminus D) under some necessary conditions on k k and D D . We derive several consequences, and we answer to a question of Barbieri-Viale and Kahn.

Bertini theorems revisited

“…For another application of the Bertini theorem for normal crossing schemes, we refer to the proof of [4, Lemma 3.3]. It has also been realized that the Bertini regularity theorem for hypersurfaces containing a 0‐cycle is a key requirement in the potential generalizations of the main results of [4] to regular schemes over imperfect base fields. For applications of the Bertini theorems for normality and $$(R_a + S_b)$$‐property, we refer the reader to the proofs of [14, Theorem 1.2, p. 33 and Theorem 8.6, p. 46].…”