𝐾-theory and 0-cycles on schemes

Abstract

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, including Bloch's formula for 0-cycles with modulus, are derived.

Murthy’s conjecture on 0-cycles

“…(3) J is a product of distinct smooth maximal ideals of A of height d. It follows from (1) and (2) The following result is a special case of [35,Theorem 7.5]. We reproduce the proof for the sake of completeness of our argument of the main results of this section.…”

“…In particular, this yields Roitman torsion theorem for the Chow group of 0-cycles with modulus. Bloch's formula for the Levine-Weibel Chow group and the 0-cycle group with modulus was shown in [35]. An application of Murthy's conjecture in the identification of a motivic spectral sequence for the relative K-theory was obtained in [47].…”

“…M. Schlichting [72] has recently defined Euler classes of projective modules of top rank over an affine algebra in a cohomology group of some Milnor K-theory sheaf and has proven an analogue of Theorem 1.2 for his Euler classes. Using Theorem 1.3, it is shown in [35] that Schlichting's cohomology group coincides with the Levine-Weibel Chow group (see below). This was known before in dimension two using [53] and [11].…”

Zero-cycles on normal varieties

“…A hypersurface section Y satisfying the above conditions will be called a 'good' hypersurface section. Since X sing is reduced, it follows from (4) that the scheme theoretic intersection Y s ∶= Y ∩ X sing is an effective Cartier divisor on X sing (see [23,Lemma 3.3]) and (Y s ) red = Y sing (see Lemma 4.12). We let…”

“…If X is defined over an algebraically closed field, Theorem 1.5 is a special case of a more general result [23,Theorem 1.8], due to Gupta and Krishna (see also [38] and [39] for earlier results).…”