The Chow ring of a singular surface

A restriction isomorphism for cycles of relative dimension zero

“…The proof of part 1), in Section 8, rests, in equal characteristic, on the Gersten conjecture for Milnor K-theory due to the first author [Ker09]; when the residue field is finite, on the étale realization using the Kato conjecture established in [KS12]; when (d − 1)! is prime to m, on algebraic K-theory and the Grothendieck-Riemann-Roch theorem [BS98].…”

Murthy’s conjecture on 0-cycles

“…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]. However, we do not know if the two Euler classes are comparable.…”

“…Furthermore, Bl Z ′ (X) \ Bl Z ′ (X N ) = π −1 (T ) and the map π −1 (T ) → T is clearly an isomorphism. Since the connected components of Bl Z ′ (X) are precisely the inverse images of the connected components of the normal surface X, we see that the intersection of π −1 (T ) with every connected component of Bl [53] and [11]…”

“…Let i : T = Y \ Y N ֒→ Y be the inclusion of the strict normal crossing divisor on Y . Using Lemma 2.4 and Bloch's formula for normal surfaces (use either [68,Theorems 2.2,8.9] or combine [53] and [11])…”

“…For singular algebras, a proof is given in an old unpublished manuscript [54] of Levine. The dimension two case of Levine's proof is available in [11]. For normal surfaces, it is also shown in [68].…”