Torsion and divisibility for reciprocity sheaves and 0-cycles with modulus (Q330177)

Let \(k\) be a field and let \(X\) be a proper variety over \(k\), equipped with an effective Cartier divisor \(D\). The Chow group CH\(_0(X/D)\) of \(0\)-cycles with modulus \(D\) is a quotient of the group \(Z_0(X)\) of \(0\)-cycles on the open complement \(Y= X-|D|\). When \(X\) is a projective curve the group CH\(_0(X/D)\) is isomorphic to the relative Picard group Pic\((X,D)\) of isomorphism classes of pairs given by a line bundle on \(X\) together with a trivialization along \(D\).NEWLINENEWLINE In this paper the authors study the nonhomotopy invariant part of the groups CH\(_0(X/D)\), and examine their divisibility and torsion properties.NEWLINENEWLINE There is a canonical surjection from the Chow group with modulus to the \(0\)th Suslin homology group NEWLINE\[NEWLINE\pi_{X,D}: \text{CH}_0(X/D)\to H^{\text{Sing}}_0(X).NEWLINE\]NEWLINE Since \(H^{\text{sing}}_0(X)\) is the maximal homotopy invariant quotient of the group \(Z_0(X)\) the kernel \(U(X/D)\) of \(\pi_{X,D}\) measures the failure of CH\(_0(X/D)\) to be homotopy invariant. For the group \(U(X/D)\) one has the following resultsNEWLINENEWLINE (1) If \(\text{char\,}k=0\) the group \(U(X/D)\) is divisible,NEWLINENEWLINE (2) if \(\text{char\,}k=p\) then \(U(X/D)\) is a \(p\)-primary torsion group.NEWLINENEWLINE The following theorem takes care of the torsion part of CH\(_0(X/D)\), in some cases.NEWLINENEWLINE Theorem 1. Suppose that:NEWLINENEWLINE (i) \(k\) is algebraically closed of exponential characteristic \(p\geq1\),NEWLINENEWLINE (ii) \(X\) is a projective variety, regular in codimension \(1\) and the open complement \(Y= X-|D|\) is smooth.NEWLINENEWLINE Let \(\alpha\in\text{CH}_0(X/D)\) be a prime-to-\(p\)-torsion cycle. Then there exists a smooth projective curve \(C\) and a morphism \(\phi:C\to X\), for which \(\phi^*(D)\) is a well-defined Cartier divisor on \(C\), and a prime-to-\(p\)-torsion cycle \(\beta\in \text{CH}_0(C/(\phi^*(D)_{\text{red}})\) such that \(\phi_*(\beta)= \alpha\).NEWLINENEWLINE The paper also contains similar results for reciprocity sheaves with transfers. The notion of reciprocity for presheaves with transfers, which is weaker than homotopy invariance, has been developed by \textit{B. Kahn} et al. [Compos. Math. 152, No. 9, 1851--1898 (2016; Zbl 1419.19001)] with the purpose of eventually constructing a new motivic triangulated category larger than Voevodsky's category \(DM^{\text{eff}}(k,\mathbb{Z})\).