Zero-cycle groups on algebraic varieties
On smooth varieties over a field, we have the motivic cohomology theory constructed in a few different way. For instance, Voevodsky's motivic cohomology of smooth varieties is isomorphic to Bloch's higher Chow groups. When the varieties have singularities, there have been a couple of new attempts, such as Levine-Weibel Chow groups \(\mathrm{CH}_0 ^{LW} (X)\) or Chow groups \(\mathrm{CH}_0 (X|D)\) with modulus, for instance, that provided certain successful consequences in a few special cases. This article is one of the recent results by the authors (possibly with other authors, as well), that compare those different attempts. The results in this article present a few omnibus results in a few different special situations in this direction. The inter-relations among those proven are not too strong, but each of them provides certain extensions of previously known results, or clarifications of certain failures of certain expectations. In the first episode of the article, Theorems 1.1 and 1.2 are regarding a natural surjection \[ \lambda_{X|D}: \mathrm{CH}_0 (X|D) \to H_0 ^{\mathrm{Sus}} (U),\] from the Chow group of \(0\)-cycles associated to the modulus pair \((X, D)\) to the \(0\)-th Suslin homology of \(U= X \setminus D\). The first of the theorems shows that when the base field \(k\) is algebraically closed and \(D\) is reduced, this map becomes an isomorphism for \(\mathrm{char} (k) > 0\). Rather interestingly this fails to be an isomorphism in general for \(\mathrm{char} (k) = 0\) case. The second theorem shows that when \(k\) is perfect and the components of \(D_{\mathrm{red}}\) are regular, the map \(\lambda_{X|D}\) becomes an isomorphism after inverting \(p\) for \(\mathrm{char} (k) = p>0\) case, while the map \(\lambda_{X|D}\) becomes an isomorphism mod \(m\) for each integer \(m \not = 0\) for \(\mathrm{char} (k) = 0\). The second episode of this omnibus paper is regarding the torsions of the Suslin homology and the étale cohomology with compact supports. More precisely, in Theorem 1.3 the authors proved that when \(k\) is algebraically closed and \(U\) is a quasi-projective variety of dimension \(d\) admitting a smooth compactification, then for each prime \(\ell \not = p\), we have an isomorphism \[ H_0 ^{\mathrm{Sus}} (U) \{ \ell \} \simeq H_{c, \acute{e}t} ^{2d-1} (U, \mathbb{Q}_{\ell} / \mathbb{Z}_{\ell} (d)).\] The third episode is regarding finite coefficients of various theories, inspired by a conjecture of Colliot-Thélène on the finiteness of the Chow groups mod \(m\) over a \(p\)-adic field. This was known by smooth projective varieties \(X\) by Saito and Sato. In Theorem 1.5, the authors study an analogue of this question replacing \(X\) with a bit more general ones. More precisely, when \(X\) is an integral quasi-projective scheme regular in codimension \(1\), with an effective Cartier divisor \(D\), the authors show that (1) \(H_0 ^{\mathrm{Sus}} (X) / m\) is finite when \(X\) is smooth, (2) \(\mathrm{CH}_0 (X|D) / m\) is finite when \(X\) is both smooth and complete, and (3) \(\mathrm{CH}^{LW}_0 (X)/m\) is finite when \(X\) is complete. The last episode is about the comparison of the Levine-Weibel Chow group with the Friedlander-Voevodsky motivic cohomology when the coefficients ring \(\Lambda\) is either \(\mathbb{Z}[ \frac{1}{p}]\) with the exponential characteristic \(p\geq 1\), or \(\mathbb{Z}/m\) where \(m\) is invertible in \(k^{\times}\). There is a natural map \[ \lambda_X: \mathrm{CH}_0 ^{LW} (X) \otimes_{\mathbb{Z}} \Lambda \to H^{2d} (X, \Lambda (d)),\] and the authors prove that when \(k\) is an infinite perfect field and \(X\) has normal crossing singularities, then the natural map is an isomorphism for the above two cases of \(\Lambda\). A couple of consequences are deduced from the theorem.
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