A note on Chow groups and intersection multiplicity of modules (Q1320256)

Let \(R\) be a complete regular local ring. If \(R\) is unramified then one knows that: (1) The Chow group \(A_ i(R)=0\) for \(i<\dim R\) [\textit{L. Claborn} and \textit{R. Fossum}, Ill. J. Math. 12, 228-253 (1968; Zbl 0159.049)]; (2) If \(M\) and \(N\) are finitely generated \(R\)-modules such that \(\ell (M \otimes N)<\infty\) then: (a) \(\chi (M,N):=\sum_{i \geq 0} (-1)^ i \ell(\text{Tor}_ i (M,N))=0\) if \(\dim M+\dim N<\dim R\) and (b) \(\chi (M,N)>0\) if \(\dim M+ \dim N=\dim R\) [\textit{J.-P. Serre}, ``Algèbre locale. Multiplicités'', Lect. Notes Math. 11 (1965; Zbl 0142.286)]; (3) For \(M,N\) as above, \(\chi_ i(M,N): = \sum_{t \geq 0} (-1)^ t\ell(\text{Tor}_{i+t} (M,N)) \geq 0\) for every \(i>0\) [\textit{S. Lichtenbaum}, Ill. J. Math. 10, 220-226 (1966; Zbl 0139.266), and \textit{M. Hochster}, ibid. Ill. J. Math. 28, 281-285 (1984; Zbl 0562.13019)]. In the ramified case, the above assertions are open problems, except for (2a) which was proved, independently, by \textit{P. Roberts} [Bull. Am. Math. Soc., New Ser. 13, 127-130 (1985; Zbl 0585.13004)] and \textit{H. Gillet} and \textit{C. Soulé} [C. R. Acad. Sci., Paris, Sér. I 300, 71- 74 (1985; Zbl 0587.13007)]. The present paper represents a contribution to these problems. Among other (more technical) things, the author proves that, for \(R\) ramified: (i) \(A_ 1(R)=0\); (ii) In order to prove (2b) one may assume that \(\text{depth} M=\dim M- 1\), \(\text{depth} N=\dim N-1\); (iii) The assertion (3) for \(i=2\) implies (2b) for \(M\) Cohen-Macaulay.