An approach to the characteristic free Dutta multiplicity
Let \((A, {\mathfrak m})\) be a noetherian local ring of dimension \(d\). Let \(F_ *\) be a perfect complex of \(A\)-modules. This means that each \(F_ n\) is free of finite rank, \(F_ 0 \neq 0\) and \(F_ n = 0\) for \(n < 0\) and \(n \gg 0\). Assume that \(F_ *\) has support in \({\mathfrak m}\) which means that it is exact outside \({\mathfrak m}\). The Dutta multiplicity \(D_ A (F_ *)\) of \(F_ *\) is defined to be the rational number \(ch^ Y_ X (F_ *) \cap [Y]\), where \(Y = \text{Spec} (A)\), \(X = \text{Spec} (A/{\mathfrak m})\), ch is the localized Chern character [see \textit{W. Fulton}, ``Intersection theory (1984; Zbl 0541.14005); chapter 18] and \([Y]\) is the element of the rational Chow group \(A_ d(Y)\) given by \[ [Y]= \sum\text{length} (A_{{\mathfrak p}}) [\text{Spec} (A/{\mathfrak p})], \] the sum being taken over prime ideals \({\mathfrak p}\) of \(A\) of dimension \(d\). This definition arose as a result of the work following the counterexample constructed by \textit{S. P. Dutta}, \textit{H. Hochster} and \textit{E. MacLaughlin} [Invent. Math. 79, 253-291 (1985; Zbl 0588.13020)] to Serre's conjecture on the non-negativity of the intersection multiplicity defined as the alternating sum of Tor lengths. In this paper the author studies the positivity of the Dutta multiplicity of \(F_ *\) and its relationship with the alternating sum of the lengths of the homologies of \(F_ *\). In particular, he proves that under certain conditions \(D_ A(F_ *) = \text{length} H_ 0 (F_ *) - \text{length} H_ 1 (F_ *) > 0\).
scientific article; zbMATH DE number 469068