The Dutta multiplicity of a bounded complex \(F\) of free modules over a local ring \((A, \mathfrak m)\) of prime characteristic with \(d = \dim A\) such that the homology of \(F\) is of finite length was introduced by S. P. Dutta [J. Algebra 85, 424-448 (1983; Zbl 0527.13014)]. In several respects it has better properties then the ordinary Euler-Poincaré characteristic. In fact, P. C. Roberts has shown [in: Commutative Algebra, Proc. Microprogram, Berkeley 1989, Publ., Math. Sci. Res. Inst. 15, 417-436 (1989; Zbl 0734.13009)], that it is positive whenever \(F\) is non-exact and of length \(d.\) This is the main ingredience of the second author’s proof of the intersection theorem in mixed characteristics (loc. cit.). More recently, the concept of Dutta multiplicity has been generalized to rings of arbitrary characteristic using localized Chern characters [see K. Kurano, J. Math. Soc. Japan 45, No. 3, 369-390 (1993; Zbl 0815.13011)]. This definition agrees with the original one for rings of positive characteristic defined in terms of limits over the Frobenius map. The main theorem of the paper is the following:
Let \((A, \mathfrak m)\) be a homomorphic image of a regular local ring. Suppose that \(A\) contains a field. Let \(F\) denote a perfect complex of length \(d = \dim A\) with \(\text{Supp} F = \{\mathfrak m\}.\) Then the Dutta multiplicity of \(F\) is positive.
It is open whether the result is true without the assumption that \(A\) contains a field. The authors’ proof requires a guide through Adams’ operations and their relations to localized Chern characters. Moreover the proof needs a reduction to a local ring that has a perfect residue field. As an application of their results the authors prove a generalization of Dutta’s result [S. P. Dutta, Proc. Am. Math. Soc. 103, No. 2, 344-346 (1988; Zbl 0653.13018)], on the positivity of intersection multiplicities.