Test modules to calculate Dutta multiplicities

Using Frobenius maps, the Dutta multiplicity chi (infinity)(F) was first de fined by Dutta for a complex F over a local ring of positive characteristic . Nowadays, using localized Chern characters or Adams operations, it is def ined for a complex not only over a local ring of positive characteristic bu t also over a ring that is a homomorphic image of an arbitrary regular loca l ring. In the paper, we shall give another characterization to Dutta multi plicity using some Galois extension. Furthermore, we shall define the notion of test modules. If a test module f or a local ring exists, then the positivity conjecture of Dutta multiplicit ies is true over the ring. In the paper, it is proved that, if the small Co hen-Macaulay modules conjecture is true, then test modules always exist for complete local domains whose field of fractions are of characteristic 0. ( C) 2001 Academic Press.