Test Modules to Calculate Dutta Multiplicities

Using Frobenius maps, the Dutta multiplicity χ∞( ) was first defined by Dutta for a complex over a local ring of positive characteristic. Nowadays, using localized Chern characters or Adams operations, it is defined for a complex not only over a local ring of positive characteristic but also over a ring that is a homomorphic image of an arbitrary regular local ring. In the paper, we shall give another characterization to Dutta multiplicity using some Galois extension. Furthermore, we shall define the notion of test modules. If a test module for a local ring exists, then the positivity conjecture of Dutta multiplicities is true over the ring. In the paper, it is proved that, if the small Cohen–Macaulay modules conjecture is true, then test modules always exist for complete local domains whose field of fractions are of characteristic 0.