Global Frobenius Betti Numbers and Frobenius Euler Characteristics

We extend the notion of Frobenius Betti numbers to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. To do so, we introduce new invariants, which we call Frobenius Euler characteristics. We prove uniform convergence and upper semicontinuity results for Frobenius Betti numbers and Euler characteristics. These invariants detect the singularities of a ring, extending two results from the local to the global setting.