Implementation of Critical Analysis of Homological Results in Commutative Algebra

Abstract

When it was first made, algebraic geometry was mostly about studying spaces that can be described by equations over real and complex numbers. When reducing these equations modulo a prime number to solve specific problems, negative characteristic questions will always come up. The characteristic of the ring R is the prime number p if and only if the equation pr = 0 is true for every r in the ring R. The most important benefit of this is that we will be able to use the Frobenius map, which is a type of ring homomorphism that moves an element from r to rp. The Frobenius map is an important part of almost all theories about the positive characteristic. Hilbert-Kunz multiplicities, also called limits of regular multiplicities over repetitions of the Frobenius map, are one of the main topics of research in the theory of positive characteristics. The most important part of Paul Monsky's argument is whether or not Hilbert-Kunz multiplicities can be thought of as logical.