Algebra Seminar: Local and Global Surjective Capacities
Local and Global Surjective Capacities
Robin Nanda Baidya, Department of Mathematics and Statistics, Georgia State University
This talk will present work that will be included in the speaker’s PhD dissertation. Let R be a commutative Noetherian ring, and let M and N be finitely generated R-modules. The global surjective capacity of M with respect to N over R is the supremum of the nonnegative integers d such that there exists a surjective R-module homomorphism from M to N^d. For every prime ideal p of R, the local surjective capacity of M with respect to N over R at p is the supremum of the nonnegative integers d such that there exists a surjective (R_p)-module homomorphism from (M_p) to (N_p)^d. Our main result states that large local surjective capacities collectively guarantee a large global surjective capacity. In the case that M is projective and N is R, our main result fully recovers, and in fact still properly generalizes, Serre’s Splitting Theorem from algebraic K-theory. In the case that R is a direct product of a combination of semilocal rings and Dedekind domains, we can supplement our main result with conditions equivalent to a given global surjective capacity. To close, we will mention several conjectures that naturally arise from this work, and we will briefly discuss some related results that will serve as the basis for future seminar talks.