Fellowship Of The Ring: Cancellation Of Finite-Dimensional Noetherian Modules
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The Module Cancellation Problem solicits hypotheses that, when imposed on modules K, L, and M over a ring S, afford the implication K⊕L≅K⊕M⟹L≅M. In a well-known paper on basic element theory from 1973, Eisenbud and Evans lament the "great scarcity of strong results" in module cancellation research, expressing the wish that, "under some general hypothesis" on finitely generated modules over a commutative Noetherian ring, cancellation could be demonstrated. Singling out cancellation theorems by Bass and Dress that feature "large" projective modules, Eisenbud and Evans contend further that, although "[s]ome criteria of 'largeness' is certainly necessary in general [. . . ,] the need for projectivity is not clear." In this talk, we will contextualize the preceding entreaty of Eisenbud and Evans, offer a response to it, and then construct a cancellation example that simultaneously eludes many observations from the module cancellation literature. Along the way, we will discuss some unanswered questions in the hopes of giving direction to this broad area of research.
Cancellation of Finite-Dimensional Noetherian Modules