The Relative Kunneth Theorem
Abstract
Let A be a unital ring and B a unital subring. In this dissertation we study the relative homological algebra arising from the pair (A, B). We introduce relative analogues of free, projective, and flat modules, and we show in which sense they generalize their absolute analogues. We systematically characterize these modules in terms of relative free modules, which play a key role in this exposition.
We introduce a section of the connecting homomorphism in the associated long exact sequence to a short exact sequence. We prove that if our original short exact sequence splits, then the associated long exact sequence also splits. We use this to prove that the expected long exact sequences of relative Tor are split. Finally, we use the splitting long exact sequences of Tor to prove a relative version of the Künneth Theorem, where the resultant short exact sequences are split.
Citation
Sanchez Ocal, Pablo (2021). The Relative Kunneth Theorem. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /195097.