Tatheer Ajani

Document Type

Honors Thesis


The Tor functor plays a large role in homological algebra and its uses include defining generalized algebraic structures such as the homology of groups and associative algebras. Many proven properties of Tor involve two distinct R-modules A and B, but self Tor concerns a single module, in other words, A = B. This work takes a look at the classical case of the vanishing of Tor when A is the quotient of R by an ideal and uses the definitions and theorems we develop in order to generalize the classical case. We begin definitions of chain complexes, homology and exactness, projective modules and resolutions, syzygy, tensor products, and functors. We use that information to construct and define the Tor functor, the key element to our classical case. The main result of this work uses the isomorphism between self Tor on the (p − 1)th syzygy module and the 1st homology module of a free resolution tensored with itself. Then using the minimal generator of the pth syzygy module, we find a 1-cycle of the free resolution tensored with itself, showing that self Tor of a finitely generated module is not 0, up to isomorphism. We conclude by demonstrating that the classical case is merely a corollary of this theorem.

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