Throughout this paper, we shall let R represent a commutative ring with a unit. It should provide a sufficient basis for subsequent proofs. We begin with the following treatment of modules and homology. These properties will be introduced and should provide a good foundation for working with Tor in further study of algebraic topology.
Given the definition of Tor, several properties can be derived easily. The definition of projective resolutions is done meticulously in order to provide a solid basis for introducing Tor precisely. Tensor product and projective resolutions are the primary tools for defining Tor. Definitions and properties of tensor products are included in this section, as they play a central role in the definition of Tor. It is not meant to serve as an introduction to basic algebra and algebraic topology, but merely to make the treatment of Tor precise and complete. This section also provides a concise reference to the basics of algebraic topology. This presentation will serve to reacquaint the reader with basic algebraic topology. In order to make the exposition of Tor complete, a short section with preliminary definitions and properties of homology is included. Tor is defined precisely and several properties are derived. This paper introduces the functor Tor and its properties. Tensor products, Hom, Tor, and Ext are basic functors of homological algebra. Homological algebra develops the algebraic tools used to study homology. Its importance comes from the fact that homology of a space is invariant under continuous transformations of the space. Homology is one of the most important concepts of modern mathematics. Parkville, VIC 3010 update: 09 March 2012 Introduction An Introduction to Tor: Definitions and Properties.
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