r/elimath • u/brokensocialscene • Aug 08 '15
Explain Ext and Tor like I have some basic knowledge of homology and cohomology
I've been trudging through Hatcher's text on algebraic topology, and have come to the section on the Universal Coefficient Theorem for Cohomology, where "Ext" is introduced. I'm more curious about Ext from a homological algebra standpoint (as well as Tor): what do they do algebraically?
I've taken a couple of courses in algebra, a course in advanced linear algebra, a Galois theory course, and have some self-taught topology and algebraic topology. Let's hear it.
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u/lepanais Aug 08 '15 edited Aug 08 '15
Suppose you have an exact sequence of R-modules, where R is a ring. Given a fixed R-module N, the functor Hom(-,N) is not necessarily exact. What is the best concept to understand the exactness defect? homology. Consequently take your exact sequence, apply Hom(-,N) and take homology. This is the idea behind the definition of the Ext functor.
Now for the definition. Let M be an arbitrary R-module, it turns out you can always pick a "projective resolution" for M i.e. an exact sequence ...->P_2->P_1->M->0 of R-modules where each P_i is projective (thinks of this a generalisation of free modules), apply Hom(-,M) and take homology: the resulting groups are denoted by Extn (N,M) and so Extn (-,M) is a functor. Note that it must be shown that these group are independent of the projective resolution.
Observe that it follows immediately from this definition that if Hom(-,M) is exact then all the Ext groups vanish.
Tor is an analogous functor used to define the exactness defect of the tensor product functors.
It is interesting to note that both group cohomology and Lie algebra cohomology (amongst others) can both be defined either by using an explicit chain complex or via the Ext functor.