18.785F17 Number Theory I Lecture 23 Notes: Tate Cohomology
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چکیده
In this lecture we introduce a variant of group cohomology known as Tate cohomology, and we define the Herbrand quotient (a ratio of cardinalities of two Tate cohomology groups), which will play a key role in our proof of Artin reciprocity. We begin with a brief review of group cohomology, restricting our attention to the minimum we need to define the Tate cohomology groups we will use. At a number of points we will need to appeal to some stan dard results from homological algebra whose proofs can be found in Section 23.6. For those seeking a more thorough introduction to group cohomology, see [1]; for general background on homological algebra, we recommend [7].
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