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  "body": "\n## Summary\n\nTate cohomology joins the cohomology and the [[homology|CFAG3VRXEA]] of a finite group $G$ into one sequence of groups $\\hat H^n(G, M)$ indexed by all integers $n$.\nIt agrees with ordinary cohomology in positive degrees and with homology in degrees at most $-2$.\nIn the two middle degrees it is modified with the norm map.\nA short exact sequence of modules gives a long exact sequence infinite in both directions.\nTate cohomology is a basic tool of class field theory and of [[Galois cohomology|9MPCZ3EAJM]], and for cyclic groups it is periodic of period $2$.\n\n## Explanations\n\n### L1 human\n\nFor a finite group, cohomology runs upward from degree $0$ and homology runs downward.\nTate noticed that the two can be glued into a single sequence that runs in both directions, after a small correction at the junction.\nThe correction uses the norm, the sum of all the group elements acting on a module.\nLet a cyclic group of order $m$ act on $\\Z$ with every element fixing every integer.\nThen the groups alternate between $\\Z/m\\Z$ and $0$.\n",
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