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I have previously experienced cohomology group and cohomology ring.

Today someone suggested me to look over the concepts of cohomology field. I think what is meant there is that the cohomology $H^n(G,C)$, the coefficient $C$ can be a group, a ring or generalized to a field.

My question is that :

what are the advantages to look at Cohomology group vs Cohomology ring vs Cohomology field?

Is it possible (always possible or when is it possible) to generalize a Cohomology group to a Cohomology ring, to a Cohomology field?

see also What are cohomology rings good for?

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annie marie cœur
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    Cohomology field is not an object I am familiar with. One can look at cohomology with coefficients in a field, which I suspect is what you are talking about, and this has many uses. For one, it tends to be easier to compute than integral cohomology. There are also well understood algebras that act on cohomology with coefficients in $\mathbb{F_p}$ that provide very useful structure on the cohomology. – Connor Malin Apr 27 '20 at 19:40
  • It is always possible to "generalise" to a ring structure (but it may be trivial). Mainly, the question has been answered at the linked post. – Dietrich Burde Apr 27 '20 at 19:40
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    You mean cohomology with coefficients in a field, which then is a vector space (or, put together with the cup product, an algebra). – Ted Shifrin Apr 27 '20 at 21:42
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    You should ask that person for a definition of a "cohomology field." – Moishe Kohan Apr 27 '20 at 21:43
  • @Ted Shifrin, yes cohomology with coefficients in a field. – annie marie cœur Apr 28 '20 at 02:01

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