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AAG (Arithmetic algebraic geometry)
Arithmetic
Last chapter of Dobbs LNM0147
Berkson and McConnell refers to their references 1, 3 and 7 for foundational material on Amitsur cohomology.
There might be a manuscript called Cech and Amitsur cohomology, by Grace Geist Orzech.
This Google Books link appears to give the basic definition of Amitsur cohomology, and mentions its description as the Cech cohomology associated to a certain Grothendieck topology.
Page 111 in Selected papers of Amitsur with commentary Google Books link describes Amitsur cohomology as a cohomology associated to a field extension, generalizing Galois cohomology. It seems like it can be described as etale cohomology and/or Cech cohomology, but this is not very clear. It is also described as a cohomology theory for commutative rings.
I think Milne states in his book that Amitsur cohomology captures a fragment of flat cohomology.
nLab page on Amitsur cohomology