By Owen Biesel

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Unpublished MIT lecture notes

**Rings, Extensions, and Cohomology**

"Presenting the lawsuits of a convention held lately at Northwestern collage, Evanston, Illinois, at the social gathering of the retirement of famous mathematician Daniel Zelinsky, this novel reference presents up to date assurance of themes in commutative and noncommutative ring extensions, specifically these concerning problems with separability, Galois thought, and cohomology.

On the middle of this brief creation to class idea is the belief of a common estate, vital all through arithmetic. After an introductory bankruptcy giving the fundamental definitions, separate chapters clarify 3 ways of expressing common houses: through adjoint functors, representable functors, and bounds.

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Bij(S, S) is the set of automorphisms of K 45 to the identity on N , namely GN , so we have a group homomorphism GK /GN ! G0 ; the fundamental theorem of Galois theory for infinite extensions (see, for example, [7, Thm. 3]) tells us that GN is an open finite-index subgroup of GK and that GK /GN ! G0 is an isomorphism, giving G0 a continuous GK -action. ¯ is a bijection 3. Note that N is its own normal closure; composition with N ,! K ⇠ ¯ The action of GK by post-composition is the G0 = AutK (N ) !

2 G-closures of ´ etale extensions are ´ etale In this section, we prove that if A is a degree-n extension of R which is ´etale as an R-algebra, then each G-closure of A over R is an ´etale degree-|G| extension of R. We 40 use the following lemma, whose present formulation is that of [2, Lem. 1. Let R be a ring, and let B be an R-algebra that is finitely generated as an R-module. Then B is an ´etale degree-n extension of R if and only if there exists an ´etale R-algebra C such that the morphism of schemes Spec(C) !

Sk and R ! R[x1 , . . ,xn ]G is injective. 3, since |G| is a non-zerodivisor the latter condition is guaranteed to hold. 4 An-closures In this section we give generators and relations for R[x1 , . . , xn ]An as an R[x1 , . . , xn ]Sn algebra, and combine this presentation with our earlier results to parametrize An closures of monogenic extensions. 1. Let n be a natural number. Then Z[x1 , . . , xn ]An is a Z[x1 , . . , xn ]Sn Q module with free basis {1, }, where is the An -orbit sum of ni=1 xii 1 .