By Tangora M.C. (ed.)
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Extra resources for Algebraic Topology: Oaxtepec 1991
3. Suppose given C a local Artin ring, J an ideal, and C = C /J. Suppose given A flat over C such that A0 = A ⊗C k is a local Cohen–Macaulay ring. And suppose given B = A/a, a quotient of A = A ×C C, flat over C, such that B0 = B ⊗C k is a complete intersection quotient of A0 of codimension r. Then 1) a can be generated by r elements a1 , . . , ar , and the Koszul complex K• (A; a1 , . . , ar ) gives a resolution of B. 2) If a1 , . . , ar are any liftings of the ai to A , then the Koszul complex K• (A ; a1 , .
Proof. Note first we cannot expect to get the exact number of generators for I, because if V, ϕ is an obstruction theory, any bigger vector space V containing V will also be one. So consider any sequence 0 → J → C → C → 0 and map u as in the definition 0 → I → P → A → 0 ¯ ↓f ↓f ↓u 0 → J → C → C → 0. 2, we lift u to a map f : P → C and get a restriction f¯ : I → J which does not depend on the choice of f , and which is zero if and only if u lifts to a map u : A → C . We also get an element ϕ(u, C ) ∈ V ⊗ J.
We say in that case that B is an extension as k-algebras of the k-algebra B by the B-module B. This discussion suggests that we consider a more general situation. Let A be a ring, let B be an A-algebra, and let M be a B-module. We define an extension of B by M as A-algebras to be an exact sequence 0→M →B →B→0 where B → B is a homomorphism of A-algebras, and M is an ideal in B with M 2 = 0. Two such extensions B , B are equivalent if there is an isomorphism B → B compatible in the exact sequences with the identity maps on B and M .
Algebraic Topology: Oaxtepec 1991 by Tangora M.C. (ed.)