By Javier Majadas
Written to enrich regular texts on commutative algebra, this brief ebook supplies entire and comparatively effortless proofs of vital effects, together with the traditional effects related to localisation of formal smoothness (M. André) and localisation of entire intersections (L. Avramov), a few very important result of D. Popescu and André on average homomorphisms, and a few effects from A. Grothendieck's EGA on gentle homomorphisms. The authors make huge use of the André-Quillen homology of commutative algebras, yet basically as much as measurement 2, that's effortless to build, they usually intentionally keep away from utilizing simplicial equipment. The publication additionally serves as an available advent to a few complicated themes and methods. the one must haves are a simple path in commutative algebra and the 1st definitions in homological algebra
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Additional resources for Smoothness, Regularity and Complete Intersection
Sample text
There exists a bijective map H 1 (A, B, M ) → ExalcomA (B, M ) satisfying: a) the element 0 ∈ H 1 (A, B, M ) goes to the class of trivial extensions b) if f : C → B is an A-algebra homomorphism, the canonical homomorphism H 1 (A, B, M ) → H 1 (A, C, M ) induced by f goes to f ∗ = ExalcomA (f, M ) : ExalcomA (B, M ) → ExalcomA (C, M ). Proof Let R be a polynomial ring over A and J an ideal of R such that B = R/J. 1) α DerA (R, M ) − → HomR (J, M ) → H 1 (A, B, M ) → 0, where for d ∈ DerA (R, M ), α(d) = d|J .
Proof If ϑ can be extended, ϑ(x) = ϑ(d(X)) = dϑ (X) ∈ B(R ). Conversely, if ϑ(x) ∈ B(R ), choose an element G ∈ R such that d(G) = ϑ(x). If deg X is odd, we have R = R ⊕ RX. For r, s ∈ R, we define ϑ (r + sX) = ϑ(r) + (−1)deg ϑ ϑ(s)X + sG. If deg X is even, we have RX (i) . R = i≥0 For r0 , . . , rm ∈ R, we define m ϑ m ri X i=0 (i) = m ϑ(ri )X i=0 (i) ri X (i−1) G. + i=1 The verification that ϑ is well defined and becomes a derivation on R is straightforward. 4 If ϑ , X and G are as above, we call ϑ the canonical extension of ϑ satisfying ϑ (X) = G.
Then B is a smooth A-algebra if and only if it is J-smooth if and only if it is n-smooth. By definition, it is enough to show that n-smooth implies smooth. 5), n-smooth implies H1 (A, B, L) = 0. d) we have then 34 Formally smooth homomorphisms TorB 1 (ΩB|A , L) = 0, and since B is noetherian and ΩB|A a B-module of finite type, we deduce that ΩB|A is a projective B-module. 7). 1) B is a smooth A-algebra. 4 Field extensions Let F |K be a field extension, M an F -module. 5) we have Hn (K, F, M ) = Hn (K, F, F ) ⊗F M H n (K, F, M ) = HomF (Hn (K, F, F ), M ) for n = 0, 1, 2, so we concentrate on the F -vector spaces Hn (K, F, F ).