[...]... case which is intended 1 Definition and first properties of (co-)homology modules In this chapter we define the Lichtenbaum–Schlessinger (co-)homology modules Hn (A, B, M ) and H n (A, B, M ), for n = 0, 1, 2, associated to a (commutative) algebra A → B and a B-module M , and we prove their main properties [LS] In Section 1.1 we give a simple definition of Hn (A, B, M ) and H n (A, B, M ), but without justifying... variables with (r ) deg Yi > 0, and the divided powers Yj t,j have integer exponents rt,j ≥ 0 (Of course, for deg Yj odd and r > 1, we understand (r) (l) Yj = 0.) Then for l > 0, the image of Xi is determined by the familiar divided power rules† (α1 ) (a) (Y1 + · · · + Yv )(l) = Y1 (αv ) · · · Yv ; and α1 +···+αv =l α1 , ,αv ≥0 (l) (b) (Y1 Y2 )(l) = Y1l Y2 (if deg Y1 and deg Y2 ≥ 2 are even) Thus R(n)... homomorphism and p : R → B and q : S → B two free DG resolutions of the A-algebra B Let f, g : R → S be two homomorphisms of augmented DG A-algebras, i.e., p = qf , p = qg Then there exist B-module homomorphisms αi : (ΩR|A ⊗R B)i → (ΩS|A ⊗S B)i+1 for i = 0, 1, 2 such that dS α0 = f0 − g0 , and dS αi + αi−1 dR = fi − gi , for i = 1, 2, 1 i+1 i where dR , dS are the differentials of R and S respectively, and denotes... ring, B an A-algebra, T a multiplicative subset of B and M a B-module Then T −1 Hn (A, B, M ) = Hn (A, B, T −1 M ), n = 0, 1, 2 and if A is noetherian and B an A-algebra of finite type, then also T −1 H n (A, B, M ) = H n (A, B, T −1 M ), n = 0, 1, 2 Proof a) follows from base change (1.4.3) by the homomorphism A → T −1 A, and c) follows from 1.4.5, a) and b) For b), note first that the case n = 0 is clear... A-algebras where A is a polynomial A-algebra By (1.4.6) and (1.4.1) we have H2 (A, B, M ) = H2 (A , B, M ) and an exact sequence 0 → H1 (A, B, M ) → H1 (A , B, M ) → H0 (A, A , M ) and the same isomorphism and exact sequence with B replaced by T −1 B So it is enough to prove the result for A instead of A Let then U be the inverse image of T in A By a) and (1.4.3) we have Hn (A , T −1 B, M ) = Hn (U −1... Ef → Eg defined by h(x, r) = (x + d(r), r) is well defined and makes the diagram 0 → 0 → M M → Ef ↓ → Eg → B → 0 → B → 0 commute We then define v : H 1 (A, B, M ) → ExalcomA (B, M ) by v([f ]) = [E] It is easy to check that u and v are inverse, and properties a) and b) 2.2 Formally smooth algebras Definition 2.2.1 Let A be a ring, B an A-algebra and J an ideal of B We say that the A-algebra B is formally... generated by the variables Xi of degree ≤ n and their divided powers (for variables of even degree > 0) We denote it by R(n) Thus R(0) = R0 , and if A → B is a surjective ring homomorphism with kernel I and R a free DG resolution of the A-algebra B with R0 = A, then R(1) is the Koszul complex associated to a set of generators of I Lemma 1.2.10 Let A be a ring and B an A-algebra Let A → S    R → B be... treatments is more complete than ours Rather, we hope that our book can serve as an introduction and motivation to study these sources We would also like to mention that we have profited from reading the interesting book by Brezuleanu, Dumitrescu and Radu [BDR] on topics similar to ours, although they do not use homological methods We are grateful to T S´nchet Giralda for interesting suggestions and a to the... · · · Xin ) = f (Xi1 · · · Xip−1 )g(Xip+2 · · · Xin )(u(x)y − u(y)x) = 0, where x = Yip+1 and y = Yip Extend λ to P by linearity It follows easily that λ is a biderivation Since f − g is also a biderivation and f (Xi ) − g(Xi ) = u(Yi ) = u(λ(Xi )), we have uλ = f − g b) Let Q be an A-algebra and M a Q-module and u : M → Q a Q-module homomorphism such that u(x)y = u(y)x for all x, y ∈ M ; suppose that... resolutions, we follow Avramov [Av1] and [Av2], taking care to avoid homology modules in dimension 3 and 4 As a by-product, we also give a proof of Kunz’s result characterizing regular local rings in positive characteristic in terms of the Frobenius homomorphism Finally, Chapters 5 and 6 study regular homomorphisms, giving in particular proofs of Theorems (III), (IV) and (V) The prerequisites for reading . Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds) 373 Smoothness, regularity and complete intersection, J. MAJADAS &. complete noetherian local rings. Theorem (II) Let A be a complete intersection ring and p a prime ideal of A. Then the localization A p is a complete intersection. 1 2