Results on projective modules, Quillen patching and locally polynomial algebras, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Isomorphisms spread — Lemma 1.1. If N, N' are finitely presented over B and S^(-1)N is isomorphic to S^(-1)N', the isomorphism comes from an isomorphism N_s -> N'_s for some s in S.
• Gluing — Lemma 1.2. For f_0, f_1 generating the unit ideal, 0 -> M -> M_(f_0) x M_(f_1) -> M_(f_0 f_1) is exact, and compatible isomorphisms over the two localizations glue.
• Quillen's lemma — Lemma 2.1. For theta in (1 + T R_f[T])^x there is k with theta(g_1 T) theta(g_2 T)^(-1) defined over R whenever g_1 - g_2 lies in f^k A.
• Quillen induction — Proposition 3.1. A property of localizations of a nonzero ring K that specializes, spreads from S^(-1)K to some K_s, holds at every maximal ideal and satisfies the sheaf condition for comaximal pairs holds for K.
• Quillen's patching theorem — Theorem 4.1. A finitely presented A[T]-module M which is extended from A_m for every maximal ideal m is isomorphic to (M/TM)[T].
• Local criteria for projectivity — Lemma 1.1. Finitely generated projective modules over local rings are free; a finitely presented module free at all primes is projective; projectivity can be tested on two comaximal multiplicative sets.
• Horrocks' theorem in rank one — Lemma 2.2. Over a local ring A, a finitely generated projective A[X]-module which becomes free of rank one after inverting the monic polynomials is free.
• Horrocks' theorem, local form — Theorem 2.3. Over a local ring A, a finitely generated projective A[X]-module which becomes free after inverting the monic polynomials is free.
• Horrocks' theorem, affine form — Theorem 3.1. If P is finitely generated projective over A[X] and S^(-1)P is extended from a finitely generated projective A-module, then P is isomorphic to (P/XP)[X].
• Quillen–Suslin theorem — Theorem 3.2. Every finitely generated projective module over a polynomial ring F[x_1,...,x_r] over a field is free.
• Unimodular rows — Corollary 3.3. Over F[x_1,...,x_r], every unimodular row is the first row of an invertible matrix.
• Algebra maps spread — Lemma 2.1. Homomorphisms and isomorphisms between finitely presented algebras over K_S come from K_s for some s in S.
• Axiom Q for automorphism groups — Proposition 2.3. Automorphism groups of a finitely presented algebra satisfy axiom Q: an automorphism over L_s[T] trivial at T = 0 becomes defined over L after the substitution T -> s^r T.
• Affine part of an automorphism — Proposition 3.3. Every automorphism of a symmetric algebra S(M_L) is a product of an automorphism tangent to the identity and an affine automorphism.
• Factorization over two localizations — Theorem 4.2. For M finitely presented and comaximal s_0, s_1, G_(M,0)(L_(s_0 s_1)) = G_(M,0)(L_(s_0))_(s_1) G_(M,0)(L_(s_1))_(s_0).
• Factorization of automorphisms — Proposition 4.4. Every automorphism of S(M) over L_(s_0 s_1) is (v_0)_(s_1) h (v_1)_(s_0) with v_i defined over L_(s_i) and h affine.
• Modules over two localizations — Lemma 1.1. Modules over comaximal localizations glue along an isomorphism, and finite presentation is local for comaximal pairs.
• Symmetric algebras determine their modules — Lemma 1.2. If S(M) and S(N) are isomorphic K-algebras, then M and N are isomorphic K-modules.
• Conormal module of an augmented algebra — Lemma 1.3. For a finitely presented algebra A with an augmentation to K of kernel I, the K-module I/I^2 is finitely presented.
• Bass–Connell–Wright theorem — Theorem 2.1. A finitely presented K-algebra which is a symmetric algebra after localizing at every maximal ideal of K is the symmetric algebra of a finitely presented K-module.
• Locally polynomial algebras — Corollary 2.2. A finitely presented locally polynomial algebra is S(P) with P finitely generated projective; over F[y_1,...,y_r] with F a field it is a polynomial ring.

Reading order

Read the lessons in order. The first proves Quillen's patching theorem and sets up Quillen induction. The second proves Horrocks' theorem and the Quillen–Suslin theorem. The third develops automorphism groups of symmetric algebras and their factorization over two localizations, which the fourth uses to prove that locally polynomial algebras are symmetric algebras.