COMMUTATIVE ALGEBRA USED IN INTRO TO A.G. COURSE (UP TO CLASS 16)

RAVI VAKIL

Topics: Rings, localization, primes, completion, dimension theory.

Class 3: If k is a field, and B is an integral domain which is a finitely-generated k-algebra. Then dim B = tr.deg_{k}K(B) (where K(B) is the quotient field of B), and for any prime ideal 𝔭 in B,

ht𝔭 + dim B/𝔭 = dim B.

(I can’t remember if this was central to anything I said.

Class 11: The tensor product of two domains is also a domain. (In case of finitely-generated domain over \overline{k}, and the tensor is over \overline{k}.)

Class 13: Transcendence degree. Krull’s Hauptidealsatz.

Class 15: If A is a noetherian local ring with maximal ideal 𝔪 and residue field \overline{k}, then dim\overline{k} 𝔪/𝔪^{2} ≥ dim A.

Class 16: Let (R, 𝔪) be a noetherian local domain of dimension one. Then the following are equivalent.

(i) R is a discrete valuation ring; (ii) R is integrally closed (I’ll speak about integral closures next day); (iii) R is a regular local ring; (iv) 𝔪 is a principal ideal.

Class 17: Theorem (Integral closure is a local property). If R is integrally closed if and only if the localization of R at each of its maximal ideals is integrally closed.

Theorem on finiteness of integral closure (in finite field extension).

Take any Dedekind domain R, and let K be its field of fractions. Let L be a finite extension of K, and let S be the integral closure of R in L. Then S is also a Dedekind domain. (Not used in this generality, but stated for the benefit of the subset of students who are thinking about schemes.) From Hartshorne I.6.3.

Date: Friday, November 5, 1999.

Class 18: I may use the following fact (but this weekend I might try to come up with a proof of the special case I’ll need). Maximality of valuation rings. If (A, 𝔪_{A}), (B, 𝔪_{B}) are local rings contained in a field K, we say that B dominates A if A ⊂ B and 𝔪_{B} ∩ A = 𝔪_{A}) Let K be a field. A local ring R contained in K is a valuation ring of K iff it is a maximal element of the set of local rings contained in K, with respect to domination.