HOMOLOGICAL DIMENSION AND THE CONTINUUM HYPOTHESIS ...

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Let R be a regular local ring (so in particular, R is a domain). Let J(R) be the maximal ideal of R (i.e., Jacobson radi
HOMOLOGICAL DIMENSION AND THE CONTINUUM HYPOTHESIS (AFTER OSOFSKY) Reference: B. Osofsky, Homological dimension and the Continuum Hypothesis. Let Rn be R[x1 , . . . , xn ] or C[x1 , . . . , xn ]. Let Qn be its quotient field. What can we say about the projective dimension pdRn Qn ? The Continuum Hypothesis will arise through the fact that |R| = 2ℵ0 . 1. Definitions and notation Let R be a ring. A directed R-module is a triple (M, M 0 , u) such that M is an R-module, M 0 is a set of R-module generators of M , and u : M 0 × M 0 → M 0 is a function (the “upper bound map”) such that (1) for all x ∈ M 0 and r ∈ R, if xr = 0 then r = 0. (2) For all x, y ∈ M 0 , we have u(x, y)R ⊃ xR + yR. For x, y ∈ M 0 define x ≤ yLif xR ⊂ yR. Define x > y to mean xR ) yR. For X ⊂ M , let Pn (X) = x0 >x1 >...>xn hx0 , . . . , xn iR. Let P−1 (X) be the submodule of M {x0 ,...,xn }⊂X

generated by X. For x ∈ M 0 , define s(x) = {y ∈ M 0 : y < x} and s¯(x) = {y ∈ M 0 : y ≤ x}. Define x∗ : Pn (s(x)) → Pn+1 (¯ s(x)) hx0 , . . . , xn i 7→ hx, x0 , . . . , xn i. For n = −1, x∗ (xr) = hxir. Define dn : Pn (X) → Pn−1 (X) as follows. Let d0 (hyi) = y. Let dn hx0 , . . . , xn i =

n−1 X

(−1)i hx0 , . . . , xbi , . . . , xn i + hx0 , . . . , xn−1 i(−1)n x−1 n−1 (xn ).

i=0 ∗



Let dn+1 (x p) = p − x dn p for n ≥ 0 and p ∈ Pn (s(x)). 2. Preliminaries Proposition 2.1. Let (M, M 0 , u) be a directed R-module. Let X ⊆ M 0 be u-closed; i.e., u(X × X) ⊂ X. Then dn+1

d

d

n · · · → Pn (X) → Pn−1 (X) → · · · → P0 (X) →0 P−1 (X) → 0

is a projective resolution of P−1 (X). Proposition 2.2. Let X ⊂ Y ⊂ M 0 be u-closed. Then Pn (X) ⊕ Pn+1 (Y )

(−dn ,id)⊕dn+1



id ⊕d

d

· · · → P0 (X) ⊕ P0 (Y ) → 1 P0 (Y ) →0

is a projective resolution. Date: April 29, 2008. 1

P−1 (Y ) →0 P−1 (X)

Theorem 2.3. Let (M, M 0 , u) be a directed R-module. Suppose that |M 0 | = ℵn for some natural number n. Then pdR (M ) ≤ n + 1. Lemma 2.4 (Auslander, “On the dimension of modules and algebras”). Let I be a wellordered set. Suppose that {Ni : i ∈ I} is a collection of submodules R-module M such  of an  S S that i ≤ j implies Ni ⊂ Nj , and M = i∈I Ni . Suppose that pdR Ni / j