Valuation Rings Rachel Chaiser May 1, 2017 University of Puget Sound Definition: Valuation • F - field • G - totally ordered additive abelian group • For all a; b 2 F, ν : F ! G [ f1g satisfies: 1. ν(ab) = ν(a) + ν(b) 2. ν(a + b) ≥ minfν(a); ν(b)g 3. ν(0) := 1 • If ν is surjective onto G = Z then ν is discrete 1 Example: p-adic Valuation • Fix a prime p 2 Z r 2 Q∗ r k a a 2 Q∗ - • Any s can be written uniquely as s = p b , b , p ab Q ! Z [ f1g r • Define νp : , νp( s ) = k • For example, 3-adic valuation: - ν3(1) = 0 1 - ν3(12) = ν3(3 · 4) = 1 5 · −2 − - ν3( 9 ) = ν3(5 3 ) = 2 2 Structures Definitions: Structures • Definition (Value Group) The subgroup of G, ν(F∗) = fν(a) j a 2 F∗g • Definition (Valuation Ring) The subring of F, V = fa 2 F j ν(a) ≥ 0g • Definition (Discrete Valuation Ring) If ν is discrete then V is a discrete valuation ring (DVR) 3 Example: p-adic Structures Q ! Z [ f1g k a - • Recall: νp : , νp(p b ) = k where p ab • The value group of νp is Z r • Assume s is in lowest terms Z f r j - g • The valuation ring of νp is (p) = s p s , the p-adic integers • The 3-adic integers: 5 62 Z 2 Z - 9 (3) while 1; 12 (3) - Z ⊂ Z(3) n 2 Z 2 Z - a (3) where n and gcd(a; 3) = 1 4 Properties Properties of Valuation Rings For general ν: • For all a; b 2 V, ν(a) ≤ ν(b) () b 2 hai • The ideals of V are totally ordered by set inclusion • V has unique maximal ideal M = fa 2 V j ν(a) > 0g For discrete ν: • t 2 V with ν(t) = 1 is a uniformizer • M = hti 5 DVRs are Noetherian Proof. Let I =6 h0i be an ideal of V. Then for some a 2 I there is a least integer k such that ν(a) = k. Let b; c 2 I and suppose b = ac. Then ν(b) = ν(a) + ν(c) = k + ν(c) ≥ k. Thus I contains every b 2 V with ν(b) ≥ k, and so the only ideals of V are Ik = fb 2 V j ν(b) ≥ kg. These ideals then form a chain V = I0 ⊃ I1 ⊃ I2 ⊃ · · · ⊃ h0i. 6 DVRs are PIDs Proof. ⟨ ⟩ Let t 2 V be a uniformizer. For x 2 tk , ν(x) = ν(atk) = ν(a) + kν(t) = ν(a) + k: ⟨ ⟩ k Thus, we can take Ik = t . Remark This illustrates that ν(a) ≤ ν(b) () b 2 hai for all a; b 2 V Corollary Every nonzero ideal of V is a power of the unique maximal ideal, hti. 7 Example: p-adic ideals • M = f r 2 Z : p j rg = hpi s⟨ ⟩ (p)⟨ ⟩ ⟨ ⟩ ⟨ ⟩ 0 1 2 3 • Z(p) = p ⊃ p ⊃ p ⊃ p ⊃ · · · ⊃ h0i • 3-adic ideals: h i - Maximal ideal 3 ⟨ ⟩ ⟨ ⟩ 2 3 - Z(3) = h1i ⊃ h3i ⊃ 3 ⊃ 3 ⊃ · · · ⊃ h0i 8 Example: Generalized p-adic Valuation • Let D be a UFD with field of fractions K • Fix a prime element p of D • Any x 2 D can be written uniquely as x = apk where p - a • Any y 2 K∗ can be written uniquely as y = qpk - q 2 K∗ is the quotient of r; s 2 D such that p - r; p - s • Define ν : K ! Z [ f1g, ν(y) = k 9 Thank you! Questions? 9.