Division rings with ranks joint work with Daniel Palacin Nadja - PowerPoint PPT Presentation

Overview Division rings Division rings with ranks joint work with Daniel Palacin Nadja Hempel UCLA Udine, July 26, 2018 Nadja Hempel Division rings with ranks

Overview Division rings 1 Overview 2 Division rings ranked/superrosy weight 1 finite burden Nadja Hempel Division rings with ranks

Overview Division rings Overview Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Division rings Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Division rings ranked/superrosy Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Rank function Let D be a division ring carrying an ordinal-valued rank function among the definable sets in the imaginary expansion, i.e. rk : { Definable sets } → Ord , such that: Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Rank function Let D be a division ring carrying an ordinal-valued rank function among the definable sets in the imaginary expansion, i.e. rk : { Definable sets } → Ord , such that: 1 rk ( A ) = 0 iff A is finite. Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Rank function Let D be a division ring carrying an ordinal-valued rank function among the definable sets in the imaginary expansion, i.e. rk : { Definable sets } → Ord , such that: 1 rk ( A ) = 0 iff A is finite. 2 The rank is preserved under definable bijections. Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Rank function Let D be a division ring carrying an ordinal-valued rank function among the definable sets in the imaginary expansion, i.e. rk : { Definable sets } → Ord , such that: 1 rk ( A ) = 0 iff A is finite. 2 The rank is preserved under definable bijections. 3 The Lascar inequalities: For a definable subgroup H of a definable group G we have that rk ( H ) + rk ( G / H ) ≤ rk ( G ) ≤ rk ( H ) ⊕ rk ( G / H ) , Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Lascar inequalities For a definable subgroup H of a definable group G we have that rk ( H ) + rk ( G / H ) ≤ rk ( G ) ≤ rk ( H ) ⊕ rk ( G / H ) , Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Lascar inequalities For a definable subgroup H of a definable group G we have that rk ( H ) + rk ( G / H ) ≤ rk ( G ) ≤ rk ( H ) ⊕ rk ( G / H ) , where the function ⊕ is the smallest symmetric strictly increasing function f among pairs of ordinals such that f ( α, β + 1 ) = f ( α, β ) + 1 . Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Lascar inequalities For a definable subgroup H of a definable group G we have that rk ( H ) + rk ( G / H ) ≤ rk ( G ) ≤ rk ( H ) ⊕ rk ( G / H ) , where the function ⊕ is the smallest symmetric strictly increasing function f among pairs of ordinals such that f ( α, β + 1 ) = f ( α, β ) + 1 . Example: See blackboard Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Lascar inequalities For a definable subgroup H of a definable group G we have that rk ( H ) + rk ( G / H ) ≤ rk ( G ) ≤ rk ( H ) ⊕ rk ( G / H ) , where the function ⊕ is the smallest symmetric strictly increasing function f among pairs of ordinals such that f ( α, β + 1 ) = f ( α, β ) + 1 . Example: See blackboard Remark The U þ rank satisfies the above properties. Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Easy consequences I Let G and H be two definable groups and let f : H → G be a definable group morphism. Then rk ( Ker f ) + rk ( Im f ) ≤ rk ( H ) ≤ rk ( Ker f ) ⊕ rk ( Im f ) . Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Easy consequences I Let G and H be two definable groups and let f : H → G be a definable group morphism. Then rk ( Ker f ) + rk ( Im f ) ≤ rk ( H ) ≤ rk ( Ker f ) ⊕ rk ( Im f ) . Consequences: if f is injective, Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Easy consequences I Let G and H be two definable groups and let f : H → G be a definable group morphism. Then rk ( Ker f ) + rk ( Im f ) ≤ rk ( H ) ≤ rk ( Ker f ) ⊕ rk ( Im f ) . Consequences: if f is injective, then rk ( H ) = rk ( G ) iff [ G : Im f ] < ∞ . Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Easy consequences I Let G and H be two definable groups and let f : H → G be a definable group morphism. Then rk ( Ker f ) + rk ( Im f ) ≤ rk ( H ) ≤ rk ( Ker f ) ⊕ rk ( Im f ) . Consequences: if f is injective, then rk ( H ) = rk ( G ) iff [ G : Im f ] < ∞ . if H < G , then Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Easy consequences I Let G and H be two definable groups and let f : H → G be a definable group morphism. Then rk ( Ker f ) + rk ( Im f ) ≤ rk ( H ) ≤ rk ( Ker f ) ⊕ rk ( Im f ) . Consequences: if f is injective, then rk ( H ) = rk ( G ) iff [ G : Im f ] < ∞ . if H < G , then rk ( H ) = rk ( G ) iff [ G : H ] < ∞ . Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Easy consequences II (DCC) There is no infinite descending of definable groups H 0 > H 1 > · · · > H n > . . . each of them having infinite index in its predecessor. (no infinite strictly descending chain of ordinals) Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden Easy consequences II (DCC) There is no infinite descending of definable groups H 0 > H 1 > · · · > H n > . . . each of them having infinite index in its predecessor. (no infinite strictly descending chain of ordinals) In particular, for infinite subdivision rings, we obtain that every descending chain stabilizes after finitely many steps. Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden wide/negligible Definition Let X be a definable set of rank ω α · n + β with β < ω α and n ∈ ω . A definable subset Y of X is Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden wide/negligible Definition Let X be a definable set of rank ω α · n + β with β < ω α and n ∈ ω . A definable subset Y of X is wide in X if rk ( Y ) ≥ ω α · n . Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden wide/negligible Definition Let X be a definable set of rank ω α · n + β with β < ω α and n ∈ ω . A definable subset Y of X is wide in X if rk ( Y ) ≥ ω α · n . negligible with respect to X if rk ( Y ) < ω α . Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden wide/negligible Definition Let X be a definable set of rank ω α · n + β with β < ω α and n ∈ ω . A definable subset Y of X is wide in X if rk ( Y ) ≥ ω α · n . negligible with respect to X if rk ( Y ) < ω α . If there is no confusion we simply say that Y is wide or respectively negligible. Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden On the way to the main result Lemma Any superrosy division ring has finite dimension (as a vector space) over any definable non-negligible subdivision ring. Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden On the way to the main result Lemma Any superrosy division ring has finite dimension (as a vector space) over any definable non-negligible subdivision ring. proof: see blackboard Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden On the way to the main result Lemma Any superrosy division ring has finite dimension (as a vector space) over any definable non-negligible subdivision ring. proof: see blackboard Lemma Any wide definable additive subgroup of a superrosy division ring has finite index. Proof uses Schlichtings’s theorem. Nadja Hempel Division rings with ranks

ranked/superrosy Overview weight 1 Division rings finite burden On the way to the main result Lemma Any superrosy division ring has finite dimension (as a vector space) over any definable non-negligible subdivision ring. proof: see blackboard Lemma Any wide definable additive subgroup of a superrosy division ring has finite index. Proof uses Schlichtings’s theorem. Corollary Let D be a superrosy division ring. If a definable group morphism from D + or D × to D + has a negligible kernel, its image has finite index in D + . Nadja Hempel Division rings with ranks