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MAP 2010 - LOGRO NO November 8-12, 2010 Chain calculus and Krull - PDF document

MAP 2010 - LOGRO NO November 8-12, 2010 Chain calculus and Krull dimension in distributive lattices Luis Espa nol Universidad de La Rioja luis.espanol@unirioja.es I. Evolution of Krull dimension of distributive lattices II. Link


  1. MAP 2010 - LOGRO˜ NO November 8-12, 2010 Chain calculus and Krull dimension in distributive lattices Luis Espa˜ nol Universidad de La Rioja luis.espanol@unirioja.es

  2. I. Evolution of Krull dimension of distributive lattices II. Link dimension of distributive lattices I always consider bounded distributive lattices L with a bottom element 0 and a top element 1, and lattice morphisms preserving 0 and 1. They form the category L . Boolean algebras: � C ( L ) centre of L C ( L ) ֒ → L ֒ → L ¬ , freely generated by L L ¬

  3. Dimension of distributive lattices Five definitions (equivalent of course) in chronological order: 1. Kdim L : Krull’s notion with chains of prime ideals of L 2. Sdim L : Simplicial notion given by Joyal 3. Bdim L : Boolean notion given by Espa˜ nol 4. Edim L : Elementary notion given by Coquand & Lombardi 5. Ldim L : Linked chain dimension given by Espa˜ nol 1. Classical definition: KdimL ≤ n if any ( n + 1)-chain of prime ideals in L is degenerated. 2-5. “Constructuive” definitions

  4. Colloque TAC, Amiens 1975. http://vbm-ehr.pagesperso-orange.fr/ChEh/

  5. Dimension of ordered sets Given an ordered set X , we have the ordered set X n of all n -chains x 0 ≤ · · · ≤ x n in X , n ≥ 0 ( n = number of ≤ ) We define dimension of ordered sets by the length of its chains: dimX ≤ n if any ( n + 1)-chain is degenerated. dimX = n if dimX ≤ n and there exists a non degenerated n -chain. In particular, dimX ≤ 0 if and only if X is a trivial poset. If we consider the simplicial object { X n } , then we can give catego- rical statements: Each X n is universal for monotone maps ( p 0 ≤ · · · ≤ p n ) : X n → X. dimX ≤ n if and only if � � s 0 , · · · , s n � : X n → X n +1 is epi. n +1

  6. Simplicial dimension We proceed by categorical duality with ordered sets. Given a distributive lattice L , We have a cosimplicial object { L n } of distributive lattices, where L n is universal for chains of morphisms ( p 0 ≤ · · · ≤ p n ) : L = L 0 → L n . Definition (1975): SdimL ≤ n if ( s 0 , · · · , s n ) : L n +1 → � n +1 L n is mono. In the classical (non-constructive) setting we have (prime ideals are morphisms L → { 0 , 1 } ): Prime ideals of L n are in bijection with n -chains p 0 ≤ · · · ≤ p n of prime ideals of L . SdimL is equivalent to Krull dimension .

  7. Boolean dimension In the next formulas the elements x k are in L . Definition (1978): (i) BdimL ≤ 2 n if for every x ∈ L ¬ , x = x 0 ∨ � n i =1 ( x i ∧ ¬ y i ). (ii) BdimL ≤ 2 n + 1 if for every x ∈ L ¬ , x = � n +1 i =1 ( x i ∧ ¬ y i ). 1. L is discrete if and only if so L ¬ is. 2. This definition can be given in terms of elements of L only. (Espa˜ nol, talks at Milano, 1987) (Espa˜ nol, notes 2001, after Coquand & Lombardi)

  8. Elementary dimension Definition: EdimL ≤ n if for any a 1 , . . . , a n +1 ∈ L there exists x 1 , . . . , x n +1 ∈ L such that  a n +1 ∨ x n +1 = 1   ( •• ) a i +1 ∧ x i +1 ≤ a i ∨ x i , i = 1 , . . . , n   a 1 ∧ x 1 = 0 n = 2 figure 1 � � � � � � � � � � � � � � � � a 2 x 2 � � �������������� � � � � � • � � � � �������������� � � � � � � � � � � • � � � � � � � a 1 x 1 � � � � � � � � � � � � � � � � 0

  9. Link dimension Definition: LdimL ≤ n if for any ( n + 1)-chain a : a 1 ≤ · · · ≤ a n +1 in L there exists a ( n + 1)-chain x : x 1 ≤ · · · ≤ x n +1 in L such that  a n +1 ∨ x n +1 = 1   a i +1 ∧ x i +1 = a i ∨ x i , i = 1 , . . . , n ( • )   a 1 ∧ x 1 = 0 ( • ) : a , x are linked . Then a , x are “relatively complemented”. LdimL ≤ n if any ( n + 1)-chain is linked LdimL ≤ 0 means that L is a Boole algebra 2-link figure 1 � � � � � � � � � � � � � � � � a 2 x 2 � � � � � � � � � � � � � � � � • � � � � � � � � � � � � � � � � a 1 x 1 � � � � � � � � � � � � � � � � 0

  10. Link dimension in the middle Kdim Edim � � ���������� � � � � � � � � � Ldim � ���������� � � � � � � � � � ( Sdim ) Bdim

  11. Linked chains 1 1 1 � � ���������� � ���������� � � � � � � � � � � � � � � � � � � p n +1 q n a n +1 x n +1 t n +1 � � ���������� � � ���������� � � � � � � � � � � � � � � � u n − 1 e n � � ���������� � ���������� � � � � � � � � � � � � � � � � p n q n − 1 a n x n t n � � � ��������� � � ���������� � � � � � � � � � � � � � � � u n − 2 e n − 1 u 1 e 1 � � ���������� � � ���������� � � � � � � � � � � � � � � � � p 1 q 1 a 1 x 1 t 1 � � ����������� � � � ����������� � � � � � � � � � � � � � � � � � 0 0 0 a ⋄ e x in L Ch n +1 Link n +1 Ch 0 = 2 = { 0 < 1 } = Link 0 , Ch − 1 = { 0 = 1 } = Link − 1 Then: a , x are linked with node chain e x i , a i are complemented in the interval [ e i − 1 , e i ], ( e 0 = 0, e i +1 = 1) For a , e given, x is unique when it exists

  12. � � � � � Linked chains as morphisms A n -chain in L is a lattice morphisms x : Ch n → L For instance p : Ch n ֒ → Link n , p ( t i ) = p i A n -link in L is a lattice morphisms x : Link n → L LdimL ≤ n is an extension property: p � Link n +1 � � Ch n +1 x L p � Link n +1 � � u u ≺ p Ch n +1 Ch n � � � � � � � � � � � � � � � � � � � � � e ≺ x � x � � � � � � � � � � � � � � L u ≺ p : u separates p

  13. Link dimension and K -dimension LdimL ≤ n ⇔ KdimL ≤ n We have cochains : morphisms P : L → Ch n P is completely determined by a chain P 0 ⊆ · · · ⊆ P n of prime ideals. a ∈ P : a : Ch n → L belongs to P : L → Ch n if P ◦ a = id : Ch n → Ch n P is onto if and only if the chain of prime ideals is non-degenerated . a is a section if and only if a is not linked . L. Espa˜ nol: “Finite chain calculus in distributive lattices and ele- mentary Krull dimension”. Contribuciones cient ´ ıficas en honor de omez . L. Lamb´ an, A. Romero, J. Rubio (eds.) Mirian Andr´ es G´ Ser. de Publ., Univ. de La Rioja, Logro˜ no 2010, pp. 273-285. http://www.unirioja.es/servicios/sp/catalogo/monografias/ F. W. Anderson, R. L. Blair. “Representations of distributive lat- tices as lattices of functions”. Math. Ann. 143 (1961) 187–211. R. Balbes & Ph. Dwinger, Distributive lattices , 1974.

  14. Linked chain calculus Elementary pieces To insert into an interval . Universal morphism l : L → [ x, y ]. y x ∨ a � ��������� � � � � � � � a : u a is inserted on l ( a ) = u ∈ [ x, y ] � � � � � � � � � � � � � � � � � � a ∧ y x � ( x ∨ a ) ∧ y Chain associativity : u = ( x ≤ y ) x ∨ ( a ∧ y ) The diamond [ p, b ] ∼ q q = a ∨ b = [ a, q ] q + b = p + a � � �������� � � � � � � a a + b = p + q To insert b � � �������� � � � � � � [ p, b ] ∼ p p = a ∧ b p + b = q + a = [ a, q ]

  15. Inserting, diamonds and links Links in an interval a ∨ y � � ��������� � � � � � � � � y x ∨ a � � ����������� � � � � � � � � u = a inserted into [ x, y ] : u � � ���������� � � � � � � � � � a ∧ y x ��������� � � � � � � � � � � x ∧ a Links with trivial diamonds: q � � � � � � � � � � � � � � � � q x � � � � � � � � � � � � � � � � � � � � � � � � � x � � � � � � � � � � � � � � � � � � � � � � � � � � a � b � � � � � � � � � � � � � � � � � � � � � � � � � • � � � � ������� � � � � � � � � � � � � � p • � � � � � � � � � � � � � � � p

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