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 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) ֒ → L ֒ → L¬,

C(L)

centre of L L¬ freely generated by L

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

Colloque TAC, Amiens 1975.

http://vbm-ehr.pagesperso-orange.fr/ChEh/

Dimension of ordered sets

Given an ordered set X, we have the ordered set Xn of all n-chains x0 ≤ · · · ≤ xn 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 {Xn}, then we can give catego- rical statements: Each Xn is universal for monotone maps (p0 ≤ · · · ≤ pn) : Xn → X. dimX ≤ n if and only if s0, · · · , sn :

• n+1

Xn → Xn+1 is epi.

Simplicial dimension

We proceed by categorical duality with ordered sets. Given a distributive lattice L, We have a cosimplicial object {Ln} of distributive lattices, where Ln is universal for chains of morphisms (p0 ≤ · · · ≤ pn) : L = L0 → Ln. Definition (1975): SdimL ≤ n if (s0, · · · , sn) : Ln+1 →

n+1 Ln is mono.

In the classical (non-constructive) setting we have (prime ideals are morphisms L → {0, 1}): Prime ideals of Ln are in bijection with n-chains p0 ≤ · · · ≤ pn

• f prime ideals of L.

SdimL is equivalent to Krull dimension.

Boolean dimension

In the next formulas the elements xk are in L. Definition (1978): (i) BdimL ≤ 2n if for every x ∈ L¬, x = x0 ∨ n

i=1(xi ∧ ¬yi).

(ii) BdimL ≤ 2n + 1 if for every x ∈ L¬, x = n+1

i=1 (xi ∧ ¬yi).

• 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)

Elementary dimension

Definition: EdimL ≤ n if for any a1, . . . , an+1 ∈ L there exists x1, . . . , xn+1 ∈ L such that (••)

    

an+1 ∨ xn+1 = 1 ai+1 ∧ xi+1 ≤ ai ∨ xi, i = 1, . . . , n a1 ∧ x1 = 0 n = 2 figure 1

• a2
• x2
• a1
• x1

Link dimension

Definition: LdimL ≤ n if for any (n + 1)-chain a : a1 ≤ · · · ≤ an+1 in L there exists a (n + 1)-chain x : x1 ≤ · · · ≤ xn+1 in L such that (•)

    

an+1 ∨ xn+1 = 1 ai+1 ∧ xi+1 = ai ∨ xi, i = 1, . . . , n a1 ∧ x1 = 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

• a2
• x2
• a1
• x1

Link dimension in the middle

Kdim

• Edim
• Ldim
• (Sdim)

Bdim

Linked chains

1 1

• tn+1

an+1

• xn+1
• en
• tn

an

• xn
• en−1

e1

• t1

a1

• x1
• 1
• pn+1
• qn
• un−1
• pn
• qn−1
• un−2

u1

• p1
• q1
• Chn+1

a ⋄e x in L

Linkn+1 Ch0 = 2 = {0 < 1} = Link0, Ch−1 = {0 = 1} = Link−1 Then: a, x are linked with node chain e xi, ai are complemented in the interval [ei−1, ei], (e0 = 0, ei+1 = 1) For a, e given, x is unique when it exists

Linked chains as morphisms

A n-chain in L is a lattice morphisms

x : Chn → L

For instance p : Chn ֒ → Linkn, p(ti) = pi A n-link in L is a lattice morphisms

x : Linkn → L

LdimL ≤ n is an extension property: Chn+1

x

• p Linkn+1
• L

Chn+1

x

• p Linkn+1
• Chn

u

• e≺x
• u ≺ p

L

u ≺ p: u separates p

Link dimension and K-dimension

LdimL ≤ n ⇔ KdimL ≤ n We have cochains: morphisms

P : L → Chn P is completely determined by a chain P0 ⊆ · · · ⊆ Pn of prime ideals. a ∈ P: a : Chn → L belongs to P : L → Chn if P ◦ a = id : Chn → Chn 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 Mirian Andr´ es G´

• mez. L. Lamb´

an, A. Romero, J. Rubio (eds.)

• 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.

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]

x a ∧ y Chain associativity: u =

(x ∨ a) ∧ y

x ∨ (a ∧ y) (x ≤ y) The diamond q

• q = a ∨ b

[p, b] ∼ = [a, q] q + b = p + a a

• b
• To insert

a + b = p + q p p = a ∧ b [p, b] ∼ = [a, q] p + b = q + a

Inserting, diamonds and links

Links in an interval a ∨ y

• y
• x ∨ a
• u = a inserted into [x, y] :

u

• x

a ∧ y x ∧ a

• Links with trivial diamonds:

q

• x
• q
• x
• a
• b
• p
• p

Conditions for linked chains

A n-chain x, n ≥ 1 is linked when: (i) [x1, xn] ∩ C(L) = ∅ 1

• z
• z′
• z′ = ¬a inserted into [y, 1]

q

• v
• v = ¬a inserted into [y, z]

a

• y
• y′
• ¬a
• y′ = ¬a inserted into [x, z]

p

• u
• u = ¬a inserted into [x, y]

x

• x′
• x′ = ¬a inserted into [0, y]

(i) A k-subchain a1 ≤ · · · ≤ ak, 1 ≤ k < n, is linked in [0, ak+1]. (ii) A subchain is linked. ui+2

• ai+1
• bi+1
• ui+1
• ai
• bi
• ui

ui+2

• ai+1
• a ∨ bi+1
• a ∨ ui+1
• a
• ui+1
• a ∧ ui+1
• ai
• a ∧ bi
• ui

1

• x
• 1
• x
• x
• x
• x
• x
• A n-chain x, n ≥ 1, is linked when it is degenerated.

Link dimension and E-dimension

LdimL ≤ n ⇔ EdimL ≤ n (⇒) Given x1, . . . , xn+1 ∈ L, take

y : yi = xi ∧ · · · ∧ xn+1.

If there exist a link y ⋄ a then x1, . . . , xn+1 and a satisfy (••). (⇐) Given x, a configuration (••) yields a link (•) Given x ≤ y: (••)

    

y ∨ b = q y ∧ b ≤ x ∨ a x ∧ a = p (•)

    

y ∨ b′ = q y ∧ b′ = x ∨ a′ x ∧ a′ = p q

• y
• b
• x
• a
• p

q

• y
• b′
• b′ = b inserted into [u, q]
• u = x ∨ a′

x

• a′
• a′ = a inserted into [p, y]

p u satisfies :

u = a inserted into [x, y]

y ∧ b ≤ u ≤ a ∨ x

Notation for sums of chains

L(n) denotes the distributive lattice of all n-chains in L. For a ∈ L, an ∈ L(n) is the chain a ≤ · · · ≤ a. C(L(n)) ∼ = C(L).

• n

: L(n) → L¬,

• n

a = a1 + · · · + an.

Image

• n

(L) ⊆ L¬

• n a ≤ an,

a1 ≤ ¬n

n a

• 2n a = 0 ⇔ a is duplicate, a = b2 : b1 ≤ b1 ≤ · · · bn ≤ bn
• 2n a = x = 0 ⇔ (0, a ˆ

2n) ∧ x2n is duplicate

(0, aˆ

i) : 0 ≤ a1 ≤ · · · ≤ ai−1 ≤ ai+1 ≤ · · · ≤ an

0a = (0, a) : 0 ≤ a1 ≤ · · · ≤ an, 1a = (a, 1) : a1 ≤ · · · ≤ an ≤ 1

ai : a1 ≤ · · · ≤ ai ≤ ai ≤ · · · ≤ an

(0, aˆ

i, 1) : 0 ≤ a1 ≤ · · · ≤ ai−1 ≤ ai+1 ≤ · · · ≤ an ≤ 1

The lattice of a chain

a ∈ L(n) :

La =

n

• i=0

[ai, ai+1] = [0a, 1a] ⊆ L(n+1)

x ∈ La :

a ≺ x

a ⊳ x : x1 ≤ a1 ≤ x2 ≤ a2 ≤ · · · ≤ xn ≤ an ≤ xn+1 x ∈ C(La) = n

i=0 C([ai, ai+1]) iff x is linked with node a

Canonical morphism ℓa : L → La, ℓa(x) = 0a ∨ (1a ∧ xn+1) ⊕ : Ln × Lm → Ln+m,

a ⊕ b = ℓa(b)

a ⊕ b = (a ⊕ b ˆ

m) ⊕ bm

(i) a = aˆ

i ⊕ ai = a1 ⊕ · · · ⊕ an

(ii) If a ≺ x then a ⊕ x = a ⊳ x (iii)

n+m(a ⊕ b) = n a + m b

Given a1, . . . , an ∈ L, (iv) a1 ⊕ · · · ⊕ an is invariant by permutations (v)

n(a1 ⊕ · · · ⊕ an) = n i=1 ai

Lattice theory’s result: L¬ =

n

• n(L)

Gr¨ atzer, General lattice theory, 2nd, 2003.

Link dimension and B-dimension

The image of

• n

: L(n) → L¬ is

• n

(L) ⊆ L¬

• n
• n(L) = L¬
• n

(L), ¬

• n

(L) ֒ →

• n+1

(L)

• n

(L) = ¬

• n

(L) ⇒

• n

(L) = L¬ LdimL ≤ n ⇔

• n+1

(L) = L¬ ⇔ BdimL ≤ n

Universal property

Universal property of ℓa : L → La, ℓa(x) = 0a ∨ (1a ∧ xn+1) ℓa ◦ a ∈ C(La) :

ℓa(aj) = aj

¬ℓa(aj) = (0, aˆ

j, 1)

Push-out in L : Chn

a

• L

ℓa

• (Chn)¬

La

By category theory, enough to prove : L

ℓa

• incl
• La

sa L¬

First: ℓa is mono and epi c

• v
• inserting x :

b

• w
• u
• a

x =

• 2n+1

ℓa ◦ (a ⊳ x) Then: . . .

Calculating with a valuation

• n

: L(n) → L¬ is a valuation

• n

0 = 0,

• n

(a ∧ b) +

• n

(a ∨ b) =

• n

a +

• n

b

ℓa(x) = 0a ∨ (1a ∧ xn+1);

• n+1

ℓa(x) = x +

• n

a

sa(x) =

2n+1 a ⊳ x

defines a morphism such that L

ℓa

• incl
• La

sa L¬

. . . the universal property follows. Moreover: Linked n-chains :

x ⋄ y

⇔

    

(0, x) ≤ (y, 1) (0, y) ≤ (x, 1)

• n x +

n y = 1

Dimension of a chain

LdimChn = n, LdimLinkn = n − 1 Definition: dima = dimLa (for all dimensions) Push-out : Chn

a

• L

ℓa

• (Chn)¬

La

Theorem: 1 + LdimL ≤ (1 + n)(1 + LdimLa) Corollary: ∃a ∈ Ln, La boolean ⇒ LdimL ≤ n

THAT’S ALL THANKS