March14,2016 GunnarFløystad Letterplaceidealsofposets:Alargeclasswithunusuallyniceproperties

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Stanley-Reisner theory Letterplace and co-letterplace ideals of posets Omnipresence Resolutions Surprise: Triangulations of spheres Unusually nice: Deformations of letterplace ideals

Letterplace ideals of posets:

A large class with unusually nice properties

Gunnar Fløystad

NCM Stockholm, March 2016

March 14, 2016

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Simplicial complexes

V finite set.

Definition

A simplicial complex ∆ on V is a family of subset of V such that if X ∈ ∆ and Y ⊆ X , then Y ∈ ∆.

Example

{1, 2, 3}, {3, 4}, {1, 2}, {2, 3}, {1, 3}, {1}, {2}, {3},

∅

Triangle with an edge attached.

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Squarefree monomial ideals

Example

x ₂ x ₄ x ₅ x ₇ is a squarefree monomial. Also written m _{2,4,5,7} . Definition

A squarefree monomial ideal I ⊆ k [x ₁ , . . . , x _n ] is generated by squarefree monomials.

Simiplicial complexes on [n] = {1, 2, 3, . . . , n} ¹⁻¹ ↔ squarefree monomial ideals I _∆ .

Definition

Stanley-Reisner ring k [∆] = k [x 1 , . . . , x n ]/I _∆ .

Note R ∈ ∆ ⇔ m _R is nonzero in k [∆].

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Squarefree monomial ideals

Example

x ₂ x ₄ x ₅ x ₇ is a squarefree monomial. Also written m _{2,4,5,7} . Definition

A squarefree monomial ideal I ⊆ k [x ₁ , . . . , x _n ] is generated by squarefree monomials.

Simiplicial complexes on [n] = {1, 2, 3, . . . , n} ¹⁻¹ ↔ squarefree monomial ideals I _∆ .

Definition

Stanley-Reisner ring k [∆] = k [x 1 , . . . , x n ]/I _∆ .

Note R ∈ ∆ ⇔ m is nonzero in [∆].

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Founding fathers

Richard Stanley, MIT Melvin Hochster, U. of Michigan

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Alexander duality

I squarefree monomial ideal.

All monomials m ∈ k[x ₁ , . . . , x _n ] with nontrivial common divisor

with every monomial n ∈ I , generate the Alexander dual ideal

J = I ^A .

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Notation

R set k [x _R ] = k [x _i ] i∈R . S ⊆ R monomial m _S = Q

i∈S x _i .

[n] = {1 < 2 < · · · < n} totally ordered poset.

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Notation

R set k [x _R ] = k [x _i ] i∈R . S ⊆ R monomial m _S = Q

i∈S x _i .

[n] = {1 < 2 < · · · < n} totally ordered poset.

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Isotone maps

P , Q finite posets.

φ : P → Q isotone map, i.e. p ≤ p ⁰ ⇒ φ(p) ≤ φ(p ⁰ ).

Hom(P , Q) set of isotone maps. Is itself a poset: φ ≤ ψ if φ(p) ≤ ψ(p) for all p ∈ P .

Graph of φ: Γφ = {(p, φ(p) | p ∈ P }.

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Isotone maps

P , Q finite posets.

φ : P → Q isotone map, i.e. p ≤ p ⁰ ⇒ φ(p) ≤ φ(p ⁰ ).

Hom(P , Q) set of isotone maps. Is itself a poset: φ ≤ ψ if φ(p) ≤ ψ(p) for all p ∈ P .

Graph of φ: Γφ = {(p, φ(p) | p ∈ P }.

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Isotone maps

P , Q finite posets.

φ : P → Q isotone map, i.e. p ≤ p ⁰ ⇒ φ(p) ≤ φ(p ⁰ ).

Hom(P , Q) set of isotone maps. Is itself a poset: φ ≤ ψ if φ(p) ≤ ψ(p) for all p ∈ P .

Graph of φ: Γφ = {(p, φ(p) | p ∈ P }.

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Letterplace and co-letterplace ideals

L(P , Q ) ⊆ k [x P×Q ] ideal generated by m _Γφ where φ ∈ Hom(P , Q).

n’th letterplace ideal

L(n, P) = L([n], P ) ⊆ k [x _[n]×P ]

is generated by x 1,p

1

x 2,p

2

· · · x n,p

n

where p 1 ≤ p 2 ≤ · · · ≤ p n . n’th co-letterplace ideal

L(P, n) = L(P, [n]) ⊆ k [x _P×[n] ] is generated by Q

p∈P x _p,i

_p

where p ≤ q ⇒ 1 ≤ i _p ≤ i _q ≤ n.

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Letterplace and co-letterplace ideals

L(P , Q ) ⊆ k [x P×Q ] ideal generated by m _Γφ where φ ∈ Hom(P , Q).

n’th letterplace ideal

L(n, P) = L([n], P ) ⊆ k [x _[n]×P ]

is generated by x 1,p

1

x 2,p

2

· · · x n,p

n

where p 1 ≤ p 2 ≤ · · · ≤ p n .

n’th co-letterplace ideal

L(P, n) = L(P, [n]) ⊆ k [x _P×[n] ] is generated by Q

p∈P x _p,i

_p

where p ≤ q ⇒ 1 ≤ i _p ≤ i _q ≤ n.

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Letterplace and co-letterplace ideals

L(P , Q ) ⊆ k [x P×Q ] ideal generated by m _Γφ where φ ∈ Hom(P , Q).

n’th letterplace ideal

L(n, P) = L([n], P ) ⊆ k [x _[n]×P ]

is generated by x 1,p

1

x 2,p

2

· · · x n,p

n

where p 1 ≤ p 2 ≤ · · · ≤ p n . n’th co-letterplace ideal

L(P, n) = L(P, [n]) ⊆ k [x _P×[n] ] is generated by Q

p∈P x _p,i

_p

where p ≤ q ⇒ 1 ≤ i _p ≤ i _q ≤ n.

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Basic properties

L(n, P ) is Cohen-Macaulay ideal of codimension |P|,

L(P , n) and L(n, P ) are Alexander dual ideals,

L(P , n) has linear resolution.

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Basic properties

L(n, P ) is Cohen-Macaulay ideal of codimension |P|, L(P , n) and L(n, P ) are Alexander dual ideals,

L(P , n) has linear resolution.

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Basic properties

L(n, P ) is Cohen-Macaulay ideal of codimension |P|,

L(P , n) and L(n, P ) are Alexander dual ideals,

L(P , n) has linear resolution.

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Letterplace and co-letterplace ideals of posets

2011 and 2015

V.Ene, J.Herzog, F.Mohammadi: Monomial ideals and toric rings of Hibi type arising from a finite poset, European Journal of

Combinatorics, 32 (2011).

G.Fløystad, J.Herzog, B.M.Greve: Letterplace and co-letterplace

ideals of posets, arxiv (2015).

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Example: L(2, 2)

Hom([2], [2]) :

2 1

→ 1

1 2

1 → 2

1 2

1 → 2

2 L(2, 2) ⊆ S = k [x _[2]×[2] ] generated by:

x ₁₁ x ₂₁ , x ₁₁ x ₂₂ , x ₁₂ x ₂₂ .

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Example: L(2, 2)

Hom([2], [2]) :

2 1

→ 1

1 2

1 → 2

1 2

1 → 2

2 L(2, 2) ⊆ S = k [x _[2]×[2] ] generated by:

x ₁₁ x ₂₁ , x ₁₁ x ₂₂ , x ₁₂ x ₂₂ .

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Resolution

S/L(2, P)

S/L(2, 2) ← S ←−−−−−−−−−−−−− ^[x

¹¹

^x

²¹

^,x

¹¹

^x

²²

^,x

¹²

^x

²²

^] S(−2) ³







x ₂₂ 0

−x ₂₁ x ₁₂ 0 −x ₁₁







←−−−−−−−−−−− S (−3) ²

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Example: L(2, 2)

Regular quotients

[2] × [2] −→ ^α [2]

(i , j ) 7→ j

k [x _[2]×[2] ] → k [x _[2] ] L(2, 2) → (x ₁ ² , x ₁ x ₂ , x ₂ ² ) x ₁₁ − x ₂₁ , x ₁₂ − x ₂₂ is a regular sequence for S/L(2, 2).

[2] × [2] −→ ^α [3]

(i, j ) 7→ i + j − 1

k [x _[2]×[2] ] → k [x _[3] ]

L(2, 2) → (x ₁ x ₂ , x ₁ x ₃ , x ₂ x ₃ )

x ₁₂ − x ₂₁ is a regular sequence for

S /L(2, 2).

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Example: L(2, 2)

Regular quotients

[2] × [2] −→ ^α [2]

(i , j ) 7→ j

k [x _[2]×[2] ] → k [x _[2] ] L(2, 2) → (x ₁ ² , x ₁ x ₂ , x ₂ ² )

[2] × [2] −→ ^α [3]

(i, j ) 7→ i + j − 1

k [x _[2]×[2] ] → k [x _[3] ]

L(2, 2) → (x ₁ x ₂ , x ₁ x ₃ , x ₂ x ₃ )

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Poset ideals

Poset ideal J ⊆ Hom(P , [n]) subidealL(J ) ⊆ L(P, n), generated by monomials m _Γφ , φ ∈ J .

L(J ) has linear resolution.

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Poset ideals

Poset ideal J ⊆ Hom(P , [n]) subidealL(J ) ⊆ L(P, n), generated by monomials m _Γφ , φ ∈ J .

L(J ) has linear resolution.

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Omnipresence

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Determinantal ideals







x ₁₁ x ₁₂ · · · x _1,n+m−1 x ₂₁ x ₂₂ · · · x 2,n+m−1

.. . .. . .. . x _n1 x _n2 · · · x _n,n+m−1





 I ⊆ k [x ₁₁ , · · · , x n,n+m−1 ] ideal generated by maximal minors.

The initial ideal of I is L([n], [m]).







x ₁₁ x ₁₂ · · · x _1,n+s−1 x ₂₁ x ₂₂ · · · x 2,n+s−1

.. . .. . .. .

x _n+r _−1,1 x _n2 · · · x _n+r _−1,n+s−1





 J ⊆ k [x ₁₁ , · · · , x _n+r −1,n+s−1 ]

ideal generated by n-minors.

The initial ideal of J is a regular

quotient of L(n, [r] × [s]).

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Determinantal ideals







x ₁₁ x ₁₂ · · · x _1,n+m−1 x ₂₁ x ₂₂ · · · x 2,n+m−1

.. . .. . .. . x _n1 x _n2 · · · x _n,n+m−1





 I ⊆ k [x ₁₁ , · · · , x n,n+m−1 ] ideal generated by maximal minors.

The initial ideal of I is L([n], [m]).







x ₁₁ x ₁₂ · · · x _1,n+s−1 x ₂₁ x ₂₂ · · · x 2,n+s−1

.. . .. . .. .

x _n+r _−1,1 x _n2 · · · x n+r−1,n+s−1





 J ⊆ k [x ₁₁ , · · · , x _n+r −1,n+s−1 ]

ideal generated by n-minors.

The initial ideal of J is a regular

quotient of L(n, [r] × [s]).

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“Free”

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Determinantal ideals

Symmetric and skew-symmetric

Ideal of 2-minors of generic symmetric matrix of size n + 1. Its initial ideal is a regular quotient of

L(2, Hom([2], [n])).

Ideal of Pfaffians of a generic skew-symmetric matrix of size 2n + 1.

Its initial ideal is a regular quotient of

L(n, V ).

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Strongly stable ideals

I = (x ₁ ² , x ₁ x ₂ , x ₁ x ₃ , x ₂ ² ) ⊆ k [x ₁ , x ₂ , x ₃ ] is strongly stable:

m = x _j n ∈ I ⇒ x _i n ∈ I for i ≤ j.

Elements of Hom([d ], [n]) ¹⁻¹ ↔

monomials in k[x ₁ , . . . , x _n ]

of degree d

by φ →

d

Y

i=1

x _φ(i)

Poset ideals J ⊆ Hom(P, [n]) ¹⁻¹ ↔

strongly stable ideals in

k[x ₁ , . . . , x _r ]

generated in degree d

If J ↔ I then I is a regular quotient of L(J ).

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Strongly stable ideals

I = (x ₁ ² , x ₁ x ₂ , x ₁ x ₃ , x ₂ ² ) ⊆ k [x ₁ , x ₂ , x ₃ ] is strongly stable:

m = x _j n ∈ I ⇒ x _i n ∈ I for i ≤ j.

Elements of Hom([d ], [n]) ¹⁻¹ ↔

monomials in k[x ₁ , . . . , x _n ]

of degree d

by φ →

d

Y

i=1

x _φ(i)

Poset ideals J ⊆ Hom(P, [n]) ¹⁻¹ ↔

strongly stable ideals in k[x ₁ , . . . , x _r ] generated in degree d

If J ↔ I then I is a regular quotient of L(J ).

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Strongly stable ideals

I = (x ₁ ² , x ₁ x ₂ , x ₁ x ₃ , x ₂ ² ) ⊆ k [x ₁ , x ₂ , x ₃ ] is strongly stable:

m = x _j n ∈ I ⇒ x _i n ∈ I for i ≤ j.

Elements of Hom([d ], [n]) ¹⁻¹ ↔

monomials in k[x ₁ , . . . , x _n ]

of degree d

by φ →

d

Y

i=1

x _φ(i)

Poset ideals J ⊆ Hom(P, [n]) ¹⁻¹ ↔

strongly stable ideals in

k[x ₁ , . . . , x _r ]

generated in degree d

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Ferrers ideals

Partition n ≥ λ 1 ≥ · · · ≥ λ n ≥ 0

λ ∈ Hom([n], [n + 1]) = Hom([n] × [n], [2]), λ(i ) = λ n+1−i + 1.

Gives a poset ideal

J ⊆ [n] × [n] = Hom(2, [n]), 2 = {•, •}

L(J ) ⊆ k [x _2×[n] ] are the Ferrers ideals of Nagel and Corso.

They are generalized by Nagel and Reiner to d -partite, d -uniform

Ferrers hypergraph ideals. These correspond to poset ideals

J ⊆ Hom(d , [n]).

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Ferrers ideals

Partition n ≥ λ 1 ≥ · · · ≥ λ n ≥ 0

λ ∈ Hom([n], [n + 1]) = Hom([n] × [n], [2]), λ(i ) = λ n+1−i + 1.

Gives a poset ideal

J ⊆ [n] × [n] = Hom(2, [n]), 2 = {•, •}

L(J ) ⊆ k [x _2×[n] ] are the Ferrers ideals of Nagel and Corso.

They are generalized by Nagel and Reiner to d -partite, d -uniform

Ferrers hypergraph ideals. These correspond to poset ideals

J ⊆ Hom(d , [n]).

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Ferrers ideals

Partition n ≥ λ 1 ≥ · · · ≥ λ n ≥ 0

λ ∈ Hom([n], [n + 1]) = Hom([n] × [n], [2]), λ(i ) = λ n+1−i + 1.

Gives a poset ideal

J ⊆ [n] × [n] = Hom(2, [n]), 2 = {•, •}

L(J ) ⊆ k [x _2×[n] ] are the Ferrers ideals of Nagel and Corso.

They are generalized by Nagel and Reiner to d -partite, d -uniform

Ferrers hypergraph ideals. These correspond to poset ideals

J ⊆ Hom(d , [n]).

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Uniform face ideals

Hom(n, [2]) identifies as the Booleans poset on n elements J ⊆ Hom(n, [2]) ¹⁻¹ ↔ simplicial complexes on {1, 2, 3, . . . , n}.

The ideals L(J ) are the uniform face ideals of D.Cook and

J.Herzog/T.Hibi.

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Resolutions

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Resolution of L(2,P)

L(2, P ) ← F ₀ ← F ₁ ← F ₂ ← · · · . Since L(2, P ) is N ^[2]×P -graded, so are the F _i .

F _i = M

r∈ N

^[2]×P

S(−r) ^β

ⁱ^,r

= M

r∈{0,1}

^[2]×P

S(−r) ^β

ⁱ^,r

= M

R⊆[2]×P

S (−R ) ^β

ⁱ^,R

.

Generators of L(2, P ) are: x _1,p

₁

x _2,p

₂

where p ₁ ≤ p ₂ , so:

β _0,R =

( 1, R = {(1, p ₁ ), (2, p ₂ ) | p ₁ ≤ p ₂ }

0 else .

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Resolution of L(2,P)

L(2, P ) ← F ₀ ← F ₁ ← F ₂ ← · · · . Since L(2, P ) is N ^[2]×P -graded, so are the F _i .

F _i = M

r∈ N

^[2]×P

S(−r) ^β

ⁱ^,r

= M

r∈{0,1}

^[2]×P

S(−r) ^β

ⁱ^,r

= M

R⊆[2]×P

S (−R ) ^β

ⁱ^,R

.

Generators of L(2, P ) are: x _1,p

₁

x _2,p

₂

where p ₁ ≤ p ₂ , so:

β _0,R =

( 1, R = {(1, p ₁ ), (2, p ₂ ) | p ₁ ≤ p ₂ }

0 else .

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Resolution of L(2, P )

Question

Let R ⊆ [2] × P . What is β _i,R ?

R _i = R ∩ ({i} × P ), R _i ⊆ P , i = 1, 2.

Bipartite graph G (R ₁ , R ₂ ) with edges p ₁ − p ₂ if p ₁ ≤ p ₂ .

β i,R = H ^|R|−i−2 (∆(G (R ₁ , R ₂ ))).

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Resolution of L(2, P )

Question

Let R ⊆ [2] × P . What is β _i,R ?

R _i = R ∩ ({i} × P ), R _i ⊆ P , i = 1, 2.

Bipartite graph G (R ₁ , R ₂ ) with edges p ₁ − p ₂ if p ₁ ≤ p ₂ .

β i,R = H ^|R|−i−2 (∆(G (R ₁ , R ₂ ))).

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Topology of bipartite graphs

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Topology of bipartite graphs

G (A, B) = {1, b}

{2, a}

{3, a}

I = (x ₁ x _b , x ₂ x _a , x ₃ x _a )

simplicial complex ∆(G (A, B)).

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Extending the order relation

Definition R ₁ ≤ R ₂ if

∀p ∈ max R ₁ , ∃q ∈ min R ₂ : p ≤ q

∀q ∈ min R ₂ , ∃p ∈ max R ₁ : p ≤ q Must be in order!

If not R ₁ ≤ R ₂ the ∆(G (R ₁ , R ₂ )) is contractible.

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Resolutions of co-letterplace ideals

Linear resolution

J ⊆ Hom(P , [n]) a poset ideal.

L(J ) has linear resolution,

⇒ Alexander dual ideal L(J ) ^A is a Cohen-Macaualy ideal, so Stanley-Reisner ring k[x _P×[n] ]/L(J ) ^A = k[∆(J )] is a

Cohen-Macaulay ring.

The linear resolution of L(J ) is explicitly determined by its

canonical module ω _k _[∆(J _)] .

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Resolutions of co-letterplace ideals

Linear resolution

J ⊆ Hom(P , [n]) a poset ideal.

L(J ) has linear resolution,

⇒ Alexander dual ideal L(J ) ^A is a Cohen-Macaualy ideal, so Stanley-Reisner ring k[x _P×[n] ]/L(J ) ^A = k[∆(J )] is a

Cohen-Macaulay ring.

The linear resolution of L(J ) is explicitly determined by its

canonical module ω _k _[∆(J _)] .

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Resolutions of co-letterplace ideals

Linear resolution

J ⊆ Hom(P , [n]) a poset ideal.

L(J ) has linear resolution,

⇒ Alexander dual ideal L(J ) ^A is a Cohen-Macaualy ideal, so Stanley-Reisner ring k[x _P×[n] ]/L(J ) ^A = k[∆(J )] is a

Cohen-Macaulay ring.

The linear resolution of L(J ) is explicitly determined by its

canonical module ω _k _[∆(J _)] .

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Resolutions of co-letterplace ideals

B(P , n) ⊆ k[x _P×[n] ] ideal generated by x _p,i x _q,j where p < q and i > j .

We describe the canonical module.

Alexander dual of ω _k [∆(J )] is image of:

L(J ) → k [x _P×[n] ] → k [x _P×[n] ]/(B (P , n) + L(J ^c ).

Equivalently ω _k _[∆(J _)] is the image of:

k [∆(J )] = k [x _[n]×P ]/L(J ) Â ← k [x _[n]×P ] ← B(P, n) Â ∩ L(J ^c ) Â .

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Triangulations of spheres

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A ball and a sphere

The canonical module

k[Σ(J )] ← k[∆(J )] ^ideal ← - ω _k [∆(J)]

∆(J ) is a (triangulated) ball of codimension |P | in the simplex on P × [n].

Σ(J ) is the boundary of ∆(J ) and so a trianguated sphere!

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Generalizing Bier spheres

Our situation:

P , J ⊆ Hom(P , [n]) simplicial sphere Σ(J ) Björner, Pfaffenholz, Sjöstrand, Ziegler ’04

m, J ⊆ Hom(m, [2]) l

∆ on {1, . . . , m}

Bier sphere B(∆)

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Deformations of letterplace ideals

Unusually nice

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Deformations

I ideal in a k -algebra R .

A a commutative local artinian k -algebra with A/m _A = k . A deformation of I over A is an ideal J ⊆ R ⊗ _k A such that:

(R ⊗ _k A)/J is a flat A-module, (R ⊗ _k A)/J ⊗ _A A/m _A = R /I . Deformation functor:

Def _I : Art → Set.

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A family of deformations

We consider L(2, P ) ⊆ S = k [x _[2]×P ] when the Hasse diagram of P is a rooted tree.

There is a polynomial ring k [U], where U a finite dimensional vector space, and an ideal

J(2, P ) ⊆ T = k[x _[2]×P ] ⊗ _k k[U],

whose generators are (write x _i,p as p _i ): p ₁ q ₂ + T (p)S _p (q),

where T (p) and S _p (q) are recursively defined polynomials in

k [x _[2]×P ] ⊗ _k k [U ].

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A family of deformations

We consider L(2, P ) ⊆ S = k [x _[2]×P ] when the Hasse diagram of P is a rooted tree.

There is a polynomial ring k[U], where U a finite dimensional vector space, and an ideal

J(2, P ) ⊆ T = k[x _[2]×P ] ⊗ _k k[U ],

whose generators are (write x _i,p as p _i ):

p ₁ q ₂ + T (p)S _p (q),

where T (p) and S _p (q) are recursively defined polynomials in

k [x _[2]×P ] ⊗ _k k [U ].

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Example of J(2, P )

a

b c d

M(a) =





u _a,b b ₁ u _c,b u _d,b u _a,c u _b,c c ₁ u _d,c



 ,

b ₁ b ₂ − a ₂ u _a,b − c ₂ u _c,b − d ₂ u _d,b c ₁ c ₂ − a ₂ u _a,c − b ₂ u _b,c − d ₂ u _d,c d ₁ d ₂ − a ₂ u _a,d − b ₂ u _b,d − c ₂ u _c,d a 1 b 2 − u _∅,a |M (a)| ^b

a ₁ c ₂ − u _∅,a |M (a)| ^c

a ₁ d ₂ − u _∅,a |M (a)| ^d

a 1 a 2 − u |M(a)| ^a

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The deformation family

T /J(2, P ) is flat over k [U],

The fiber T /J(2, P ) ⊗ _k _[U] k = S/L(2, P ),

Every deformation of L(2, P ) is obtained from a coordinate

change of J(2, P ) ⊆ T , and then taking a regular quotient.

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The deformation family

T /J(2, P ) is flat over k [U],

The fiber T /J(2, P ) ⊗ _k _[U] k = S/L(2, P ),

Every deformation of L(2, P ) is obtained from a coordinate

change of J(2, P ) ⊆ T , and then taking a regular quotient.

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The deformation functor

The deformation functor of L(2, P ) is unobstructed.

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Rigid ideal

J(2, P )

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The Hilbert scheme

Let k [x _[2]×P ] ⊗ _k k [U] be positively graded by an abelian group A, U → U ⁰ an A-graded map of vector spaces.

Consider the multigraded Hilbert scheme H _A ^h where h is the Hilbert function of k [x _[2]×P ] ⊗ _k k [U ⁰ ]/L(2, P ).

The ideal L(2, P) is a smooth point on the Hilbert scheme H _A ^h .

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The Hilbert scheme

Let k [x _[2]×P ] ⊗ _k k [U] be positively graded by an abelian group A, U → U ⁰ an A-graded map of vector spaces.

Consider the multigraded Hilbert scheme H _A ^h where h is the Hilbert function of k [x _[2]×P ] ⊗ _k k [U ⁰ ]/L(2, P ).

The ideal L(2, P) is a smooth point on the Hilbert scheme H _A ^h .

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The Hilbert scheme

Let k [x _[2]×P ] ⊗ _k k [U] be positively graded by an abelian group A, U → U ⁰ an A-graded map of vector spaces.

Consider the multigraded Hilbert scheme H _A ^h where h is the Hilbert function of k [x _[2]×P ] ⊗ _k k [U ⁰ ]/L(2, P ).

The ideal L(2, P) is a smooth point on the Hilbert scheme H _A ^h .

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References

A.D’Ali, G.Fløystad, A.Nematbakhsh, Resolutions of letterplace ideals of posets, arxiv (2016).

A.D’Ali, G.Fløystad, A.Nematbakhsh, Resolutions of co-letterplace ideals and triangulations of Bier spheres, arxiv (2016).

G.Fløystad, A.Nematbakhsh, Deformations of quadratic letterplace ideals, in preparation.

These slides available on my home page.

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Symmetry is very much used and studied in mathematics.

Maybe partially ordered sets are underused in mathematics.

March14,2016 GunnarFløystad Letterplaceidealsofposets:Alargeclasswithunusuallyniceproperties

Letterplace ideals of posets:

A large class with unusually nice properties

Gunnar Fløystad

NCM Stockholm, March 2016

March 14, 2016

Simplicial complexes

V finite set.

Definition

A simplicial complex ∆ on V is a family of subset of V such that if X ∈ ∆ and Y ⊆ X , then Y ∈ ∆.

Example

{1, 2, 3}, {3, 4}, {1, 2}, {2, 3}, {1, 3}, {1}, {2}, {3},

∅

Triangle with an edge attached.

Squarefree monomial ideals

Example

x 2 x 4 x 5 x 7 is a squarefree monomial. Also written m {2,4,5,7} . Definition

A squarefree monomial ideal I ⊆ k [x 1 , . . . , x n ] is generated by squarefree monomials.

Simiplicial complexes on [n] = {1, 2, 3, . . . , n} 1−1 ↔ squarefree monomial ideals I ∆ .

Definition

Stanley-Reisner ring k [∆] = k [x 1 , . . . , x n ]/I ∆ .

Note R ∈ ∆ ⇔ m R is nonzero in k [∆].

Squarefree monomial ideals

Example

x 2 x 4 x 5 x 7 is a squarefree monomial. Also written m {2,4,5,7} . Definition

A squarefree monomial ideal I ⊆ k [x 1 , . . . , x n ] is generated by squarefree monomials.

Simiplicial complexes on [n] = {1, 2, 3, . . . , n} 1−1 ↔ squarefree monomial ideals I ∆ .

Definition

Stanley-Reisner ring k [∆] = k [x 1 , . . . , x n ]/I ∆ .

Note R ∈ ∆ ⇔ m is nonzero in [∆].

Founding fathers

Richard Stanley, MIT Melvin Hochster, U. of Michigan

Alexander duality

I squarefree monomial ideal.

All monomials m ∈ k[x 1 , . . . , x n ] with nontrivial common divisor

with every monomial n ∈ I , generate the Alexander dual ideal

J = I A .

Notation

R set k [x R ] = k [x i ] i∈R . S ⊆ R monomial m S = Q

i∈S x i .

[n] = {1 < 2 < · · · < n} totally ordered poset.

Notation

R set k [x R ] = k [x i ] i∈R . S ⊆ R monomial m S = Q

i∈S x i .

[n] = {1 < 2 < · · · < n} totally ordered poset.

Isotone maps

P , Q finite posets.

φ : P → Q isotone map, i.e. p ≤ p 0 ⇒ φ(p) ≤ φ(p 0 ).

Hom(P , Q) set of isotone maps. Is itself a poset: φ ≤ ψ if φ(p) ≤ ψ(p) for all p ∈ P .

Graph of φ: Γφ = {(p, φ(p) | p ∈ P }.

Isotone maps

P , Q finite posets.

φ : P → Q isotone map, i.e. p ≤ p 0 ⇒ φ(p) ≤ φ(p 0 ).

Hom(P , Q) set of isotone maps. Is itself a poset: φ ≤ ψ if φ(p) ≤ ψ(p) for all p ∈ P .

Graph of φ: Γφ = {(p, φ(p) | p ∈ P }.

Isotone maps

P , Q finite posets.

φ : P → Q isotone map, i.e. p ≤ p 0 ⇒ φ(p) ≤ φ(p 0 ).

Hom(P , Q) set of isotone maps. Is itself a poset: φ ≤ ψ if φ(p) ≤ ψ(p) for all p ∈ P .

Graph of φ: Γφ = {(p, φ(p) | p ∈ P }.

Letterplace and co-letterplace ideals

L(P , Q ) ⊆ k [x P×Q ] ideal generated by m Γφ where φ ∈ Hom(P , Q).

n’th letterplace ideal

L(n, P) = L([n], P ) ⊆ k [x [n]×P ]

is generated by x 1,p

x 2,p

· · · x n,p

where p 1 ≤ p 2 ≤ · · · ≤ p n . n’th co-letterplace ideal

L(P, n) = L(P, [n]) ⊆ k [x P×[n] ] is generated by Q

p∈P x p,i

where p ≤ q ⇒ 1 ≤ i p ≤ i q ≤ n.

Letterplace and co-letterplace ideals

L(P , Q ) ⊆ k [x P×Q ] ideal generated by m Γφ where φ ∈ Hom(P , Q).

n’th letterplace ideal

L(n, P) = L([n], P ) ⊆ k [x [n]×P ]

is generated by x 1,p

x 2,p

· · · x n,p

where p 1 ≤ p 2 ≤ · · · ≤ p n .

n’th co-letterplace ideal

x ₂ x ₄ x ₅ x ₇ is a squarefree monomial. Also written m _{2,4,5,7} . Definition

A squarefree monomial ideal I ⊆ k [x ₁ , . . . , x _n ] is generated by squarefree monomials.

Simiplicial complexes on [n] = {1, 2, 3, . . . , n} ¹⁻¹ ↔ squarefree monomial ideals I _∆ .

Stanley-Reisner ring k [∆] = k [x 1 , . . . , x n ]/I _∆ .

Note R ∈ ∆ ⇔ m _R is nonzero in k [∆].

x ₂ x ₄ x ₅ x ₇ is a squarefree monomial. Also written m _{2,4,5,7} . Definition

A squarefree monomial ideal I ⊆ k [x ₁ , . . . , x _n ] is generated by squarefree monomials.

Simiplicial complexes on [n] = {1, 2, 3, . . . , n} ¹⁻¹ ↔ squarefree monomial ideals I _∆ .

Stanley-Reisner ring k [∆] = k [x 1 , . . . , x n ]/I _∆ .

All monomials m ∈ k[x ₁ , . . . , x _n ] with nontrivial common divisor

J = I ^A .

R set k [x _R ] = k [x _i ] i∈R . S ⊆ R monomial m _S = Q

i∈S x _i .

R set k [x _R ] = k [x _i ] i∈R . S ⊆ R monomial m _S = Q

i∈S x _i .

φ : P → Q isotone map, i.e. p ≤ p ⁰ ⇒ φ(p) ≤ φ(p ⁰ ).

φ : P → Q isotone map, i.e. p ≤ p ⁰ ⇒ φ(p) ≤ φ(p ⁰ ).

φ : P → Q isotone map, i.e. p ≤ p ⁰ ⇒ φ(p) ≤ φ(p ⁰ ).

L(P , Q ) ⊆ k [x P×Q ] ideal generated by m _Γφ where φ ∈ Hom(P , Q).

L(n, P) = L([n], P ) ⊆ k [x _[n]×P ]

L(P, n) = L(P, [n]) ⊆ k [x _P×[n] ] is generated by Q

p∈P x _p,i

where p ≤ q ⇒ 1 ≤ i _p ≤ i _q ≤ n.

L(P , Q ) ⊆ k [x P×Q ] ideal generated by m _Γφ where φ ∈ Hom(P , Q).

L(n, P) = L([n], P ) ⊆ k [x _[n]×P ]

L(P, n) = L(P, [n]) ⊆ k [x _P×[n] ] is generated by Q

p∈P x _p,i

where p ≤ q ⇒ 1 ≤ i _p ≤ i _q ≤ n.

L(P , Q ) ⊆ k [x P×Q ] ideal generated by m _Γφ where φ ∈ Hom(P , Q).

L(n, P) = L([n], P ) ⊆ k [x _[n]×P ]

L(P, n) = L(P, [n]) ⊆ k [x _P×[n] ] is generated by Q

p∈P x _p,i

where p ≤ q ⇒ 1 ≤ i _p ≤ i _q ≤ n.

L(2, 2) ⊆ S = k [x _[2]×[2] ] generated by:

x ₁₁ x ₂₁ , x ₁₁ x ₂₂ , x ₁₂ x ₂₂ .

L(2, 2) ⊆ S = k [x _[2]×[2] ] generated by:

x ₁₁ x ₂₁ , x ₁₁ x ₂₂ , x ₁₂ x ₂₂ .

S/L(2, 2) ← S ←−−−−−−−−−−−−− ^[x

^x

^,x

^x

^,x

^x

^] S(−2) ³