Generic deformations of matroid idealsuserpage.fu-berlin.de/aconstant/Pdfs/Slides/Alex_Syz.pdf ·...

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Generic deformations of matroid ideals Alexandru Constantinescu (joint work with Thomas Kahle and Matteo Varbaro) Universit´ e de Neuchˆ atel, Switzerland 1

Transcript of Generic deformations of matroid idealsuserpage.fu-berlin.de/aconstant/Pdfs/Slides/Alex_Syz.pdf ·...

Generic deformations of matroid ideals

Alexandru Constantinescu(joint work with Thomas Kahle and Matteo Varbaro)

Universite de Neuchatel, Switzerland

1

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

2

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

3

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

4

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

[n]

5

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

[n]

x1, . . . , xn

6

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

2[n]

x1, . . . , xn

7

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

2[n]

S = K[x1, . . . , xn]

8

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ matroid complex

S = K[x1, . . . , xn]

9

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ matroid complex

S = K[x1, . . . , xn] ⊇ I∆ monomial ideal

10

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs)

S = K[x1, . . . , xn] ⊇ I∆ monomial ideal

11

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs)

S = K[x1, . . . , xn] ⊇ I∆ −→ Hilbert series of S/I∆: h0+h1t+···+hs ts

(1−t)d

12

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs)

S = K[x1, . . . , xn] ⊇ I∆ −→ Hilbert series of S/I∆: h0+h1t+···+hs ts

(1−t)d

13

A matroid is a simplicial complex ∆ on [n], such that

∀ F ,G ∈ ∆ with |F | < |G |, ∃ v ∈ G \ F , such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs)

S = K[x1, . . . , xn] ⊇ I∆ −→ Hilbert series of S/I∆: h0+h1t+···+hs ts

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

14

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

15

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

16

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

17

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

18

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay19

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay20

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay21

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay22

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay23

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay24

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay25

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay26

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay27

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay28

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay29

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay30

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

1CM= Cohen Macaulay31

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

Theorem (-,Kahle, Varbaro ’12)If ∆ is the (d–1)-skeleton of a d-dim, CM complex, then βp(S/I∆) = βp(S/ gin(I∆)).

1CM= Cohen Macaulay32

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

Theorem (-,Kahle, Varbaro ’12)If ∆ is the (d–1)-skeleton of a d-dim, CM complex, then βp(S/I∆) = βp(S/ gin(I∆)).

In general: βi,j (S/I ) ≤ βi,j (S/ gin(I )).

1CM= Cohen Macaulay33

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

Theorem (-,Kahle, Varbaro ’12)If ∆ is the (d–1)-skeleton of a d-dim, CM complex, then βp(S/I∆) = βp(S/ gin(I∆)).

In general: βi,j (S/I ) ≤ βi,j (S/ gin(I )).

1CM= Cohen Macaulay34

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

WANT: weakgin(I ) S/(weakgin(I∆) + (xi − xj ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

Theorem (-,Kahle, Varbaro ’12)If ∆ is the (d–1)-skeleton of a d-dim, CM complex, then βp(S/I∆) = βp(S/ gin(I∆)).

In general: βi,j (S/I ) ≤ βi,j (S/ gin(I )).

1CM= Cohen Macaulay35

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

WANT: weakgin(I ) S/(weakgin(I∆) + (xi − xj ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

Theorem (-,Kahle, Varbaro ’12)If ∆ is the (d–1)-skeleton of a d-dim, CM complex, then βp(S/I∆) = βp(S/ gin(I∆)).

In general: βi,j (S/I ) ≤ βi,j (S/ gin(I )).

1CM= Cohen Macaulay36

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

WANT: weakgin(I ) S/(weakgin(I∆) + (xi − xj ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

Theorem (-,Kahle, Varbaro ’12)If ∆ is the (d–1)-skeleton of a d-dim, CM complex, then βp(S/I∆) = βp(S/ gin(I∆)).

In general: βi,j (S/I ) ≤ βi,j (S/ gin(I )).

1CM= Cohen Macaulay37

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

WANT: weakgin(I ) S/(weakgin(I∆) + (xi − xj ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

Theorem (-,Kahle, Varbaro ’12)If ∆ is the (d–1)-skeleton of a d-dim, CM complex, then βp(S/I∆) = βp(S/ gin(I∆)).

In general: βi,j (S/I ) ≤ βi,j (S/ gin(I )).

1CM= Cohen Macaulay38

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

WANT: weakgin(I ) S/(weakgin(I∆) + (xi − xj ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

Theorem (-,Kahle, Varbaro ’12)If ∆ is the (d–1)-skeleton of a d-dim, CM complex, then βp(S/I∆) = βp(S/ gin(I∆)).

In general: βi,j (S/I ) ≤ βi,j (S/ gin(I )).

1CM= Cohen Macaulay39

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

WANT: weakgin(I ) S/(weakgin(I∆) + (xi − xj ))

A graded, CM1 algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

Wish (-,Kahle, Varbaro ’12)If ∆ is a matroid complex, then βp(S/I∆) = βp(S/weakgin(I∆)).

1CM= Cohen Macaulay40

A matroid is a simplicial complex ∆ on [n], such that ∀ F, G ∈ ∆ with |F| < |G|, ∃ v ∈ G \ F, such that F ∪ {v} ∈ ∆.

2[n] ⊇ ∆ −→ h-vector of ∆: (h0, . . . , hs )

S = K[x1, . . . , xn ] ⊇ I∆ −→ Hilbert series of S/I∆:h0+h1t+···+hs t

s

(1−t)d

Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence.

A pure O-sequence is the Hilbert function of some artinian, monomial, level algebra.

Stanley-Reisner ring S/I∆

Artinian reduction S/(I∆ + (`i ))

Artinian reduction of gin(I ) S/(gin(I∆) + (xi ))

WANT: weakgin(I ) S/(weakgin(I∆) + (xi − xj ))

A graded, CM algebra S/I is level if: 0 −→ S(−a)βp −→ · · · −→ F0 −→ S/I −→ 0

Wish (-,Kahle, Varbaro ’12)If ∆ is a matroid complex, then βp(S/I∆) = βp(S/weakgin(I∆)).

Thank you for your attention!41