Generic deformations of matroid ideals Alexandru Constantinescu - - PowerPoint PPT Presentation

Generic deformations of matroid ideals

Alexandru Constantinescu

(joint work with Thomas Kahle and Matteo Varbaro)

Universit´ e de Neuchˆ atel, 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+···+hsts (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+···+hsts (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+···+hsts (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 ts (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 ts (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 ts (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 ts (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 ts (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 Macaulay 19

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. 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 Macaulay 20

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. 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 Macaulay 21

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. 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 Macaulay 22

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. 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 Macaulay 23

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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

1CM= Cohen Macaulay 24

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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

1CM= Cohen Macaulay 25

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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

1CM= Cohen Macaulay 26

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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

1CM= Cohen Macaulay 27

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 28

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 29

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 30

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 31

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 32

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 33

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 34

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 35

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 36

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 37

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 38

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 39

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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 Macaulay 40

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

Stanley-Reisner ring

S/I∆

Artinian reduction

S/(I∆ + (ℓi))

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!

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