Post on 28-Mar-2020
transcript
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