The lex-plus-power Conjecture holds
for extremal Betti numbers
Enrico Sbarra
University of Pisa, Italy
Outline
I Macaulay Theorem and its generalizations
I The Eisenbud-Green-Harris Conjecture
I The Lex-Plus-Power Conjecture
I Our question and toolkit
I Results
generalizations of Macaulay theorem
Let A = K [X1 , . . . , Xn ] be a polynomial ring over a field K , with deg Xi = 1 for all i.
Let Hilb(M)• and Hilb(M) denote the Hilbert function and the Hilbert Series of a graded
module M, respectively.
Let a = (0) and R = A/a = A.
generalizations of Macaulay theorem
Let A = K [X1 , . . . , Xn ] be a polynomial ring over a field K , with deg Xi = 1 for all i.
Let Hilb(M)• and Hilb(M) denote the Hilbert function and the Hilbert Series of a graded
module M, respectively.
Let a = (0) and R = A/a = A.
Theorem: (Macaulay ’27) Let H : N −→ N be a function with H(0) = 1. TFAE:
• There exists a homogeneous ideal I of A such that H = Hilb(A/I )• .
• H(d + 1) ≤ H(d)<d> for all d ≥ 1.
generalizations of Macaulay theorem
Let A = K [X1 , . . . , Xn ] be a polynomial ring over a field K , with deg Xi = 1 for all i.
Let Hilb(M)• and Hilb(M) denote the Hilbert function and the Hilbert Series of a graded
module M, respectively.
Let a = (0) and R = A/a = A.
Theorem: (Macaulay ’27) Let H : N −→ N be a function with H(0) = 1. TFAE:
• There exists a homogeneous ideal I of A such that H = Hilb(A/I )• .
• H(d + 1) ≤ H(d)<d> for all d ≥ 1.
Let a be a homogeneous ideal of A.
Let R = A/a and I be a homogeneous ideal of R.
Q: What are the possible Hilbert functions of R/I ?
generalizations of Macaulay theorem
Let A = K [X1 , . . . , Xn ] be a polynomial ring over a field K , with deg Xi = 1 for all i.
Let Hilb(M)• and Hilb(M) denote the Hilbert function and the Hilbert Series of a graded
module M, respectively.
Let a = (0) and R = A/a = A.
Theorem: (Macaulay ’27) Let H : N −→ N be a function with H(0) = 1. TFAE:
• There exists a homogeneous ideal I of A such that H = Hilb(A/I )• .
• H(d + 1) ≤ H(d)<d> for all d ≥ 1.
Let a be a homogeneous ideal of A.
Let R = A/a and I be a homogeneous ideal of R.
Q: What are the possible Hilbert functions of R/I ?
• (Kruskal ’63, Katona ’68) When a = (X12 , . . . , Xn2 ).
• (Clements-Lindström ’69) When a = (X1d1 , . . . , Xndn ), 2 ≤ d1 ≤ d2 ≤ . . . ≤ dn .
Let >lex be the lexicographic monomial order on Mon(A), with X1 > X2 > . . . > Xn .
A monomial ideal L of A is called (lexicographic, lex-segment, or simply) lex-ideal if the
following property holds: For all d,
u, v monomials of degree d, with u ∈ L and v >lex u =⇒ v ∈ L.
Let >lex be the lexicographic monomial order on Mon(A), with X1 > X2 > . . . > Xn .
A monomial ideal L of A is called (lexicographic, lex-segment, or simply) lex-ideal if the
following property holds: For all d,
u, v monomials of degree d, with u ∈ L and v >lex u =⇒ v ∈ L.
Remarks:
• A function H : N −→ N with H(0) = 1 is admissible if and only if there exists a lex-ideal
L such that H = H(A/L).
Let >lex be the lexicographic monomial order on Mon(A), with X1 > X2 > . . . > Xn .
A monomial ideal L of A is called (lexicographic, lex-segment, or simply) lex-ideal if the
following property holds: For all d,
u, v monomials of degree d, with u ∈ L and v >lex u =⇒ v ∈ L.
Remarks:
• A function H : N −→ N with H(0) = 1 is admissible if and only if there exists a lex-ideal
L such that H = H(A/L).
• Let a = (X1d1 , . . . , Xndn ) and R = A/a. All the possible Hilbert functions in R are
attained by quotients R/LR, where L is a lex-ideal of A and L ⊇ a.
Let >lex be the lexicographic monomial order on Mon(A), with X1 > X2 > . . . > Xn .
A monomial ideal L of A is called (lexicographic, lex-segment, or simply) lex-ideal if the
following property holds: For all d,
u, v monomials of degree d, with u ∈ L and v >lex u =⇒ v ∈ L.
Remarks:
• A function H : N −→ N with H(0) = 1 is admissible if and only if there exists a lex-ideal
L such that H = H(A/L).
• Let a = (X1d1 , . . . , Xndn ) and R = A/a. All the possible Hilbert functions in R are
attained by quotients R/LR, where L is a lex-ideal of A and L ⊇ a.
A lex-ideal which contains a pure-powers ideal a = (X1d1 , . . . , Xndn ) is called a lex-plus-powers
ideal (with respect to d1 , . . . , dn ).
the EGH conjecture
Let d = (d1 , . . . , dn ) ∈ Nn , with 2 ≤ d1 ≤ d2 ≤ . . . ≤ dn .
Let a = (f), where f = f1 , . . . , fn , is a regular sequence with deg fi = di .
Conjecture (EGH ’92)
the EGH conjecture
Let d = (d1 , . . . , dn ) ∈ Nn , with 2 ≤ d1 ≤ d2 ≤ . . . ≤ dn .
Let a = (f), where f = f1 , . . . , fn , is a regular sequence with deg fi = di .
Conjecture (EGH ’92)
Let A=K [X1 , . . . , Xn ], where K is a field. Let I be a homogeneous ideal in A containing a.
Then,
the EGH conjecture
Let d = (d1 , . . . , dn ) ∈ Nn , with 2 ≤ d1 ≤ d2 ≤ . . . ≤ dn .
Let a = (f), where f = f1 , . . . , fn , is a regular sequence with deg fi = di .
Conjecture (EGH ’92)
Let A=K [X1 , . . . , Xn ], where K is a field. Let I be a homogeneous ideal in A containing a.
Then,
d
I has the same Hilbert function as an ideal containing {Xi i : 1 ≤ i ≤ n}.
the EGH conjecture
Let d = (d1 , . . . , dn ) ∈ Nn , with 2 ≤ d1 ≤ d2 ≤ . . . ≤ dn .
Let a = (f), where f = f1 , . . . , fn , is a regular sequence with deg fi = di .
Conjecture (EGH ’92)
Let A=K [X1 , . . . , Xn ], where K is a field. Let I be a homogeneous ideal in A containing a.
Then,
d
I has the same Hilbert function as an ideal containing {Xi i : 1 ≤ i ≤ n}.
Known only in a few cases, e.g.
• in two variables
• in the monomial case
• (Caviglia-Mac Lagan, ’08) when the degrees di of the regular sequence “quickly increase”
• Abedelfatah, C.-S., Chen, Cuong, Francisco, ...
generalizations of Macaulay theorem
A
Recall that, if M is a graded A-module, βijA (M) := dimK TorA
i (M, k)j = Hilb Tori (M, k) j .
Remark:
• Let L be the lex-ideal with the same Hilbert function as I . Then
A
A
β1j
(I ) ≤ β1j
(L), for all j.
generalizations of Macaulay theorem
A
Recall that, if M is a graded A-module, βijA (M) := dimK TorA
i (M, k)j = Hilb Tori (M, k) j .
Remark:
• Let L be the lex-ideal with the same Hilbert function as I . Then
A
A
β1j
(I ) ≤ β1j
(L), for all j.
Other extremal properties of lex-ideals:
• (Bigatti, Hulett ’93, Pardue ’96) The above inequality holds for all βij .
generalizations of Macaulay theorem
A
Recall that, if M is a graded A-module, βijA (M) := dimK TorA
i (M, k)j = Hilb Tori (M, k) j .
Remark:
• Let L be the lex-ideal with the same Hilbert function as I . Then
A
A
β1j
(I ) ≤ β1j
(L), for all j.
Other extremal properties of lex-ideals:
• (Bigatti, Hulett ’93, Pardue ’96) The above inequality holds for all βij .
i (M) denote the jth graded component of the ith local cohomology module of M with
Let Hm
j
support in m.
• (S. ’01) Let L be the lex-ideal with the same Hilbert function as I . Then
i
i
Hilb Hm
(A/I ) ≤ Hilb Hm
(A/L) , for all i, j.
j
j
the LPP conjecture
Let a = (f) = (f1 , . . . , fr ), where f1 , . . . , fr is a regular sequence with deg fi = di .
LPP Conjecture (Evans ’02)
If f satisfies EGH, then for all homogeneous ideals I of A containing a and for all i, j,
βijA (A/I ) ≤ βijA (A/(L + (X1d1 , . . . , Xrdr ))).
the LPP conjecture
Let a = (f) = (f1 , . . . , fr ), where f1 , . . . , fr is a regular sequence with deg fi = di .
LPP Conjecture (Evans ’02)
If f satisfies EGH, then for all homogeneous ideals I of A containing a and for all i, j,
βijA (A/I ) ≤ βijA (A/(L + (X1d1 , . . . , Xrdr ))).
the LPP conjecture
Let a = (f) = (f1 , . . . , fr ), where f1 , . . . , fr is a regular sequence with deg fi = di .
LPP Conjecture (Evans ’02)
If f satisfies EGH, then for all homogeneous ideals I of A containing a and for all i, j,
βijA (A/I ) ≤ βijA (A/(L + (X1d1 , . . . , Xrdr ))).
Results:
• (Mermin-Peeva-Stillman, ’08) when f is monomial, deg fi = 2 for all i.
• (Mermin-Murai, ’11) when f is monomial.
the LPP conjecture
Let a = (f) = (f1 , . . . , fr ), where f1 , . . . , fr is a regular sequence with deg fi = di .
LPP Conjecture (Evans ’02)
If f satisfies EGH, then for all homogeneous ideals I of A containing a and for all i,
d1
A
dr
Hilb TorA
i (A/I , K ) ≤ Hilb Tori (A/(L + (X1 , . . . , Xr )), K ) .
Results:
• (Mermin-Peeva-Stillman, ’08) when f is monomial, deg fi = 2 for all i.
• (Mermin-Murai, ’11) when f is monomial.
Our question:
the LPP conjecture
Let a = (f) = (f1 , . . . , fr ), where f1 , . . . , fr is a regular sequence with deg fi = di .
LPP Conjecture (Evans ’02)
If f satisfies EGH, then for all homogeneous ideals I of A containing a and for all i,
d1
A
dr
Hilb TorA
i (A/I , K ) ≤ Hilb Tori (A/(L + (X1 , . . . , Xr )), K ) .
Results:
• (Mermin-Peeva-Stillman, ’08) when f is monomial, deg fi = 2 for all i.
• (Mermin-Murai, ’11) when f is monomial.
Our question:
the LPP conjecture
Let a = (f) = (f1 , . . . , fr ), where f1 , . . . , fr is a regular sequence with deg fi = di .
LPP Conjecture (Evans ’02)
If f satisfies EGH, then for all homogeneous ideals I of A containing a and for all i,
d1
A
dr
Hilb TorA
i (A/I , K ) ≤ Hilb Tori (A/(L + (X1 , . . . , Xr )), K ) .
Results:
• (Mermin-Peeva-Stillman, ’08) when f is monomial, deg fi = 2 for all i.
• (Mermin-Murai, ’11) when f is monomial.
Our question:
If f satisfies EGH, is it true that the analogous inequality holds for the Hilbert series of the
respective local cohomology modules?
Toolkit:
I
embeddings of posets of Hilbert functions;
I
Z -stable ideals;
I
distractions;
Toolkit:
I
embeddings of posets of Hilbert functions;
I
Z -stable ideals;
I
distractions;
I
relaxing BnB in Tuscany.
Embeddings of Hilbert functions.
Let B = K [X1 , . . . , Xn ], b ⊆ B and S = B/b.
Embeddings of Hilbert functions.
Let B = K [X1 , . . . , Xn ], b ⊆ B and S = B/b.
Let I S = {J : J is a homogeneous S-ideal}, ⊆
Let HS = {Hilb (J)• : J ∈ I S }, ≤
Embeddings of Hilbert functions.
Let B = K [X1 , . . . , Xn ], b ⊆ B and S = B/b.
Let I S = {J : J is a homogeneous S-ideal}, ⊆
Let HS = {Hilb (J)• : J ∈ I S }, ≤
Definition
An embedding of S is an order preserving injection : HS −→ I S such that, if (H) = J, then
Hilb(J)• = H.
Notation: (S, ).
Also, for I ∈ I S , we let (I ) := (Hilb(I )• ).
Embeddings of Hilbert functions.
Let B = K [X1 , . . . , Xn ], b ⊆ B and S = B/b.
Let I S = {J : J is a homogeneous S-ideal}, ⊆
Let HS = {Hilb (J)• : J ∈ I S }, ≤
Definition
An embedding of S is an order preserving injection : HS −→ I S such that, if (H) = J, then
Hilb(J)• = H.
Notation: (S, ).
Also, for I ∈ I S , we let (I ) := (Hilb(I )• ).
Definition
A K -algebra R embeds into (S, ) if HR ⊆ HS .
Notation: (R, S, ).
Clearly, (S, ) embeds into itself and if R embeds into (S, ) then Hilb (R) = Hilb (S).
Embeddings of Hilbert functions.
The notion of embedding captures the key property of rings for which an analogous of
Macaulay Theorem holds.
Examples
(i) Let S = B and define : HS −→ IS , (Hilb (I )• ) := L.
(ii) Let S = B/b, where b = (X1d1 , . . . , Xrdr ) and define (Hilb (I )• ) := LS.
We denote this embedding by CL .
(iii) Let m ∈ N>0 , and let S be the mth-Veronese subring ofB. By GMP,
given I ∈ I S ,
L
there exists a lex-ideal L ⊆ B such that Hilb (I ) = Hilb
d≥0 Lmd .
L
Define (Hilb (I )• ) := d≥0 Lmd .
Embeddings of Hilbert functions.
The notion of embedding captures the key property of rings for which an analogous of
Macaulay Theorem holds.
Examples
(i) Let S = B and define : HS −→ IS , (Hilb (I )• ) := L.
(ii) Let S = B/b, where b = (X1d1 , . . . , Xrdr ) and define (Hilb (I )• ) := LS.
We denote this embedding by CL .
(iii) Let m ∈ N>0 , and let S be the mth-Veronese subring ofB. By GMP,
given I ∈ I S ,
L
there exists a lex-ideal L ⊆ B such that Hilb (I ) = Hilb
d≥0 Lmd .
L
Define (Hilb (I )• ) := d≥0 Lmd .
Re-formulation of the conjectures in terms of embeddings
Let A, a = (f) and R be as before. Let B, b and S be as in Ex. (ii).
• EGH: R embeds into (S, CL ).
• LPP: Assume that f satisfies EGH. Then, for all homogeneous ideals I of R and for all i, j,
βij (R/I ) ≤ βij (S/CL (I )).
Embeddings of Hilbert functions.
We say that (R, S, ) is (local) cohomology extremal if for all homogeneous ideals I of R and
all i
i
i
Hilb Hm
(R/I ) ≤ Hilb Hm
(S/(I )) .
R
S
Embeddings of Hilbert functions.
We say that (R, S, ) is (local) cohomology extremal if for all homogeneous ideals I of R and
all i
i
i
Hilb Hm
(R/I ) ≤ Hilb Hm
(S/(I )) .
R
S
Main Theorem (C.-S., ’15)
If (R, S, ) is cohomology extremal then (R[Z ], S[Z ], 1 ) is cohomology extremal.
Embeddings of Hilbert functions.
We say that (R, S, ) is (local) cohomology extremal if for all homogeneous ideals I of R and
all i
i
i
Hilb Hm
(R/I ) ≤ Hilb Hm
(S/(I )) .
R
S
Main Theorem (C.-S., ’15)
If (R, S, ) is cohomology extremal then (R[Z ], S[Z ], 1 ) is cohomology extremal.
1 : HS[Z ] −→ IS[Z ] embedding such that
1 (I ) = ⊕h Jhhi Z h is a Z -stable ideal with (Jhhi ) = Jhhi .
results
results
Corollary 1 (C.-S.)
Let m ∈ N+ . If (R, S, ) is cohomology extremal then (R[Z1 , . . . , Zm ], S[Z1 , . . . , Zm ], m ) is
cohomology extremal.
results
Corollary 1 (C.-S.)
Let m ∈ N+ . If (R, S, ) is cohomology extremal then (R[Z1 , . . . , Zm ], S[Z1 , . . . , Zm ], m ) is
cohomology extremal.
The analogue for local cohomology of MM inequality:
Corollary 2 (C.-S.)
Let A = K [X1 , . . . , Xn ], a = (X1d1 , . . . , Xrdr ). Let I ⊆ A be a homogeneous ideal containing a
and let L + a be the lex-plus-powers ideal associated to I with respect to d1 , . . . , dr . Then,
i
i
Hilb Hm
(A/I ) ≤ Hilb Hm
(A/L + a) , for all i.
results
Corollary 1 (C.-S.)
Let m ∈ N+ . If (R, S, ) is cohomology extremal then (R[Z1 , . . . , Zm ], S[Z1 , . . . , Zm ], m ) is
cohomology extremal.
The analogue for local cohomology of MM inequality:
Corollary 2 (C.-S.)
Let A = K [X1 , . . . , Xn ], a = (X1d1 , . . . , Xrdr ). Let I ⊆ A be a homogeneous ideal containing a
and let L + a be the lex-plus-powers ideal associated to I with respect to d1 , . . . , dr . Then,
i
i
Hilb Hm
(A/I ) ≤ Hilb Hm
(A/L + a) , for all i.
The local cohomology analogue of the LPP conjecture holds true:
Theorem 3 (C.-S.)
Assume that f satisfies EGH. Then, for all homogeneous ideals I of R,
i
i
Hilb Hm
(R/I ) ≤ Hilb Hm
(S/CL (I )) , for all i.
R
S
In other words, (R, S, CL ) is cohomology extremal.
Back to the original LPP inequality.
Back to the original LPP inequality.
Recall the definition of extremal Betti number for a finitely generated graded module:
A non-zero βij (M) such that βrs (M) = 0 whenever r ≥ i, s ≥ j + 1 and s − r ≥ j − i is called
an extremal Betti number of M.
A pair of indexes (i, j − i) such that βij (M) is extremal is called a corner of M.
i
j−i
βij (M)
Back to the original LPP inequality.
Recall the definition of extremal Betti number for a finitely generated graded module:
A non-zero βij (M) such that βrs (M) = 0 whenever r ≥ i, s ≥ j + 1 and s − r ≥ j − i is called
an extremal Betti number of M.
A pair of indexes (i, j − i) such that βij (M) is extremal is called a corner of M.
i
j−i
βij (M)
Known: If (i, j − i) is a corner of M, then
n−i
βij (M) = Hilb Hm
(M)
j−n
.
Back to the original LPP inequality.
Theorem 4 (C.-S.): LPP for extremal Betti numbers
Assume that f satisfies EGH. Then, for all homogeneous ideals I of R,
βij (R/I ) ≤ βij (S/CL (I )),
when (i, j − i) is a corner of S/CL (I ).
Back to the original LPP inequality.
Theorem 4 (C.-S.): LPP for extremal Betti numbers
Assume that f satisfies EGH. Then, for all homogeneous ideals I of R,
βij (R/I ) ≤ βij (S/CL (I )),
when (i, j − i) is a corner of S/CL (I ).
Under the same assumptions:
Theorem 5 (C.-S.): Inclusion of Betti regions
Let (i, j − i) be a corner of R/I . Then there exists a corner (i 0 , j 0 − i 0 ) of S/CL (I ) such that
i ≤ i 0 and j − i ≤ j 0 − i 0 .
i
i’
j−i
βij(R/I)
j’−i’
βij’’ (S/εCL(I))
essential bibliography
G. Caviglia and M. Kummini: Poset embeddings of Hilbert functions and Betti numbers. J.
Algebra 410, 244-257 (2014)
G. Caviglia and E. Sbarra: The lex-plus-power inequality for local cohomology modules.
Math. Annalen 2015. doi: 10.1007/s00208-015-1180-5
D. Eisenbud, M. Green and J. Harris: Higher Castelnuovo theory. In: Journées de Gèométrie
Algébrique d’Orsay (Orsay, 1992). Astérisque, no. 218, pp. 187-202 (1993)
C. A. Francisco and B. P. Richert: Lex-plus-powers ideals. In: Syzygies and Hilbert Functions.
Lect. Notes Pure Appl. Math. 254, pp. 113-144. Chapman & Hall/CRC, Boca Raton (2007)
J. Mermin and S. Murai: The lex-plus-powers conjecture holds for pure powers. Adv. Math.
226(4), 3511-3539 (2011)
Thank you!