Generalizing the Borel condition
Chris Francisco
Oklahoma State University
Joint work with Jeff Mermin and Jay Schweig
Lincoln, NE
October 2011
Motivation: Borel ideals
Let S = k [x1 , . . . , xn ], k a field.
Motivation: Borel ideals
Let S = k [x1 , . . . , xn ], k a field.
Definition: A monomial ideal M ⊂ S is a Borel ideal if
Motivation: Borel ideals
Let S = k [x1 , . . . , xn ], k a field.
Definition: A monomial ideal M ⊂ S is a Borel ideal if
I
given any monomial m ∈ M,
I
a variable xj dividing m, and
I
an index i < j,
Motivation: Borel ideals
Let S = k [x1 , . . . , xn ], k a field.
Definition: A monomial ideal M ⊂ S is a Borel ideal if
I
given any monomial m ∈ M,
I
a variable xj dividing m, and
an index i < j,
xi
then m · ∈ M.
xj
I
Motivation: Borel ideals
Let S = k [x1 , . . . , xn ], k a field.
Definition: A monomial ideal M ⊂ S is a Borel ideal if
I
given any monomial m ∈ M,
I
a variable xj dividing m, and
an index i < j,
xi
then m · ∈ M.
xj
I
Also known as strongly stable or 0-Borel ideals.
Q-Borel ideals
Let Q be a naturally-labeled poset on {x1 , . . . , xn }.
(So xi <Q xj implies i < j.)
Definition: A monomial ideal M ⊂ S is a Q-Borel ideal if
Q-Borel ideals
Let Q be a naturally-labeled poset on {x1 , . . . , xn }.
(So xi <Q xj implies i < j.)
Definition: A monomial ideal M ⊂ S is a Q-Borel ideal if
I
given any monomial m ∈ M,
I
a variable xj dividing m, and
Q-Borel ideals
Let Q be a naturally-labeled poset on {x1 , . . . , xn }.
(So xi <Q xj implies i < j.)
Definition: A monomial ideal M ⊂ S is a Q-Borel ideal if
I
given any monomial m ∈ M,
I
a variable xj dividing m, and
I
an index i < j
Q-Borel ideals
Let Q be a naturally-labeled poset on {x1 , . . . , xn }.
(So xi <Q xj implies i < j.)
Definition: A monomial ideal M ⊂ S is a Q-Borel ideal if
I
given any monomial m ∈ M,
I
a variable xj dividing m, and
I
an index i < j
a variable xi <Q xj ,
Q-Borel ideals
Let Q be a naturally-labeled poset on {x1 , . . . , xn }.
(So xi <Q xj implies i < j.)
Definition: A monomial ideal M ⊂ S is a Q-Borel ideal if
I
given any monomial m ∈ M,
I
a variable xj dividing m, and
I
an index i < j
then m ·
xi
∈ M.
xj
a variable xi <Q xj ,
Q-Borel example
Let Q be the poset with relations a <Q b and a <Q c.
Q-Borel example
Let Q be the poset with relations a <Q b and a <Q c.
bt
@
@
ct
@t
a
Q-Borel example
Let Q be the poset with relations a <Q b and a <Q c.
bt
@
@
ct
@t
a
Let I = Q(bc), the smallest Q-Borel ideal containing bc.
Q-Borel example
Let Q be the poset with relations a <Q b and a <Q c.
bt
@
@
ct
@t
a
Let I = Q(bc), the smallest Q-Borel ideal containing bc.
Monomials in I: bc,
Q-Borel example
Let Q be the poset with relations a <Q b and a <Q c.
bt
@
@
ct
@t
a
Let I = Q(bc), the smallest Q-Borel ideal containing bc.
Monomials in I: bc, ac (b → a),
Q-Borel example
Let Q be the poset with relations a <Q b and a <Q c.
bt
@
@
ct
@t
a
Let I = Q(bc), the smallest Q-Borel ideal containing bc.
Monomials in I: bc, ac (b → a), ab (c → a),
Q-Borel example
Let Q be the poset with relations a <Q b and a <Q c.
bt
@
@
ct
@t
a
Let I = Q(bc), the smallest Q-Borel ideal containing bc.
Monomials in I: bc, ac (b → a), ab (c → a), a2 (b → a, c → a).
Q-Borel example
Let Q be the poset with relations a <Q b and a <Q c.
bt
@
@
ct
@t
a
Let I = Q(bc), the smallest Q-Borel ideal containing bc.
Monomials in I: bc, ac (b → a), ab (c → a), a2 (b → a, c → a).
So I = (a2 , ab, ac, bc).
Q-Borel example
Let Q be the poset with relations a <Q b and a <Q c.
bt
@
@
ct
@t
a
Let I = Q(bc), the smallest Q-Borel ideal containing bc.
Monomials in I: bc, ac (b → a), ab (c → a), a2 (b → a, c → a).
So I = (a2 , ab, ac, bc).
This is not an ordinary Borel ideal because b2 ∈
/ I (c 6→ b).
Extremal cases
Chain C of length n
Antichain A
t
xn
t
.. xn−1
.
tx
2
t
x1
t
t
x1 x2
...
t
xn
Extremal cases
Chain C of length n
Antichain A
t
xn
t
.. xn−1
.
tx
2
t
t
x1 x2
t
x1
I
C-Borel ideals are the usual Borel ideals.
...
t
xn
Extremal cases
Chain C of length n
Antichain A
t
xn
t
.. xn−1
.
tx
2
t
t
x1 x2
t
x1
I
C-Borel ideals are the usual Borel ideals.
I
Every monomial ideal is A-Borel.
...
t
xn
Extremal cases
Chain C of length n
Antichain A
t
xn
t
.. xn−1
.
tx
2
t
t
x1 x2
...
t
xn
t
x1
I
C-Borel ideals are the usual Borel ideals.
I
Every monomial ideal is A-Borel.
Guiding idea: The closer Q is to C, the more a Q-Borel ideal
should behave like a Borel ideal.
Associated primes of Q-Borel ideals
Borel ideals (Bayer-Stillman): If B is a Borel ideal, then any
associated prime of S/B is of the form (x1 , x2 , . . . , xi ).
Associated primes of Q-Borel ideals
Borel ideals (Bayer-Stillman): If B is a Borel ideal, then any
associated prime of S/B is of the form (x1 , x2 , . . . , xi ).
Q-Borel ideals: If I is a Q-Borel ideal, and p ∈ Ass(S/I), then p
is generated by an order ideal in Q.
Associated primes of Q-Borel ideals
Borel ideals (Bayer-Stillman): If B is a Borel ideal, then any
associated prime of S/B is of the form (x1 , x2 , . . . , xi ).
Q-Borel ideals: If I is a Q-Borel ideal, and p ∈ Ass(S/I), then p
is generated by an order ideal in Q.
Proof: Say m ∈
/ I but xj m ∈ I. Then for any xi <Q xj , xi m ∈ I as
well.
Associated primes of Q-Borel ideals
Borel ideals (Bayer-Stillman): If B is a Borel ideal, then any
associated prime of S/B is of the form (x1 , x2 , . . . , xi ).
Q-Borel ideals: If I is a Q-Borel ideal, and p ∈ Ass(S/I), then p
is generated by an order ideal in Q.
Proof: Say m ∈
/ I but xj m ∈ I. Then for any xi <Q xj , xi m ∈ I as
well.
Goal: Compute irredundant primary decomposition of Q-Borel
ideals from Q-Borel generators and poset structure of Q.
Associated primes of Q-Borel ideals
Borel ideals (Bayer-Stillman): If B is a Borel ideal, then any
associated prime of S/B is of the form (x1 , x2 , . . . , xi ).
Q-Borel ideals: If I is a Q-Borel ideal, and p ∈ Ass(S/I), then p
is generated by an order ideal in Q.
Proof: Say m ∈
/ I but xj m ∈ I. Then for any xi <Q xj , xi m ∈ I as
well.
Goal: Compute irredundant primary decomposition of Q-Borel
ideals from Q-Borel generators and poset structure of Q.
Special case: Principal Q-Borel ideals, I = Q(m).
Principal Q-Borel ideals
Principal Q-Borel ideals are the products of monomial primes.
Principal Q-Borel ideals
Principal Q-Borel ideals are the products of monomial primes.
Theorem A: Suppose
Y
I=
pep ,
p⊂S
where the p are all monomial primes of S, and ep ≥ 0. Then
\
I=
pap ,
p⊂S
where
ap =
X
p0 ⊂p
ep0 .
Principal Q-Borel ideals
Principal Q-Borel ideals are the products of monomial primes.
Theorem A: Suppose
Y
I=
pep ,
p⊂S
where the p are all monomial primes of S, and ep ≥ 0. Then
\
I=
pap ,
p⊂S
where
ap =
X
ep0 .
p0 ⊂p
Get a primary decomposition consisting of powers of monomial
primes.
Irredundant primary decomposition
For a monomial m0 , let A(m0 ) = {xi : xi ≤Q xj for some xj | m0 }.
In English: Variables below any element of supp(m0 ).
Irredundant primary decomposition
For a monomial m0 , let A(m0 ) = {xi : xi ≤Q xj for some xj | m0 }.
In English: Variables below any element of supp(m0 ).
Theorem B: Let I = Q(m). Let p be a prime ideal. Then
p ∈ Ass(S/I) if and only if
I
Gens(p) = A(m0 ) for some monomial m0 | m, and
I
A(m0 ) is connected.
Irredundant primary decomposition
For a monomial m0 , let A(m0 ) = {xi : xi ≤Q xj for some xj | m0 }.
In English: Variables below any element of supp(m0 ).
Theorem B: Let I = Q(m). Let p be a prime ideal. Then
p ∈ Ass(S/I) if and only if
I
Gens(p) = A(m0 ) for some monomial m0 | m, and
I
A(m0 ) is connected.
With Theorem A, this gives an irredundant primary
decomposition of I = Q(m).
Irredundant primary decomposition
For a monomial m0 , let A(m0 ) = {xi : xi ≤Q xj for some xj | m0 }.
In English: Variables below any element of supp(m0 ).
Theorem B: Let I = Q(m). Let p be a prime ideal. Then
p ∈ Ass(S/I) if and only if
I
Gens(p) = A(m0 ) for some monomial m0 | m, and
I
A(m0 ) is connected.
With Theorem A, this gives an irredundant primary
decomposition of I = Q(m).
Method: Compute all order ideals corresponding to divisors of
m. For the connected ones, use Theorem A to compute
exponents.
Principal Q-Borel example
e
Poset Q:
t
b
t
@@t
c
f
t
@@t
d
ta
Principal Q-Borel example
e
Poset Q:
t
b
t
@@t
c
f
t
@@t
d
ta
I = Q(def ) = (d, a)1 (e, b, c)1 (f , c, d, a)1
Principal Q-Borel example
e
Poset Q:
t
b
t
@@t
c
f
t
@@t
d
ta
I = Q(def ) = (d, a)1 (e, b, c)1 (f , c, d, a)1
Candidates for primes
I
A(d) ↔ (d, a) connected
Exponent: 1 from A(d)
Principal Q-Borel example
e
Poset Q:
t
b
t
@@t
c
f
t
@@t
d
ta
I = Q(def ) = (d, a)1 (e, b, c)1 (f , c, d, a)1
Candidates for primes
I
A(d) ↔ (d, a) connected
I
A(e) ↔ (e, b, c) connected
Exponent: 1 from A(d)
Exponent: 1 from A(e)
Principal Q-Borel example
e
Poset Q:
t
b
t
@@t
c
f
t
@@t
d
ta
I = Q(def ) = (d, a)1 (e, b, c)1 (f , c, d, a)1
Candidates for primes
I
A(d) ↔ (d, a) connected
I
A(e) ↔ (e, b, c) connected
I
A(f ) = A(df ) ↔ (f , c, d, a) connected
from A(f ), A(d)
Exponent: 1 from A(d)
Exponent: 1 from A(e)
Exponent: 1 each
Principal Q-Borel example
e
Poset Q:
t
b
t
@@t
c
f
t
@@t
d
ta
I = Q(def ) = (d, a)1 (e, b, c)1 (f , c, d, a)1
Candidates for primes
I
A(d) ↔ (d, a) connected
I
A(e) ↔ (e, b, c) connected
I
A(f ) = A(df ) ↔ (f , c, d, a) connected
from A(f ), A(d)
I
A(de) ↔ (d, a, e, b, c) not connected
Exponent: 1 from A(d)
Exponent: 1 from A(e)
Exponent: 1 each
Principal Q-Borel example
e
Poset Q:
t
b
t
@@t
c
f
t
@@t
d
ta
I = Q(def ) = (d, a)1 (e, b, c)1 (f , c, d, a)1
Candidates for primes
I
A(d) ↔ (d, a) connected
I
A(e) ↔ (e, b, c) connected
I
A(f ) = A(df ) ↔ (f , c, d, a) connected
from A(f ), A(d)
I
A(de) ↔ (d, a, e, b, c) not connected
I
A(ef ) = A(def ) ↔ (f , c, d, a, e, b) connected
each from A(d), A(e), A(f )
Exponent: 1 from A(d)
Exponent: 1 from A(e)
Exponent: 1 each
Exponent: 1
Principal Q-Borel example
e
Poset Q:
t
b
t
@@t
c
f
t
@@t
d
ta
I = Q(def ) = (d, a)1 (e, b, c)1 (f , c, d, a)1
Candidates for primes
I
A(d) ↔ (d, a) connected
I
A(e) ↔ (e, b, c) connected
I
A(f ) = A(df ) ↔ (f , c, d, a) connected
from A(f ), A(d)
I
A(de) ↔ (d, a, e, b, c) not connected
I
A(ef ) = A(def ) ↔ (f , c, d, a, e, b) connected
each from A(d), A(e), A(f )
Exponent: 1 from A(d)
Exponent: 1 from A(e)
Exponent: 1 each
I = (d, a) ∩ (e, b, c) ∩ (f , c, d, a)2 ∩ (f , c, d, a, e, b)3
Exponent: 1
Resolutions of principal Q-Borels
Theorem (Conca-Herzog): Principal Q-Borel ideals are
polymatroidal and thus have linear resolution.
Resolutions of principal Q-Borels
Theorem (Conca-Herzog): Principal Q-Borel ideals are
polymatroidal and thus have linear resolution.
Theorem: I = Q(m), Q maximal poset stabilizing I, and all
maximal elements of Q divide m. Then
pd(S/I) = n − #(connected components of Q) + 1.
Resolutions of principal Q-Borels
Theorem (Conca-Herzog): Principal Q-Borel ideals are
polymatroidal and thus have linear resolution.
Theorem: I = Q(m), Q maximal poset stabilizing I, and all
maximal elements of Q divide m. Then
pd(S/I) = n − #(connected components of Q) + 1.
Theorem: If I = Q(m), codim I = min |A(xi )|.
xi |m
Resolutions of principal Q-Borels
Theorem (Conca-Herzog): Principal Q-Borel ideals are
polymatroidal and thus have linear resolution.
Theorem: I = Q(m), Q maximal poset stabilizing I, and all
maximal elements of Q divide m. Then
pd(S/I) = n − #(connected components of Q) + 1.
Theorem: If I = Q(m), codim I = min |A(xi )|.
xi |m
Under above hypotheses, recover part of a Herzog-Hibi result:
Corollary: I = Q(m) Cohen-Macaulay if and only if
I
Q is the chain, m = xnan (I = man ), or
I
Q is the antichain (I is a principal ideal)
Y -Borel ideals
Let Y be the poset:
t
t
y @
z
@tx
t
t
..
.
xt−1
tx
2
tx
1
Y -Borel ideals
Let Y be the poset:
t
t
y @
z
@tx
t
t
..
.
xt−1
tx
2
tx
1
“Close” to the chain C, but Y -Borel ideals may not be
componentwise linear.
Minimal free resolution of Y -Borel ideal I
Basis: Throughout, m is a minimal generator, α a squarefree
monomial in k [x1 , . . . , xt ] with max α < max m.
Minimal free resolution of Y -Borel ideal I
Basis: Throughout, m is a minimal generator, α a squarefree
monomial in k [x1 , . . . , xt ] with max α < max m.
Two forms of symbols:
I
Eliahou-Kervaire: [m, α]
Minimal free resolution of Y -Borel ideal I
Basis: Throughout, m is a minimal generator, α a squarefree
monomial in k [x1 , . . . , xt ] with max α < max m.
Two forms of symbols:
I
Eliahou-Kervaire: [m, α]
I
I
homological degree deg α
multidegree mα
Minimal free resolution of Y -Borel ideal I
Basis: Throughout, m is a minimal generator, α a squarefree
monomial in k [x1 , . . . , xt ] with max α < max m.
Two forms of symbols:
I
Eliahou-Kervaire: [m, α]
I
I
I
homological degree deg α
multidegree mα
Other symbols: [m, α · y rm ]
Minimal free resolution of Y -Borel ideal I
Basis: Throughout, m is a minimal generator, α a squarefree
monomial in k [x1 , . . . , xt ] with max α < max m.
Two forms of symbols:
I
Eliahou-Kervaire: [m, α]
I
I
I
homological degree deg α
multidegree mα
Other symbols: [m, α · y rm ]
I
I
I
homological degree 1 + deg α
multidegree mαy rm
rm
rm minimal such that m · yz ∈ I.
Minimal free resolution of Y -Borel ideal I
Basis: Throughout, m is a minimal generator, α a squarefree
monomial in k [x1 , . . . , xt ] with max α < max m.
Two forms of symbols:
I
Eliahou-Kervaire: [m, α]
I
I
I
homological degree deg α
multidegree mα
Other symbols: [m, α · y rm ]
I
I
I
homological degree 1 + deg α
multidegree mαy rm
rm
rm minimal such that m · yz ∈ I.
Induction using Mayer-Vietoris.
If z divides no generator, ideal is Borel in k [x1 , . . . , xt , y ].
Y -Borel example
S = k [x1 , x2 , y , z], I = Y (x1 , y 2 , z 2 )
Y -Borel example
S = k [x1 , x2 , y , z], I = Y (x1 , y 2 , z 2 ) = (x1 , x22 , x2 y , x2 z, y 2 , z 2 ).
Y -Borel example
S = k [x1 , x2 , y , z], I = Y (x1 , y 2 , z 2 ) = (x1 , x22 , x2 y , x2 z, y 2 , z 2 ).
total:
0:
1:
2:
1
1
.
.
6
1
5
.
11
.
10
1
8
.
6
2
2
.
1
1
(Betti diagram of S/I)
Y -Borel example
S = k [x1 , x2 , y , z], I = Y (x1 , y 2 , z 2 ) = (x1 , x22 , x2 y , x2 z, y 2 , z 2 ).
total:
0:
1:
2:
1
1
.
.
6
1
5
.
11
.
10
1
8
.
6
2
2
.
1
1
(Betti diagram of S/I)
First syzygies
Eliahou-Kervaire symbols: [x22 , x1 ], [x2 y , x1 ], [x2 y , x2 ], [x2 z, x1 ],
[x2 z, x2 ], [y 2 , x1 ], [y 2 , x2 ], [z 2 , x1 ], [z 2 , x2 ]
Y -Borel example
S = k [x1 , x2 , y , z], I = Y (x1 , y 2 , z 2 ) = (x1 , x22 , x2 y , x2 z, y 2 , z 2 ).
total:
0:
1:
2:
1
1
.
.
6
1
5
.
11
.
10
1
8
.
6
2
2
.
1
1
(Betti diagram of S/I)
First syzygies
Eliahou-Kervaire symbols: [x22 , x1 ], [x2 y , x1 ], [x2 y , x2 ], [x2 z, x1 ],
[x2 z, x2 ], [y 2 , x1 ], [y 2 , x2 ], [z 2 , x1 ], [z 2 , x2 ]
Other symbols: [x2 z, 1 · y ] ,
| {z }
usual EK symbol
Y -Borel example
S = k [x1 , x2 , y , z], I = Y (x1 , y 2 , z 2 ) = (x1 , x22 , x2 y , x2 z, y 2 , z 2 ).
total:
0:
1:
2:
1
1
.
.
6
1
5
.
11
.
10
1
8
.
6
2
2
.
1
1
(Betti diagram of S/I)
First syzygies
Eliahou-Kervaire symbols: [x22 , x1 ], [x2 y , x1 ], [x2 y , x2 ], [x2 z, x1 ],
[x2 z, x2 ], [y 2 , x1 ], [y 2 , x2 ], [z 2 , x1 ], [z 2 , x2 ]
Other symbols: [x2 z, 1 · y ] , [z 2 , 1 · y 2 ] (multidegree y 2 z 2 )
| {z }
usual EK symbol