Rees algebras of square-free
monomial ideals
Louiza Fouli
New Mexico State University
University of Nebraska
AMS Central Section Meeting
October 15, 2011
Rees algebras
This is joint work with Kuei-Nuan Lin.
Rees algebras
This is joint work with Kuei-Nuan Lin.
Let R be a Noetherian ring and I an ideal.
Rees algebras
This is joint work with Kuei-Nuan Lin.
Let R be a Noetherian ring and I an ideal.
The Rees algebra of I is R(I) = R[It] = ⊕ I i t i ⊂ R[t].
i≥0
Rees algebras
This is joint work with Kuei-Nuan Lin.
Let R be a Noetherian ring and I an ideal.
The Rees algebra of I is R(I) = R[It] = ⊕ I i t i ⊂ R[t].
i≥0
The Rees algebra is the algebraic realization of the
blowup of a variety along a subvariety.
Rees algebras
This is joint work with Kuei-Nuan Lin.
Let R be a Noetherian ring and I an ideal.
The Rees algebra of I is R(I) = R[It] = ⊕ I i t i ⊂ R[t].
i≥0
The Rees algebra is the algebraic realization of the
blowup of a variety along a subvariety.
It is an important tool in the birational study of
algebraic varieties, particularly in the study of
desingularization.
Rees algebras
This is joint work with Kuei-Nuan Lin.
Let R be a Noetherian ring and I an ideal.
The Rees algebra of I is R(I) = R[It] = ⊕ I i t i ⊂ R[t].
i≥0
The Rees algebra is the algebraic realization of the
blowup of a variety along a subvariety.
It is an important tool in the birational study of
algebraic varieties, particularly in the study of
desingularization.
The Rees algebra facilitates the study of the
asymptotic behavior of I.
Rees algebras
This is joint work with Kuei-Nuan Lin.
Let R be a Noetherian ring and I an ideal.
The Rees algebra of I is R(I) = R[It] = ⊕ I i t i ⊂ R[t].
i≥0
The Rees algebra is the algebraic realization of the
blowup of a variety along a subvariety.
It is an important tool in the birational study of
algebraic varieties, particularly in the study of
desingularization.
The Rees algebra facilitates the study of the
asymptotic behavior of I.
Integral Closure: R(I) = ⊕ I i t i = R(I).
i≥0
Rees algebras
This is joint work with Kuei-Nuan Lin.
Let R be a Noetherian ring and I an ideal.
The Rees algebra of I is R(I) = R[It] = ⊕ I i t i ⊂ R[t].
i≥0
The Rees algebra is the algebraic realization of the
blowup of a variety along a subvariety.
It is an important tool in the birational study of
algebraic varieties, particularly in the study of
desingularization.
The Rees algebra facilitates the study of the
asymptotic behavior of I.
Integral Closure: R(I) = ⊕ I i t i = R(I). So I = [R(I)]1 .
i≥0
An alternate description
We will consider the following construction for the Rees
algebra
An alternate description
We will consider the following construction for the Rees
algebra
Let I = (f1 , . . . , fn ), S = R[T1 , . . . , Tn ], where Ti are
indeterminates.
An alternate description
We will consider the following construction for the Rees
algebra
Let I = (f1 , . . . , fn ), S = R[T1 , . . . , Tn ], where Ti are
indeterminates.
There is a natural map φ : S −→ R(I) that sends Ti to
fi t.
An alternate description
We will consider the following construction for the Rees
algebra
Let I = (f1 , . . . , fn ), S = R[T1 , . . . , Tn ], where Ti are
indeterminates.
There is a natural map φ : S −→ R(I) that sends Ti to
fi t.
Then R(I) ≃ S/ ker φ.
An alternate description
We will consider the following construction for the Rees
algebra
Let I = (f1 , . . . , fn ), S = R[T1 , . . . , Tn ], where Ti are
indeterminates.
There is a natural map φ : S −→ R(I) that sends Ti to
fi t.
Then R(I) ≃ S/ ker φ. Let J = ker φ.
An alternate description
We will consider the following construction for the Rees
algebra
Let I = (f1 , . . . , fn ), S = R[T1 , . . . , Tn ], where Ti are
indeterminates.
There is a natural map φ : S −→ R(I) that sends Ti to
fi t.
Then R(I) ≃ S/ ker φ. Let J = ker φ.
∞
L
Ji is a graded ideal.
Then J =
i=1
An alternate description
We will consider the following construction for the Rees
algebra
Let I = (f1 , . . . , fn ), S = R[T1 , . . . , Tn ], where Ti are
indeterminates.
There is a natural map φ : S −→ R(I) that sends Ti to
fi t.
Then R(I) ≃ S/ ker φ. Let J = ker φ.
∞
L
Ji is a graded ideal. A minimal
Then J =
i=1
generating set of J is often referred to as the defining
equations of the Rees algebra.
Example in degree 2
Example
Let R = k [x1 , x2 , x3 , x4 ] and I = (x1 x2 , x2 x3 , x3 x4 , x1 x4 ).
Example in degree 2
Example
Let R = k [x1 , x2 , x3 , x4 ] and I = (x1 x2 , x2 x3 , x3 x4 , x1 x4 ).
Then R(I) ≃ R[T1 , T2 , T3 , T4 ]/J and J is minimally
generated by
Example in degree 2
Example
Let R = k [x1 , x2 , x3 , x4 ] and I = (x1 x2 , x2 x3 , x3 x4 , x1 x4 ).
Then R(I) ≃ R[T1 , T2 , T3 , T4 ]/J and J is minimally
generated by
x 3 T1 − x 1 T2 , x 4 T4 − x 2 T3 , x 1 T3 − x 3 T4 ,
x 4 T1 − x 2 T4 , T1 T3 − T2 T4 .
Example in degree 2
Example
Let R = k [x1 , x2 , x3 , x4 ] and I = (x1 x2 , x2 x3 , x3 x4 , x1 x4 ).
Then R(I) ≃ R[T1 , T2 , T3 , T4 ]/J and J is minimally
generated by
x 3 T1 − x 1 T2 , x 4 T4 − x 2 T3 , x 1 T3 − x 3 T4 ,
x 4 T1 − x 2 T4 , T1 T3 − T2 T4 .
I is the edge ideal of a graph:
Example in degree 2
Example
Let R = k [x1 , x2 , x3 , x4 ] and I = (x1 x2 , x2 x3 , x3 x4 , x1 x4 ).
Then R(I) ≃ R[T1 , T2 , T3 , T4 ]/J and J is minimally
generated by
x 3 T1 − x 1 T2 , x 4 T4 − x 2 T3 , x 1 T3 − x 3 T4 ,
x 4 T1 − x 2 T4 , T1 T3 − T2 T4 .
I is the edge ideal of a graph:
x1
x2
x4
b
b
b
b
x3
Example in degree 2
Example
Let R = k [x1 , x2 , x3 , x4 ] and I = (x1 x2 , x2 x3 , x3 x4 , x1 x4 ).
Then R(I) ≃ R[T1 , T2 , T3 , T4 ]/J and J is minimally
generated by
x 3 T1 − x 1 T2 , x 4 T4 − x 2 T3 , x 1 T3 − x 3 T4 ,
x 4 T1 − x 2 T4 , T1 T3 − T2 T4 .
I is the edge ideal of a graph:
x1
x2
x4
b
b
b
b
x3
Notice that f1 f3 = f2 f4 = x1 x2 x3 x4 and that the degree
2 relation “comes” from the “even closed walk”, in this
case the square.
The degree 2 case
Theorem (Villarreal)
Let k be a field and let
I = (f1 , . . . , fn ) ⊂ R = k [x1 , . . . , xd ] be a square-free
monomial ideal generated in degree 2.
The degree 2 case
Theorem (Villarreal)
Let k be a field and let
I = (f1 , . . . , fn ) ⊂ R = k [x1 , . . . , xd ] be a square-free
monomial ideal generated in degree 2.
Let S = R[T1 , . . . , Tn ] and R(I) ≃ S/J.
The degree 2 case
Theorem (Villarreal)
Let k be a field and let
I = (f1 , . . . , fn ) ⊂ R = k [x1 , . . . , xd ] be a square-free
monomial ideal generated in degree 2.
Let S = R[T1 , . . . , Tn ] and R(I) ≃ S/J.
Let Is denote the set of all non-decreasing
sequences of integers α = (i1 , . . . , is ) of length s.
The degree 2 case
Theorem (Villarreal)
Let k be a field and let
I = (f1 , . . . , fn ) ⊂ R = k [x1 , . . . , xd ] be a square-free
monomial ideal generated in degree 2.
Let S = R[T1 , . . . , Tn ] and R(I) ≃ S/J.
Let Is denote the set of all non-decreasing
sequences of integers α = (i1 , . . . , is ) of length s.
Let fα = fi1 · · · fis ∈ I and Tα = Ti1 · · · Tis ∈ S.
The degree 2 case
Theorem (Villarreal)
Let k be a field and let
I = (f1 , . . . , fn ) ⊂ R = k [x1 , . . . , xd ] be a square-free
monomial ideal generated in degree 2.
Let S = R[T1 , . . . , Tn ] and R(I) ≃ S/J.
Let Is denote the set of all non-decreasing
sequences of integers α = (i1 , . . . , is ) of length s.
Let fα = fi1 · · · fis ∈ I and Tα = Ti1 · · · Tis ∈ S.
∞
S
Then J = SJ1 + S( Ps ), where
i=2
Ps = {Tα − Tβ | fα = fβ , for some α, β ∈ Is }
The degree 2 case
Theorem (Villarreal)
Let k be a field and let
I = (f1 , . . . , fn ) ⊂ R = k [x1 , . . . , xd ] be a square-free
monomial ideal generated in degree 2.
Let S = R[T1 , . . . , Tn ] and R(I) ≃ S/J.
Let Is denote the set of all non-decreasing
sequences of integers α = (i1 , . . . , is ) of length s.
Let fα = fi1 · · · fis ∈ I and Tα = Ti1 · · · Tis ∈ S.
∞
S
Then J = SJ1 + S( Ps ), where
i=2
Ps = {Tα − Tβ | fα = fβ , for some α, β ∈ Is }
= {Tα − Tβ | α, β form an even closed walk }
A generating set for the defining ideal
Theorem (D. Taylor)
Let R be a polynomial ring over a field and I be a
monomial ideal.
A generating set for the defining ideal
Theorem (D. Taylor)
Let R be a polynomial ring over a field and I be a
monomial ideal.
Let f1 , . . . , fn be a minimal monomial generating set of
I and let S = R[T1 , . . . , Tn ].
A generating set for the defining ideal
Theorem (D. Taylor)
Let R be a polynomial ring over a field and I be a
monomial ideal.
Let f1 , . . . , fn be a minimal monomial generating set of
I and let S = R[T1 , . . . , Tn ]. Let R(I) ≃ S/J.
A generating set for the defining ideal
Theorem (D. Taylor)
Let R be a polynomial ring over a field and I be a
monomial ideal.
Let f1 , . . . , fn be a minimal monomial generating set of
I and let S = R[T1 , . . . , Tn ]. Let R(I) ≃ S/J.
fβ
fα
Let Tα,β =
Tα −
Tβ , where
gcd(fα , fβ )
gcd(fα , fβ )
α, β ∈ Is .
A generating set for the defining ideal
Theorem (D. Taylor)
Let R be a polynomial ring over a field and I be a
monomial ideal.
Let f1 , . . . , fn be a minimal monomial generating set of
I and let S = R[T1 , . . . , Tn ]. Let R(I) ≃ S/J.
fβ
fα
Let Tα,β =
Tα −
Tβ , where
gcd(fα , fβ )
gcd(fα , fβ )
α, β ∈ Is .
∞
S
Then J = SJ1 + S · ( Ji ), where
i=2
Js = {Tα,β | α, β ∈ Is }.
A degree 3 example
Example (Villarreal)
Let R = k [x1 , . . . , x7 ] and let I be the ideal of R
generated by f1 = x1 x2 x3 , f2 = x2 x4 x5 , f3 = x5 x6 x7 ,
f 4 = x3 x6 x7 .
A degree 3 example
Example (Villarreal)
Let R = k [x1 , . . . , x7 ] and let I be the ideal of R
generated by f1 = x1 x2 x3 , f2 = x2 x4 x5 , f3 = x5 x6 x7 ,
f 4 = x3 x6 x7 .
Then the defining equations of R(I) are generated by
x 3 T3 − x 5 T4 , x 6 x 7 T1 − x 1 x 2 T4 , x 6 x 7 T2 − x 2 x 4 T3 ,
x 4 x 5 T1 − x 1 x 3 T2 , x 4 T1 T3 − x 1 T2 T4 .
A degree 3 example
Example (Villarreal)
Let R = k [x1 , . . . , x7 ] and let I be the ideal of R
generated by f1 = x1 x2 x3 , f2 = x2 x4 x5 , f3 = x5 x6 x7 ,
f 4 = x3 x6 x7 .
Then the defining equations of R(I) are generated by
x 3 T3 − x 5 T4 , x 6 x 7 T1 − x 1 x 2 T4 , x 6 x 7 T2 − x 2 x 4 T3 ,
x 4 x 5 T1 − x 1 x 3 T2 , x 4 T1 T3 − x 1 T2 T4 .
Notice that other than the linear relations there is also
a relation in degree 2.
The Generator Graph
Construction
Let R = k [x1 , . . . , xd ] be a polynomial ring over a field
k.
The Generator Graph
Construction
Let R = k [x1 , . . . , xd ] be a polynomial ring over a field
k.
Let I be a square-free monomial ideal in R generated
in the same degree.
The Generator Graph
Construction
Let R = k [x1 , . . . , xd ] be a polynomial ring over a field
k.
Let I be a square-free monomial ideal in R generated
in the same degree.
Let f1 , . . . , fn be a minimal monomial generating set of
I.
The Generator Graph
Construction
Let R = k [x1 , . . . , xd ] be a polynomial ring over a field
k.
Let I be a square-free monomial ideal in R generated
in the same degree.
Let f1 , . . . , fn be a minimal monomial generating set of
I.
e
We construct the following graph G(I):
The Generator Graph
Construction
Let R = k [x1 , . . . , xd ] be a polynomial ring over a field
k.
Let I be a square-free monomial ideal in R generated
in the same degree.
Let f1 , . . . , fn be a minimal monomial generating set of
I.
e
We construct the following graph G(I):
For each fi monomial generator of I we associate a
vertex yi .
The Generator Graph
Construction
Let R = k [x1 , . . . , xd ] be a polynomial ring over a field
k.
Let I be a square-free monomial ideal in R generated
in the same degree.
Let f1 , . . . , fn be a minimal monomial generating set of
I.
e
We construct the following graph G(I):
For each fi monomial generator of I we associate a
vertex yi .
When gcd(fi , fj ) 6= 1, then we connect the vertices yi
and yj and create the edge {yi , yj }.
The Generator Graph
Construction
Let R = k [x1 , . . . , xd ] be a polynomial ring over a field
k.
Let I be a square-free monomial ideal in R generated
in the same degree.
Let f1 , . . . , fn be a minimal monomial generating set of
I.
e
We construct the following graph G(I):
For each fi monomial generator of I we associate a
vertex yi .
When gcd(fi , fj ) 6= 1, then we connect the vertices yi
and yj and create the edge {yi , yj }.
e
We call the graph G(I)
the generator graph of I.
Degree 3 Example revisited
Example
Recall that R = k [x1 , . . . , x7 ] and I be the ideal of R
generated by f1 = x1 x2 x3 , f2 = x2 x4 x5 , f3 = x5 x6 x7 ,
f 4 = x3 x6 x7 .
Degree 3 Example revisited
Example
Recall that R = k [x1 , . . . , x7 ] and I be the ideal of R
generated by f1 = x1 x2 x3 , f2 = x2 x4 x5 , f3 = x5 x6 x7 ,
f 4 = x3 x6 x7 .
We construct the generator graph of I:
Degree 3 Example revisited
Example
Recall that R = k [x1 , . . . , x7 ] and I be the ideal of R
generated by f1 = x1 x2 x3 , f2 = x2 x4 x5 , f3 = x5 x6 x7 ,
f 4 = x3 x6 x7 .
We construct the generator graph of I:
f 1 = x1 x2 x3
b
b
f 2 = x2 x4 x5
f 4 = x3 x6 x7
b
b
f 3 = x5 x6 x7
Degree 3 Example revisited
Example
Recall that R = k [x1 , . . . , x7 ] and I be the ideal of R
generated by f1 = x1 x2 x3 , f2 = x2 x4 x5 , f3 = x5 x6 x7 ,
f 4 = x3 x6 x7 .
We construct the generator graph of I:
f 1 = x1 x2 x3
b
b
f 2 = x2 x4 x5
f 4 = x3 x6 x7
b
b
f 3 = x5 x6 x7
Degree 3 Example revisited
Example
Recall that R = k [x1 , . . . , x7 ] and I be the ideal of R
generated by f1 = x1 x2 x3 , f2 = x2 x4 x5 , f3 = x5 x6 x7 ,
f 4 = x3 x6 x7 .
We construct the generator graph of I:
f 1 = x1 x2 x3
b
b
f 2 = x2 x4 x5
f 4 = x3 x6 x7
b
b
f 3 = x5 x6 x7
Recall that the defining equations of R(I) are
generated by J1 and x4 T1 T3 − x1 T2 T4 .
Degree 3 Example revisited
Example
Recall that R = k [x1 , . . . , x7 ] and I be the ideal of R
generated by f1 = x1 x2 x3 , f2 = x2 x4 x5 , f3 = x5 x6 x7 ,
f 4 = x3 x6 x7 .
We construct the generator graph of I:
f 1 = x1 x2 x3
b
b
f 2 = x2 x4 x5
f 4 = x3 x6 x7
b
b
f 3 = x5 x6 x7
Recall that the defining equations of R(I) are
generated by J1 and x4 T1 T3 − x1 T2 T4 .
Let α = (1, 3) and β = (2, 4). Then
Tα,β = x4 T1 T3 − x1 T2 T4 ∈ J.
Results
Theorem (F-K.N. Lin)
Let G be the generator graph of I, where I is a
square-free monomial ideal generated in the same
degree.
Results
Theorem (F-K.N. Lin)
Let G be the generator graph of I, where I is a
square-free monomial ideal generated in the same
degree.
We assume that the connected components of G are
all subgraphs of G with at most 4 vertices.
Results
Theorem (F-K.N. Lin)
Let G be the generator graph of I, where I is a
square-free monomial ideal generated in the same
degree.
We assume that the connected components of G are
all subgraphs of G with at most 4 vertices.
Then the defining ideal of R(I) is generated by J1 and
binomials Tα,β that correspond to the squares of G.
Results
Theorem (F-K.N. Lin)
Let G be the generator graph of I, where I is a
square-free monomial ideal generated in the same
degree.
We assume that the connected components of G are
all subgraphs of G with at most 4 vertices.
Then the defining ideal of R(I) is generated by J1 and
binomials Tα,β that correspond to the squares of G.
Remark
By Taylor’s Theorem we know that Tα,β ∈ J.
Results
Theorem (F-K.N. Lin)
Let G be the generator graph of I, where I is a
square-free monomial ideal generated in the same
degree.
We assume that the connected components of G are
all subgraphs of G with at most 4 vertices.
Then the defining ideal of R(I) is generated by J1 and
binomials Tα,β that correspond to the squares of G.
Remark
By Taylor’s Theorem we know that Tα,β ∈ J.
We show that for all α, β ∈ Is , where s ≥ 3 then
Tα,β ∈ J1 ∪ J2 .
Ideals of linear type
Let R be a Noetherian ring and I an ideal of R.
Ideals of linear type
Let R be a Noetherian ring and I an ideal of R.
The ideal I is said to be of linear type if the defining
ideal J of the Rees algebra of I is generated in
degree one.
Ideals of linear type
Let R be a Noetherian ring and I an ideal of R.
The ideal I is said to be of linear type if the defining
ideal J of the Rees algebra of I is generated in
degree one.In other words, R(I) ≃ S/J and J = J1 S.
Ideals of linear type
Let R be a Noetherian ring and I an ideal of R.
The ideal I is said to be of linear type if the defining
ideal J of the Rees algebra of I is generated in
degree one.In other words, R(I) ≃ S/J and J = J1 S.
Theorem (Villarreal)
Let I be the edge ideal of a connected graph G.
Ideals of linear type
Let R be a Noetherian ring and I an ideal of R.
The ideal I is said to be of linear type if the defining
ideal J of the Rees algebra of I is generated in
degree one.In other words, R(I) ≃ S/J and J = J1 S.
Theorem (Villarreal)
Let I be the edge ideal of a connected graph G.
Then I is of linear type if and only if G is the graph of a
tree or contains a unique odd cycle.
Classes of ideals of linear type
A tree is a graph with no cycles.
Classes of ideals of linear type
A tree is a graph with no cycles.
b
b
b
b
b
Classes of ideals of linear type
A tree is a graph with no cycles.
b
b
b
b
b
A forest is a disjoint union of trees.
Classes of ideals of linear type
A tree is a graph with no cycles.
b
b
b
b
b
A forest is a disjoint union of trees.
Theorem (F-K.N. Lin)
Let I be a square-free monomial ideal generated in
the same degree.
Classes of ideals of linear type
A tree is a graph with no cycles.
b
b
b
b
b
A forest is a disjoint union of trees.
Theorem (F-K.N. Lin)
Let I be a square-free monomial ideal generated in
the same degree.
Suppose that the generator graph of I is a forest or a
disjoint union of odd cycles.
Classes of ideals of linear type
A tree is a graph with no cycles.
b
b
b
b
b
A forest is a disjoint union of trees.
Theorem (F-K.N. Lin)
Let I be a square-free monomial ideal generated in
the same degree.
Suppose that the generator graph of I is a forest or a
disjoint union of odd cycles.
Then I is of linear type.
Example in degree 4
Example
Let R = k [x1 , . . . , x15 ] and let I be generated by
f1 = x1 x2 x3 x4 , f2 = x1 x5 x6 x7 , f3 = x2 x8 x9 x10 ,
f4 = x5 x6 x11 x12 , f5 = x7 x13 x14 x15 .
Example in degree 4
Example
Let R = k [x1 , . . . , x15 ] and let I be generated by
f1 = x1 x2 x3 x4 , f2 = x1 x5 x6 x7 , f3 = x2 x8 x9 x10 ,
f4 = x5 x6 x11 x12 , f5 = x7 x13 x14 x15 .
We create the generator graph of I:
Example in degree 4
Example
Let R = k [x1 , . . . , x15 ] and let I be generated by
f1 = x1 x2 x3 x4 , f2 = x1 x5 x6 x7 , f3 = x2 x8 x9 x10 ,
f4 = x5 x6 x11 x12 , f5 = x7 x13 x14 x15 .
We create the generator graph of I:
f4 = x5 x6 x11 x12
f 2 = x1 x5 x6 x7
f 1 = x1 x2 x3 x4
b
b
b
f5 = x7 x13 x14 x15
b
b
f3 = x2 x8 x9 x10
Example in degree 4
Example
Let R = k [x1 , . . . , x15 ] and let I be generated by
f1 = x1 x2 x3 x4 , f2 = x1 x5 x6 x7 , f3 = x2 x8 x9 x10 ,
f4 = x5 x6 x11 x12 , f5 = x7 x13 x14 x15 .
We create the generator graph of I:
f4 = x5 x6 x11 x12
f 2 = x1 x5 x6 x7
f 1 = x1 x2 x3 x4
b
b
b
f5 = x7 x13 x14 x15
b
b
f3 = x2 x8 x9 x10
Example in degree 4
Example
Let R = k [x1 , . . . , x15 ] and let I be generated by
f1 = x1 x2 x3 x4 , f2 = x1 x5 x6 x7 , f3 = x2 x8 x9 x10 ,
f4 = x5 x6 x11 x12 , f5 = x7 x13 x14 x15 .
We create the generator graph of I:
f4 = x5 x6 x11 x12
f 2 = x1 x5 x6 x7
f 1 = x1 x2 x3 x4
b
b
b
f5 = x7 x13 x14 x15
b
b
f3 = x2 x8 x9 x10
Then by the previous Theorem, I is of linear type as
its generator graph is the graph of a tree.
Sketch of the Proof
Suppose the generator graph G is the graph of a tree.
Sketch of the Proof
Suppose the generator graph G is the graph of a tree.
Let us assume that s = 2 and let α = (a1 , a2 ) and
β = (b1 , b2 ).
Sketch of the Proof
Suppose the generator graph G is the graph of a tree.
Let us assume that s = 2 and let α = (a1 , a2 ) and
β = (b1 , b2 ). Then fα = fa1 fa2 and fβ = fb1 fb2 .
Sketch of the Proof
Suppose the generator graph G is the graph of a tree.
Let us assume that s = 2 and let α = (a1 , a2 ) and
β = (b1 , b2 ). Then fα = fa1 fa2 and fβ = fb1 fb2 .
We consider the following scenario:
Sketch of the Proof
Suppose the generator graph G is the graph of a tree.
Let us assume that s = 2 and let α = (a1 , a2 ) and
β = (b1 , b2 ). Then fα = fa1 fa2 and fβ = fb1 fb2 .
We consider the following scenario:
y a1
y a2
y b1
b
b
b
b
y b2
Sketch of the Proof
Suppose the generator graph G is the graph of a tree.
Let us assume that s = 2 and let α = (a1 , a2 ) and
β = (b1 , b2 ). Then fα = fa1 fa2 and fβ = fb1 fb2 .
We consider the following scenario:
y a1
y a2
y b1
b
b
b
b
y b2
Then gcd(fα , fβ ) =
C ∈ R.
gcd(fa1 , fb1 ) gcd(fa2 , fb2 )
for some
C
Sketch of the Proof
Suppose the generator graph G is the graph of a tree.
Let us assume that s = 2 and let α = (a1 , a2 ) and
β = (b1 , b2 ). Then fα = fa1 fa2 and fβ = fb1 fb2 .
We consider the following scenario:
y a1
y a2
y b1
b
b
b
b
y b2
Then gcd(fα , fβ ) =
gcd(fa1 , fb1 ) gcd(fa2 , fb2 )
for some
C
C ∈ R.
Then it is straightforward to see that
fb2 CTa2
fa1 CTb1
Tα,β =
[Ta1 ,b1 ] +
[Ta ,b ] ∈ J1
gcd(fa2 , fb2 )
gcd(fa1 , fb1 ) 2 2
Ideals of fiber type
Let R be a Noetherian local ring with residue field k
and let I be an ideal of R.
Ideals of fiber type
Let R be a Noetherian local ring with residue field k
and let I be an ideal of R.
The ring F (I) = R(I) ⊗R k is called the special fiber
ring of I.
Ideals of fiber type
Let R be a Noetherian local ring with residue field k
and let I be an ideal of R.
The ring F (I) = R(I) ⊗R k is called the special fiber
ring of I. Notice that F (I) ≃ k [T1 , . . . , Tn ]/H, for some
ideal H, where n = µ(I).
Ideals of fiber type
Let R be a Noetherian local ring with residue field k
and let I be an ideal of R.
The ring F (I) = R(I) ⊗R k is called the special fiber
ring of I. Notice that F (I) ≃ k [T1 , . . . , Tn ]/H, for some
ideal H, where n = µ(I).
When R(I) ≃ S/(J1 , H) then I is called an ideal of
fiber type.
Ideals of fiber type
Let R be a Noetherian local ring with residue field k
and let I be an ideal of R.
The ring F (I) = R(I) ⊗R k is called the special fiber
ring of I. Notice that F (I) ≃ k [T1 , . . . , Tn ]/H, for some
ideal H, where n = µ(I).
When R(I) ≃ S/(J1 , H) then I is called an ideal of
fiber type.
Edge ideals of graphs are ideals of fiber type.
Ideals of fiber type
Let R be a Noetherian local ring with residue field k
and let I be an ideal of R.
The ring F (I) = R(I) ⊗R k is called the special fiber
ring of I. Notice that F (I) ≃ k [T1 , . . . , Tn ]/H, for some
ideal H, where n = µ(I).
When R(I) ≃ S/(J1 , H) then I is called an ideal of
fiber type.
Edge ideals of graphs are ideals of fiber type.When I
is the edge ideal of a square, the degree 2 relation is
given by T1 T3 − T2 T4 ∈ H.
Ideals of fiber type
Let R be a Noetherian local ring with residue field k
and let I be an ideal of R.
The ring F (I) = R(I) ⊗R k is called the special fiber
ring of I. Notice that F (I) ≃ k [T1 , . . . , Tn ]/H, for some
ideal H, where n = µ(I).
When R(I) ≃ S/(J1 , H) then I is called an ideal of
fiber type.
Edge ideals of graphs are ideals of fiber type.When I
is the edge ideal of a square, the degree 2 relation is
given by T1 T3 − T2 T4 ∈ H.
In the degree 3 Example of Villarreal, the generator
graph was a square and the degree 2 relation was
x4 T1 T3 − x1 T2 T4 6∈ H.
Ideals of fiber type
Let R be a Noetherian local ring with residue field k
and let I be an ideal of R.
The ring F (I) = R(I) ⊗R k is called the special fiber
ring of I. Notice that F (I) ≃ k [T1 , . . . , Tn ]/H, for some
ideal H, where n = µ(I).
When R(I) ≃ S/(J1 , H) then I is called an ideal of
fiber type.
Edge ideals of graphs are ideals of fiber type.When I
is the edge ideal of a square, the degree 2 relation is
given by T1 T3 − T2 T4 ∈ H.
In the degree 3 Example of Villarreal, the generator
graph was a square and the degree 2 relation was
x4 T1 T3 − x1 T2 T4 6∈ H. Hence I is not of fiber type.