Os ideais de tipo linear são amplamente estudados na álgebra comutativa com os anéis de polinômios e sua conexão direta com as álgebra de Rees por conta de sua definição.
Definição (Ideal de tipo linear) Seja
F
=
{
f
1
,
…
,
f
r
}
⊂
k
[
x
1
,
…
,
x
n
]
=
R
{\displaystyle F=\{f_{1},\dots ,f_{r}\}\subset k[x_{1},\dots ,x_{n}]=R}
I
=
(
F
)
{\displaystyle I=(F)}
φ
:
R
[
y
1
,
…
,
y
r
]
⟶
R
[
f
1
t
,
…
,
f
r
t
]
y
i
↦
f
i
t
a
∈
R
↦
a
{\displaystyle {\begin{matrix}\varphi :&R[y_{1},\dots ,y_{r}]&\longrightarrow &R[f_{1}t,\dots ,f_{r}t]\\&y_{i}&\mapsto &f_{i}t\\&a\in R&\mapsto &a\end{matrix}}}
Então
I
{\displaystyle I}
ker
(
φ
)
{\displaystyle \ker(\varphi )}
1
{\displaystyle 1}
y
i
{\displaystyle y_{i}}
editar
Seja
F
=
{
X
α
1
,
…
,
X
α
n
}
⊂
k
[
x
1
,
…
,
x
n
]
{\displaystyle F=\{X^{\alpha _{1}},\dots ,X^{\alpha _{n}}\}\subset k[x_{1},\dots ,x_{n}]}
α
i
=
(
α
i
1
,
…
,
α
i
n
)
∈
Z
+
n
{\displaystyle \alpha _{i}=(\alpha _{i1},\dots ,\alpha _{in})\in \mathbb {Z} _{+}^{n}}
X
α
i
=
x
1
α
i
1
x
2
α
i
2
⋯
x
n
α
i
n
{\displaystyle X^{\alpha _{i}}=x_{1}^{\alpha _{i1}}x_{2}^{\alpha _{i2}}\cdots x_{n}^{\alpha _{in}}}
A
F
=
[
α
1
|
α
2
|
…
|
α
n
]
{\displaystyle A_{F}=[\alpha _{1}|\alpha _{2}|\dots |\alpha _{n}]}
matriz-log
Exemplo: Note que para
F
=
{
x
y
,
y
z
,
x
z
}
⊂
k
[
x
,
y
,
z
]
{\displaystyle F=\{xy,yz,xz\}\subset k[x,y,z]}
A
F
=
(
1
1
0
0
1
1
1
0
1
)
{\displaystyle A_{F}={\begin{pmatrix}1&1&0\\0&1&1\\1&0&1\end{pmatrix}}}
F
{\displaystyle F}
Proposição: Seja
F
⊂
k
[
x
1
,
…
,
x
n
]
{\displaystyle F\subset k[x_{1},\dots ,x_{n}]}
det
(
A
F
)
≠
0
{\displaystyle \det(A_{F})\neq 0}
(
F
)
{\displaystyle (F)}
Simis, A.;Villarreal, R, Linear syzygies and birational combinatorics , Results Math. 48 (2005), 326-343 
![{\displaystyle F=\{f_{1},\dots ,f_{r}\}\subset k[x_{1},\dots ,x_{n}]=R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc266611abd70912a180d48099b017d299838414)
![{\displaystyle {\begin{matrix}\varphi :&R[y_{1},\dots ,y_{r}]&\longrightarrow &R[f_{1}t,\dots ,f_{r}t]\\&y_{i}&\mapsto &f_{i}t\\&a\in R&\mapsto &a\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e72a636cb8096e5775fb51978b0563a25d79e7)
![{\displaystyle F=\{X^{\alpha _{1}},\dots ,X^{\alpha _{n}}\}\subset k[x_{1},\dots ,x_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab5e4ca212194239ac59bdd21022d279533e3b0)
![{\displaystyle A_{F}=[\alpha _{1}|\alpha _{2}|\dots |\alpha _{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/168d73c9cbd5912bc7d195b9d07f25b446c15e1c)
![{\displaystyle F=\{xy,yz,xz\}\subset k[x,y,z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/302977055e527a14f75bbfae1f1b8d95b4f5d9ab)
![{\displaystyle F\subset k[x_{1},\dots ,x_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c93a153f215d544cea982550681cd34db75a8533)
