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Artikel: Locally nilpotent derivation GudangMovies21 Rebahinxxi

• Source: Locally nilpotent derivation

In mathematics, a derivation



∂


{\displaystyle \partial }

of a commutative ring



A


{\displaystyle A}

is called a locally nilpotent derivation (LND) if every element of



A


{\displaystyle A}

is annihilated by some power of



∂


{\displaystyle \partial }

.
One motivation for the study of locally nilpotent derivations comes from the fact that some of the counterexamples to Hilbert's 14th problem are obtained as the kernels of a derivation on a polynomial ring.
Over a field



k


{\displaystyle k}

of characteristic zero, to give a locally nilpotent derivation on the integral domain



A


{\displaystyle A}

, finitely generated over the field, is equivalent to giving an action of the additive group



(
k
,
+
)


{\displaystyle (k,+)}

to the affine variety



X
=
Spec
⁡
(
A
)


{\displaystyle X=\operatorname {Spec} (A)}

. Roughly speaking, an affine variety admitting "plenty" of actions of the additive group is considered similar to an affine space.




Definition


Let



A


{\displaystyle A}

be a ring. Recall that a derivation of



A


{\displaystyle A}

is a map



∂
:

A
→
A


{\displaystyle \partial \colon \,A\to A}

satisfying the Leibniz rule



∂
(
a
b
)
=
(
∂
a
)
b
+
a
(
∂
b
)


{\displaystyle \partial (ab)=(\partial a)b+a(\partial b)}

for any



a
,
b
∈
A


{\displaystyle a,b\in A}

. If



A


{\displaystyle A}

is an algebra over a field



k


{\displaystyle k}

, we additionally require



∂


{\displaystyle \partial }

to be



k


{\displaystyle k}

-linear, so



k
⊆
ker
⁡
∂


{\displaystyle k\subseteq \ker \partial }

.
A derivation



∂


{\displaystyle \partial }

is called a locally nilpotent derivation (LND) if for every



a
∈
A


{\displaystyle a\in A}

, there exists a positive integer



n


{\displaystyle n}

such that




∂

n


(
a
)
=
0


{\displaystyle \partial ^{n}(a)=0}

.
If



A


{\displaystyle A}

is graded, we say that a locally nilpotent derivation



∂


{\displaystyle \partial }

is homogeneous (of degree



d


{\displaystyle d}

) if



deg
⁡
∂
a
=
deg
⁡
a
+
d


{\displaystyle \deg \partial a=\deg a+d}

for every



a
∈
A


{\displaystyle a\in A}

.
The set of locally nilpotent derivations of a ring



A


{\displaystyle A}

is denoted by



LND
⁡
(
A
)


{\displaystyle \operatorname {LND} (A)}

. Note that this set has no obvious structure: it is neither closed under addition (e.g. if




∂

1


=
y



∂

∂
x






{\displaystyle \partial _{1}=y{\tfrac {\partial }{\partial x}}}

,




∂

2


=
x



∂

∂
y






{\displaystyle \partial _{2}=x{\tfrac {\partial }{\partial y}}}

then




∂

1


,

∂

2


∈
LND
⁡
(
k
[
x
,
y
]
)


{\displaystyle \partial _{1},\partial _{2}\in \operatorname {LND} (k[x,y])}

but



(

∂

1


+

∂

2



)

2


(
x
)
=
x


{\displaystyle (\partial _{1}+\partial _{2})^{2}(x)=x}

, so




∂

1


+

∂

2


∉
LND
⁡
(
k
[
x
,
y
]
)


{\displaystyle \partial _{1}+\partial _{2}\not \in \operatorname {LND} (k[x,y])}

) nor under multiplication by elements of



A


{\displaystyle A}

(e.g.






∂

∂
x




∈
LND
⁡
(
k
[
x
]
)


{\displaystyle {\tfrac {\partial }{\partial x}}\in \operatorname {LND} (k[x])}

, but



x



∂

∂
x




∉
LND
⁡
(
k
[
x
]
)


{\displaystyle x{\tfrac {\partial }{\partial x}}\not \in \operatorname {LND} (k[x])}

). However, if



[

∂

1


,

∂

2


]
=
0


{\displaystyle [\partial _{1},\partial _{2}]=0}

then




∂

1


,

∂

2


∈
LND
⁡
(
A
)


{\displaystyle \partial _{1},\partial _{2}\in \operatorname {LND} (A)}

implies




∂

1


+

∂

2


∈
LND
⁡
(
A
)


{\displaystyle \partial _{1}+\partial _{2}\in \operatorname {LND} (A)}

and if



∂
∈
LND
⁡
(
A
)


{\displaystyle \partial \in \operatorname {LND} (A)}

,



h
∈
ker
⁡
∂


{\displaystyle h\in \ker \partial }

then



h
∂
∈
LND
⁡
(
A
)


{\displaystyle h\partial \in \operatorname {LND} (A)}

.




Relation to Ga-actions


Let



A


{\displaystyle A}

be an algebra over a field



k


{\displaystyle k}

of characteristic zero (e.g.



k
=

C



{\displaystyle k=\mathbb {C} }

). Then there is a one-to-one correspondence between the locally nilpotent



k


{\displaystyle k}

-derivations on



A


{\displaystyle A}

and the actions of the additive group





G


a




{\displaystyle \mathbb {G} _{a}}

of



k


{\displaystyle k}

on the affine variety



Spec
⁡
A


{\displaystyle \operatorname {Spec} A}

, as follows.
A





G


a




{\displaystyle \mathbb {G} _{a}}

-action on



Spec
⁡
A


{\displaystyle \operatorname {Spec} A}

corresponds to a



k


{\displaystyle k}

-algebra homomorphism



ρ
:
A
→
A
[
t
]


{\displaystyle \rho \colon A\to A[t]}

. Any such



ρ


{\displaystyle \rho }

determines a locally nilpotent derivation



∂


{\displaystyle \partial }

of



A


{\displaystyle A}

by taking its derivative at zero, namely



∂
=
ϵ
∘



d

d
t




∘
ρ
,


{\displaystyle \partial =\epsilon \circ {\tfrac {d}{dt}}\circ \rho ,}

where



ϵ


{\displaystyle \epsilon }

denotes the evaluation at



t
=
0


{\displaystyle t=0}

.
Conversely, any locally nilpotent derivation



∂


{\displaystyle \partial }

determines a homomorphism



ρ
:
A
→
A
[
t
]


{\displaystyle \rho \colon A\to A[t]}

by



ρ
=
exp
⁡
(
t
∂
)
=

∑

n
=
0


∞





t

n



n
!




∂

n


.


{\displaystyle \rho =\exp(t\partial )=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}\partial ^{n}.}


It is easy to see that the conjugate actions correspond to conjugate derivations, i.e. if



α
∈
Aut
⁡
A


{\displaystyle \alpha \in \operatorname {Aut} A}

and



∂
∈
LND
⁡
(
A
)


{\displaystyle \partial \in \operatorname {LND} (A)}

then



α
∘
∂
∘

α

−
1


∈
LND
⁡
(
A
)


{\displaystyle \alpha \circ \partial \circ \alpha ^{-1}\in \operatorname {LND} (A)}

and



exp
⁡
(
t
⋅
α
∘
∂
∘

α

−
1


)
=
α
∘
exp
⁡
(
t
∂
)
∘

α

−
1




{\displaystyle \exp(t\cdot \alpha \circ \partial \circ \alpha ^{-1})=\alpha \circ \exp(t\partial )\circ \alpha ^{-1}}





The kernel algorithm


The algebra



ker
⁡
∂


{\displaystyle \ker \partial }

consists of the invariants of the corresponding





G


a




{\displaystyle \mathbb {G} _{a}}

-action. It is algebraically and factorially closed in



A


{\displaystyle A}

. A special case of Hilbert's 14th problem asks whether



ker
⁡
∂


{\displaystyle \ker \partial }

is finitely generated, or, if



A
=
k
[
X
]


{\displaystyle A=k[X]}

, whether the quotient



X

/



/



G


a




{\displaystyle X/\!/\mathbb {G} _{a}}

is affine. By Zariski's finiteness theorem, it is true if



dim
⁡
X
≤
3


{\displaystyle \dim X\leq 3}

. On the other hand, this question is highly nontrivial even for



X
=


C


n




{\displaystyle X=\mathbb {C} ^{n}}

,



n
≥
4


{\displaystyle n\geq 4}

. For



n
≥
5


{\displaystyle n\geq 5}

the answer, in general, is negative. The case



n
=
4


{\displaystyle n=4}

is open.
However, in practice it often happens that



ker
⁡
∂


{\displaystyle \ker \partial }

is known to be finitely generated: notably, by the Maurer–Weitzenböck theorem, it is the case for linear LND's of the polynomial algebra over a field of characteristic zero (by linear we mean homogeneous of degree zero with respect to the standard grading).
Assume



ker
⁡
∂


{\displaystyle \ker \partial }

is finitely generated. If



A
=
k
[

g

1


,
…
,

g

n


]


{\displaystyle A=k[g_{1},\dots ,g_{n}]}

is a finitely generated algebra over a field of characteristic zero, then



ker
⁡
∂


{\displaystyle \ker \partial }

can be computed using van den Essen's algorithm, as follows. Choose a local slice, i.e. an element



r
∈
ker
⁡

∂

2


∖
ker
⁡
∂


{\displaystyle r\in \ker \partial ^{2}\setminus \ker \partial }

and put



f
=
∂
r
∈
ker
⁡
∂


{\displaystyle f=\partial r\in \ker \partial }

. Let




π

r


:

A
→
(
ker
⁡
∂

)

f




{\displaystyle \pi _{r}\colon \,A\to (\ker \partial )_{f}}

be the Dixmier map given by




π

r


(
a
)
=

∑

n
=
0


∞





(
−
1

)

n




n
!




∂

n


(
a
)



r

n



f

n






{\displaystyle \pi _{r}(a)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\partial ^{n}(a){\frac {r^{n}}{f^{n}}}}

. Now for every



i
=
1
,
…
,
n


{\displaystyle i=1,\dots ,n}

, chose a minimal integer




m

i




{\displaystyle m_{i}}

such that




h

i


:
=

f


m

i





π

r


(

g

i


)
∈
ker
⁡
∂


{\displaystyle h_{i}\colon =f^{m_{i}}\pi _{r}(g_{i})\in \ker \partial }

, put




B

0


=
k
[

h

1


,
…
,

h

n


,
f
]
⊆
ker
⁡
∂


{\displaystyle B_{0}=k[h_{1},\dots ,h_{n},f]\subseteq \ker \partial }

, and define inductively




B

i




{\displaystyle B_{i}}

to be the subring of



A


{\displaystyle A}

generated by



{
h
∈
A
:
f
h
∈

B

i
−
1


}


{\displaystyle \{h\in A:fh\in B_{i-1}\}}

. By induction, one proves that




B

0


⊂

B

1


⊂
⋯
⊂
ker
⁡
∂


{\displaystyle B_{0}\subset B_{1}\subset \dots \subset \ker \partial }

are finitely generated and if




B

i


=

B

i
+
1




{\displaystyle B_{i}=B_{i+1}}

then




B

i


=
ker
⁡
∂


{\displaystyle B_{i}=\ker \partial }

, so




B

N


=
ker
⁡
∂


{\displaystyle B_{N}=\ker \partial }

for some



N


{\displaystyle N}

. Finding the generators of each




B

i




{\displaystyle B_{i}}

and checking whether




B

i


=

B

i
+
1




{\displaystyle B_{i}=B_{i+1}}

is a standard computation using Gröbner bases.




Slice theorem


Assume that



∂
∈
LND
⁡
(
A
)


{\displaystyle \partial \in \operatorname {LND} (A)}

admits a slice, i.e.



s
∈
A


{\displaystyle s\in A}

such that



∂
s
=
1


{\displaystyle \partial s=1}

. The slice theorem asserts that



A


{\displaystyle A}

is a polynomial algebra



(
ker
⁡
∂
)
[
s
]


{\displaystyle (\ker \partial )[s]}

and



∂
=



d

d
s






{\displaystyle \partial ={\tfrac {d}{ds}}}

.
For any local slice



r
∈
ker
⁡
∂
∖
ker
⁡

∂

2




{\displaystyle r\in \ker \partial \setminus \ker \partial ^{2}}

we can apply the slice theorem to the localization




A

∂
r




{\displaystyle A_{\partial r}}

, and thus obtain that



A


{\displaystyle A}

is locally a polynomial algebra with a standard derivation. In geometric terms, if a geometric quotient



π
:

X
→
X

/


/



G


a




{\displaystyle \pi \colon \,X\to X//\mathbb {G} _{a}}

is affine (e.g. when



dim
⁡
X
≤
3


{\displaystyle \dim X\leq 3}

by the Zariski theorem), then it has a Zariski-open subset



U


{\displaystyle U}

such that




π

−
1


(
U
)


{\displaystyle \pi ^{-1}(U)}

is isomorphic over



U


{\displaystyle U}

to



U
×


A


1




{\displaystyle U\times \mathbb {A} ^{1}}

, where





G


a




{\displaystyle \mathbb {G} _{a}}

acts by translation on the second factor.
However, in general it is not true that



X
→
X

/


/



G


a




{\displaystyle X\to X//\mathbb {G} _{a}}

is locally trivial. For example, let



∂
=
u



∂

∂
x




+
v



∂

∂
y




+
(
1
+
u

y

2


)



∂

∂
z




∈
LND
⁡
(

C

[
x
,
y
,
z
,
u
,
v
]
)


{\displaystyle \partial =u{\tfrac {\partial }{\partial x}}+v{\tfrac {\partial }{\partial y}}+(1+uy^{2}){\tfrac {\partial }{\partial z}}\in \operatorname {LND} (\mathbb {C} [x,y,z,u,v])}

. Then



ker
⁡
∂


{\displaystyle \ker \partial }

is a coordinate ring of a singular variety, and the fibers of the quotient map over singular points are two-dimensional.
If



dim
⁡
X
=
3


{\displaystyle \dim X=3}

then



Γ
=
X
∖
U


{\displaystyle \Gamma =X\setminus U}

is a curve. To describe the





G


a




{\displaystyle \mathbb {G} _{a}}

-action, it is important to understand the geometry



Γ


{\displaystyle \Gamma }

. Assume further that



k
=

C



{\displaystyle k=\mathbb {C} }

and that



X


{\displaystyle X}

is smooth and contractible (in which case



S


{\displaystyle S}

is smooth and contractible as well) and choose



Γ


{\displaystyle \Gamma }

to be minimal (with respect to inclusion). Then Kaliman proved that each irreducible component of



Γ


{\displaystyle \Gamma }

is a polynomial curve, i.e. its normalization is isomorphic to





C


1




{\displaystyle \mathbb {C} ^{1}}

. The curve



Γ


{\displaystyle \Gamma }

for the action given by Freudenburg's (2,5)-derivation (see below) is a union of two lines in





C


2




{\displaystyle \mathbb {C} ^{2}}

, so



Γ


{\displaystyle \Gamma }

may not be irreducible. However, it is conjectured that



Γ


{\displaystyle \Gamma }

is always contractible.




Examples






= Example 1

=
The standard coordinate derivations






∂

∂

x

i








{\displaystyle {\tfrac {\partial }{\partial x_{i}}}}

of a polynomial algebra



k
[

x

1


,
…
,

x

n


]


{\displaystyle k[x_{1},\dots ,x_{n}]}

are locally nilpotent. The corresponding





G


a




{\displaystyle \mathbb {G} _{a}}

-actions are translations:



t
⋅

x

i


=

x

i


+
t


{\displaystyle t\cdot x_{i}=x_{i}+t}

,



t
⋅

x

j


=

x

j




{\displaystyle t\cdot x_{j}=x_{j}}

for



j
≠
i


{\displaystyle j\neq i}

.




= Example 2 (Freudenburg's (2,5)-homogeneous derivation)

=
Let




f

1


=

x

1



x

3


−

x

2


2




{\displaystyle f_{1}=x_{1}x_{3}-x_{2}^{2}}

,




f

2


=

x

3



f

1


2


+
2

x

1


2



x

2



f

1


+

x

5




{\displaystyle f_{2}=x_{3}f_{1}^{2}+2x_{1}^{2}x_{2}f_{1}+x^{5}}

, and let



∂


{\displaystyle \partial }

be the Jacobian derivation



∂
(

f

3


)
=
det


[




∂

f

i




∂

x

j






]


i
,
j
=
1
,
2
,
3




{\textstyle \partial (f_{3})=\det \left[{\tfrac {\partial f_{i}}{\partial x_{j}}}\right]_{i,j=1,2,3}}

. Then



∂
∈
LND
⁡
(
k
[

x

1


,

x

2


,

x

3


]
)


{\displaystyle \partial \in \operatorname {LND} (k[x_{1},x_{2},x_{3}])}

and



rank
⁡
∂
=
3


{\displaystyle \operatorname {rank} \partial =3}

(see below); that is,



∂


{\displaystyle \partial }

annihilates no variable. The fixed point set of the corresponding





G


a




{\displaystyle \mathbb {G} _{a}}

-action equals



{

x

1


=

x

2


=
0
}


{\displaystyle \{x_{1}=x_{2}=0\}}

.




= Example 3

=
Consider



S

l

2


(
k
)
=
{
a
d
−
b
c
=
1
}
⊆

k

4




{\displaystyle Sl_{2}(k)=\{ad-bc=1\}\subseteq k^{4}}

. The locally nilpotent derivation



a



∂

∂
b




+
c



∂

∂
d






{\displaystyle a{\tfrac {\partial }{\partial b}}+c{\tfrac {\partial }{\partial d}}}

of its coordinate ring corresponds to a natural action of





G


a




{\displaystyle \mathbb {G} _{a}}

on



S

l

2


(
k
)


{\displaystyle Sl_{2}(k)}

via right multiplication of upper triangular matrices. This action gives a nontrivial





G


a




{\displaystyle \mathbb {G} _{a}}

-bundle over





A


2


∖
{
(
0
,
0
)
}


{\displaystyle \mathbb {A} ^{2}\setminus \{(0,0)\}}

. However, if



k
=

C



{\displaystyle k=\mathbb {C} }

then this bundle is trivial in the smooth category




LND's of the polynomial algebra


Let



k


{\displaystyle k}

be a field of characteristic zero (using Kambayashi's theorem one can reduce most results to the case



k
=

C



{\displaystyle k=\mathbb {C} }

) and let



A
=
k
[

x

1


,
…
,

x

n


]


{\displaystyle A=k[x_{1},\dots ,x_{n}]}

be a polynomial algebra.




= n = 2 (Ga-actions on an affine plane)

=




= n = 3 (Ga-actions on an affine 3-space)

=




= Triangular derivations

=
Let




f

1


,
…
,

f

n




{\displaystyle f_{1},\dots ,f_{n}}

be any system of variables of



A


{\displaystyle A}

; that is,



A
=
k
[

f

1


,
…
,

f

n


]


{\displaystyle A=k[f_{1},\dots ,f_{n}]}

. A derivation of



A


{\displaystyle A}

is called triangular with respect to this system of variables, if



∂

f

1


∈
k


{\displaystyle \partial f_{1}\in k}

and



∂

f

i


∈
k
[

f

1


,
…
,

f

i
−
1


]


{\displaystyle \partial f_{i}\in k[f_{1},\dots ,f_{i-1}]}

for



i
=
2
,
…
,
n


{\displaystyle i=2,\dots ,n}

. A derivation is called triangulable if it is conjugate to a triangular one, or, equivalently, if it is triangular with respect to some system of variables. Every triangular derivation is locally nilpotent. The converse is true for



≤
2


{\displaystyle \leq 2}

by Rentschler's theorem above, but it is not true for



n
≥
3


{\displaystyle n\geq 3}

.

Bass's example
The derivation of



k
[

x

1


,

x

2


,

x

3


]


{\displaystyle k[x_{1},x_{2},x_{3}]}

given by




x

1





∂

∂

x

2






+
2

x

2



x

1





∂

∂

x

3








{\displaystyle x_{1}{\tfrac {\partial }{\partial x_{2}}}+2x_{2}x_{1}{\tfrac {\partial }{\partial x_{3}}}}

is not triangulable. Indeed, the fixed-point set of the corresponding





G


a




{\displaystyle \mathbb {G} _{a}}

-action is a quadric cone




x

2



x

3


=

x

2


2




{\displaystyle x_{2}x_{3}=x_{2}^{2}}

, while by the result of Popov, a fixed point set of a triangulable





G


a




{\displaystyle \mathbb {G} _{a}}

-action is isomorphic to



Z
×


A


1




{\displaystyle Z\times \mathbb {A} ^{1}}

for some affine variety



Z


{\displaystyle Z}

; and thus cannot have an isolated singularity.




Makar-Limanov invariant



The intersection of the kernels of all locally nilpotent derivations of the coordinate ring, or, equivalently, the ring of invariants of all





G


a




{\displaystyle \mathbb {G} _{a}}

-actions, is called "Makar-Limanov invariant" and is an important algebraic invariant of an affine variety. For example, it is trivial for an affine space; but for the Koras–Russell cubic threefold, which is diffeomorphic to





C


3




{\displaystyle \mathbb {C} ^{3}}

, it is not.




References






Further reading


A Nowicki, the fourteenth problem of hilbert for polynomial derivations

Kata Kunci Pencarian:

• Locally nilpotent derivation
• Hilbert's fourteenth problem
• Lie algebra
• Lie group
• Glossary of algebraic geometry
• Zariski's finiteness theorem
• List of group theory topics
• Heisenberg group
• Scheme (mathematics)
• List of abstract algebra topics