Table Of ContentLND-FILTRATIONS AND SEMI-RIGID DOMAINS
BACHAR ALHAJJAR
Abstract. Weinvestigatethefiltrationcorrespondingtothedegreefunctioninducedbyanon-zerolocally
nilpotent derivation and its associated graded algebra. We show that this kind of filtration, referred to as
5 theLND-filtration,istheidealcandidatetostudythestructureofsemi-rigid k-domains,thatis,k-domains
1 for which every non-zero locally nilpotent derivation gives rise to the same filtration. Indeed, the LND-
0 filtration gives a very precise understanding of these structure, it is impeccable for the computation of the
2 Makar-Limanovinvariant,anditisanefficienttooltodeterminetheirisomorphismtypesandautomorphism
groups. Then, we construct a new interesting class of semi-rigid k-domains in which we elaborate the
n fundamental requirementofLND-filtrations. Theimportanceofthesenew examples isduetothefactthat
a theypossessarelativelybigsetofinvariantsub-algebras,whichcannotberecoverdbyknowninvariantssuch
J
astheMakar-LimanovandtheDerkseninvariants. Also,wedefineanewfamilyofinvariantsub-algebrasasa
2 generalizationoftheDerkseninvariant. Finally,weintroduceanalgorithmtoestablishexplicitisomorphisms
between cylinders over non-isomorphic members of the new class, providing by that new counter-examples
] tothecancellationproblem.
C
A
.
h
t
a Introduction
m
Let k be a field of characteristic zero and let B be a commutative k-domain. A k-derivation ∂ ∈Der (B)
k
[ is said to be locally nilpotent if for every a∈B, there is an integer n≥0 such that ∂n(a)=0.
1 An important invariantof k-domains B admitting non-trivial locally nilpotent derivations is the so called
v Makar-Limanov invariant ML(B) which was defined by Makar-Limanovas the intersectionof the kernels of
5 alllocallynilpotent derivationsonB ([ML3]). This invariantwasinitially introducedas atooltodistinguish
4
certain k-domains from polynomial rings but it has many other applications for the study of k-algebras and
4
their automorphismgroups ([ML2]). One of the main difficulties in applications is to compute this invariant
0
0 without a prior knowledge of all locally nilpotent derivations of a given k-domain.
. In [K-ML1], S. Kaliman and L. Makar-Limanov developed general techniques to determine the ML-
1
0 invariantfor a class of finitely generatedk-domains B =k[X1,...,Xn]/I. The idea, referredto as “homoge-
5 nization of derivations”, is to reduce the problem to the study of homogeneous locally nilpotent derivations
1 on gradedalgebrasGr(B) associatedto B. For this, one considers appropriatefiltrations F ={Fi}i∈R on B
v: generatedbyreal-valuedweightdegreefunctions ω ∈Rn, insucha waythateverynon-zerolocallynilpotent
i derivation on B induces a non-zero homogeneous locally nilpotent derivation on the associated graded alge-
X
bra Gr (B). Unfortunately, these techniques only work if the associated graded algebra Gr(B) is in fact a
F
ar k-domain itself, which will only occur if the ideal Iˆ, generated by top homogenous components relative to ω
of all elements in I, is prime.
Finding a new way to tackle similar complications became an inevitable necessity. Therefore, we start
inspecting real-valued weight degree functions on k[N] with the following new perspective. The positive
integerN ischosentobe biggerthann,the dimensionofthe ambientspace. The consideredringB =k[n]/I
is identified in a specific “twisting” way to k[N]/J ≃B. This different point of view allows us to avoidthese
kind of difficulties. Furthermore, it simplifies the study of homogenous locally nilpotent derivation even in
these cases where classical techniques work.
We present a new class of examples for which the “homogenization” method can be effectively applied
withthealternativeperspective,whileallotherapproachesfail. Thenewclasscomestobeaveryinteresting
object due to the fact that it possesses a huge set of invariant sub-algebras, which can not be recoverd by
any known invariant such as the Makar-Limanov and the Derksen invariants.
2000 Mathematics Subject Classification. Primary: 14R20. Secondary: 13N15,
Key words and phrases. Locally nilpotent derivation, degree functions, Makar-Limanov invariant, Derksen invariant, affine
space,affinevariety,gradedring,automorphismgroup.
1
As a modest outcome, the alternative approachdelivers a full description of the filtration induced by any
locallynilpotentderivation,withafinitelygeneratedkernel,anditsassociatedgradedalgebra. Inparticular,
anon-zerolocallynilpotentderivation∂givesrisetoaproperN-filtrationF ={Fi}i∈N ofB bythesub-spaces
F =ker∂i+1, i∈N. We call it the ∂-filtration, which corresponds to the N-degree function deg induced by
i ∂
∂. Itturnsoutthatdeg isnothingbutthedegreefunctioninducedbyanN-weightdegreefunctionω ∈N[N]
∂
defined on k[N] for suitable choices of ω and N.
Inturn,the∂-filtrationcomesouttobetheidealcandidatetostudythestructureofsemi-rigid k-domains,
that is, k-domains for which every non-zero locally nilpotent derivation gives rise to the same filtration that
we call the unique LND-filtration. This unique LND-filtration gives a very precise understanding of the
structure of semi-rigid k-domains, it is impeccable for the computation of the ML-invariant, and it is an
efficient tool to determine isomorphism types and automorphism groups. Nevertheless, the computation of
the ML-invariant, isomorphism types, and automorphism groups of similar classically known structures can
be simplified and reduced considering this new point of view.
Another important tool for the study of non-rigid k-domains is the Derksen invariant D(B) which is
defined to be the sub-algebra of B generated by ker∂ for all non-zero locally nilpotent derivations. We
generalize this invariant to obtain a new family {AL (B)} of invariant sub-algebras of B, where for each
i i∈N
i∈NwedefineAL (B)tobethealgebrageneratedbyker∂i+1 forallnon-zerolocallynilpotentderivationof
i
B. We are interested in one particular member of this family of invariants that corresponds to i=1, which
we call the ring of all local slices of B and which we denote AL-invariant. We show that the new class of
k-domains can be realized as an affine modification of the AL-invariant with center (f,I) for certain ideal I
in AL and some f ∈I.
Finally,weproposeanalgorithmtoconstructexplicitisomorphismsbetweencylindersovernon-isomorphic
members of the new class, providing by that new counter-examples to the cancellation problem.
1. Preliminaries
In this section we briefly recall basic facts on filtered algebra and their relation with derivation in a form
appropriate to our needs.
Inthesequel,unlessotherwisespecifiedB willdenoteacommutativedomainoverafieldkofcharacteristic
zero. The set Z of non-negative integers will be denoted by N.
>0
1.1. Filtrations and associated graded algebras.
Definition 1.1. An N-filtration of B is a collection {Fi}i∈N of k-sub-vector-spaces of B with the following
properties:
1- F ⊂F for all i∈N .
i i+1
2- B =∪i∈NFi .
3- F .F ⊂F for all i,j ∈N .
i j i+j
The filtration is called proper if the following additional property holds:
4- If a∈F \F and b∈F \F , then ab∈F \F .
i i−1 j j−1 i+j i+j−1
There is a one-to-one correspondence between proper N-filtrations and so called N-degree functions:
Definition 1.2. An N-degree function on B is a map deg : B −→N∪{−∞} such that, for all a,b∈ B, the
following conditions are satisfied:
(1) deg(a)=−∞ ⇔ a=0.
(2) deg(ab)=deg(a)+deg(b).
(3) deg(a+b)≤max{deg(a),deg(b)}.
Iftheequalityin(2)replacedbytheinequalitydeg(ab)≤deg(a)+deg(b),wesaythatdegisanN-semi-degree
function.
Indeed, for an N-degree function on B, the sub-sets F = {b ∈ B | deg(b) ≤ i} are k-subvector spaces of
i
B that give rise to a proper N-filtration {Fi}i∈N. Conversely, every proper N-filtration {Fi}i∈N, yields an
N-degree function ω :B −→N∪{−∞} defined by ω(0)=−∞ and ω(b)=i if b∈F \F .
i i−1
Definition 1.3. Given a k-domain B = ∪i∈NFi equipped with a proper N-filtration, the associated graded
algebra Gr(B) is the k-vector space
Gr(B)=⊕i∈NFi/Fi−1
2
equipped with the unique multiplicative structure for which the productof the elements a+F ∈F /F
i−1 i i−1
and b+F ∈F /F , where a∈F and b∈F , is the element
j−1 j j−1 i j
(a+F )(b+F ):=ab+F ∈F /F .
i−1 j−1 i+j−1 i+j i+j−1
Property 4 for a proper filtration in Definition 1.1 ensures that Gr(B) is a commutative k-domain when B
is an integral domain. Since for each a ∈ B \{0} the set {n ∈ N | a ∈ F } has a minimum, there exists i
n
such that a ∈F and a ∈/ F . So we can define a k-linear map gr : B −→ Gr(B) by sending a to its class
i i−1
in F /F , i.e a 7→ a+F , and gr(0) = 0. We will frequently denote gr(a) simply by a. Observe that
i i−1 i−1
gr(a)=0 if and only if a=0.
Denote by deg the N-degree function deg : B −→ N∪{−∞} corresponding to the proper N-filtration
{Fi}i∈N. We have the following properties.
Lemma 1.4. Given a,b∈B the following holds:
P1) ab=ab, that is gr is a multiplicative map.
P2) If deg(a)>deg(b), then a+b=a.
P3) If deg(a)=deg(b)=deg(a+b), then a+b=a+b.
P4) If deg(a)=deg(b)>deg(a+b), then a+b=0. In particular, gr is not an additive map in general.
Proof. Letusassumethatdeg(a)=ianddeg(b)=j. Bydefinition,deg(ab)=i+j meansthatab∈F and
i+j
ab∈/ F ,soab=ab+F :=(a+F )(b+F )=ab. WhichgivesP1. ForP2weobservethatsince
i+j−1 i+j−1 i−1 j−1
deg(a+b)=deg(a),wehavea+b=(a+b)+F =(a+F )+(b+F ),andsinceF ⊂F ⊆F as
i−1 i−1 i−1 j−1 j i−1
i>j, we get b+F =0. P3) is immediate, by definition. Finally, assume by contradiction that a+b6=0,
i−1
thena+b=(a+F )+(b+F )=((a+b)+F )6=0,whichmeansthata+b∈/ F anddeg(a+b)=i,
i−1 i−1 i−1 i−1
which is absurd. So P4 follows. (cid:3)
1.2. Derivations.
Byak-derivation ofB,wemeanak-linearmapD :B −→B whichsatisfiestheLeibnizrule: Foralla,b∈B;
D(ab)=aD(b)+bD(a). The set of all k-derivations of B is denoted by Der (B).
k
The kernel of a derivation D is the subalgebra kerD ={b∈B;D(b)=0} of B.
The plinth ideal of D is the ideal pl(D)=kerD∩D(B) of kerD, where D(B) denotes the image of B.
An element s∈B such that D(s)∈ker(D)\{0} is called a local slice for D.
Definition 1.5. Given a k-algebra B =∪i∈NFi equipped with a proper N-filtration, a k-derivation D of B
is saidto respect the filtrationif there exists aninteger dsuch that D(F )⊂F for all i∈N. The smallest
i i+d
integer d, such that D(Fi) ⊂ Fi+d for all i ∈ N, is called the degree of D with respect to F = {Fi}i∈N and
denoted by deg D.
F
Note that if D respects the filtration F = {Fi}i∈N then degFD is well-defined. Indeed, denote by deg the
N-degreefunctioncorrespondingto F ={Fi}i∈N andletU be the non-emptysubsetofZ∪{−∞}definedby
U :={deg(D(b))−deg(b) ; b∈B\{0}}. Since D respectsthe filtrationF ={Fi}i∈N, the set U is bounded
above by d. Thus it has a greatest element d which is exactly deg D by definition.
0 F
Suppose that D respects the filtration F = {Fi}i∈N and let d = degFD, we define a k-linear map D :
Gr(B)−→Gr(B) as follows: If D =0, then D =0 the zero map. Otherwise, if D 6=0 then we define
D :F /F −→F /F
i i−1 i+d i+d−1
by the rule D(a+F ) = D(a)+F . Now extend D to all of Gr(B) by linearity. One checks that D
i−1 i+d−1
satisfies the Leibniz rule, therefore it is a homogeneous k-derivation of Gr(B) of degree d, that is, D sends
homogeneous elements of degree i to zero or to homogeneous elements of degree i+d .
Observe that D =0 if and only if D =0. In addition, gr(kerD)⊂kerD.
2. LND-Filtrations and New Invariant sub-algebras
In this section we introduce the ∂-filtration associated with a locally nilpotent derivation ∂. We explain
how to compute this filtration and its associated graded algebra in certain situations. Also we present new
invariants that generalize the Derksen invariant.
3
Definition 2.1. A k-derivation ∂ ∈ Der (B) is said to be locally nilpotent if for every a ∈ B, there exists
k
n ∈ N (depending of a) such that ∂n(a) = 0. The set of all locally nilpotent derivations of B is denoted by
LND(B).
In particular, every locally nilpotent derivation ∂ of B gives rise to a proper N-filtration of B by the sub-
spaces F = ker∂i+1, i ∈ N, that we call the ∂-filtration. It is straightforward to check (see [F, Prop. 1.9])
i
that the ∂-filtration corresponds to the N-degree function deg :B −→N∪{−∞} defined by
∂
deg (a):=min{i∈N|∂i+1(a)=0}, and deg (0):=−∞.
∂ ∂
Note that by definition F =ker∂ and that F \F consists of all local slices for ∂.
0 1 0
Let Gr∂(B)=⊕i∈NFi/Fi−1 denote the associatedgradedalgebrarelative to the ∂-filtration{Fi}i∈N. Let
gr
gr : B −→ Gr (B); a 7−→∂ a be the natural map between B and Gr (B) defined in 1.3, where a denote
∂ ∂ ∂
gr (a).
∂
The nextProposition,whichis due to Daigle ([F, Theorem2.11],seealso[D3, Theorem1.7andCorollary
4.12]),impliesinparticularthatifBisoffinitetranscendencedegreeoverk,theneverynon-zeroD ∈LND(B)
respects the ∂-filtration and therefore induces a non-zero homogeneous locally nilpotent derivation D of
Gr (B).
∂
Proposition 2.2. (Daigle) Suppose that B is a commutative domain, of finite transcendence degree over
k. Then for every pair D ∈ Der (B) and ∂ ∈ LND(B), D respects the ∂-filtration. Consequently, D is a
k
well defined homogeneous derivation of the integral domain Gr (B) relative to this filtration, and it is locally
∂
nilpotent if D is locally nilpotent.
2.1. New Invariants (AL -invariants).
i
Definition 2.3. Let ∂ ∈ LND(B) be non-zero and let {Fi}i∈N be the ∂-filtration. We denote by L∂ the
sub-algebra of B generated by F =ker∂2 and we call it the ring of local slices for ∂. The sub-algebra of B
1
generated by L for all non-zero locally nilpotent derivation of B will be called the ring of all local slices of
∂
B. It will be denoted by AL(B) and referred to as the AL-invariant, which is manifested by the fact that
AL(B) is invariant by all algebraic k-automorphisms of B.
In a sense, the AL-invariant is a generalization of the Derksen invariant D(B) which is defined to be the
sub-algebra of B generated by ker∂ for all non-zero LND(B).
In a more general way we define the following invariants for non-rigid k-domains. Let AL (B) denotes
i
the sub-algebra of B generated by ker∂i+1 for all non-zero locally nilpotent derivation of B, then it is
clear that AL (B) is invariant by all algebraic k-automorphisms of B. Indeed, let ∂ ∈ LND(B)\{0} and
i
α ∈ Aut (B), then (α−1∂α)n = α−1∂nα for every n ∈ N, which implies that ∂ := α−1∂α ∈ LND(B).
k α
Therefore, α.∂n =∂n.α so we get deg (b)=deg (α(b)) for any b∈B. In other words, α respects deg and
α ∂α ∂ ∂
deg , that is, α sends an element of degree i relative to deg , to an element of the same degree i relative
∂α ∂α
to deg .
∂
Note that AL (B)=D(B), AL (B)=AL(B), and AL (B)⊆AL (B) for all i.
0 1 i i+1
In the case where B = k[X ,...,X ]/I = k[x ,...,x ] is a finitely generated k-domain that admits a
1 n 1 n
non-zero ∂ ∈ LND(B), the chain of inclusions AL (B) ֒→ AL (B) ֒→ AL (B) ֒→ ··· ֒→ AL (B) ֒→ ··· will
0 1 2 i
eventually stabilize, that is, there exists d ∈ N such that AL (B) = B. Indeed, by definition of LND, there
d
existd1,...,dn ∈Nsuchthat∂di+1(xi)=0. Denoteby{Fi}i∈Nthe∂-filtrationandletd=maxi∈{1,...,n}{di},
then x ∈F for all i. Since F ⊂AL (B) by definition, we see that B ⊆AL (B) and we are done.
i d d d d
Recall that the Makar-Limanov invariant ML(B) is defined to be the intersection of the kernels of all
locally nilpotent derivations on B. One might think that ALi∈N-invariants, in addition to the ML-invariant,
cover all invariant sub-algebras of B. This, however,is not correct, see §4.1.3 below.
2.2. Computing the ∂-filtration and its associated graded algebra.
Here, given a finitely generated k-domain B, we describe a general method which enables the computation
of the ∂-filtration for a locally nilpotent derivation with finitely generated kernel. First we consider a more
general situation where the plinth ideal pl(∂) is finitely generated as an ideal in ker∂ then we deal with the
case where ker∂ is itself finitely generated as a k-algebra.
4
2.2.1. LetB =k[X ,...,X ]/I =k[x ,...,x ]beafinitelygeneratedk-domain,andlet∂ ∈LND(B)besuch
1 n 1 n
that pl(∂) is generated by precisely m elements say f1,...,fm as an ideal in ker∂. Denote by F = {Fi}i∈N
the ∂-filtration, then: By definition F = ker∂. Furthermore, given elements s ∈ F such that ∂(s ) = f
0 i 1 i i
for every i∈{1,...,m}, it is straightforwardto check that
F = F s +...+F s +F .
1 0 1 0 m 0
Letting deg (x )= d , we denote by H the F -sub-module in B generated by elements of degree j relative
∂ i i j 0
to deg of the form su1...sumxv1...xvn, that is,
∂ 1 m 1 n
H := F (su1...sumxv1...xvn)
j 0 1 m 1 n
Plul+XPidi.vi=j
whereul,vi ∈Nforalliandl. Theinteger lul+ idivi isnothingbutdeg∂(su11su22....summxv11xv22.....xvnn).
Then we define a new N-filration G ={Gi}i∈N of B by setting
P P
G = H .
i j
j≤i
X
By construction G ⊆ F for all i ∈ N, with equality for i = 0 and i = 1. The following result provides a
i i
characterizationof when these two filtrations coincide:
Lemma 2.4. The filtrations F and G are equal if and only if G is proper.
Proof. One direction is clear since F is proper. Conversely,suppose that G is proper with the corresponding
N-degree function ω on B (see §1.1). Given f ∈F \F , i>1, for every local slice s∈F \F , there exist
i i−1 1 0
f 6=0,a 6= 0,a ,...,a ∈F such that f f =a si+a si−1+···+a (see [ML1, Proof of Lemma 4]).
0 i i−1 0 0 0 i i−1 0
Since ω(g)=0 (resp. ω(g)=1) for every g ∈F (resp. g ∈F \F ), we obtain
0 1 0
ω(f)=ω(f f)=ω(a si+a si−1+···+a )=max{ω(a si),...,ω(a )}=i,
0 i i−1 0 i 0
and so f ∈G . (cid:3)
i
2.2.2. The twisting technique.
Next, we determine the ∂-filtration, for a locally nilpotent derivation ∂ with finitely generated kernel, by
giving an effective criterion to decide when the N-filtration G defined above is proper.
Hereafter, we assume that 0 ∈ Spec(B) and that ker(∂) is generated by elements z ∈ B such that
j
z (0,...,0)= 0, j ∈ {1,...,r}. Since ker(∂) is finitely generated k-algebra, the plinth ideal pl(∂) is finitely
j
generated. So there exist s , ..., s ∈F such that F =F s +...+F s +F . We can also assume that
1 m 1 1 0 1 0 m 0
s is irreducible and s (0,...,0)=0 for all i.
i i
LettingJ ⊂k[r+n+m] =k[Z ,...,Z ][X ,...,X ][S ,...,S ]betheidealgeneratedbyI andtheelements
1 r 1 n 1 m
Z =z (X ,...,X ), j ∈{1,...,r}, S =s (X ,...,X ), i∈{1,...,m}, then we have
j j 1 n i i 1 n
B =k[Z ,...,Z ][X ,...,X ][S ,...,S ]/J.
1 r 1 n 1 m
Note that by construction 0∈Spec(k[r+n+m]/J).
We define an N-weight degree function ω on k[r+n+m] by declaring that ω(Z ) = 0 = deg (z ) for all
i ∂ i
i∈{1,...,r}, ω(S )=1 =deg (s ) for all i ∈{1,...,m}, and ω(X )= deg (x )= d for all i∈ {1,...,n}.
i ∂ i i ∂ i i
The corresponding proper N-filtration Q := {P ∈ k[n] | ω(P) ≤ i}, i ∈ N, on k[r+n+m] has the form
i
Q =⊕ H where
i j≤i j
Hj :=⊕Pjuj+Pidivi=jk[Z1,...,Zr]S1u1...SmumX1v1...Xnvn.
By construction π(Q )=G where π :k[r+n+m] −→B denotes the natural projection. Indeed, since
i i
π(Q )= π(H )
i j
j≤i
X
and π(Hj)= Pjuj+Pidivi=jk[z1,...,zr]su11....summxv11...xvnn, we get
P π(H )= (ker∂)su1....sumxv1....xvn
j 1 m 1 n
Pjuj+XPidivi=j
which means precisely that π(Q )=G .
i i
5
Let Jˆ⊂ k[r+n+m] be the homogeneous ideal generated by the highest homogeneous components relative
to ω ofallelements in J. Then we havethe following result,which is inspiredby the technique developedby
S. Kaliman and L. Makar-Limanov:
Proposition 2.5. The N-filration G is proper if and only if Jˆis prime.
Proof. It is enough to show that G ={π(Qi)}i∈N coincides with the filtration corresponding to the N-semi-
degree function ω on B defined by
B
ω (p):= min {ω(P)}.
B
P∈π−1(p)
Indeed, if so, the result will follow from [K-ML1, Lemma 3.2] which asserts in particular that ω is an
B
N-degree function on B if and only if Jˆis prime. Let {Gi′}i∈N be the filtration corresponding to ωB. Given
′ ′
f ∈ G there exists F ∈ Q such that π(F) = f, which means that G ⊂ π(Q ). Conversely, it is clear that
i i i i
ω (z )=ω(Z )=0foralli∈{1,...,r}. Furthermoreω (s )=ω(S )=1foralli∈{1,...,m},forotherwise
B i i B i i
si ∈ker∂ which is absurd. Finally, if di 6=0 and ωB(xi)<ω(Xi)=di 6=0, then xi ∈π(Qdi−1)⊂ker∂di−1
which implies that deg (x ) < d , a contradiction. So ω (x ) = d . Thus ω (f) ≤ i for every f ∈ π(Q )
∂ i i B i i B i
which means that π(Q )⊂G′. (cid:3)
i i
The next Proposition, which is a reinterpretation of [K-ML1, Prop. 4.1], describes the associated graded
algebra Gr (B) of the filtered algebra (B,F) in the case where the N-filtration G is proper:
∂
Proposition 2.6. If the N-filtration G is proper then Gr (B)≃k[r+n+m]/Jˆ.
∂
Proof. By virtue of ([K-ML1, Prop. 4.1]) the graded algebra associated to the filtered algebra (B,G) is
isomorphic to k[r+n+m]/Jˆ. So the assertion follows from Lemma 2.4. (cid:3)
3. Semi-Rigid k-Domains
3.1. Definitions and basic properties.
In [F-M], D. Finston and S. Maubach considered rings B whose sets of locally nilpotent derivations are
“one-dimensional” in the sense that LND(B) = ker(∂).∂ for some non-zero ∂ ∈LND(B). They called them
almost-rigid rings. Hereafter, we consider the following definition which seems more natural in our context
(see Prop. 3.3 below for a comparison between the two notions).
Definition 3.1. A commutativedomainB overa fieldk of characteristiczerois calledsemi-rigid if allnon-
zero locally nilpotent derivations of B induce the same proper N-filtration (equivalently, the same N-degree
function).
The unique proper N-filtration of a semi-rigid k-domain B, that corresponds to any non-zero ∂ ∈ LND(B),
will be referred to and called the unique LND-filtration of B.
Semi-rigid k-domains B can be equivalently characterized in terms of their Makar-Limanov invariant
ML(B):=∩ ker(D) as follows:
D∈LND(B)
Proposition3.2. A k-domain B is semi-rigid ifand onlyif ML(B)=ker(∂) for any non-zero∂ ∈LND(B).
Proof. Given D,E ∈LND(B)\{0} such that A:=ker(D)= ker(E), there exist non-zero elements a,b∈A
such that aD = bE ([F, Principle 12]) which implies that the D-filtration is equal to the E-filtration. So
if ML(B) = ker(∂) for any non-zero ∂ ∈ LND(B) then B is semi-rigid. The other implication is clear by
definition. (cid:3)
Recall that D ∈Der (B) is irreducible if and only if D(B) is contained in no proper principal ideal of B,
k
and that B is said to satisfy the ascending chain condition (ACC) on principal ideals if and only if every
infinite chain (b )⊂(b )⊂(b )⊂··· of principal ideals of B eventually stabilizes. B is said to be a highest
1 2 3
common factor ring, or HCF-ring, if and only if the intersection of any two principal ideals of B is again
principal.
Proposition 3.3. Let B be a semi-rigid k-domain satisfying the ACC on principal ideals. If ML(B) is an
HCF-ring, then there exists a unique irreducible ∂ ∈ LND(B) up to multiplication by unit. Consequently,
every D ∈LND(B) has the form D =f∂ for some f ∈ker(∂), and so B is almost rigid.
6
Proof. Existence follows from the fact that since B satisfies the ACC on principal ideals, then for every
non-zero T ∈ LND(B), there exists an irreducible T ∈ LND(B) and c ∈ ker(T) such that T = cT . ([F,
0 0
Prop. 2.2 and Principle 7]). The argument for uniqueness is similar to that in [F, Prop. 2.2.b], but with
a little difference, that is, in [F] it is assumed that B itself is an HCF-ring while here we only require that
ML(B) is an HCF-ring. Namely, let D,E ∈ LND(B) be irreducible derivations, and let A = ML(B). By
hypothesis ker(D) = ker(E) = A, so there exist non-zero a,b ∈ A such that aD = bE ([F, Principle 12]).
Herewecanassumethata,barenotunitsotherwisewearedone. SetT =aD =bE. SinceAisanHCF-ring,
there exists c∈A such that aA∩bA=cA. Therefore, T(B)⊂cB, and there exists T ∈LND(B) such that
0
T =cT . Write c=as=bt for s,t∈B. Then cT =asT =aD implies D =sT , and likewise E =tT . By
0 0 0 0 0
irreducibility, s and t are units of B, and we are done. (cid:3)
3.2. Elementary examples of semi-rigid k-domains.
3.2.1. Polynomial rings in one variable over rigid k-domains. Recall that a k-domain A is called
rigid if the zero derivation is the only locally nilpotent k-derivation of A. Equivalently, A is rigid if and
only if ML(A)=A. The next Proposition, which is due to Makar-Limanov ([ML1, Lemma 21], also [C-ML,
Theorem 3.1]), presents the simplest examples of semi-rigid k-domains.
Proposition 3.4. (Makar-Limanov) Let A be a rigid domain of finite transcendence degree over a field k
of characteristic zero. Then the polynomial ring A[x] is semi-rigid.
Proof. Fortheconvenienceofthereader,weprovideanargumentformulatedintheLND-filtrationlanguage.
Let∂ be the locally nilpotent derivationofA[x] defined by ∂(a)=0 for everya∈A and∂(x)=1. Then the
∂-filtration {Fi}i∈N is given by Fi =Axi⊕Fi−1 where F0 =ker(∂)=A, and the associated graded algebra
is Gr(A[x]) = ⊕i∈NAxi, where x := gr(x) and A = A. Since A[x] is of finite transcendence degree over k,
Proposition 2.2 implies that every non-zero D ∈LND(A[x]) respects the ∂-filtration and induces a non-zero
homogeneous locally nilpotent derivation D of Gr(A[x]) of a certain degree d=deg (D)≥−1. It is enough
∂
to check that in fact d = −1. Indeed, if so then D = a∂, for some a ∈ A which implies the semi-rigidity of
A[x]. So suppose for contradiction that d ≥ 0, then D(x) ∈ F = Axd+1+F . Therefore, D sends x to
d+1 d
0
zero or to axd+1. Either way, we have x ∈ ker(D), see Corollary 1.20 [F]. Furthermore, D(a) = ,
(a0xd
0
soD =xdE where E(a)= andE(x)=0. This assertsthatE ∈LND(A[x]) by virtue of [F, Principle
(a0
7]. Clearly, E restricts to LND(A), so by hypothesis E =0 which yields D =0, a contradiction. (cid:3)
3.2.2. Danielewski k-domains. Let
B =k[X,S,Y]/hXnY −P(X,S)i
n,P
where n ≥ 1, d ≥ 2, P(X,S) = Sd +f (X)Sd−1 +···+f (X), and f (X) ∈ k[X]. We call B the
d−1 0 i n,P
Danielewski k-domain corresponding to the pair (n,P). Let x, s, y be the class of X, S, and Y in B .
n,P
It is well known (see [ML1, Section 4] for the case P ∈ k[S]; and [P, Section 2.4] for the case, where
P(X,S)∈ k[X,S]) that if n ≥ 2, then ML(B ) = k[x] and LND(B ) = k[x].∂, for the locally nilpotent
n,P n,P
derivation ∂ of B defined by
∂P
∂ =xn∂ + ∂ ,
s y
∂s
where ∂P =dsd−1+(d−1)f (x)sd−2+...+f (x). Hence, B is almost rigid.
∂s d−1 1 n,P
We easily recover these previous results using the LND-filtration method as follows:
The ∂-filtration {Fi}i∈N of Bn,P is given by:
F =k[x]sjyi+F .
di+j di+j−1
wherei∈Nandj ∈{0,...d−1}. TheassociatedgradedalgebraisGr (B )=k[X,S,Y]/hXnY −Sdi,and
∂ n,P
B :=F /F =k[x]sjyi where i∈N and j ∈{0,...d−1} (see §4.3 for more details). Corollary
[di+j] di+j di+j−1
4.6below provides,inparticular,analternativeargumentformulatedinthe LND-filtrationlanguageproving
directly that {Fi}i∈N is indeed the unique LND-filtration of B.
7
3.3. Algebraic isomorphisms between semi-rigid k-domains.
Let Ψ : A −→ B be an algebraic isomorphism between two k-domains. Given ∂ ∈ LND(B), then for any
n∈N we have (Ψ−1∂Ψ)n =Ψ−1∂nΨ. So we see that ∂ :=Ψ−1∂Ψ∈LND(A). An immediate consequence
Ψ
is that Ψ(ML(A)) = ML(B). Furthermore, Ψ{ker(∂ )} = ker(∂), and more generally, Ψ sends elements of
Ψ
degree n, relative to ∂ , to elements of the same degree n relative to ∂, that is, deg (a) = deg (Ψ(a)) for
Ψ ∂Ψ ∂
all a ∈ A. So in particular, Ψ(AL(A)) = AL(B). These properties, combined with Definition 3.1, give the
following result.
Proposition3.5. LetΨ:A−→B bean isomorphism between twosemi-rigid k-domains. Let{Fi}i∈N (resp.
{Gi}i∈N ) be the unique LND-filtration of A (resp. B). Then: Ψ(Fi)=Gi for every i.
In the case where A = B, we obtain an action of the group Aut (B) of algebraic k-automorphisms of
k
B by conjugation on LND(B). As a consequence of Proposition 3.5, every k-automorphism of a semi-rigid
k-domain B preserves its unique LND-filtration {Fi}i∈N. Letting Autk(B,ML(B)) be the sub-group of
Aut (B) consisting of elements whose induced action on ML(B) is trivial, we have the following Corollary
k
which describes the structure of Aut (B).
k
Corollary 3.6. For every semi-rigid k-domain B, there exists an exact sequence
0→Aut (B,ML(B))→Aut (B)→Aut (ML(B)).
k k k
Furthermore, every element of Aut (B,ML(B)) induces for every i ≥ 1 an automorphism of F -module of
k 0
each F .
i
4. A new class of semi-rigid k-domains
In this section, we introduce a new family of domains R of the form
n,e,P,Q
R :=k[X,Y,Z]/hXnY −P (X,Q(X,Y)−XeZ)i
n,e,P,Q
where e≥0, n≥1, (n,e)6=(1,0), d,m≥2,
P(X,S)=Sd+f (X)Sd−1+···+f (X)S+f (X) , and
d−1 1 0
Q(X,Y)=Ym+g (X)Ym−1+···+g (X)Y +g (X).
m−1 1 0
The trivial case (e=0), corresponds to the Danielewski k-domains B =k[X,S,Y]/hXnY −P(X,S)i.
n,P
Indeed, the ring R = k[X,Y,Z]/hXnY −P(X,Q(X,Y)−Z)i is isomorphic to B via an isomor-
n,0,P,Q n,P
phism induced by Φ:k[X,S,Y]−→k[X,Y,Z],where Φ(X)=X, Φ(S)=−Z+Q(X,Y), andΦ(Y)=Y. It
is clear that Φ∗ =π ◦Φ is surjective,where π :k[X,Y,Z]→R
XnY−P(X,Q(X,Y)−Z) XnY−P(X,Q(X,Y)−Z) n,e,P,Q
is the natural projection. Thus R = ImΦ∗ ≃ k[X,S,Y]/kerΦ∗. This yields, in particular, that the
n,0,P,Q
ideal kerΦ∗ ⊂ k[3] is principal. But since Φ∗(XnY −P(X,S)) = 0, hFi ⊂ kerΦ∗, and XnY −P(X,S) is
irreducible, we conclude that hXnY −P(X,S)i = kerΦ∗. Therefore, Φ∗ induces an isomorphism between
the two rings.
Remark 4.1. Computing the ML-invariant for these examples using known techniques up to date is rather
a hopeless task. Indeed, for the non-trivial case of R where e 6= 0, a real-valued weight degree
n,e,P,Q
function ω on k[3] has to be of the form ω = ( md−1 λ, n λ,λ), where λ ∈ R, to induce a degree
nm−md+1 nm−md+1
functionω onR . Hence, the associatedgradedalgebra,correspondingto ω -filtration,takesthe form
0 n,e,P,Q 0
Gr (R ) = k[X,Y,Z]/hXnY −(Ym −XeZ)di. The latter ring is again another member of the new
ω n,e,P,Q
family that corresponds to R = Gr (R ). So any hope of simplifying the study of locally
n,e,Sd,Ym ω n,e,P,Q
nilpotent derivation of R , by studying the homogenous locally nilpotent derivation on the associated
n,e,P,Q
graded algebra R , collapses. On the other hand, the remaining choices of ω in R[3] induces a semi-
n,e,Sd,Ym
degree function on R with the associated graded algebra Gr (R )=k[X,Y,Z]/h(Ym−XeZ)di.
n,e,P,Q ω n,e,P,Q
This is not an integral domain, which complicates the situation even more.
Nevertheless, the LND-filtration method allows us to pass through these complications as will be shown
in the rest of this paper. Indeed, consider ω ∈ N[4] the N-weight degree function on k[4] defined by
ω(X,S,Y,Z)=(0,1,d,md), then it induces ω a degree function on
Rn,e,P,Q
R ≃k[X,S,Y,Z]/hXnY −P (X,S),Q(X,Y)−XeZ−Si.
n,e,P,Q
8
It turns out that the degree function ω coincides with deg for any non-zero ∂ ∈LND(R ).
Rn,e,P,Q ∂ n,e,P,Q
4.1. Properties of the new class.
Here, we point out some properties of R that we will establish in the rest of this section:
n,e,P,Q
4.1.1. Algebraic construction: Consider the following k-domain
B =k[X,S,Y]/hXnY −P(X,S)i,
n,P
which is the Danielewski k-domain corresponding to the pair (n,P). Let us extend this ring by taking the
sub-algebra of k[X±1,S] generated by B ⊂k[X±1,S] and z ∈k[X±1,S], where z is an algebraic element
n,P
over k[X,S] that has a dependence relation of the form
Xnm+eZ−[P(X,S)]m−Xg (X)[P(X,S)]m−1−···−Xm−1g (X)P(X,S)−Xmg (X)+XnmS.
m−1 1 0
By sending S to Q(X,Y)−XeZ we immediately see that:
B ⊂R , and B ⊂R ,
n,P n,e,P,Q nm+e,F n,e,P,Q
where B is the Danielewski k-domain corresponding to the pair (nm+e,F):
nm+e,F
B =k[X,S,Z]/hXnm+eZ−F(X,S)i,
nm+e,F
and F(X,S)=[P(X,S)]m+Xg (X)[P(X,S)]m−1+···+Xm−1g (X)P(X,S)+Xmg (X).
m−1 1 0
Clearly, we have B .B = R , which simply means that R can be realized as the
n,P nm+e,F n,e,P,Q n,e,P,Q
sub-algebra of k[X±1,S] generated by both B and B .
n,P nm+e,F
These new rings R , for e 6= 0, are not isomorphic to any of Danielewski rings, see Proposition
n,e,P,Q
4.13. Nevertheless,theysharewiththemthepropertytocomenaturallyequippedwithanirreduciblelocally
nilpotent derivation. But in contrast with the Danielewski rings, the corresponding derivation on k[X,Y,Z]
are no longer triangular, in fact not even triangulable by virtue of the characterization due to Daigle [D1].
For instance: let D be the locally nilpotent (triangular) derivation of k[X,S,Y,Z] defined by:
∂(X)=0, ∂(S)=Xn+e, ∂(Y)=Xe∂P, and ∂(Z)= ∂Q ∂P −Xn
∂S ∂Y ∂S
where ∂P =dSd−1+(d−1)f (X)Sd−2+···+f (X),and ∂Q =mYm−1+(m−1)g (X)Ym−2+···+g (X).
∂S d−1 1 ∂Y m−1 1
Then ∂ induces a non-zero irreducible locally nilpotent derivation of R . Let x,s,y,z be the class of
n,e,P,Q
X, S :=Q(X,Y)−XeZ, Y, and Z in R , then:
n,e,P,Q
∂P ∂Q ∂P
∂ =xe ∂ +( −xn)∂ ∈LND(R )
y z n,e,P,Q
∂s ∂y ∂s
Furthermore, ML(R ) = k[x] whenever (n,e) 6= (1,0), see Corollary 4.6. This implies that R
n,e,P,Q n,e,P,Q
is semi-rigid, even almost rigid by virtue of Proposition 3.3. Hence AL(R ) = k[X,S]. In addition,
n,e,P,Q
every non-zero locally nilpotent derivation of R restricts to a non-zero locally nilpotent derivation
n,e,P,Q
on B . And most importantly, every k-automorphism of R restricts to an automorphism of B .
n,P n,e,P,Q n,P
Also, it restricts to an k-automorphism of AL(R ) (resp. ML(R )). So in particular, every k-
n,e,P,Q n,e,P,Q
automorphismof B (≃R ) restricts to a k-automorphismof AL(R ) (resp. ML(R )). But
n,P n,0,P,Q n,e,P n,e,P,Q
of course this is not the full picture, see 4.1.3.
4.1.2. Affine modification of the AL-invariant: Here, we present another point of view about the
construction of the new class.
The affine modification of the AL-invariant AL(R )=k[X,S] along Xnm+e with center
n,e,P,Q
I =hXnm+e,Xn(m−1)+eP(X,S),F(X,S)i,
1
see [K-Z, Definition 1.1], coincides by virtue of [K-Z, Proposition 1.1] with
k[X,S] I /Xnm+e =k[X,S][P(X,S)/Xn,F(X,S)/Xnm+e]=k[X,S,y,z]≃R .
1 n,e,P,Q
Also,theaffinemo(cid:2)dificationo(cid:3)ftheAL-invariantAL(Rn,e,P,Q)=k[X,S]alongXnwithcenterI2 =hXn,P(X,S)i
coincides with
k[X,S][I /Xn]=k[X,S][P(X,S)/Xn]=k[X,S,y]≃B .
2 n,P
Finally, the affine modification of B along Xe with center I =hXe,Q(X,Y)−Si coincides with
n,P 3
B [I /Xe]=B [(Q(X,Y)−S)/Xe])=B [z]≃R .
n,P 3 n,P n,P n,e,P,Q
9
We put together previous observations in the following Proposition.
Proposition 4.2. with the above notation we have:
(1) R is the affine modification of the AL-invariant along Xnm+e with center I .
n,e,P,Q 1
(2) B is the affine modification of the AL-invariant along Xn with center I .
n,P 2
(3) R is the affine modification of B along Xe with center I .
n,e,P,Q n,P 3
4.1.3. Invariant sub-algebras of Rn,e,P,Q: For simplicity let Q(X,Y) = Ym. Denote Rn,e,P :=
R and B ≃R . Consider the two chains of inclusions:
n,e,P,Ym n,P n,0,P
The first chain of inclusions, which is realized by sending S to Ym −XZ for the first inclusion and by
sending Z to XZ for the rest steps
B ֒→R ֒→···֒→R .
n,P n,1,P n,e,P
The second chain of inclusions, which is realized by sending Y to XY for every step
B ֒→B ֒→···֒→B .
1,P 2,P n,P
Together they produce the following chain of inclusions
B ֒→B ֒→···֒→B ֒→R ֒→···֒→R
1,P 2,P n,P n,1,P n,e,P
with the following properties.
Theorem 4.3. With the above notation the following holds:
(a) Every non-zero ∂ ∈LND(R ) restricts to a non-zero LND(R ) for any e ∈{1,...,e}. Also,
n,e,P n,e0,P 0
it restricts to a non-zero LND(B ) for any n ∈{0,...,n}.
n0,P 0
(b) Every algebraic k-automorphism of R restricts to an algebraic k-automorphism of R for any
n,e,P n,e0,P
e ∈{1,...,e}. Also, it restricts to a k-automorphism of B for any n ∈{1,...,n}.
0 n0,P 0
(c) R ≃ R if an only if n = n and e = e . Hence these k-domains are not algebraically
n1,e1,P n2,e2,P 1 2 1 2
isomorphic to each other (pairwise).
(d) Every element of the set
{ML(R )=k[X], AL(R )=k[X,S], B , R ; n ∈{1,...,n},e ∈{1,...,e}}
n,e,P n,e,P n0,P n,e0,P 0 0
represents an invariant sub-algebra of R .
n,e,P
(e) AL (R ) = D(B) = ML(R ) ֒→ AL(R )= AL (R )= ··· = AL (R )֒→ B =
0 n,e,P n,e,P n,e,P 1 n,e,P d−1 n,e,P n,P
AL (R )=···=AL (R )֒→AL (R )=R .
d n,e,P md−1 n,e,P md n,e,P n,e,P
Proof. (a) Is immediate by Corollary 4.6 below. (b) is an immediate consequence of Theorem 4.10 below.
(c) a consequence of Proposition 4.12 and Proposition 4.13 below. (a), (b), and Corollary 4.6 implies (d).
Finally, (e) is a trivial consequence of Theorem 4.5, and Definition 2.3. (cid:3)
4.2. A toy example.
We will begin with a very elementary example illustrating the steps needed to determine the LND-filtration
and its associated graded algebra, and then we proceed to the general case. We let
R=k[X,Y,Z]/hX2Y −(Y2−XZ)2i
and we let x, y, z be the class of X, Y, and Z in R. A direct computation reveals that the derivation
2XS∂ +(4YS−X2)∂
Y Z
of k[X,Y,Z] where S := Y2−XZ is locally nilpotent and annihilates the polynomial X2Y −(Y2−XZ)2.
Therefore, it induces a locally nilpotent derivation ∂ of R for which we have ∂(x)=0, ∂3(y)=0,∂5(z)=0.
Furthermore, the element s = y2 − xz is a local slice for ∂ with ∂(s) = x3. So we have deg (x) = 0,
∂
deg (y) = 2, deg (z) = 4, deg (s) = 1. The kernel of ∂ is k[x] and the plinth ideal is the principal ideal
∂ ∂ ∂
generated by x3.
Proposition 4.4. With the notation above, we have:
(1) The ∂-filtration {Fi}i∈N is given by :
F =k[x]slyjzi+F
4i+2j+l 4i+2j+l−1
where i∈N, j ∈{0,1}, l∈{0,1}.
10
