Last updated: 21 January 2015
It is advisable to view the action of certain finite groups of automorphisms on local regular rings or on graded algebras of polynomials over a field (i.e. on acts linearly.
One often has, in effect, a correspondence between the theorems over local regular rings and those overs graded rings of finite type over a field. The passage "local graded" is generally easy; the passage "graded local" less easy, it requires at least the "associated graded".
Situation: We are given a local regular ring and a finite group of automorphisms acting on We denote by the subring of formed by the invariant elements under The problem is to know under which hypotheses the ring is regular.
|-||In the case where one must determine in which cases is an algebra of polynomials.|
|-||In the language of scheme, the problem is formulated as: Given an affine scheme regular over being a local ring with residue field and a finite group of automorphisms acting on in which cases is a regular scheme?|
The passage of translating from one to the other is given by the following lemma:
Lemma. Let be a graded algebra of finite type over a field The following three assertions are equivalent:
|1.||is regular (as a scheme over|
|2.||is regular denotes the maximal ideal formed by the homogeneous elements of positive degree);|
|3.||is a graded algebra of polynomials.|
"Graded case" signifies: is a graded algebra of polynomials over a field
Definition. Let be a finite group acts on Assume where denotes the characteristic exponent of One says that an element of is a pseudo-reflection if is of rank less than or equal to 1, (i.e. if for all where is a fixed vector and is a linear form on
Example of a pseudo-reflection:
If the every pseudo-reflection can be put in the form (1) where is a root of unity.
Theorem 1. Let and a finite group of automorphisms of such that where is the characteristic exponent of The following assertions are equivalent:
|(a)||is an algebra of polynomials;|
|(b)||is generated by pseudo-reflections.|
This theorem has been proved by SHEPHARD-TODD, in , in the case
In the direction (a) (b), their proof is applicable to a field of arbitrary characteristic
In the direction (b) (a), their proof is the following: they make a list of the all the groups having this property, and they check that is indeed a polynomial algebra.
In 1955, CHEVALLEY has given a case free proof of the theorem.
Theorem 2. Let and a finite group of automorphisms of then: is a polynomial algebra) implies is generated by pseudo-reflections).
The converse is false starting at the case For and the orthogonal group for the quadratic form
Outline of the proof of SHEPHARD-TODD. The theorems of Shephard-Todd over pass to characteristic 0, even to characteristic such that
1st case: There exists a real structure on invariant under (i.e. imbeds in and the elements of may be written as matrices with real entries).
We have seen that, if was a pseudo-reflection, in the case (even here where it is always the case), was put in the form where was a root of unity. Since we thus have forced here.
Thus the elements of are in this case reflections (a reflection is a pseudo-reflection of order 2).
These groups are the groups studied in the works of Elie CARTAN, and are, with a few exceptions, the Weyl groups of semisimple Lie groups.
When is irreducible, it is represented by a Coxeter diagram; and the list of possible diagrams is well known.
2nd case: is truly a "complex" group.
SHEPHARD and TODD then draw up the list of all possible
For this, they send into In these transformations correspond to "homologies".
Then MITCHELL, in 1914, has made the list of finite groups generated by homologies. To obtain the finite subgroups of generated by reflections it suffices to lift these groups into
Among the irreducible groups one finds two kinds:
|The imprimitives: These are the groups where the dimension of the vector space, and is an integer such that divides These groups are variants of the symmetric group In effect, they acts as follows: is a primitive root of unity, and|
|(a)||In dimension 2, there are a large number, around 12.|
|(b)||In dimension one finds only a finite number.|
|In dimension 3, one finds:|
|Two groups of order 648 and 1296 which correspond to the same projective group;|
|One group of order 2160 (which is a 6-fold cover of its projective group which, itself, is isomorphic to|
|In dimension , one finds two groups of orders 46080 and 7680, and one group of order 25920.|
|In dimension , one finds one group of order|
|In dimension , one finds one group of order|
In the following, we make the following assumptions:
is a regular local ring with maximal ideal a finite subgroup of denotes the subring of invariant of by (we know that is local), and assume in addition:
|(ii)||is of finite type over and|
|(iii)||and have the same residue field of characteristic (totally ramified case)|
Let be the Zariski tangent space relative to Then acts linearly on and we denote by the map
Theorem If then is regular if and only if is generated by pseudoreflections.
Theorem If is regular then is generated by pseudoreflections.
Proof of Theorem
Let be the subgroups of generated by the elements such that is a pseudoreflection. Let thus we have the following inclusions: We will show that which will lead to, by Galois theory, that and Theorem
Lemma. is unramified over in codimension 1 (i.e. divisorially unramified).
Said another way, after localization by the prime ideals of height 1, the extension is unramified.
Said another way again, the ramification locus of the cover is of codimension greater that or equal to 2.
We show that the inertia group of all prime ideals of height 1 over is zero is the Galois group of the extension over For this, we show that if is a prime ideal of height 1 in such that is a prime ideal of height 1 in since is and integral domain, integral over and is integrally closed), we have which will give the desired result.
Now the inertia group of in is formed by the elements such that and that acts trivially on (same definition for Clearly we have the inclusion
It remains to prove the reverse inclusion. So, let so that for all and acts trivially on
Let Now the image of in (i.e. is a space of dimension or . In the case that there exists equivalently, there exists such that the ideal is prime.
(In effect, being of finite type over a noetherian it itself is noetherian; being local, noetherian, regular, and and then is regular and so is an integral domain, and is a prime ideal.)
Now Since is the prime ideal of height 1, it follows that Thus (taking images in and thus Reducing relation (2) modulo we obtain Thus is of dimension and is a pseudoreflection. It follows that
This lemma being proved, using the purity theorem of "Nagata-Auslander" , since is normal (as it is a ring of invariants), and is regular by hypothesis, the nonramification of over in codimension 1 implies the nonramification of over Since and are both local rings having the same residue field (since, by hypothesis, and have the same residue field) it follows that is Thus and is generated by pseudoreflections.
Proof of Theorem
Remark. The problem occurs equally when one considers the quotient of a bounded domain by a discrete subgroup If denotes a point of and its stabilizer (this is a finite group), then:
an analytic variety without singularities) (The are generated by pseudo-reflections).
Example of the singular case: Fix the point If is generated by the symmetries then is not generated by pseudo-reflections, and the quotient space is a quadratic cone, which thus has a singularity at the origin.
It is thus necessary to prove that, if and if is generated by pseudoreflections, then is regular (the other implication being given by theorem
Fix and two local rings such that is contained in
is of finite type over and integral over
is noetheriam and regular. Then the following assertions are equivalent:
is a free
(a) (b) As is a local regular ring, and an of finite type, one has: (since is integral over Now, we are in the following situation: and local noetherian, integral over the injection homomorphism is a local homomorphism; thus, as is an of finite type, we have Further, being local regular, is a Macaulay ring, whence By comparing the relations (3) and (4), we find: Therefore, is a projective since, by hypothesis, is an of finite type and is a local noetherian ring, this implies that is a free
(b) (a). Assume is a free We will show that is regular, and for that that is finite. Now, being free over is flat over whence being regular, has finite cohomological dimension, and thus also
It remains to show that is a free and theorem will be proved.
For this, it suffices to show that is zero (since is of finite type over Now is an Thus acts on it. Further, is an of finite type (since it is a finite dimensional vector space over Thus, if one shows that by Nakayama's lemma this will imply that But if:
|(a)||Every element of invariant under is zero, and|
|(b)||acts trivally on (i.e. every pseudo-reflection of acts trivially on since is generated by pseudo-reflections), then we will indeed have|
(a) Because is invertible in This thus makes the functor "invariants" exact, that is to say that the exact sequence of
(In fact, the only point to show is the surjectivity of Thus, let Because there exists such that (exact sequence (5)). If we then consider and since for all
Note (since is free over itself). Thus and due to the exact sequence (6),
(b) One may reduce to the case where and are complete. (In effect, for a local noetherian ring we have the equivalence: noetherian noetherian.)
Then, if an element of is a pseudo-reflection, there exist elements of such that where is a root of unity in And these elements form a basis of the space (see the definition of a pseudo-reflection).
We may lift the into elements of and then further and further, to elements of satisfying: and this, for any fixed positive integer
In effect, for example, we will sow how to execute the lift of to
We consider the following diagram where is the kernel of the following map
This diagram is formed of exact rows and columns, since and this is indeed the same diagram as in Cartan-Eilenberg (, exercise 1, chap. I), where and
Next, the elements which belong in fact to can be lifted to elements of thus of satisfying, On may execute the same operation for the element by replacing by the map and writing the analogous diagram.
Finally, as is complete, this proves to us that there exist elements such that As, in addition, one may lift a root of unity in to an element of Thus there exists and such that
We see then that In effect, let If let be the class of in thus there exists such that Then and where One is thus brought to study the case where
The case can be treated as follows: Since is complete: where (mod We put where is the class of in (then where and the lift of obtained by lifting the etc.
This shows us, after the completion of and from the fact that and have the same residue field, that Now this brings us to study the case Now, in all cases, if we have Next, for all can be put in the form By passing to the limit, since is complete, we see thus that is written as a formal power series in with coefficients in But being complete and the integral over thus belongs in fact to Next, we indeed have
We now consider acts trivially on (by construction of the In other words: and the map factorizes: where denotes multiplication by in Now the maps, are maps which operate on and we also have the following diagram Whence the inclusions As the image of acting on is zero, that is to say, acts trivially on (b) is thus proved, which permits us to conclude that is zero, and thus and complete the proof.
 AUSLANDER (Maurice). – On the purity of the branch locus, Amer. J. of Math., v. 84, 1962, p. 116-125.
 BOURBAKI (Nicolas). – Algèbre commutative. Chapitre 2: Localisation. – Paris, Hermann, 1961 (Act. scient. et ind., 1290; Bourbaki, 27)
 CARTAN (Henri) and EILENBERG (Samuel). – Homological algebra. – Princeton, Princeton University Press, 1956 (Princeton mathematical Series, 19).
 GROTHENDIECK (Alexander) and DIEUDONN\'E (Jean). – Eléments de géométrie algébrique. Chapitre 4: Etude locale des schémas et des morphismes de schémas. – Paris, Presses Universitaires de France, 1964 (Institut des hautes Etudes scientifiques. Publications mathématiques, 20).
 JACOBSON (Nathan). – Lie algebras. – New York, Interscience Publishers, 1962 (Interscience Tracts in pure and applied Mathematics, 10).
 SHEPHARD (G.C.) and TODD (J.A.). – Finite unitary reflection groups, Canad. J. of Math., v. 6, 1954, p. 274-304.
This is a translation of the paper
Ecole Normale Supérieure de Jeunes Filles
Colloque d'Algèbre [1967. Paris], no. 8, 11 p. Lecture Notes taken by Marie-José BERTIN by J. P. Serre. Translation to English by Arun Ram, January 2015.