Partial normalizations of Coxeter arrangements and discriminants

0.4 MB·English

by  Michel Granger

PARTIAL NORMALIZATIONS OF
COXETER ARRANGEMENTS AND DISCRIMINANTS
MICHELGRANGER,DAVIDMOND,ANDMATHIASSCHULZE
Tothe memory of V.I. Arnol’d
Abstract. WestudynaturalpartialnormalizationspacesofCoxeter arrangementsanddiscriminants
andrelatetheirgeometrytorepresentationtheory. TheunderlyingringstructuresarisefromDubrovin’s
Frobenius manifold structure which is lifted (without unit) to the space of the arrangement. We also
describeanindependentapproachtothesestructuresviadualityofmaximalCohen–Macaulayfractional
2
ideals. In the process, we find 3rd order differential relations for the basic invariants of the Coxeter
1
group. Finally,weshowthatourpartialnormalizationsgiverisetonewfreedivisors.
0
2

n
a
Contents
J

5 Introduction 1
2
Acknowledgments 4

1. Review of Coxeter groups 4
]
G 2. F-manifold-structures 6
A 3. Algebra structures on cokernels of square matrices 9
3.1. Rank condition 9
.
h
3.2. Rings associated to free divisors 10
t
a 4. Ring structures associated with Coxeter groups 12
m
4.1. Rank conditions and associated rings 12
[ 4.2. Relation of rings for A and D 13

4.3. Local trivialization 14
2
4.4. Relation with the normalization 16
v
8 4.5. Example 0.2 revisited 17
1 5. Dual and Hessian rank conditions 18
7 6. Free and adjoint divisors 20
0
References 23
.
8
0
1
1
Introduction
:
v
V.I. Arnol’d was the first to identify the singularities of type ADE, that is A , D , E , E or E , as
i ℓ ℓ 6 7 8
X the simple singularities – those that are adjacent to only finitely many other types. He also uncovered
r the links between the Coxeter groups of type B , C and F and boundary singularities, see [Arn79].
ℓ ℓ 4
a
In the latter paper his formulæ for generators of the module of logarithmic vector fields Der(−logD)
along the discriminant D parallels K. Saito’s definition of free divisors. Along with Brieskorn, Dynkin,
Gelfan’d, and Gabriel, Arnol’d revealed the ADE list as one of the central piazzas in mathematical
heaven, where representation theory, algebra, geometry and topology converge. As with so many of
Arnol’d’s contributions, his work on this topic has given rise to a huge range of further work by others.
Letf: X =(Cn,0)→(C,0)=SbeacomplexfunctionsingularityoftypeADEandletF: X×B →S
be aminiversaldeformationoff withbaseB =(Cµ,0). Writing f :=F(−,u),the discriminantD ⊂B
u
is the set of parameter values u ∈ B such that f−1(0) is singular. It is isomorphic to the discriminant
u
of the Coxeter group W of the same name. Here the discriminant is the set of exceptional orbits in
the orbit space V/W. This is only the most superficial feature of the profound link between singularity
theory and the geometry of Coxeter groups which Arnol’d helped to make clear.
Date:January26,2012.
1991 Mathematics Subject Classification. 20F55,17B66, 13B22.
Keywords and phrases. Coxeter group,logarithmicvector field,freedivisor.
1
The starting point of this paper is the fact, common to Coxeter groups and singularities, that D is
a free divisor (see e.g. [Her02, §4.3]) with a symmetric Saito matrix K whose cokernel is a ring in the
singularity case. By definition the Saito matrix K is the µ×µ-matrix whose columns are the coefficient
vectors of a basis of Der(−logD) with respect to a basis of the module DerB := DerC(OB) of vector
fields on B.
On the singularity theory side these two roles are well known. Let h be a defining equation for D.
Then K appears in the exact sequence
0 //Oµ K // Der dh //J // 0
B B D
which defines Der(−logD) as the vector fields which preserve the ideal of D. Let π : Σ → B denote
the restriction of the projection X ×B →B. If Σ ⊂ X ×B is the relative critical locus defined by the
Jacobian ideal Jrel of F relative to B, and Σ0 := Σ∩V(F) so that D =π(Σ0), then K also appears in
F
the exact sequence
(0.1) 0 //OBµ K //DerB dF //π∗OΣ0 // 0
in which dF maps a vector field η ∈Der to the function dF(η˜) on Σ0, where η˜is a lift of η to X×B,
B
and π : Σ →B is the restriction of the projection X ×B →B. As π O is free over O of rank µ, we
∗ Σ B
can make the identifications
π∗OΣ ∼=OBµ ∼=DerB,
and reinterpret K as the matrix of the O -linear operator induced on π O by multiplication by F,
B ∗ Σ
whose cokernel is also, evidently, π∗OΣ0.
Similar to the case of ADE singularities and corresponding Coxeter groups, Coxeter groups of type
B and F are linked with boundary singularities,for which a similar argumentshows that the cokernel
k 4
ofK isnaturallyaring. AlsofortheseandtheremainingCoxetergroupsI (k),H andH ,thecokernel
2 3 4
of K carries a natural ring structure. The simplest way to see this involves the Frobenius structure
constructed on the orbit space by Dubrovin in [Dub98], following K. Saito. Here the key ingredient is
a fiber-wise multiplication on the tangent bundle, which coincides with the multiplication coming from
O in the ADE singularity case. We recall the necessary details of Dubrovin’s construction, following
Σ
C.Hertling’saccountin[Her02],inSection2,inpreparationfortheproofofourmainresult. Thisstates
that also the cokernel of a transposed Saito matrix for the reflection arrangement of a Coxeter group
carries a natural ring structure.
Theorem 0.1.
(1) LetA be thereflection arrangement of a Coxeter group W acting on thevector space V ∼=Cℓ, let
p ,...,p be generators of the ring of W-invariant polynomials, homogeneous in each irreducible
1 ℓ
component of V, and let J be the Jacobian matrix of the map (p ,...,p ), which is a transposed
1 ℓ
Saito matrix for A. Then cokerJ has a natural structure of C[V]-algebra.
(2) Denoting SpeccokerJ by A˜, we have
(i) A˜is finite and birational over A (and thus lies between A and its normalization).
(ii) For x ∈ A, let W be the stabilizer of x in W and let X be the flat of A containing
x
x. There is a natural bijection between the geometric fiber of A˜ over x and the set of
irreducible summands in the representation of W on V/X.
x
(iii) Under the bijection of (2ii), smooth points of A˜correspond to representations of type A .
1
Example 0.2.
(1) In the case of A , the arrangementA consists of three concurrent coplanar lines. In this case A˜
2
is isomorphic to the union of the three coordinate axes in 3-space, for this is the only connected curve
singularity regular and birational over A but not isomorphic to it. More generally, in the case of A ,
ℓ
ℓ+1
with reflecting hyperplanes, A˜ is isomorphic to the codimension-2 subspace arrangement in
2
(cid:18) (cid:19)
(ℓ+1)-space consisting of the (ℓ−1)-planes L :={x =x =0} for 1≤i<j ≤ℓ+1. The projection
i,j i j
x 7→ x−x♯, where x♯ is x averaged by the action of the symmetric group S permuting coordinates,
ℓ
gives an S -equivariant map of A˜ to the standard arrangement A ⊂ { ℓ+1x = 0}, sending L
ℓ i=1 i i,j
isomorphically to {x =x }. We return to this example, and prove these assertions, in Subsection 4.5.
i j
(2) Figure1showsa2-dimensionalsectionofthehyperplanearrangemenPtA forA ,onthe left, and,
3
on the right, a topologically accurate view of the preimage of this section in A˜.
2
Figure 1. A and A˜for the Coxeter group A
3
A A˜
x4
=
x
x1 3
=
x
4 x
= x 4 x 1=
x 2 2=x x3
3
x1=x2
The planes {x =x } and {x =x } meet orthogonallyif i , i , i and i are all different, and the
i1 i2 i3 i4 1 2 3 4
reflections in these planes commute; it follows that at a point x in the stratum {x =x 6=x =x },
i1 i2 i3 i4
the representation is of type A ⊕A and by (2ii) of Theorem 0.1 above, the fiber of A˜over x consists
1 1
of two points. In each of these pictures there are four nodes of valency three. In the left hand picture,
eachlies ina1-dimensionalstratuminA˜wherethe localrepresentationisoftype A , sothatlocallyA
2
consists of three planes in 3-space, meeting along a common line. The preimage of this stratum in A˜is
a line, along which A˜is locally isomorphic to the union of the three planes he ,e i,he ,e i and he ,e i
1 4 2 4 3 4
in 4-space.
It would be interesting to find explicit embeddings of the space A˜in the remaining cases.
Toprovethetheorem,beginningwiththemultiplicativestructureonDer andcoker(K)comingfrom
B
Dubrovin’s Frobenius structure, we endow both Der and cokerJ with a multiplication, and Der with
V V
a DerB-module structure, whose crucial feature is that the derivative tp: DerV →DerB⊗OBOV of p is
Der -linear. On Der , but not on cokerJ, this multiplication lacks a neutral element.
B V
Nevertheless, the first evidence for the theorem was found by an entirely different route not involv-
ing Dubrovin’s Frobenius structure. This was based on the fact that the cokernel of the linear map
Sℓ Λ //Sℓ defined by a square matrix Λ has a natural S-algebra structure if and only if the so-called
rank condition (rc) holds. This is a purely algebraic condition on the adjugate matrix of Λ, which can
be checked by explicit calculation. We explain this in general in Section 3.
In Section 4, we then specialize to the case where Λ is the Jacobian matrix J of the basic invariants
of a Coxeter group A, or the Saito matrix of the discriminant D of a Coxeter group. The space D˜ =
SpeccokerK is normal (indeed smooth) exactly in the ADE-case; on the other hand A˜=SpeccokerJ
is normal only in the case of A . We discuss the geometry of these two spaces, and their link with
1
the representation theory. In particular we compare them with the normalizations of D and A in
Subsection 4.4.
In Section 5, our earlier approach to the main theorem lead to an interesting problem on Coxeter
groups. The algebra of the fiber over 0 of the projection p: V →V/W carries two structures: that of a
zero-dimensionalGorenstein algebra and that of the regularW-representation. It is not clear how these
two structures are related: which irreducible components of the same W-isomorphism type admit an
isomorphisminduced by the algebra structure? The following consequence of Theorem 0.1, whose proof
is completed by Proposition 5.7, answers this question in a special case.
Corollary 0.3. Let W be an irreducible Coxeter group in GL(V) with homogeneous basic invariants
p ,...,p , ordered by increasing degree, and let F be the ideal in C[V] generated by p ,...,p . Then for
1 ℓ 1 ℓ
each j =1,...,ℓ, there exists an ℓ×ℓ-matrix A with entries in C[V] such that
j
∂p ∂p ∂p ∂p
ℓ,..., ℓ = j,..., j A mod F ·(C[V])ℓ.
j
∂x ∂x ∂x ∂x
(cid:18) 1 ℓ(cid:19) (cid:18) 1 ℓ(cid:19)
InallcasesexceptforE , E andE ,wegiveanexplicitformulaforthematricesA inCorollary0.3:
6 7 8 j
they are Hessians of basic invariants. This statement is a 3rd order partial differential condition on
the basic invariants which we call the Hessian rank condition (Hrc). Besides the missing proof for the
E-types, which would lead to a self contained algebraic proof of Theorem 0.1, it would be interesting to
know whether (Hrc) is a new condition or can be explained in the framework of Frobenius manifolds.
In our final Section 6, we show that by adding to D a divisor which pulls back to the conductor of
the ring extension O →cokerK, we obtain a new free divisor (Theorem 6.5). This was already shown
D
3
on the singularity side in [MS10]. The preimage in V of this free divisor is a free divisor containing the
reflection arrangement(Corollary 6.6).
Acknowledgments. Wethankthe“MathematischesForschungsinstitutOberwolfach”fortwotwo-week
“Researchin Pairs”stays in 2010and 2011. The authors are grateful to the referee for forceful, detailed
and helpful comments on an earlier version.
1. Review of Coxeter groups
Formoredetailsonthematerialreviewedinthissection,werefertothebookofHumphreys[Hum90].
Let VR be an ℓ-dimensional R-vector space and let V =VR⊗RC. Consider a finite group W ⊂GL(V)
generated by reflections defined over R. Any such representation W decomposes into a direct sum of
irreducible representations, and W is irreducible if and only if the corresponding root system is. The
irreducible isomorphism types are A , B , D , E , E , E , F , G =I (6), H , H , and I (k).
ℓ ℓ ℓ 6 7 8 4 2 2 3 4 2
ThegroupW actsnaturallyonthesymmetricalgebraS :=C[V]bythecontragredientaction,andwe
denote by R:=SW the corresponding graded ring of invariants. By a choice of linear basis, we identify
S with C[x ,...,x ]. The natural inclusion R ⊂ S turns S into a finite R-module of rank #W. The
1 ℓ
averaging operator
1
(1.1) #: S →R, g 7→g# := gw
#W
w∈W
X
defines a section of this inclusion.
By Chevalley’s theorem ([Hum90, Thm. 3.5]), R is a polynomial algebra R = C[p ,...,p ] where
1 ℓ
p ,...,p are homogeneous W-invariant polynomials in S. We set
1 ℓ
(1.2) degp =m +1=w
i i i
and assume that m ≤ ··· ≤ m . Then the degrees w , or the exponents m , are uniquely determined
1 ℓ i i
and
ℓ
(1.3) m =#A
i
i=1
X
where A is the arrangementof reflection hyperplanes of W ([Hum90, Thm. 3.9]).
We make this more precise in the case W is irreducible. Then the eigenvaluesof any Coxeter element
are exp(2πimi) where h is the Coxeter number ([Hum90, Thm. 3.19]). Moreover,
h
(1.4) 1=m <m ≤···≤m <m =h−1,
1 2 ℓ−1 ℓ
(1.5) m +m =h.
i ℓ−i+1
In particular, this implies that ℓ m = ℓh. For m =1, the W-invariant 2-form p is unique up to a
i=1 i 2 1 1
constant factor. By a choice of a positive multiple of p , it determines a unique W-invariant Euclidean
1
inner product (·,·) on VR, whicPh turns W into a subgroup of O(VR) and serves to identify VR and VR∗.
With respect to dual bases of VR and VR∗ we notice that the two corresponding inner products have
mutually inverse matrices. At the level of V∗, we denote by
Γ:=((x ,x ))=((dx ,dx ))
i j i j
the (symmetric) matrix of (·,·) with respect to coordinates x ,...,x . In suitable coordinates
1 ℓ
ℓ ℓ
(1.6) p = x2, (x,y)= x y , Γ=(δ ).
1 i i i i,j
i=1 i=1
X X
Werefertosuchcoordinatesasstandardcoordinates. IncaseW isreducible,wehavetheabovesituation
on each of the irreducible summands separately.
Geometrically the finiteness of S over R means that the map
(1.7) V =SpecS p //SpecR=V/W
is finite of degree #W. We identify the reflection arrangement A of W with its underlying variety
H. Let ∆ be a reduced defining equation for A, and denote by D =p(A) the discriminant. An
H∈A
anti-invariantofW isarelativeinvariantf ∈S withassociatedcharacterdet−1,thatis,wf =det−1(w)f
S
for all w ∈ W. The following crucial fact due to Solomon [Sol63, §3, Lem.] (see also ([Hum90, Prop.
3.13(b)]) implies that ∆2 is a reduced defining equation for D.
4
Theorem 1.1 (Solomon). R∆ is the set of all anti-invariants. (cid:3)
A second fundamental fact, due to K. Saito [Sai93, §3], is the following
Theorem 1.2 (Saito). For irreducible W, ∆2 is a monic polynomial in p of degree ℓ, that is,
ℓ
ℓ
∆2 = a (p ,...,p )pk, with a =1. (cid:3)
ℓ−k 1 ℓ−1 ℓ 0
k=0
X
We denote by Der and Der the modules of vector fields on V = SpecS and V/W = SpecR
S R
respectively. The group W acts naturally on Der . Terao [Ter83] showed that each θ ∈ Der(−logD)
S
has a unique lifting p−1(θ) to V and that the set of lifted vector fields is
p−1Der(−logD)=(Der )W, p∗Der(−logD)=(Der )W ⊗ S =Der(−logA),
S S R
and both A and D are free divisors. This can be seen as follows: We denote by
(1.8) J :=(∂ (p ))
xj i
the Jacobian matrix of p in (1.7) with respect to the coordinates x ,...,x and p ,...,p . Via the
1 ℓ 1 ℓ
identification of the 1-form dp with a vector field η such that (dp ,v)=hη ,vi,
i i i i
ℓ ℓ ℓ
(1.9) dp = ∂ (p )dx ↔η = hη ,dx i∂ = (dp ,dx )∂
i xj i j i i j xj i j xj
j=1 j=1 j=1
X X X
ℓ ℓ
= ∂ (p )(dx ,dx )∂ = ∂ (p )(x ,x )∂ ,
xk i k j xj xk i k j xj
k,j=1 k,j=1
X X
thebasicinvariantsdefineinvariantvectorfieldsη ,...,η ∈(Der )W,whichmustthenbeinDer(−logA).
1 ℓ S
By (1.9), their Saito matrix reads
(1.10) (η (x ))=ΓJt
j i
Now detJ is an anti-invariantbecause J is the differential of the invariant map p=(p ,...,p ). Hence,
1 ℓ
detJ ∈ C∗∆ by Theorem 1.1, (1.3), and the algebraic independence of the p . By scaling p, we can
i
therefore assume that
(1.11) detJ =∆.
Saito’s criterion([Sai80, ]) then showsthat A is free with basisη ,...,η . Applying the tangentmaptp
1 ℓ
(see (2.3)) gives vector fields δ ,...,δ ∈Der such that δ ◦p=tp(η ) with (symmetric) Saito matrix
1 ℓ R j j
(1.12) K =(Ki):=(δ (p ))=JΓJt
j j i
with det(JΓJt)∈C∗∆2. At generic points of A, p is a fold map and hence
(1.13) δ ,...,δ ∈Der(−logD).
1 ℓ
Again Saito’s criterionshows that D is a free divisor with basis δ ,...,δ . In standardcoordinates as in
1 ℓ
(1.6), this proves
Lemma 1.3. D admits a symmetric Saito matrix K =JJt.
If W is irreducible then, in standard coordinates as in (1.6),
ℓ
1
(1.14) χ := δ = w p ∂ .
w 2 1 i i pi
i=1
X
We shall refer to the grading defined by this semisimple operator as the w-grading. In particular, δ is
k
w-homogeneous of degree w −w . If W is reducible, we have a homogeneity such as (1.14) for each
k 1
irreducible summand.
Throughout the paper we will abbreviate
SA :=S/S∆, RD :=R/R∆2.
5
2. F-manifold-structures
In this section we prove Theorem 0.1. We will make use of the Frobenius manifold structure on
V/W, constructed by Dubrovin in [Dub98]. However our main reference for background on Frobenius
manifolds(includingthisresult)isthebookofHertling[Her02]. InfacttheonlyaspectsoftheFrobenius
structure we use are the existence of an integrable structure of commutative associative C-algebras on
the fibers of the tangent bundle; a manifold with this structure is called by Hertling and Manin an
F-manifold. This notion is much simpler than that of Frobenius manifold, omitting as it does all of the
metric properties,andthe connections,whichmakethe definition ofFrobenius manifoldsocomplicated.
Following Hertling, we use local analytic methods, and in particular local analytic coordinate changes,
in order to make use of normal forms. Such analytic methods will be justified in Remark 2.5, and we
pass to the analytic category without changing our notation.
The following accountsummarizes parts of [Her02, Ch. 2]. For any n-dimensionalF-manifold M, the
multiplication on TM is encoded by an n-dimensional subvariety of T∗M, the analytic spectrum L, as
follows: for each point p∈M, points in T∗M determine C-linear maps T M →C; among these, a finite
p p
number areC- algebrahomomorphisms. These finitely many points in eachfiber of T∗M piece together
to form L. Thus the composite
(2.1) DerM →π∗OT∗M →π∗OL
is an isomorphism of C-algebras ([Her02, Thm. 2.3]).
The multiplication ◦ in TM satisfies the integrability property
Lie (◦)=X ◦Lie (◦)+Y ◦Lie (◦).
X◦Y Y X
Provided the multiplication is generically semi-simple, as is the case for the structure constructed by
Dubrovin and Hertling, this implies that L is Lagrangian ([Her02, Theorem 3.2]). This in turn means
that the restriction to L of the canonical action form α on T∗M is closed and therefore exact. A
generating function forLis anyfunctionF ∈O suchthatdF =α| . Ageneratingfunctiondetermines
L L
an Euler field E on M, namely a vector field mapped to F by the isomorphism (2.1). The discriminant
of M is defined by any of the following equivalent characterizations:
(1) D =π(F−1(0)),
(2) D is the set of points x∈M where the endomorphism E◦: T M →T M is not invertible.
x x
Similarly, the module Der(−logD) may be viewed as either
(1) the set of vector fields whose image under the isomorphism(2.1) vanishes on F−1(0), or equiva-
lently as
(2) the image in Der of multiplication by E.
M
This yields the well-known
Lemma 2.1. The cokernel R˜ = cokerK of the Saito matrix K of D acquires an R-algebra structure
D
as quotient of the Frobenius manifold multiplication in Der .
R
Proof. The matrix of multiplication by E with respect to the basis ∂ ,...,∂ of Der is K. Thus
x1 xℓ R
(2.2) 0 // Rℓ K // Rℓ //R˜ //0
D
∼= ∼= ∼=
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // Der E◦ // Der //Der /Der (−logD) //0
R R R R
isapresentationofDerR/E◦DerR =DerR/DerR(−logD),whichisitselfisomorphictoπ∗OF−1(0). (cid:3)
We will denote SpecR˜ by D˜.
D
Recall from (1.8) that J: Sℓ →Sℓ is the matrix of the morphism
n n n
(2.3) tp:Der →p∗Der =Der ⊗ S, tp( η ∂ )= η ∂ (p )∂ ,
S R R R j xj j xj i pi
j=1 i=1j=1
X XX
definedby left composition(ofvectorfields assections ofTV)with dp. The followingdiagram,in which
the vertical arrows are bundle projections, helps to keep track of these morphisms. Sections of p∗Der
R
6
are maps from bottom left to top right making the lower triangle in the diagram commute.
dp//
(2.4) TV T(V/W)
(cid:15)(cid:15) (cid:15)(cid:15)
//
V V/W
p
Both tp : Der → p∗Der and and ωp : Der → p∗Der , defined by right composition with p, are
S R R R
familiar in singularity theory. By definition,
(2.5) χ∈Der lifts to η ∈Der ⇐⇒ tp(η)=ωp(χ).
R S
UsingLemma1.3,(2.2),andtheobviousidentifications,thereisacommutativediagramofS-modules
(2.6) J .
∆OOOO
0 //DeOOrS tp // DerROO⊗RS //S˜AOOOO // 0
Jt =
0 // Rℓ⊗ S K⊗1 // Der ⊗ S //R˜ ⊗ S // 0
R R R D R
Both rows here are exact: the upper row defines S˜A, and the lower row is the tensor product with the
flat R-module S of the short exact sequence defining R˜ . Now R˜ ⊗ S, as a tensor product of rings,
D D R
has a natural ring structure; to show that S˜A is a ring, it will be enough to show
Lemma 2.2. The image of tp is an ideal of Der ⊗ S.
R R
We proveLemma 2.2 by showing that the Frobenius multiplication in Der lifts to a p∗Der -module
R R
structure on Der , and that tp:Der →Der ⊗ S is Der -linear.
S S R R R
Proposition 2.3.
(1) The Frobenius multiplication in Der can be lifted to Der , though without multiplicative unit.
R S
(2) The same procedure makes Der into a Der -module.
S R
(3) The map tp in (2.3) is Der -linear, with respect to the structure in (2) and Frobenius multipli-
R
cation induced on Der ⊗ S.
R R
Proof. By (2.5), for a multiplication in Der , (1) means that
S
(2.7) tp(η ◦η )=ωp(χ ◦χ )
1 2 1 2
whereη ∈Der isaliftofχ ∈Der fori=1,2. Similarly,thescalarmultiplicationfor(2)mustsatisfy
i S i R
(2.8) tp(χ·η)=ωp(χ◦ξ)
where χ∈Der and η ∈Der is a lift of ξ ∈Der .
R S R
Locally,atapointv ∈V\A,p,tpandωpareisomorphisms,sothereisnothingtoprove. Nowsuppose
v ∈H is agenericpointona reflectinghyperplane H ∈A,with p(v) outsidethe bifurcationsetB. Ina
neighborhoodofp(v)inV/W,wemaytakecanonicalcoordinatesu ,...,u (cf.[Her02,2.12.(ii)]). These
1 ℓ
are characterized by the property that the vector fields e := ∂ , i = 1,...,ℓ satisfy e ◦e = δ ·e .
i ui i j i,j i
By [Her02, Cor.4.6],the tangentspaceT D is spannedbyℓ−1ofthese idempotent vectorfields, and
p(v)
the remaining idempotent, which we label e , is normal to it. The map p : (V,v) → (V/W,p(v)) has
1 v
multiplicity 2, critical set H and set of critical values D, from which it follows that d p: T H →T D
v v p(v)
is an isomorphism. Since we have fixed our coordinate system on (V/W,p), we are free to choose only
the coordinates on (V,v). Define x =u ◦p for i=2,...,ℓ. To extend these to a coordinate system on
i i
(V,v), we may take as x any function whose derivative at v is linearly independent of d x ,...,d x .
1 v 2 v ℓ
This means we may take as x any defining equation of the critical set (the hyperplane H) of p at v.
1
With respect to these coordinates, p takes the form
(2.9) p (x ,...,x )=(f(x ,...,x ),x ,...,x ).
v 1 ℓ 1 ℓ 2 ℓ
As p has critical set {x = 0} and discriminant {u = 0}, both f and ∂ (f) vanish along {x = 0}.
v 1 1 x1 1
Thus f(x) = x2g(x) for some g ∈ O . Since p has multiplicity 2 at v, g(0) 6= 0. Now replace the
1 V,v
7
coordinate x by x g(x)1/2. With respect to these new coordinates, which we still call x ,...,x , p
1 1 1 ℓ v
becomes a standard fold:
p (x ,...,x )=(x2,x ,...,x ).
v 1 ℓ 1 2 ℓ
We can now explicitly calculate the multiplication in Der , locally at v:
S
tp (x ∂ )=ωp (2u ∂ ),
v 1 x1 v 1 u1
(tpv(∂xi)=ωpv(∂ui), for i=2,...,ℓ.
So (2.7) implies
tp ((x ∂ )◦(x ∂ ))=ωp ((2u ∂ )◦(2u ∂ ))
v 1 x1 1 x1 v 1 u1 1 u1
=ωp (4u2∂ )=ωp (2u (2u ∂ ))=tp ((2x2)x ∂ ),
v 1 u1 v 1 1 u1 v 1 1 x1
and hence x ∂ ◦x ∂ =2x3∂ . So in order that (2.7) should hold, we are forced to define
1 x1 1 x1 1 x1
2x ∂ , for i=j =1,
∂ ◦∂ = 1 x1
xi xj (δi,j ·∂xi, otherwise.
Since the multiplication in Der is uniquely defined by (2.7) outside codimension 2, it extends to V by
V
Hartog’s Extension Theorem. This proves (1); (2) is obtained by an analogous argument using (2.8).
Finally, (3) follows from (2.5) and (2.8) on V \A, and therefore holds everywhere. (cid:3)
Proof of Lemma 2.2. Let ξ ∈ Der , g ∈ S and η ∈ Der . By Proposition 2.3.(3) and the evident
S R
S-linearity of the lifted Frobenius multiplication,
(η⊗ g)·tp(ξ)=tp(η◦gξ).
R
(cid:3)
We have proved the following result, which implies (1) of Theorem 0.1.
Theorem 2.4. The cokernel S˜A =cokerJ of the transposed Saito matrix of A is an SA-algebra. (cid:3)
Remark 2.5. Eventhoughourproofusescomplexanalyticmethods,suchascanonicalcoordinatesinthe
proofofProposition2.3,the conclusionisvalidoveranyfieldoverwhichthebasicinvariantsaredefined.
We show this in Section 3 below by proving that the fact that cokerJ is an S-algebra is equivalent to a
condition on ideal membership, the so-called rank condition (rc).
WeendthissectionbyclarifyingtherelationshipbetweenS˜A andR˜D⊗RSA. Ingeneraltheyarenot
isomorphic,andthespaceSpecS˜A isnotthefiberproductSpec(R˜D×DA). ForR˜D⊗RSA isthecokernel
of 1⊗∆: R˜ ⊗ S →R˜ ⊗ S, and using the epimorphism Der ⊗ S ։R˜ ⊗ S we find that there
D R D R R R D R
is an epimorphism DerR⊗RS ։R˜D⊗RSA, whose kernelis equalto DerR⊗RS∆+Der(−logD)⊗RS.
Both summands here are contained in the image of tp: Der → Der ⊗ S, the first by Cramer’s rule
S R R
and the second because every vector field η ∈ Der(−logD) is liftable via p. Thus S˜A is a quotient of
R˜D⊗RSA. The kernel N of the projection R˜D⊗SA →S˜A is the quotient
N :=tp(Der )/ Der(−logD)⊗ S+Der ⊗ S∆ .
S R R R
At a generic point x∈A this vanishes: h(cid:0)ere p is a fold map, right-left-equiv(cid:1)alent to
(x ,...,x )7→(x ,...,x ,x2)
1 ℓ 1 ℓ−1 ℓ
and an easy local calculation shows that in this case N = 0. However, if p has multiplicity > 2 at x
x
then N 6=0. For example at an A point, p is right-left equivalent to
x 2
(x ,...,x )7→(x2+x x +x2,x x (x +x ),x ,...,x );
1 ℓ 1 1 2 2 1 2 1 2 3 ℓ
tp(Der ) is generated by ∂ ,...,∂ together with
S p3 pℓ
(2x +x )∂ +(2x x +x2)∂ ,(x +2x )∂ +(x2+2x x )∂ ,
1 2 p1 1 2 2 p2 1 2 p1 1 1 2 p2
while the coefficients of ∂ in the generatorsof Der(−logD)⊗ S+Der ⊗ S∆ are at least quadratic
p1 R R R
in x ,...,x . In fact, assuming Lemma 2.2, we have
1 ℓ
Theorem 2.6. A˜=(D˜ × A)
D red
8
Proof. S˜A = cokertp, with tp as in (2.6), is a maximal Cohen–Macaulay SA-module of rank 1. This
means that at a smooth point of A, S˜A is isomorphic to SA, and is thus reduced. As S˜A is finite over
SA, its depth over itself (assuming it is a ring) is equal to its depth over SA. Since it is therefore a
Cohen–Macaulay ring, generic reducedness implies reducedness. (cid:3)
For later use we note that by [MP89, Cor. 3.15], we have
Theorem 2.7. A˜is Cohen–Macaulay and D˜ is Gorenstein. (cid:3)
3. Algebra structures on cokernels of square matrices
3.1. Rank condition. In this subsection we recall a condition on the rows of the adjugate of a square
matrix over a ring R, which is equivalent to that matrix presenting an R-algebra, at least in the local
and local graded cases. It is the key to proving Corollary 0.3 in the Introduction.
Let R be an ℓ-dimensional (graded) local Cohen–Macaulay ring with maximal (graded) ideal m. In
the gradedlocalcase,weassume thatall R-modulesaregradedandall R-linearmapsarehomogeneous.
LetAbe anℓ×ℓ-matrixoverR withtransposeΛ:=At. We considerboth AandΛ asR-linearmaps
Rℓ → Rℓ. Assume that ∆ := detA is a reduced non-zero-divisor and set D = V(∆). By Cramer’s rule
∆annihilatesM :=cokerAwhichishenceamoduleoverR :=R/R∆. ForanyidealI ⊆R,wedenote
D
by I :=R I its image in R . By Q :=Q(R ), we denote the total ring of fractions of R .
D D D D D D
The k-th Fitting ideal ofM over R, Fk(M),is the idealofR generatedby the (ℓ−k)×(ℓ−k)-minors
of A. It is an invariantof M, andindependent of the presentationA. We denote by mi the generatorof
j
F1(M) obtained from A by deleting row i and column j. Note that Fk(M) is the k’th Fitting ideal of
D
M over R . For properties of Fitting ideals, see e.g. [?, Ch. 20].
D
Definition 3.1. Wesaythattherank condition (rc)holdsforAifgradeF1(M)≥2andF1(M)isequal
to the ideal of maximal minors of the matrix obtained from A by deleting one of its rows, possibly after
left multiplication of A by some invertible matrix over R.
Note that (rc) implies that F1 (M) is a maximalCohen–MacaulayR -module, by the Hilbert–Burch
D D
theorem.
Itturnsoutthat(rc)dependsonlyonthemoduleM =cokerA,andnotonthe choiceofpresentation
A. Thisisaconsequenceofthefollowingtwotheorems,whichalsomakeclearthereasonforourinterest
in the condition (rc).
Theorem 3.2 ([MP89, Thm. 3.4]). If M is an R -algebra then (rc) holds for A. (cid:3)
D
The proofin [MP89]shows thatif M is anR -algebraby e,m ,...,m , where e is the multiplicative
D 2 ℓ
identity of M, and A is a presentation of M with respect to these generators, then F1(M) is equal to
the ideal of maximal minors of A with its first row deleted.
The converse theorem also holds. A proof, due to de Jong and van Straten, can be found in [MP89,
Prop. 3.14]. We will use some of the notions introduced there, however, and so we give a sketch, based
on the accounts there and in [dJvS90].
Recallthatafractional ideal U (overR )isafinitelygeneratedR -submoduleofQ whichcontains
D D D
a non-zero-divisor and that
(3.1) Hom (U,V)=[V : U]
RD QD
is a fractionalideal, for any two fractionalideals U and V. We shalluse this identificationimplicitly. In
particular, the duality functor
(−)∨ :=Hom (−,R )
RD D
preservesfractionalideals. ItisinclusionreversingandadualityonmaximalCohen–Macaulayfractional
ideals (see [dJvS90, Prop. 1.7]).
Theorem 3.3. If (rc) holds for A then M is a fractional ideal generated by ϕ ,...,ϕ ∈Q where
1 ℓ D
(3.2) ϕ mℓ =mi, i,j =1,...,ℓ,
i j j
and an R -subalgebra of Q isomorphic to End (F1 (M)).
D D RD D
Proof. Using (rc) for A, Lemma 3.4 (below) yields a presentation
(3.3) 0 // Rℓ Λ //Rℓ (mℓ1,...,mℓℓ) //F1(M) // 0.
D
9
In particular,F1 (M) is a maximalCohen–MacaulayR -module of rank 1,and therefore canbe viewed
D D
as a fractional ideal. As F1 (M) is contained in R , F1 (M)∨ is a fractional ideal containing R .
D D D D
Dualizing (3.3) with respect to R gives the exact sequence
D
0 //F1(M)∨ // Rℓ A //Rℓ .
D D D
There is also a 2-periodic exact sequence
··· // Rℓ A // Rℓ adA //Rℓ A // ··· .
D D D
Therefore,
coker A=M,
F1 (M)∨ ∼=ker A∼= RD
D RD (imRDadA=F1D(M).
and hence F1 (M)∨ ∼=End (F1 (M)). From this all the statements follow. (cid:3)
D RD D
In Subsection 4.2 we identify the generators in Theorem 3.3 in the case that D is the reflection
arrangementor discriminant of an irreducible Coxeter group.
Lemma 3.4 ([dJvS90, Prop. 1.10]). Suppose that the ideal I (generated by the maximal minors of the
matrix A with one row deleted) has grade 2. Then there is a free resolution
(3.4) 0 //Rℓ Λ // Rℓ (mℓ1,...,mℓℓ) //I // 0. (cid:3)
D
We cannow make goodthe promise we made in Remark 2.5: that Theorem2.4 is validoverany field
over which the basic invariants p ...,p are defined. From Theorems 2.4 and 3.2 it follows that (rc)
1 ℓ
holds for cokerA: for each i,j ∈{1,...,ℓ}, the equation
(3.5) mi =C mℓ +···+C mℓ
j 1 1 ℓ ℓ
in unknown functions C ,...,C has a solution in which the C are germs of complex analytic functions
1 ℓ i
at 0.
Let K be a subfield of C containing the coefficients of the basic invariants p , so that the coefficients
j
of the polynomials mi all lie in K. We claim that (3.5) has solutions C ∈ K[V]. From this claim, the
j i
existence of the S-algebra structure on cokerA follows by Theorem 3.3.
To prove the claim, first note that since the mi are all homogeneous, each C can be replaced by its
j i
graded part of degree D −D (see (4.6)). Let K[V] ⊂ K[V] be the vector space of all polynomials of
i ℓ d
degree d. The map
ℓ
A: K[V] ℓ →K[V] , A(C ,...,C )= C mℓ,
Di−Dℓ Di 1 ℓ j j
j=1
(cid:0) (cid:1) X
is K-linear. Therefore the solvability of (3.5) in K[V] reduces to a simple theorem of linear algebra,
which can be rephrased more abstractly as follows: Let A: Km → Kn be a K-linear map, and suppose
K⊂L is a field extension. Then
im(A⊗K1L)∩Kn =im(A).
We leave the proof of this to the reader.
3.2. Rings associated to free divisors. In this subsection we make some generalobservations about
thealgebrapresentedbythetransposeofaSaitomatrixofafreedivisor. LetD =V(∆)beafreedivisor
in (Cℓ,0) with Saito matrix A. Then we have an exact sequence
(3.6) 0 //Rℓ A // Rℓ (∆1,...,∆ℓ) // R //R /J //0,
D D D
where ∆ := ∂∆/∂x for j = 1,...,ℓ, and J := R J is the Jacobian ideal of D. Now assume also
j j D D ∆
that D is Euler homogeneous. By adding multiples of the Euler vector field χ = δ to the remaining
1
members δ ,...,δ of a Saito basis of D, we may assume that these annihilate ∆. We shall assume that
2 ℓ
A is obtained from such a basis. We say that D satisfies (rc) if (rc) holds for Λ = At. In this case, we
write
R˜ :=M =cokerΛ⊂Q
D D
for the ring of Theorem 3.3.
It is well known that for any algebraic or analytic space D satisfying Serre’s condition S2, the frac-
tional ideal End (J∨) is naturally contained in the integral closure of R in Q , and the inclusion
RD D D D
10