Discriminantal bundles

# Discriminantal bundles, arrangement groups, and subdirect products of free groups

Daniel C. Cohen Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803 www.math.lsu.edu/~cohen Michael Falk Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, Arizona 86011-5717 www.cefns.nau.edu/~falk  and  Richard Randell Department of Mathematics, University of Iowa, Iowa City, Iowa 52242 www.math.uiowa.edu/~randell
###### Abstract.

The Lawrence-Krammer-Bigelow representation of the braid group arises from the monodromy representation on the twisted homology of the fiber of a certain fiber bundle in which the base and total space are complements of braid arrangements, and the fiber is the complement of a discriminantal arrangement. We present a more general version of this construction and use it to construct nontrivial bundles on the complement of an arbitrary arrangement  whose fibers are products of discriminantal arrangements.

This leads us to consider the natural homomorphism from the arrangement group to the product of groups corresponding to a set  of rank-two flats. Generalizing an argument of T. Stanford, we describe the kernel in terms of iterated commutators, when generators of can be chosen compatibly. We use this to derive a test for injectivity of We show is injective for several well-studied decomposable arrangements.

If  is central, the homomorphism induces a natural homomorphism from the projectivized group into the product whose factors are free groups. We show is injective if and only if is. In this case is isomorphic to a specific finitely-presented, combinatorially-determined subdirect product of free groups. In particular is residually free, residually torsionfree nilpotent, a-T-menable, and linear. We show the image of is a normal subgroup with free abelian quotient, and compute the rank of the quotient in terms of the incidence graph of  with . When is injective, we conclude is of type and not of type

###### Key words and phrases:
arrangement, braid group, monodromy representation, multinet, right-angled Artin group, subdirect product, Bestvina-Brady group
###### 2010 Mathematics Subject Classification:
20F36, 32S22, 52C35, 55R80

## 1. Introduction

Suppose that is an arrangement of affine hyperplanes in For each let be a linear polynomial with zero locus Let . Let denote the complement , where In this paper we present a general construction of nontrivial fiber bundles over . The fibers are the complements of affine discriminantal arrangements, in the sense of Schechtman and Varchenko [SV91]. One may then construct representations of the arrangement group via the monodromy action on the homology of the fiber with coefficients in certain local systems, generalizing the Lawrence-Krammer-Bigelow representation of the pure braid group as described in [PP02].

These bundles are pullbacks of the Fadell-Neuwirth projection fiber bundles of ordered configuration spaces [FN62, Bir75], determined by an integer and a collection of generating functions continuous functions on having the property that the zero locus of is contained in for each The fiber is the ordered configuration space of points in a plane with punctures, realized as the complement of the affine discriminantal arrangement in [SV91]. We call the pullback a discriminantal bundle over

When and the are linear, one obtains a strictly linear fibration over (with punctured plane fiber) as in the definition of fiber-type arrangements. In this case our construction coincides with the “root map” construction of [CS97, Coh01], used to produce the braid-monodromy presentation of the fundamental group of the total space. The fact that bundles involving strictly linearly fibered arrangements may be realized as pullbacks of configuration space bundles was established in [Coh01]. This was used to show that fundamental groups of complements of fiber-type arrangements are linear in [CCP07].

When we obtain analogues of the bundles which arise in the Lawrence and Lawrence-Krammer-Bigelow (LKB) representations of braid groups. Restricted to the pure braid group, the LKB representations are rationally equivalent to the monodromy representations on the second homology of the fiber of the bundle with coefficients in a certain rank-one -local system [PP02]. These LKB representations are generally faithful and are related to several polynomial invariants of knots and links [Big02, Big07]. Our construction could be used in principle to define analogous polynomials for elements of other arrangement groups.

If extends continuously and is not identically zero on the resulting bundle over will have trivial monodromy around The support of a generating set is the set of hyperplanes in  such that the associated discriminantal bundle has nontrivial monodromy around Any rank-two arrangement supports a generating set. If  is a rank-three arrangement supporting a multinet structure [FY07] then  supports a generating set. For arbitrary , we have no method to construct a generating set supported by . But by taking Whitney sums of discriminantal bundles over we obtain bundles with nontrivial monodromy around every hyperplane of .

To understand when the resulting representation of is faithful, we are led to study kernels of cartesian products of inclusion-induced homomorphisms of Let  be a set of rank-two flats whose union is , and be the product of inclusion-induced homomorphisms, where is the group of the subarrangement Then is generated by elements dual to the hyperplanes of , uniquely defined up to conjugacy, and is the quotient of obtained by killing the generators corresponding to hyperplanes outside of When has a generating set with the property that the subgroup of of maps isomorphically to for all we say is adapted to . In this case we generalize an argument of T. Stanford [Sta99] to show the kernel of is generated by iterated commutators of generators and their inverses, whose supports are not contained in any As a corollary we obtain a criterion for to be injective. If  is central, we show that is injective if and only if its restriction to the group of the decone of  is injective. Using this we show injectivity of for the rank-three wheel arrangement, labelled in [FR86] and considered in unpublished work by Arvola [Arv92], for the group of the Kohno arrangement of seven lines, labelled in [FR86], and for a pair of seven-line arrangements that appear in [Fal97].

When  is central, the homomorphism induces a well-defined homomorphism where and are the projectivized fundamental groups. Since is a rank-two flat (of multiplicity greater than two), is a (nonabelian) free group, and so the image is a combinatorially-determined subdirect product of free groups, in the terminology of [BM09]. Then is residually free, residually torsionfree nilpotent, has a linear representation, and satisfies the Haagerup property (i.e., is a-T-menable in the sense of Gromov). Hence has these properties when is injective. (Injectivity of is not a priori combinatorially determined, however.) We show is a normal subgroup, with free abelian quotient. We compute the rank of the quotient in terms of the incidence graph of  with . Using [MMW98] we obtain precise information about the finiteness type of We deduce that, for the rank-three wheel arrangement, is isomorphic to the Stallings’ group [Sta63], as originally observed by Matei and Suciu [MS04] (this observation provided motivation for the current project); see also Question 2.10 in Bestvina’s problem list [Bes04]. We also conclude that the group of the Kohno arrangement is of type and not of type and the groups of the two seven-line arrangements from [Fal97] are but not The example has recently been generalized by Artal-Bartolo, Cogolludo-Augustin, and Matei to a large family of arrangements whose groups are Bestvina-Brady groups, as reported in [Mat07]. We reproduce their result using our approach. If  is a decomposable arrangement [PS06], is the set of all rank-two flats, and has a generating set adapted to , our result implies that the kernel of is precisely the nilpotent residue of In all our examples of decomposable arrangements, is injective; we conjecture that this is always the case, that is, that all decomposable arrangement groups embed in products of free groups.

## 2. Constructing Discriminantal Bundles over Arrangement Complements

In this section we will construct a number of bundles, starting with any arrangement complement. Our construction will mimic how one might construct the pure braid space for strings, given the space for strings.

The pure braid space is the complement in of the arrangement defined by

 ∏1≤i

Its fundamental group is the pure braid group Then is the set

 {(z1,…,zℓ,zℓ+1,…,zℓ+k)∣zi≠zj}.

To see how this can be built from we write the last variables as Then

 PBℓ+k={(z1,…,zℓ,w1,…,wk)∣zi≠zj,zi≠wj,wi≠wj}.

The point here is that we can define the total space by introducing new variables, and prohibiting the variables from taking the values given by our “generating set” or each other. With the pure braid space as the base the construction is clear; for a general hyperplane arrangement the problem is in finding an analogue of the set .

### 2.1. Generating Sets

Fix a hyperplane arrangement in with the zero locus of the linear polynomial for Let and

###### Definition 2.1.

A set is called a generating set for the arrangement provided that each is a continuous function on and each difference is nowhere zero on

So defines a generating set for the braid arrangement in Functions in a generating set need not be linear in general. In most examples the are rational functions on regular on In this case  forms a generating set for  if and only if the irreducible components of the (possibly non-reduced) quasi-affine hypersurfaces defined by are (unions of) hyperplanes of , for each .

Let be a (labelled) generating set for and let Introduce new variables and consider the topological space

 V⊂M×Ck⊆Cℓ+k

defined by

 Q(z1,…,zℓ)⋅∏1≤i≤k1≤j≤μ(wj−fi(z))⋅∏1≤i

The space is then defined to be the complement

If the are holomorphic on then is a quasi-affine analytic subset of .

By the discussion above, where is the braid arrangement in and for The projection is the Fadell-Neuwirth bundle In general we have the following.

###### Theorem 2.2.

Let be an arrangement with generating set and let Then projection is the projection map of a fiber bundle. This bundle is the pullback of the Fadell-Neuwirth bundle via the function .

###### Proof.

Let be the coordinates on , and let be coordinates on Then the total space of the pullback of via is the set of all points which satisfy It is readily checked that the map

 (z1,z2,…,zℓ,wℓ+1,…,wℓ+k)⟼(z1,…,zℓ,f1(z),…,fμ(z),w1,…,wk)

from to the total space of the pullback is a bundle equivalence. ∎

The function is called a generating function for the bundle. (Different generating functions for the same generating set differ by permutation of coordinates in )

###### Definition 2.3.

A discriminantal arrangement of type is the arrangement defined by the polynomial

 ∏i,j(wi−mj)⋅∏i

where are fixed distinct complex numbers and are complex variables.

Different choices of lead to lattice-isotopic arrangements. Thus the complement of is determined up to homeomorphism by and [Ran89]. We denote the complement of by The arrangement is an affine supersolvable arrangement, hence is itself a fiber-type arrangement. In particular is aspherical (see [Ter86], [FR85]).

The complement of may be realized as the configuration space of ordered points in . Since is the fiber of the Fadell-Neuwirth bundle , the fundamental group is a subgroup of the pure braid group Note that is a free group of rank .

###### Proposition 2.4.

The fiber of is homeomorphic to .

In light of the preceding observation, we call such a bundle a discriminantal bundle.

###### Corollary 2.5.

The fiber of is aspherical with fundamental group isomorphic to the pure braid subgroup of type .

Write and The bundle map is the restriction of a linear projection. If the are linear and then the total space is the complement of an arrangement . When  is central, the map is the bundle projection associated with the modular flat  of , see [Par00, FP02]. If then is a strictly linear fibration [FR85, Ter86], and is the associated root map as defined in [CS97, Coh01].

Much of the topology of fiber-type arrangements carries over. The results below all follow from the characterization of these discriminantal bundles as pullbacks of the Fadell-Neuwirth bundle, together with standard results for fiber bundles - see [FR85].

###### Theorem 2.6.

The bundle has a section, and the action of the fundamental group of the base on the fiber is trivial on the first homology.

###### Corollary 2.7.

The homology of is the tensor product of the homology of the base with that of the fiber

###### Corollary 2.8.

If the base arrangement complement is aspherical, then so is the total space .

###### Corollary 2.9.

The fundamental group of is the semidirect product of the fundamental group of with

In particular, if the fundamental group of is an iterated semidirect product of free groups, or more stringently an almost-direct product of free groups, see [FR85, CS98], then so is the fundamental group of In the latter instance, the cohomology ring of the group may be calculated from the almost-direct product structure, see [Coh10]. Additionally, we have the following, as noted in [CCP07, Lem. 6.2].

###### Corollary 2.10.

If the fundamental group of the base is linear, then so is the fundamental group of the total space .

###### Proof.

The group is a subgroup of the product , which is linear since both factors are. ∎

The fiber of the Fadell-Neuwirth bundle is a copy of  with punctures. The monodromy of the bundle is the (faithful) Artin representation

 PBμ→Aut(Fμ)

of the pure braid group in the group of automorphisms of the free group. With this identification, the pure braid group acts diagonally on for any since the diagonal hyperplanes are preserved. The bundle associated to via this action.

###### Corollary 2.11.

The structure group of the bundle reduces to the pure braid group on strings, and is associated with the bundle via the diagonal action of on

###### Definition 2.12.

Let  be a generating set for  and We say  is trivial on if extends continuously and is not identically zero on for all The support of  is the set of hyperplanes on which  is not trivial.

The monodromy of the bundle is nontrivial around if and only if is in the support of . If denotes the support of , then  is a generating set for and is a subbundle of the pullback by the inclusion map

###### Example 2.13.

Let be the Coxeter arrangement of type with defining equations . Let where Then  is a generating set on , with support , and is the complement of the union of hyperplanes and affine quadrics in

###### Example 2.14.

Let be the Coxeter arrangement of type , with defining equations , , and . The set is a generating set for , with support . Here denotes the zero function. This linear generating set arises from the structure of the projection as a strictly linear fibration; this is an instance of the root map construction. The corresponding generating function realizes the bundle as the pullback of the Fadell-Neuwirth bundle . This is used to determine the structure of the type pure braid group as an almost-direct product of free groups in [Coh01, Thm. 1.4.3].

Another generating set for the arrangement is given by , where for The support of is the entire arrangement . Note that the hyperplanes and are poles of of multiplicity two.

###### Example 2.15.

Let  be the arrangement consisting of the origin in Let Then is a generating set for . The total space is the complement in of the real linear subspaces and This complement of four 2-planes in does not have the homotopy type of the complement of a complex hyperplane arrangement, by [Zie93].

While nonlinear generating functions yield total spaces which are not arrangement complements (as sets), we have not found an example of a generating set consisting of (nonlinear) holomorphic functions on for which the total space does not have the homotopy type of an arrangement complement.

### 2.2. Existence of Generating Sets

To construct faithful representations of the group we need to know which subarrangements of  support generating sets. It turns out the conditions are somewhat restrictive. But one can always construct generating sets supported on rank-two subarrangements.

Let  be an arbitrary arrangement, and let be a rank-two flat of , an intersection of hyperplanes in  of codimension two in Let denote the set of hyperplanes of  containing We explicitly construct a generating set for  supported by The reader may notice a similarity with the description of a configuration space of distinct points in  as a hyperplane complement. This construction gives an indication of our original ideas for producing fibered families of hyperplanes.

We may label the hyperplanes of  so that We wish to consider the one parameter family of hyperplanes containing .

Since and are distinct, is the transverse intersection of and . We consider the family of hyperplanes, where has defining equation This family includes all the hyperplanes of except Note that gives , and would correspond to . There are distinct nonzero constants so that

We then define the generating set of size by

 FX={0,α2,m3α1,…,mμα1}.

Then is a generating set for , with support equal to

Next we show that, under a mild hypothesis, a polynomial generating set of size three supported on a rank-three arrangement corresponds to a pencil of Čeva type, as studied in [FY07], after a linear change of coordinates in We identify (possibly non-reduced) projective plane curves with their defining polynomials, and say a curve is completely reducible if its defining polynomial splits into linear factors (possibly with mulitplicities).

###### Definition 2.16.

A pencil of Čeva type (or Čeva pencil) is a 1-dimensional linear system of projective plane curves (a rational map ) with no fixed components, connected generic fiber, and three or more completely reducible fibers.

We denote the projectivization of a central arrangement  by The set of irreducible components of completely reducible fibers in a Čeva pencil forms a projective line arrangement which inherits a natural partition and multiplicity function from the pencil. It is shown in [FY07] that a projective line arrangement arises in this way from a Čeva pencil if and only if the associated partition forms a multinet for the multiplicity function (See also [MB09].) Say is primitive if the values of are mutually relatively prime.

###### Definition 2.17.

Suppose is primitive. A -multinet on the multiarrangement consists of a partition of  into blocks, with the associated “base locus” being the set of intersection points of lines from different blocks, satisfying

1. each block of  has lines, counting multiplicity;

2. each point of  contains the same number of lines from each block, counting multiplicity;

3. is connected for each

###### Theorem 2.18.

Suppose is primitive and supports a -multinet. Then there is a generating set with support equal to .

###### Proof.

By [FY07], there are completely reducible polynomials and pairwise relatively prime, whose zero loci are the blocks and of the multinet structure, and for some Without loss of generality, Then one easily checks that is a generating set with support equal to . ∎

Given a generating set consisting of homogeneous polynomials of the same degree we may set for Then the -dimensional linear system corresponding to the rational mapping

 [Q1:⋯:Qμ−1]:CPℓ−1→CPμ−2

has completely reducible fibers This linear system may have fixed components and/or disconnected general fiber.

###### Theorem 2.19.

Let  be an arrangement of rank three with generating set consisting of homogeneous polynomial functions of degree on Suppose the support of  is , and the polynomials and are relatively prime. Then  supports a -multinet structure for some

###### Proof.

With notation as above, the pencil determined by has three completely reducible fibers and whose irreducible components comprise . Since and are relatively prime, the pencil has no fixed components. The associated partition of  satisfies (i) and (ii) of the definition of multinet. We can then refine this partition to a multinet, by [FY07, Remark 2.6]. ∎

### 2.3. Linear generating sets

Any set of linear forms is a generating set. Indeed, if is a set of distinct linear forms, then  is a generating set whose support  has defining polynomial In this case, as observed earlier, is the complement of an arrangement which contains  as a modular flat in the case where  is central, and is the associated bundle projection. Rescaling the may result in a different supporting arrangement , so we cannot replace the generating set of linear forms  with its associated arrangement (or matroid), and retain a well-defined operation

If  consists of the coordinate functions then its support  is the braid arrangement. If  consists of the natural defining forms for the braid arrangement, then  is the center-of-mass arrangement defined in [CK07], whose complement parametrizes the labelled configurations of distinct points in with pairwise distinct midpoints. In fact, if  consists of the natural defining forms

 p∑k=1zik−p∑k=1zjk,

for the -fold center-of-mass arrangement on points, then the associated arrangement  is the -fold center-of mass arrangement on points.

The correspondence has a nice interpretation at the level of the Grassmannian. Suppose and  contains linearly independent forms. Then the image of is an -dimensional linear subspace of The arrangement of hyperplanes determined by , with defining polynomial is linearly isomorphic to the arrangement cut out on by the coordinate hyperplanes in The support arrangement  with generating set  is isomorphic to the arrangement cut out on the same subspace by the hyperplanes of the braid arrangement in (The fact that  is not determined by the arrangement means that this operation is not invariant under the torus action on subspaces of )

If are distinct linear forms, then the set of reciprocals is also a generating set, whose support is the arrangement defined by

 ∏1≤i≤μfi⋅∏1≤i

## 3. Products of localization homomorphisms

Given an arbitrary arrangement , we would like to build a bundle with base which is sufficiently twisted to yield a faithful representation of For  itself to support a discriminantal bundle requires fairly special circumstances, as we have seen, but  may have several proper subarrangements supporting such bundles. Indeed, any rank-two subarrangement, and any rank-three subarrangement supporting a multinet, will have that property. We propose to pull back the product of all such discriminantal bundles supported on subarrangements, to obtain a bundle over

More precisely, let  denote the set of subarrangements of  supporting generating sets, and let

 φD:M(A)→∏S∈DM(S)

be the product of inclusion maps. Choosing a generating set of size and a positive integer for each we have discriminantal bundles and hence a product bundle

 ∏pS:∏S∈DEkS(S,FS)→∏S∈DM(S).

(Note: the codomain is also an arrangement complement.) The pullback gives a bundle over whose fiber is This bundle will have nontrivial monodromy around every hyperplane of , since

To use the product bundle constructed above to produce faithful representations of one would first build faithful representations of for using the monodromy of discriminantal bundles, and then show that induces an injection on fundamental groups. We can carry out the first step at least in case comes from a rank-two lattice element.

### 3.1. Monodromy representations associated to rank-two lattice elements

Fix now a single lattice element of rank two. Let denote the arrangement consisting of just those hyperplanes which contain and let denote the complement of We have an inclusion-induced homomorphism

 iX:π1(M)→π1(MX).

Let be the generating set with support constructed in the preceding section. Let and be the associated generating function. Let and let be the associated discriminantal bundle.

###### Proposition 3.1.

The induced homomorphism is injective.

###### Proof.

Recalling the construction of from §2.2, we may assume that and we can choose coordinates so that is defined by by and is defined by for Then is given by Let be the projection Then is a discriminantal bundle projection; in particular the kernel of is the fundamental group of the fiber of a free group on generators.

For let be a loop in dual to Choosing the base point in the hyperplane we may assume that the loops lie in the subspace Then sends to for Then any element of the kernel of lies in a free group of rank Moreover, sends this subgroup to the fundamental group of the fiber of Then is injective by the Hopfian property of free groups. ∎

###### Corollary 3.2.

The LKB-type representation arising from the bundle is faithful.

###### Proof.

Since is the pullback of the Fadell-Neuwirth bundle via and is injective, it suffices that the LKB-type representation of Fadell-Neuwirth bundles is faithful. This is true by [Big01, Kra02, PP02]. ∎

### 3.2. The kernel of (φS)∗

Next we consider a general product mapping where  is an arbitrary set of subarrangements of , and is the inclusion. We pick a base point in and obtain an induced homomorphism,

 (φS)∗:π1(M)→∏S∈Sπ1(MS).

For simplicity we denote by Let us also denote by so that For consistency with [MKS04] and [Fal93], for the remainder of this section we adopt the conventions and for group elements and

Recall that is generated by small loops around the hyperplanes of . For each kills the generators corresponding to hyperplanes in For the pure braid group, and a certain sets of flats , this is the effect of deleting strands. So elements in the kernel of the product mapping are analogous to Brunnian braids, braids that become trivial upon deletion of any strand.

###### Example 3.3.

Let  be the braid arrangement in so that , the -string pure braid group. Denote the pure braid generators by for corresponding to the hyperplanes given by By considering the projection to along the axis, we see that the subgroup generated by and is a free subgroup on three generators.

Let be the set of rank-two flats of  of multiplicity greater than two: Consider the commutator . Then, for every Indeed, one of or lies outside of hence at least one factor of the commutator is sent to 1 by Thus Clearly , since is a nontrivial reduced word lying in the free subgroup . Consequently, is not injective in general. Interpreted as a map on pure braids, the homomorphism has the effect of deleting, in turn, each of the four strands, Thus corresponds to a nontrivial Brunnian pure braid on four strands. (The closure of this braid is the Borromean rings link.)

The same argument used in this example shows that is not injective for complement of any strictly linearly-fibered arrangement which is not a product, and any set of flats .

Stanford showed that any braid (necessarily pure) which becomes trivial upon deletion of the strands outside a set is generated by the pure braid generators for along with iterated commutators of pure braid generators and their inverses which include at least one factor of this type. His argument can be cast in a more general setting so as to apply to other groups, including some arrangement groups.

Let be a group with finite generating set and let  be a family of subsets of

###### Definition 3.4.

The support of an element is relative to  is

We will write for the support of  being understood. The support of is which may be empty. Nonidentity elements of may also have empty support. In particular, need not hold.

###### Definition 3.5.

A monic commutator in is an element of defined recursively as follows:

• if then and are monic commutators;

• if and are monic commutators, then is a monic commutator.

In other words, a monic commutator is an iterated commutator of generators and their inverses.

Let be the family of all subsets of linearly ordered so that implies In particular,

###### Lemma 3.6.

Every element of can be written in the form where each is a product of monic commutators with support equal to , or

###### Proof.

The proof follows Stanford [Sta99] mutatis mutandis. We start with with support Assume inductively that where is a product of monic commutators whose support is or for and is or a product of monic commutators whose support is greater than or equal to in the linear order. Assume and fix a factorization of into monic commutators. Suppose some monic commutator factor occurring in has support equal to We can then reduce by one the number of monic commutator factors in with support equal to as follows. Write as a product of monic commutators where each is a monic commutator with support greater than , is a monic commutator with support , and is a product of monic commutators with support greater than or equal to Then

 s=yx1[x1,y]x2[x2,y]x3⋯xt[xt,y]z.

We replace by and by Then with a product of monic commutators with support and with having one fewer monic commutator factors with support than Iterating the process, we finally may write as above, but with a product of monic commutators having support strictly greater than in the linear order. Then, setting we have with a product of monic commutators with support greater than or equal to This completes the inductive step. Setting yields the theorem. ∎

Note, in the theorem above, if is not an intersection of elements of , then

For a subset of let be the quotient of by the normal closure of Let be the canonical projection. Thus kills generators not in

###### Definition 3.7.

A subset of is retractive if restricts to an injection A retractive family is a family of retractive subsets of all of whose intersections are also retractive.

When is retractive, we may tacitly identify with Then is a retraction in the usual sense: for By convention, is retractive.

###### Definition 3.8.

A subset of is transverse to a family  of subsets of if or equivalently, for every

Let

 ρS=∏S∈SρS:G→∏S∈SGS.
###### Theorem 3.9.

Suppose  is a retractive family. Then the kernel of is generated by monic commutators whose support is transverse to .

###### Proof.

Again we adapt Stanford’s argument. It is easy to show by induction that if is a monic commutator whose support meets since if or

Conversely, suppose Write as in the preceding lemma. Fix It suffices to show if We prove this by induction on We may assume is an intersection of elements of , by our earlier observation. If then where kills the images of the elements of Then we have since By the inductive hypothesis, if is a proper subset of On the other hand, if