Regularity of FI-modules and local cohomology
aa r X i v : . [ m a t h . A C ] M a r REGULARITY OF FI-MODULES AND LOCAL COHOMOLOGY
ROHIT NAGPAL, STEVEN V SAM, AND ANDREW SNOWDEN
Abstract.
We resolve a conjecture of Ramos and Li that relates the regularity of an FI -module to its local cohomology groups. This is an analogue of the familiar relationshipbetween regularity and local cohomology in commutative algebra. Introduction
Let S be a standard-graded polynomial ring in finitely many variables over a field k , andlet M be a non-zero finitely generated graded S -module. It is a classical fact in commutativealgebra that the following two quantities are equal (see [Ei, § • The minimum integer α such that Tor Si ( M, k ) is supported in degrees ≤ α + i for all i . • The minimum integer β such that H i m ( M ) is supported in degrees ≤ β − i for all i .Here H i m is local cohomology at the irrelevant ideal m . The quantity α = β is called the (Castelnuovo–Mumford) regularity of M , and is one of the most important numericalinvariants of M . In this paper, we establish the analog of the α = β identity for FI -modules.To state our result precisely, we must recall some definitions. Let FI be the categoryof finite sets and injections. Fix a commutative noetherian ring k . An FI-module over k is a functor from FI to the category of k -modules. We write Mod FI for the category of FI -modules. We refer to [CEF] for a general introduction to FI -modules.Let M be an FI -module. Define Tor ( M ) to be the FI -module that assigns to S thequotient of M ( S ) by the sum of the images of the M ( T ), as T varies over all proper subsetsof S . Then Tor is a right-exact functor, and so we can consider its left derived functorsTor • . In §
2, we explain how Tor • is the derived functor of a tensor product. We note thatthe FI -module homology considered in [CE] is the same as our Tor • . We let t i ( M ) be themaximum degree occurring in Tor i ( M ) (using the convention t i ( M ) = −∞ if Tor i ( M ) = 0),and define the regularity of M , denoted reg( M ), to be the minimum integer ρ such that t i ( M ) ≤ ρ + i for all i . We note that, while most FI -modules have infinite projective (and Tor)dimension, every finitely generated FI -module has finite regularity; see [CE, Theorem A] orCorollary 2.5 below.An element x ∈ M ( S ) is torsion if there exists an injection f : S → T such that f ∗ ( x ) = 0.Let H m ( M ) be the maximal torsion submodule of M . Then H m is a left-exact functor, andso we can consider its right derived functors H • m , which we refer to as local cohomology . If M is finitely generated then each H i m ( M ) is finitely generated and torsion, and H i m ( M ) = 0for i ≫ h i ( M ) be the maximum degree occurring in H i m ( M ),with the convention that h i ( M ) = −∞ if H i m ( M ) = 0.We can now state the main result of this paper: Date : March 21, 2017.2010
Mathematics Subject Classification.
Theorem 1.1.
Let M be a finitely generated FI -module. Then (1.1a) reg( M ) = max (cid:0) t ( M ) , max i ≥ ( h i ( M ) + i ) (cid:1) . Moreover, we have t n ( M ) = n + max i ≥ ( h i ( M ) + i ) for all n ≫ . In particular, max n> ( t n ( M ) − n ) = max i ≥ ( h i ( M ) + i ) . Remark 1.2. If M is a module over a polynomial ring in finitely many variables then onecan omit the t ( M ) on the right side of (1.1a). However, it is necessary in the case of FI -modules. Indeed, if M is the FI -module given by M ( S ) = k for all S and all injections actas the identity, then all local cohomology groups of M vanish, so h i ( M ) = −∞ for all i , butreg( M ) = t ( M ) = 0. (cid:3) Remark 1.3.
Theorem 1.1 can be proved for FI -modules presented in finite degrees. Wehave restricted ourselves to finitely generated modules to keep the paper less technical. (cid:3) Remark 1.4.
The theorem was first conjectured by Li and Ramos [LR, Conjecture 1.3]. Infact, they conjectured the result for FI G -modules, where G is a finite group. The version for FI G -modules follows immediately from the version for FI -modules, since local cohomologyand regularity do not depend on the G -action. (cid:3) Overview of proof.
Using the structure theorem for FI -modules (Theorem 2.4), an easyspectral sequence argument shows that the regularity of M is at most the maximum of h i ( M ) + i . Theorem 1.1 essentially says that there is not too much cancellation in thisspectral sequence.In characteristic 0, one can see this as follows. Let M λ be the irreducible representationof S n corresponding to the partition λ . Let ℓ ( λ ) be the number of parts in λ . For arepresentation V of S n , define ℓ ( V ) to be the maximum ℓ ( λ ) over those λ for which M λ occursin V . Now consider the relevant spectral sequence. One can directly observe that variousterms in the spectral sequence have different ℓ values, and so some representations mustalways survive on the subsequent page. This proves that there is not too much cancellation.In positive characteristic, there does not seem to be a complete analog of ℓ . However, weconstruct an invariant ν that has some of the same properties. This is one of the key insightsof this paper. The invariant ν is strong enough to distinguish terms in the spectral sequence,and thus allows the characteristic 0 argument to be carried out. Outline of paper. In §
2, we review some basic results on local cohomology of FI -modules.In §
3, we define the invariant ν mentioned above and establish some of its basic properties.These results are combined in § Acknowledgments.
We thank Eric Ramos for pointing our an error in an earlier versionof this paper.
EGULARITY OF FI -MODULES AND LOCAL COHOMOLOGY 3 Preliminaries on FI -modules We fix a commutative noetherian ring k for the entirety of the paper. Let Rep( S ⋆ ) be thecategory of sequences of representations of the symmetric groups over k . Given V • and W • in Rep( S ⋆ ), we define their tensor product by( V • ⊗ W • ) n = M i + j = n Ind S n S i × S j ( V i ⊗ W j ) . Then ⊗ endows Rep( S ⋆ ) with a monoidal structure (this is easier to see using the equivalencedescribed in [SS2, (5.1.6), (5.1.8)]). Furthermore, there is a symmetry of this monoidalstructure by switching the order of V and W and conjugating S i × S j to S j × S i via theelement τ ij ∈ S n which swaps the order of the two subsets 1 , . . . , i and i + 1 , . . . , n . We thushave notions of commutative algebra and module objects in Rep( S ⋆ ).Let A = k [ t ], where t has degree 1. We regard A as an object of Rep( S ⋆ ) by letting S n act trivially on A n = k . In this way, A is a commutative algebra object of Rep( S ⋆ ). Byan A -module, we will always mean a module object for A in Rep( S ⋆ ). We write Mod A forthe category of A -modules. As shown in [SS3, Proposition 7.2.5], the categories Mod A andMod FI are equivalent. We pass freely between the two points of view. We regard k as an A -module in the obvious way ( t acts by 0). We denote by Tor i ( − ) the i th left derived functorof k ⊗ A − on the category of A -modules. One easily sees that this definition coincides withthe one from the introduction.There is essentially only one Tor computation that we will use, namely Tor • ( k ). Letsgn n be the sign representation of S n , which we regard as an object of Rep( S ⋆ ) supported indegree n . There is an inclusion of k -modules sgn → A . We can consider the resulting Koszulcomplex V • (sgn ) ⊗ A . One easily sees that V n (sgn ) = sgn n . In degree n , this complex isthe usual complex that calculates the reduced homology of the standard n -simplex. It is wellknown that the standard n -simplex has no nontrivial reduced homology unless n = 0. Thisimplies that the Koszul complex above is exact in degrees >
0, and that its 0th homologyis just k . We thus have a resolution A ⊗ sgn • → k , and it is minimal in the sense that afterapplying − ⊗ A k , all differentials vanish. Proposition 2.1. If T is an A / A + -module, then Tor p ( T ) = T ⊗ sgn p .Proof. Apply − ⊗ T to the Koszul complex to conclude that Tor p ( T ) = T ⊗ sgn p . (cid:3) The restriction functor from Mod FI to Rep( S ⋆ ) admits a left adjoint denoted I . Wecall FI -modules of the form I ( V ) induced FI -modules. In terms of A -modules, we have I ( V ) = A ⊗ V . For a representation V of S d we have I ( V ) n = k [Hom FI ([ d ] , [ n ])] ⊗ k [ S d ] V = V ⊗ A n − d , where [ k ] = { , . . . , k } and [0] = ∅ . See [CEF, Definition 2.2.2] for more details on I ( V ); notethat there the notation M ( V ) is used in place of I ( V ). We say that an FI -module M is semi-induced if it has a finite length filtration where the quotients are induced. (Semi-inducedmodules have also been called ♯ -filtered modules in the literature.) In characteristic 0,induced modules are projective, and so semi-induced modules are induced.In the introduction, we defined H m ( M ) to be the maximal torsion submodule of an FI -module M . We now introduce Γ m as a synonym for H m , as it is better suited to the derivedfunctor notation RΓ m . Note that R i Γ m is exactly the same as H i m . Theorem 2.2.
Let M be a finitely generated FI -module. Then the following are equivalent: ROHIT NAGPAL, STEVEN V SAM, AND ANDREW SNOWDEN (a) M is semi-induced. (b) RΓ m ( M ) = 0 . (c) Tor i ( M ) = 0 for i > .Proof. The equivalence (a) ⇐⇒ (b) is proven in [LR, Proposition 5.12], and the equivalence(a) ⇐⇒ (c) is established in [R, Theorem B]. (cid:3) Lemma 2.3.
Let M be a bounded complex of FI -modules. Suppose all cohomology groupsare finitely generated torsion FI -modules. Then M is quasi-isomorphic to a bounded complexof finitely generated torsion FI -modules.Proof. For an FI -module N , let N ≤ n be the natural FI -module defined by( N ≤ n ) k = ( N k if k ≤ n k > n . It is clear that the functor ( − ) ≤ n is exact. We note that there is a natural surjection N → N ≤ n .Over a noetherian ring, a bounded complex with finitely generated cohomology is quasi-isomorphic to a complex with finitely generated terms (this can be proven via an elementaryargument that inductively lifts generators for cohomology groups). So we may assume thatthe terms of M are finitely generated. Let n be large enough so that that all cohomologygroups of M are supported in degrees ≤ n . Then M → M ≤ n is a quasi-isomorphism, and M ≤ n is a bounded complex of finitely generated torsion modules. (cid:3) Theorem 2.4 (Structure theorem for FI -modules) . Let M be a finitely generated FI -moduleover a noetherian ring k . Then, in the derived category of FI -modules, there is an exacttriangle T → M → F → such that (a) T is a bounded complex of finitely generated torsion modules supported in nonnegativedegrees. (b) F is a bounded complex of finitely generated semi-induced modules supported in non-negative degrees.Proof. This follows from [N, Theorem A] and the previous lemma. In characteristic 0, thistheorem was proved in [SS1]. (cid:3)
Corollary 2.5.
A finitely generated FI -module has finite regularity.Proof. Using the theorem and a d´evissage argument, one is reduced to the case of induced FI -modules, which obviously have finite regularity, and A / A + -modules, which have finiteregularity by Proposition 2.1. (cid:3) Proposition 2.6.
Let M be a finitely generated FI -module, and let T → M → F → be thetriangle in Theorem 2.4. Then H i m ( M ) = H i ( T ) . In particular, H i m ( T ) is finitely generatedfor all i and vanishes for i ≫ .Proof. This follows from Theorem 2.2 and the fact that RΓ m ( N ) = N if N is a torsion FI -module. See also [LR, Theorem E]. (cid:3) For a non-zero graded k -module M , we let maxdeg( M ) be the maximum degree in which M is non-zero, or ∞ if M is non-zero in arbitrarily high degrees. We also put maxdeg( M ) = −∞ if M = 0. With this notation, we havereg( M ) = max i ≥ [maxdeg(Tor i ( M )) − i ] , h i ( M ) = maxdeg H i m ( M ) . EGULARITY OF FI -MODULES AND LOCAL COHOMOLOGY 5 A result on symmetric group representations
Over a field of characteristic 0, representations of symmetric groups decompose as a directsum of simple representations, and the simples are indexed by partitions. Often, the numberof rows in the partitions that appear gives useful information about the representation. Ourgoal is to extend the notion of “the number of rows” to a more general ring. Call a two-sidedideal I ⊆ k [ S p ] good if the following properties hold:(a) I is idempotent,(b) I annihilates sgn p ,(c) I does not annihilate Ind S p S p − (sgn p − ) ⊗ k M for any nonzero k -module M ,(d) I is k -flat (and thus k -projective).We show that if k [ S p ] has a good ideal then for a k [ S n ]-module M and n − np ≤ k ≤ n , wecan make sense of the number of rows in M being equal to k . Proposition 3.1. If is invertible in k , then there is a good ideal in k [ S ] .Proof. Let N = 1 + (1 ,
2) be the norm element of k [ S ], and let I be the two-sided idealgenerated by N. We verify that I is good:(a) We have N = 2N, and so, since 2 is invertible, I is idempotent.(b) It is clear that N annihilates sgn , and so I does as well.(c) If M is a k -module then Ind S S (sgn ) ⊗ k M = k [ S ] ⊗ k M , which is clearly notannihilated by N.(d) As a k -module, I is free of rank 1, and thus k -flat. (cid:3) . Proposition 3.2. If is invertible in k , then there is a good ideal in k [ S ] .Proof. Let N = P σ ∈ S σ be the norm element of k [ S ]. Note that it is central. Let I be thetwo-sided ideal generated by τ = (1 + (1 , , − N . Note that makes sense as we have assumed 3 to be invertible in k . We now verify that I is good:(a) A straightforward computation shows that τ = τ , and so I is idempotent.(b) Both (1 + (1 , , so the same is true for I .(c) We have Ind S S (sgn ) ⊗ k M ∼ = M ⊕ where σ · ( m , m , m ) = sgn( σ )( m σ − (1) , m σ − (2) , m σ − (3) ) . Let x ∈ M be any nonzero element. Then τ · ( x, ,
0) = ( x, − x, = 0, so I does notannihilate Ind S S (sgn ) ⊗ k M .(d) We first claim that I is equal to the ideal J generated by the differences of two trans-positions. The sum of the coefficients of the odd (or even) permutations appearingin 3 τ is zero. This shows that I ⊂ J . The reverse inclusion J ⊂ I follows from thefollowing identity (1 , − (1 ,
2) = (1 , τ − τ (1 , . This establishes the claim. Clearly, we have k [ S ] /J ∼ = k . This implies that k [ S ] /I ∼ = k . Thus, as a k -module, I is a summand of k [ S ], and therefore k -flat. (cid:3) ROHIT NAGPAL, STEVEN V SAM, AND ANDREW SNOWDEN
Throughout the rest of this section, we fix an integer p ≥ I of k [ S p ].If pr ≤ n , we define I n ( r ) to be the two-sided ideal of k [ S n ] generated by I ⊠ r under theinclusion k [ S × rp ] ֒ → k [ S n ]. For convenience, we set I n ( r ) = 0 if pr > n . It is clear that I n ( r )is idempotent. Definition 3.3.
Let M be a k [ S n ]-module. We define ν ( M ) = n − r if M is not annihilatedby I n ( r ) but is annihilated by I n ( s ) for all r < s . (cid:3) Proposition 3.4.
Consider an exact sequence → M → M → M → of k [ S n ] -modules. Then M is annihilated by I n ( r ) if and only if both M and M are.Consequently, ν ( M ) = min( ν ( M ) , ν ( M )) . Proof. If M is annihilated by I n ( r ) then obviously M and M are. Suppose that M and M are annihilated by I n ( r ). Then the image of I n ( r ) M in M vanishes, and so I n ( r ) M ⊂ M ,and so I n ( r ) M = 0. But I n ( r ) = I n ( r ), and so M is annihilated by I n ( r ). (cid:3) Lemma 3.5.
Let N be any nonzero k -module. Then the ideal I ⊠ r of k [ S × rp ] does not anni-hilate (Ind S p S p − sgn p − ) ⊠ r ⊗ k N .Proof. This follows by induction on r and the definition of good. (cid:3) The following proposition is motivated by [CE, Proposition 3.1].
Proposition 3.6.
Let M be a representation of S d , let ( p − d ≤ k , put n = k + d . Thenwe have ν ( M ⊗ sgn k ) = k .Proof. Set ν = ν ( M ⊗ sgn k ). We first show that I n ( r ) does not annihilate M ⊗ sgn k for r ≤ d . The Mackey decomposition theorem givesRes S n S × rp ( M ⊗ sgn k ) = M g ∈ ( S d × S k ) \ S n /S × rp Ind S × rp S × rp ∩ ( S d × S k ) g Res ( S d × S k ) g S × rp ∩ ( S d × S k ) g ( M ⊗ sgn k ) g where the sum is over double coset representatives, and ( − ) g means conjugation by g . Taking g so that S × rp ∩ ( S d × S k ) g = S × rp − , we see that it contains (Ind S p S p − sgn p − ) ⊠ r ⊗ k M as a directsummand. Since I n ( r ) is generated by I ⊠ r , it suffices to show that I ⊠ r does not annihilatethis direct summand. But this follows from Lemma 3.5.Now we show that I n ( r ) annihilates M ⊗ sgn k if n/p ≥ r > d . Note that Res S n S × rp ( M ⊗ sgn k ) decomposes naturally into a finite direct sum of k [ S × rp ]-modules of the form ⊠ ri =1 M i .Since r > d , at least one M i is isomorphic to sgn p for each such direct summand. Thus I ⊠ r annihilates each such direct summand. This shows that I n ( r ) annihilates M ⊗ sgn k ,completing the proof. (cid:3) Remark 3.7.
Our invariant ν is an attempt to extend the notion of “minimum number ofrows in a simple object” away from characteristic zero. To see this, let notation be as inProposition 3.6. In characteristic 0, the partitions in M have d boxes. Thus, by the Pierirule, every partition appearing in M ⊗ sgn k has at least k rows, and some have exactly k rows.Since I n ( r ) = 0 for r > n/p , our invariant ν can’t distinguish between partitions with atmost n − np rows. (cid:3) EGULARITY OF FI -MODULES AND LOCAL COHOMOLOGY 7 The main theorem
The aim of this section is to prove Theorem 1.1. Before beginning we note that if M isa graded k -module and M [ ] and M [ ] are the localizations of M obtained by inverting 2and 3, respectively, thenmaxdeg( M ) = max(maxdeg( M [ ]) , maxdeg( M [ ])) . (Proof: the kernel of the localization map M → M [ p ] is the set of elements annihilatedby a power of p ; if x ∈ M is annihilated by both 2 n and 3 m then x = 0, since 2 n and 3 m are coprime.) Localization commutes with Tor and local cohomology, so it suffices to proveTheorem 1.1 assuming that either 2 or 3 is invertible in k . In particular, in the remainderof this section, we may assume that k [ S p ] has a good ideal for either p = 2 or p = 3.For a complex M of FI -modules, we defineTor n ( M ) = H − n ( M ⊗ L A k ) . (We use cohomological indexing throughout this section.) The regularity of M is theminimal ρ so that maxdeg(Tor n ( M )) ≤ n + ρ for all n ∈ Z . Lemma 4.1.
Let M be a finite length complex of finitely generated torsion FI -modules. Let m be minimal such that M m = 0 . Then ν (Tor n ( M )) ≥ n + m for all n ≫ .Proof. Using the Koszul complex, we see that Tor n ( M ) is a subquotient of M j ≥ M j − n ⊗ sgn j . Only the terms with j ≥ n + m contribute. Each of these has ν ≥ n + m by Proposition 3.6,and this passes to subquotients by Proposition 3.4. (cid:3) Lemma 4.2.
Let M be a finitely generated torsion FI -module, and let ρ = maxdeg( M ) .Then the regularity of M is ρ , and for n ≫ we have ν (Tor n ( M ) n + ρ ) = n. Proof.
Let M be the degree ρ piece of M = M , and let M = M /M . By induction on ρ ,we can assume reg( M ) < ρ . We have an exact sequenceTor n +1 ( M ) n + ρ → Tor n ( M ) n + ρ → Tor n ( M ) n + ρ → . Note that Tor n ( M ) n + ρ = 0 by the bound on the regularity of M , which is why we have a 0on the right above. Since M is concentrated in one degree, we have Tor n ( M ) = M ⊗ sgn n .So by Proposition 3.6, the lemma is true for M . By Lemma 4.1, the leftmost term abovehas ν = n + 1. Since the middle term has ν = n , we see (from Proposition 3.4) that therightmost term is non-zero and has ν = n , which completes the proof. (cid:3) Proposition 4.3.
Let M be a finite length complex of finitely generated torsion FI -modules.Put ρ = max i ∈ Z ( i + maxdeg H i ( M )) . Then the regularity of M is ρ . Moreover, if r is minimal so that ρ = r + maxdeg H r ( M ) then ν (Tor n ( M ) n + ρ ) = n + r for all n ≫ . ROHIT NAGPAL, STEVEN V SAM, AND ANDREW SNOWDEN
Proof.
Let j be the minimal index so that H j ( M ) = 0; we may as well assume that M i = 0for i < j . Let M be the kernel of d : M j → M j +1 , regarded as a complex concentrated indegree j , let M = M , and let M = M /M , so that we have a short exact sequence ofcomplexes. Note that H j ( M ) → H j ( M ) is an isomorphism and H i ( M ) → H i ( M ) is anisomorphism for all i > j . Since M has fewer non-zero cohomology groups than M , we canassume (by induction) that the proposition holds for M . The proposition holds for M byLemma 4.2. We have an exact sequenceTor n +1 ( M ) n + ρ → Tor n ( M ) n + ρ → Tor n ( M ) n + ρ → Tor n ( M ) n + ρ → . Note that Tor n − ( M ) n + ρ = 0, since the regularity of M is at most ρ , which is why we havea 0 on the right. We now consider two cases: • Case 1: j = r . We then have that ν (Tor n ( M ) n + ρ ) = n + r . By Lemma 4.1, ν (Tor n +1 ( M ) n + ρ ) > n + r . If there exists s > r such that ρ = s + maxdeg H s ( M )then M has regularity ρ and ν (Tor n ( M ) n + ρ ) = n + s > n + r ; otherwise, M hasregularity < ρ and Tor n ( M ) n + ρ = 0. Thus the two outside terms in the above 4-term sequence have ν > n + r (or vanish), and so Tor ( M ) n + ρ is non-zero and has ν = n + r . • Case 2: j = r . In this case, M has regularity < ρ , and so Tor n ( M ) n + ρ = 0. ThusTor n ( M ) n + ρ = Tor n ( M ) n + ρ , and the result follows by the inductive hypothesis. (cid:3) We now prove our main result.
Proof of Theorem 1.1.
Let T → M → F → be the exact triangle as in Theorem 2.4. Bytaking Tor we get a long exact sequence · · · → Tor n ( T ) → Tor n ( M ) → Tor n ( F ) → · · · . Note that F is represented by a bounded complex of semi-induced modules and higherTor groups of semi-induced modules are zero. Hence F ⊗ L A k is computed by the usualtensor product F ⊗ A k . Since F is concentrated in non-negative cohomological degrees,this shows that Tor n ( F ) = 0 for n >
0. Thus, by the long exact sequence above, we haveTor n ( T ) = Tor n ( M ) for n >
0. Thusreg( M ) = max( t ( M ) , reg( T )) . By Proposition 2.6, we have H i ( T ) = H i m ( M ) for all i , and so maxdeg(H i ( T )) = h i ( M ). Thetheorem therefore follows from Proposition 4.3. (cid:3) References [CE] Thomas Church, Jordan S. Ellenberg, Homology of FI-modules,
Geom. Topol. , to appear, arXiv:1506.01022v2 .[CEF] Thomas Church, Jordan S. Ellenberg, Benson Farb, FI-modules and stability for representations ofsymmetric groups,
Duke Math. J. (2015), no. 9, 1833–1910. arXiv:1204.4533v4 .[Ei] David Eisenbud,
The Geometry of Syzygies , Graduate Texts in Mathematics , Springer–Verlag,2005.[LR] Liping Li, Eric Ramos, Depth and the local cohomology of FI G -modules, arXiv:1602.04405v3 .[N] Rohit Nagpal, FI-modules and the cohomology of modular S n -representations, arXiv:1505.04294v1 .[R] Eric Ramos, Homological invariants of FI -modules and FI G -modules, arXiv:1511.03964v3 [SS1] Steven V Sam, Andrew Snowden, GL-equivariant modules over polynomial rings in infinitely manyvariables, Trans. Amer. Math. Soc. (2016), 1097–1158, arXiv:1206.2233v3 .[SS2] Steven V Sam, Andrew Snowden, Introduction to twisted commutative algebras, arXiv:1209.5122v1 . EGULARITY OF FI -MODULES AND LOCAL COHOMOLOGY 9 [SS3] Steven V Sam, Andrew Snowden, Gr¨obner methods for representations of combinatorial categories, J. Amer. Math. Soc. (2017), 159–203, arXiv:1409.1670v3 . Department of Mathematics, University of Chicago, Chicago, IL
E-mail address : [email protected] URL : http://math.uchicago.edu/~nagpal/ Department of Mathematics, University of Wisconsin, Madison, WI
E-mail address : [email protected] URL : http://math.wisc.edu/~svs/ Department of Mathematics, University of Michigan, Ann Arbor, MI
E-mail address : [email protected] URL ::