Arrangements and

Computations III:

Λ(V ) and BGG

∑xi 0 0 0

0∑

xi 0 0

0 0∑

xi 0

0 0 0∑

xi

x1 + x4 + x5 0 0 00 x2 + x3 + x5 0 00 0 x0 + x3 + x4 00 0 0 x0 + x1 + x2

−x3 0 0 x3

0 x4 0 x4

0 0 −x5 x5

x0 x0 0 0x2 0 −x2 00 x1 x1 0

.

4 10 15 20 25 · · ·

Hal Schenck

Mathematics Department

University of Illinois

August 19, 2009

1

Let A be the Orlik-Solomon algebra of Cℓ \ A,

with |A| = n. For each a =∑

aiei ∈ A1, we

consider the complex (A, a).

The ith term is Ai, and differential is ∧a:

(A, a): 0 // A0a

// A1a

// A2a

// · · · a// Aℓ

// 0 .

Arose in

• hypergeometric functions (Aomoto)

• cohomology with local system coefficients

–Esnault, Schechtman, Viehweg

–Schechtman, Terao, Varchenko

The resonance varieties of A are the loci of

points a =∑n

i=1 aiei ↔ (a1 : · · · : an) ∈ Pn−1 for

which (A, a) fails to be exact, that is:.

Definition 1 For each k ≥ 1,

Rk(A) = {a ∈ Pn−1 | Hk(A, a) 6= 0}.

Yuzvinsky: for generic a, (A, a) is exact.

2

Definition 2 Π partition of A is neighborly if

∀Y ∈ L2(A), π block of Π,

µ(Y ) ≤ |Y ∩ π| −→ Y ⊆ π.

Falk: proved that components of R1(A) arise

from neighborly partitions, and conjectured that

R1(A) is a union of linear components.

This was proved by

• Cohen–Suciu and by

• Libgober–Yuzvinsky R1(A) = ∐L+i

• Cohen–Orlik also true for R≥2(A)

• Falk can fail if characteristic 6= 0.

Libgober–Yuzvinsky connects R1(A) to pen-

cils/nets/webs; recent work in this area by:

• Falk–Yuzvinsky

• Pereira–Yuzvinsky

Recall conjectural connection to LCS ranks φk:

Conjecture 3 (Suciu) Under certain conditions,

∏

k≥1

(1 − tk)φk =∏

Li∈R1(A)

(1 − (dim(Li)t)

3

Example 4 Let A = V (xy(x − y)z) ⊆ P2, and

E = Λ(C4), with generators e1, . . . , e4. The

Orlik-Solomon algebra

A = E/〈∂(e1e2e3), ∂(e1e2e3e4)〉, with

∂(e1e2e3) = e1 ∧ e2 − e1 ∧ e3 + e2 ∧ e3

To compute R1(A), we need only the first

two differentials in the Aomoto complex. Use

e13, e14, e23, e24, e34 as a basis for A2.

e1 7→ e1 ∧ (4∑

i=1

aiei) = a2e12 + a3e13 + a4e14.

Since e12 = e13 − e23, a2e12 = a2(e13 − e23),

giving (a2 + a3)e13 + a4e14 − a2e23. compute!

0 −→ C1

a1

a2

a3

a4

−−−−−→ C4

a2 + a3 −a1 −a1 0a4 0 0 −a1

−a2 a1 + a3 −a2 00 a4 0 −a2

0 0 a4 −a3

−−−−−−−−−−−−−−−−−−−−−−−−−−→ C5

4

Letting a =n∑

i=1aiei, we have

R1(A) ↔ H1(A,∧a)↔ ∃b ∈ E1 | a ∧ b vanishes in A2↔ ∃b ∈ E1 | a ∧ b ∈ I2↔ decomposable 2-tensors in I2↔ P(I2) ∩ Gr(2, E1) ⊆ P(

∧2 E1)

I2 is determined by the intersection lattice L(A)

in rank ≤ 2, so to study R1(A), let A ⊆ P2.

Grassmannian gives fastest compution of R1(A).

Problem Code up for R≥2(A) (Segre map).

Note interesting connection to syzygies. Since

a ∧ b ∈ I2 −→ a ∧ b =∑

cifi, ci ∈ C, fi ∈ I2, the

relations a ∧ a ∧ b = 0 = b ∧ a ∧ b yield linear

syzygies on I2:∑

acifi = 0 =∑

bcifi.

That is,

R1(A) is related to TorE2 (A, C)3

5

Example 5 For A = V (xy(x − y)z) ⊆ P2, the

Orlik-Solomon algebra is just

A = E/∂(e1e2e3),

since the relation ∂(e1e2e3e4) is redundant:

∂(e1e2e3e4) = e1 ∧ ∂(e1e2e3) − e4∂(e1e2e3)

Observe that

e1 ∧ e2 − e1 ∧ e3 + e2 ∧ e3 = (e1 − e2)∧ (e2 − e3)

This means that the line

s(e1 − e2) + t(e2 − e3) ⊆ R1(A) ⊆ P(E1)

Parametrically, this may be written

(s : t − s : −t : 0) = V (a4, a1 + a2 + a3)

Such components of R1(A) are called local.

(Compute) the corresponding linear syzygies.

6

WHO CARES? Conjecturally, R1(A) is (some-

times) connected to the LCS ranks. But it is

always connected to the Chen ranks! Intro-

duced by K.T. Chen, these are the LCS ranks

of the maximal metabelian quotient of G:

θk(G) := φk(G/G′′),

where G′ = [G, G].

Conjecture 6 (Suciu) Let G = G(A) be an

arrangement group, and let hr be the number

of components of R1(A) of dimension r. Then,

for k ≫ 0:

θk(G) = (k − 1)∑

r≥1

hr

(r + k − 1

k

).

For Example 3, R1(A) ≃ P1 and thus

θk(G) = (k − 1).

7

How to determine the Chen ranks? The Alexan-

der invariant G′/G′′ is a module over Z[G/G′].

For arrangements, Z[G/G′] = Laurent polyno-

mials in n-variables.

Massey:∑

k≥0

θk+2 tk = HS(gr G′/G′′ ⊗ Q, t)

Easier to work with is the linearized Alexander

invariant B of Cohen-Suciu

(A2 ⊕ E3) ⊗ S ∆// E2 ⊗ S // B // 0 , where

∆ is built from Koszul diff. and (E2 → A2)t.

Theorem 7 (Cohen-Suciu)

V (ann B) = R1(A)

Theorem 8 (Papadima-Suciu) For k ≥ 2,

∑

k≥2

θk tk = HS(B, t).

In particular, the Chen ranks are combinatori-

ally determined, and depend only on L(A) in

rank ≤ 2.

8

Example 9 Recall the matroid for A3 is:

@@

@@

@@

@

��

��

��

�

PPPPPPPPPPP

�����������v v

v v

v

v

5

4

1

3

2

0

For A3, B is the cokernel of the matrix on the

first slide. (compute) R1(A3) =

V (x1 + x4 + x5, x0, x2, x3)∐

V (x2 + x3 + x5, x0, x1, x4)∐

V (x0 + x3 + x4, x1, x2, x4)∐

V (x0 + x1 + x2, x3, x4, x5)∐

V (x0 + x1 + x2, x0 − x5, x1 − x3, x2 − x4).

and (compute) the Hilbert Series of B:

(4t2+2t3−t4)/(1−t)2 = 4t2+10t3+15t4+20t5+· · ·

Magic Trick! (compute) TorEi (A3, C)i+1

Magic Trick! (compute) free resolution of the

cokernel of last map in the Aomoto complex.

9

Theorem 10 (Eisenbud-Popescu-Yuzvinsky)

For an arrangement A, the Aomoto complex

is exact, as a sequence of S-modules:

0 // A0 ⊗ S ·a// A1 ⊗ S ·a

// · · · ·a // Aℓ ⊗ S // F(A) // 0 .

Theorem 11 (–, Suciu) The linearized Alexan-

der invariant B is functorially determined by

the Orlik-Solomon algebra:

B ∼= Extℓ−1S (F(A), S).

Use this, localization, and the result of Libgober-

Yuzvinsky that R1(A) = ∐Li to obtain:

Theorem 12 (–, Suciu) For k ≫ 0,

θk(G) ≥ (k − 1)∑

Li∈R1(A)

(dimLi + k − 1

k

).

Problem Prove the remaining inequality! Note:

θk(G) is polynomial in k, of degree = dimR1(A).

10

WHAT MAKES ALL THIS WORK IS BGG:

the Bernstein-Gelfand-Gelfand correspondence.

Let S = Sym(V ∗) and E =∧(V ). BGG is an

isomorphism between derived categories of

• bounded cpxs of coherent sheaves on P(V ∗).

• bounded cpxs of f.gen’d, graded E–modules.

From this, can extract functors

R: f.gen’d, graded S-modules −→ linear free

E-complexes.

L: f.gen’d, graded E-modules −→ linear free

S-complexes.

Point: can translate problems to possibly sim-

pler setting. For example, we’ll see this gives a

fast way to compute sheaf cohomology, using

Tate resolutions.

11

P a f’gend, graded E-module, then L(P) is the

complex

· · · // S ⊗ Pi+1·a

// S ⊗ Pi·a

// S ⊗ Pi−1·a

// · · · ,

where a =n∑

i=1xi⊗ei, so that 1⊗p 7→

∑xi⊗ei∧p

Note: elts of V ∗ deg = 1, elts of V deg = −1.

Example 13 P = E =∧

C3. Then we have

0 // S ⊗ E0// S ⊗ E1

// S ⊗ E2// S ⊗ E3

// 0 .

Clearly 1 7→∑3

1 xi ⊗ ei. For d1

e1 7→ −x2e12 − x3e13

e2 7→ x1e12 − x3e23

e3 7→ x1e13 + x2e23

d2 : e12 7→ x3e123, e13 7→ −x2e123 e23 7→ x1e123

Thus, L(E) is

S1

x1

x2

x3

−−−−−→ S3

−x2 x1 0−x3 0 x1

0 −x3 x2

−−−−−−−−−−−−−−−→ S3

[x3 −x2 x1

]−−−−−−−−−−−−−→ S1

The Koszul complex!

12

M a f’gend, graded S-module, then R(M) is

the complex

· · · // E ⊗ Mi−1·a

// E ⊗ Mi·a

// E ⊗ Mi+1·a

// · · · ,

where a =n∑

i=1ei ⊗ xi, so 1 ⊗ m 7→

∑ei ⊗ xi · m,

and E is the C-dual of E:

E ≃ E(n) = HomC(E, C).

Just as L(P) = S ⊗C P , R(M) = HomC(E, M).

Example 14 M = C[x0, x1]/〈x0x1, x20〉. Then

0 // E ⊗ M0// E ⊗ M1

// E ⊗ M2// E ⊗ M3

// · · ·

1 7→ e0 ⊗ x0 + e1 ⊗ x1

x0 7→ e0 ⊗ x20 + e1 ⊗ x0x1

x1 7→ e0 ⊗ x0x1 + e1 ⊗ x21

xn1 7→ e0 ⊗ x0xn

1 + e1 ⊗ xn+11

Thus, R(M) is

E(2)1

[e0

e1

]

−−−−→ E(3)2

[0 e1

]−−−−−−−→ E(4)1

[e1

]−−−−→ E(5)1

[e1

]−−−−→ · · ·

13

This complex is exact, except at the second

step. Obviously the kernel of[0 e1

]

is generated by α = [1,0] and β = [0, e1], with

relations im(d1) = β + e0α = 0, e1β = 0, so

that

H1(R(M)) ≃ E(3)/e0 ∧ e1

Compute this, and compute the free resolution

of M . This illustrates

Theorem 15 (Eisenbud-Fløystad-Schreyer)

Hj(R(M))i+j = TorSi (M, C)i+j.

Corollary 16 The Castelnuovo-Mumford reg-

ularity of M is ≤ d iff Hi(R(M)) = 0 for all

i > d.

14

What can be said about higher resonance va-

rieties? Cohen–Orlik proved that for k ≥ 2,

Rk(A) =⋃

Li linear.

Suciu showed union need not be disjoint.

Theorem 17 (Eisenbud-Popescu-Yuzvinsky)

Resonance persists: p ∈ Rk(A) −→ p ∈ Rk+1(A).

The key observation is a ∈ Rk(A) ⊆ P(E) means

Hk(A, a) 6= 0 ↔ TorSℓ−k(F(A), S/I(p)) 6= 0.

The result follows from interpreting this in terms

of Koszul cohomology.

Theorem 18 (Denham, –) As for R1(A), higher

resonance may be interpreted via Ext:

Rk(A) =⋃

k′≤k

V (annExtℓ−k′(F(A), S)).

Differentials in free resolution can be analyzed

using BGG and Grothendieck spectral sequence

(work in progress, Denham, –).

15

For a coherent sheaf F on Pd, there is a f’gend,

graded S-module M whose sheafification is F.

If F has Castelnuovo-Mumford regularity r, then

the Tate resolution of F is obtained by splic-

ing the complex R(M≥r):

0 // E ⊗ Mrdr

// E ⊗ Mr+1// E ⊗ Mr+2

// · · · ,

with a free resolution P• for the kernel of dr:

· · · // P1// P0

//

%%KKKKKK E ⊗ Mr// E ⊗ Mr+1

// · · ·

ker(dr)

66nnnnnn

))SSSSSSSSS

0

88ppppppp

0

By Corollary 16, R(M≥r) is exact except at

the first step, so this yields an exact complex

of free E-modules.

Example 19 Since M = S has regularity zero,

we obtain Cartan resolutions in both direc-

tions, with splice map E → E = E(d+1) multi-

plication by e0∧e1∧· · ·∧ed = ker

[e0, · · · , ed

]t.

16

Theorem 20 (Eisenbud-Fløystad-Schreyer)

The ith free module T i in a Tate resolution for

F satisfies

T i =⊕

jE ⊗ Hj(F(i − j)).

Example 21 Twisted cubic I ⊆ S = C[x, y, z, w]

0 −→ S(−3)2

[−z wy −z−x y

]

−−−−−−−−−−→ S(−2)3

[y2−xz yz−xw z2−yw

]−−−−−−−−−−−−−−−−−−−−−−−−→ S −→ S/I

Display as a betti table:

bij = dimC TorSi (M, C)i+j.

total 1 3 20 1 – –1 – 3 2

This has regularity one, so now we can (com-

pute) the Tate resolution:

17

Plugging these numbers into Theorem 20, we

see that

i −3 −2 −1 0 1 2

h1(F(i)) 8 5 2 0 0 0

h0(F(i)) 0 0 0 1 4 7

Does this make sense?

F = OX = OP1(3)

so

h1(F(i)) = h1(OP1(3i)) = h0(OP1(−3i − 2))

and

h0(F(i)) = h0(OP1(3i)) = 3i + 1, i ≥ 0

THE END! THANK YOU!

18

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