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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
K. Aomoto, Un theoreme du type de Matsushima-Murakami con-
cernant l’integrale des fonctions multiformes, J. Math. Pures Appl.
52 (1973), 1–11.
K.T. Chen, Iterated integrals of differential forms and loop space
cohomology, Ann. of Math. 97 (1973), 217-246.
D. Cohen, P. Orlik, Arrangements and local systems, Math. Res.
Lett. 7 (2000), 299–316.
D. Cohen, A. Suciu, Alexander invariants of complex hyperplane
arrangements, Trans. Amer. Math. Soc. 351 (1999), 4043–4067.
D. Cohen, A. Suciu, Characteristic varieties of arrangements, Math.
Proc. Cambridge Phil. Soc. 127 (1999), 33–53.
W. Decker, D. Eisenbud, Sheaf algorithms using the exterior al-
gebra, in: Computations in Algebraic Geometry using Macaulay 2,
Springer-Verlag, Berlin-Heidelberg-New York, 2002.
G. Denham, H. Schenck, The double Ext spectral sequence and
rank varieties, preprint, 2009.
G. Denham, S. Yuzvinsky, Annihilators of Orlik-Solomon relations,
Adv. in Appl. Math. 28 (2002), 231–249.
D. Eisenbud, The geometry of syzygies, Springer-Verlag, Berlin-
Heidelberg-New York, 2004.
19
D. Eisenbud, G. Fløystad, F.-O. Schreyer, Sheaf cohomology and
free resolutions over exterior algebras, Trans. Amer. Math. Soc.
355 (2003), 4397–4426.
D. Eisenbud, S. Popescu, S. Yuzvinsky, Hyperplane arrangement
cohomology and monomials in the exterior algebra, Trans. Amer.
Math. Soc. 355 (2003), 4365–4383.
H. Esnault, V. Schechtman, E. Viehweg, Cohomology of local
systems on the complement of hyperplanes, Invent. Math. 109
(1992), 557-561.
M. Falk, Arrangements and cohomology, Ann. Combin. 1 (1997),
135–157.
M. Falk, Resonance varieties over fields of positive characteristic,
IMRN 3 (2007)
M. Falk, S. Yuzvinsky, Multinets, resonance varieties, and pencils
of plane curves, Compos. Math. 143 (2007), 1069–1088
A. Libgober, S. Yuzvinsky, Cohomology of the Orlik-Solomon alge-
bras and local systems, Compositio Math. 121 (2000), 337–361.
P. Orlik, L. Solomon, Combinatorics and topology of complements
of hyperplanes, Invent. Math. 56 (1980), 167–189.
P. Orlik, H. Terao, Arrangements of hyperplanes, Grundlehren
Math. Wiss., Bd. 300, Springer-Verlag, Berlin-Heidelberg-New
York, 1992.
J. Pereira, S. Yuzvinsky, Completely reducible hypersurfaces in a
pencil, Adv. Math. 219 (2008), 672–688.
V. Schechtman, H. Terao, A. Varchenko, Local systems over com-
plements of hyperplanes and the Kac-Kazhdan conditions for sin-
gular vectors, J. Pure Appl. Algebra 100 (1995), 93–102.
H. Schenck, A. Suciu, Lower central series and free resolutions of
hyperplane arrangements, Trans. Amer. Math. Soc. 354 (2002),
3409–3433.
H. Schenck, A. Suciu, Resonance, linear syzygies, Chen groups,
and the Bernstein-Gelfand-Gelfand correspondence, Trans. Amer.
Math. Soc. 358 (2006), 2269-2289.
A. Suciu, Fundamental groups of line arrangements: Enumerative
aspects, Contemporary Math., vol. 276, Amer. Math. Soc, Provi-
dence, RI, 2001, pp. 43–79.
S. Yuzvinsky, Cohomology of Brieskorn-Orlik-Solomon algebras,
Comm. Algebra 23 (1995), 5339–5354.
S. Yuzvinsky, Realization of finite Abelian groups by nets in P2,
Compos. Math. 140 (2004), 1614–1624.
S. Yuzvinsky, A new bound on the number of special fibers in a
pencil of plane curves, Proc. Amer. Math. Soc. 137 (2009),
1641–1648.