Euler characteristic of coherent sheaves on simplicial torics via the Stanley-Reisner ring

0.17 MB·English

by  Hal Schenck

EULER CHARACTERISTIC OF COHERENT SHEAVES ON
SIMPLICIAL TORICS VIA THE STANLEY-REISNER RING
1
HALSCHENCK
1
0
2 Abstract. We combine work of Cox on the total coordinate ring of a toric
n varietyandresultsofEisenbud-Musta¸tˇa-StillmanandMusta¸tˇaoncohomology
a oftoricandmonomialidealstoobtainaformulaforcomputingχ(OX(D))for
J aWeildivisorDonacompletesimplicialtoricvarietyXΣ. Themainpointis
touseAlexanderdualitytopassfromthetoricirrelevantideal,whichappears
1
inthe computation of χ(OX(D)), to the Stanley-Reisner ideal of Σ, whichis
] usedindefiningtheChowringofXΣ.
G
A
.
h 1. Introduction
t
a
ForadivisorD onasmoothcomplete varietyX,the Hirzebruch-Riemann-Roch
m
theoremdescribestheEulercharacteristicofO (D)intermsofintersectiontheory:
X
[

1 χ(OX(D))= ch(D)·Td(X).
Z
v
3 ThedivisorD correspondstoaclass[D]intheChowringofX,andch(D)consists
5 of the first n = dim(X) terms of the formal Taylor expansion of eD. The Todd
3 classof D is defined similarly,but using the Taylorexpansionfor D . To define
0 1−e−D
theToddclassofX,filterthe tangentbundle T bylinebundles O(D ). Thenone
. X i
1 shows that Td(X)= n Td(D ) is independent of the filtration.
0 LetΣ⊆RnbeacoQmip=le1tesimpilicialrationalpolyhedralfanwithd=|Σ(1)|rays,
1
1 XΣtheassociatedtoricvariety,andD ∈Cl(XΣ)aWeildivisoronXΣ. Wecombine
: AlexanderdualityandtheCoxringwithresultsofMusta¸tˇa[16]onmonomialideals
v
to obtain a formula for the Euler characteristicof the associatedrank one reflexive
i
X sheaf O (D). Put Z ={0,1}d and 1={1}d. Then for l ≫0,
XΣ
ar (1) χ(OX(D))= (−1)|m|−d+ndimK(S/IΣ)1−m·dimKSl·φ(m)+D.
mX∈Z\0
Here I denotes the Stanley-Reisner ideal, and Zd →φ Cl(X ) is the standard sur-
Σ Σ
jection of Zd onto the class group. The Cox ring S is a polynomial ring, gradedby
Cl(X ); on S/I we use the Zd grading. We recall the definitions of these objects
Σ Σ
in§2. Anycoherentsheafonanondegeneratetoricvarietycorrespondstoafinitely
generatedCl(X )-gradedS–module (see [3]for the simplicialcase,and[17]for the
Σ
general case), so such a sheaf has a resolution by rank one reflexive sheaves, and
Equation 1 yields a formula for χ(F) for any coherent sheaf F. Bounds on l are
determined by Eisenbud-Musta¸tˇa-Stillman in [6], and are discussed in §2.
2000 Mathematics Subject Classification. 14M25,14C40.
Key words and phrases. Toricvariety,Coxring,Chowring,cohomology.
Schenck supportedbyNSF07–07667, NSA904-03-1-0006.
1
2 HALSCHENCK
Connections to physics and some history. The methods which are used
to prove Equation 1 have applications to computations arising in mathematical
physics: in a recent preprint [1], Blumenhagen, Jurke, Rahn and Roschy conjec-
turedanalgorithmforcomputingthecohomologyoflinebundlesonatoricvariety.
TheirmotivationwastocomputemasslessmodesinTypeIIB/Fandheteroticcom-
pactifications, on a complete intersection in a toric variety. A strong form of the
algorithmisestablishedbyMaclaganandSmithinCorollary3.4of[14];laterproofs
appearinJow[9]andRahn-Roschy[20]. Inallthese papersAlexanderduality and
results of [6] play a key role, as they do in the proof of Equation 1. The original
motivationforthisworkwastofindatoricprooffortheHirzebruch-Riemann-Roch
theorem.
The first toric interpretation of Hirzebruch-Riemann-Roch is due to Khovanskii
[11]. In[12],[13],Pukhlikov-Khovanskiistudyadditivemeasuresonvirtualpolyhe-
dra,and obtain a Riemann-Rochformula for integrating sums of quasipolynomials
on virtual polytopes. Pommersheim [18] and Pommersheim and Thomas [19] ob-
tain results on Todd classes of simplicial torics, and in [2], Brion-Vergne prove an
equivariant Riemann-Roch for simplicial torics.
The results of Eisenbud-Musta¸tˇa-Stillman in [6] show that in the toric setting,
χ(O (D)) may be calculated via certain Ext modules overthe Cox ring of X. On
X
the other hand, evaluating the expression ch(D)·Td(X) involves a computation
in the Chow ring of X, and the Cox andRChow rings of a simplicial toric variety
are connected by Alexander duality.
The paper is structured as follows: in §2 we recall the results of [6] and the
computation of cohomology via the Cox ring. In §3 we introduce the Chow ring,
recallthattheStanley-ReisneridealofΣistheAlexanderdualofthetoricirrelevant
idealofΣ,anduseresultsofMusta¸tˇaandStanleytoconnecttheparts. Equation1
is proved in §4, and illustrated on the Hirzebruch surface H .
2
Toric facts. Let N ≃Zn be a lattice, with dual lattice M, and let Σ⊆N⊗ZR≃
Rn be a complete simplicial rational polyhedral fan (henceforth, simply fan), with
Σ(i)denotingthesetofi-dimensionalfacesofΣ,andletX betheassociatedtoric
Σ
variety. A Weil divisor on X is of the form
Σ
D = a D , with a ∈Z.
ρ ρ ρ
ρ∈XΣ(1)
Let d=|Σ(1)|. The class group of X has a presentation
Σ
0−→M −ψ→Zd −φ→Cl(X )−→0,
Σ
where ψ is defined by
χm 7→ hm,v iD , where v is a minimal lattice generator for ρ.
ρ ρ ρ
ρ∈XΣ(1)
In [3], Cox introducedthe totalcoordinate ring (henceforth called the Cox ring) of
X . This is a polynomial ring, graded by the class group Cl(X ).
Σ Σ
Definition 1.1.
S =K[x |ρ∈Σ(1)]= S .
ρ α
α∈CMl(XΣ)
The utility of this grading is that for α ≃ D ∈ Cl(X ), H0(O (D)) ≃ S . For
Σ X α
more background on toric varieties, see [4], [5], or [7].
EULER CHARACTERISTIC OF COHERENT SHEAVES ON SIMPLICIAL TORICS 3
2. Cohomology and the Cox ring
The Cox ring has a distinguished ideal, the toric irrelevant ideal
B(Σ)=hxσˆ |σ ∈Σi, where xσˆ = x .
ρ
ρ6∈Yσ(1)
Note that B(Σ) is generated by monomials corresponding to the complements of
the maximal faces of Σ. For an ideal I =hf ,...,f i let
1 m
I[l] =hfl,...,fl i.
1 m
In [6], Eisenbud-Musta¸tˇa-Stillman show that for D ∈Cl(X ), i≥1 and l≫0,
Σ
(2) Hi(O (D))≃Exti+1(S/B(Σ)[l],S(D)) ,
X S 0
They also obtain a bound for l. Fix a basis for M, and let A be a d×n matrix
with a row for each ray u ∈Σ(1), written with respect to the fixed basis. Define
ρ
a=max(|entries of A|)
(3) b=max(|(n−1)×(n−1) minors of A|)
c=min(|nonzero n×n minors of A|).
Corollary 3.3 of [6] shows that if D = a D , then Equation 2 holds for
ρ ρ ρ
(4) l ≥n2 mPax (|a |)ab/c
ρ
ρ∈Σ(1)
For brevity, we use lower case to denote dimK of an object, e.g. sα =dimKSα.
Lemma 2.1. For l ≫0 and D ∈Cl(X ),
Σ
n n+1
(5) χ(O (D))= (−1)ihi(D)=s − (−1)iexti(S/B(Σ)[l],S(D)) .
X D S 0
Xi=0 Xi=0
Proof. Ext0(S/B(Σ)[l],S)=Ext1(S/B(Σ)[l],S)=0, so this follows from [6]. (cid:3)
S S
Lemma 2.2. If F is a free resolution for S/B(Σ)[l], then
•
n+1 d
(−1)iextiS(S/B(Σ)[l],S(D))0 = (−1)idimKFi∨(D)0
Xi=0 Xi=0
(6)
d
= (−1)idimK(Fi)∨D.
Xi=0
Proof. Take Euler characteristics. (cid:3)
Lemma 2.3. If F is a minimal free resolution for S/B(Σ)[l], then
•
dimK(Fi)∨D = toriS(S/B(Σ)[l],K)D′ ·sD′+D.
D′∈XCl(XΣ)
Proof. Let F be a minimal free resolution for S/B(Σ)[l], and
•
ri(D′)=toriS(S/B(Σ)[l],K)D′.
Then
F = S(−D′)ri(D′).
i
D′∈MCl(XΣ)
Now dualize and take the shift by D into account. (cid:3)
4 HALSCHENCK
3. Combinatorial commutative algebra
Taylor resolution. Wenowobservethatthemultigradedbettinumbersr (D′)of
i
S/B(Σ)[l]canbereplacedwithrelatednumberswhicharisefromaTaylorresolution
forS/B(Σ). TheTaylorresolution[23]ofamonomialidealisavariantoftheKoszul
complex, which takes into account the LCM’s of the monomials involved.
Let I = hm ,...,m i be a monomial ideal, and consider a complete simplex
1 k
with vertices labelled by the m , and each n-face F labelled with the LCM of the
i
n+1 monomials correspondingto vertices ofF. Define a chain complex where the
differential on an n-face F =[v ,...,v ] is
i0 in
n
m
d(F)= (−1)j F F \v ,
m ij
Xj=0 F\vij
withm denotingthemonomiallabellingfaceF. AsshownbyTaylor,thiscomplex
F
isactuallyaresolution(thoughoftennonminimal)ofI. Whenthem aresquarefree,
i
the LCM of a subset of lth powers is the lth power of the LCM of the original
monomials, hence the Taylor resolution for I[l] is given by the lth power of the
Taylor resolution for I, in the sense that a summand S(−α) in the free resolution
for I is replaced with S(−l·α) in the resolution for I[l].
Thus, the Taylor resolution of S/B(Σ) determines the Taylor resolution of
S/B(Σ)[l]. The formula in Lemma 2.3 requires a minimal free resolution, which
the Taylor resolution is generally not. However, this is no obstacle:
Lemma 3.1. If F is a free resolution for S/B(Σ), then
•
n+1 d
(−1)iextiS(S/B(Σ)[l],S(D))0 = (−1)i toriS(S/B(Σ),K)D′·sl·D′+D.
Xi=0 Xi=0 D′∈XCl(XΣ)
Proof. If F is a minimal resolution of S/B(Σ)[l], then Lemmas 2.2 and 2.3 yield
•
n+1 d
(−1)iextiS(S/B(Σ)[l],S(D))0 = (−1)i toriS(S/B(Σ)[l],K)D′·sD′+D.
Xi=0 Xi=0 D′∈XCl(XΣ)
Lemma 2.2 shows that the lth power of a Taylor resolution for S/B(Σ) can be
used to compute the left-hand side. Furthermore, when F is non-minimal, in the
•
expression
d
(−1)idimK(Fi)∨D
Xi=0
the nonminimal summands cancel out, hence we may pass back to the description
in terms of Tor, yielding the result. (cid:3)
Alexander duality and monomial ideals. Let ∆ be a simplicial complex on
vertex set {1,...,d}. Let S = Z[x ,...,x ] be a polynomial ring, with variables
1 d
corresponding to the vertices of ∆.
Definition 3.2. The Stanley-Reisner ideal I ⊆ S is the ideal generated by all
∆
monomials corresponding to nonfaces of ∆:
I =hx ···x |[i ,...,i ] is not a face of ∆i.
∆ i1 ik 1 k
EULER CHARACTERISTIC OF COHERENT SHEAVES ON SIMPLICIAL TORICS 5
σ
u = (−1,2) 4
1
σ
u 1
2
σ
3
u
3
u4 σ
2
Figure 1. The fan for H
2
The Stanley-Reisner ring is S/I . The intersection of a complete simplicial fan
∆
Σ ⊆ Rn with the unit sphere Sn−1 gives a simplicial complex we denote by P ;
Σ
define I as the Stanley-Reisner ideal of P .
Σ Σ
Definition 3.3. If ∆is a simplicial complex on [d]={1,...d}, then theAlexander
dual ∆∨ is a simplicial complex consisting of the complements of the nonfaces of
∆:
∆∨ ={[d]\σ|σ 6∈∆}.
Example3.4. TheHirzebruchsurfaceH correspondstothefaninFigure1. Since
2
[u ,u ] and [u ,u ] are nonfaces of Σ, and every other nonface such as [u ,u ,u ]
2 4 1 3 1 2 4
contains them, the Stanley-Reisner ideal is
I =hx x ,x x i.
Σ 1 3 2 4
The Alexander dual Σ∨ contains all ρ ∈ Σ(1). Since u[u = [u ,u ] and u[u =
1 3 2 4 2 4
[u ,u ], Σ∨(2)={[u ,u ],[u ,u ]}. So
1 3 2 4 1 3
IΣ∨ =hx1x2,x1x4,x2x3,x3x4i.
Lemma 3.5. The toric irrelevant ideal B(Σ) is Alexander dual to the Stanley-
Reisner ideal I .
Σ
Proof. The Alexander dual IΣ∨ to IΣ is obtained by monomializing ([15], Proposi-
tion 1.35) a primary decomposition for I . If MC(Σ) denotes the set of minimal
Σ
cofaces of Σ, then the primary decomposition of I is
Σ
I = hx ,...,x i.
Σ i1 ik
[i1,...,ik\]∈MC(Σ)
The ideal IΣ∨ is generated by monomials corresponding to minimal cofaces, which
are complements to maximal faces, hence IΣ∨ =B(Σ). (cid:3)
Theorem 3.6 (Danilov [5], Jurkiewicz [10]). For a complete simplicial fan Σ, let
J = hdiv(χm)|m ∈ Mi. The rational Chow ring Ch(X ) is the rational Stanley-
Σ
Reisner ring of Σ, modulo J.
The ideal J is minimally generated by a regular sequence; it is these linear
forms which encode the geometry of Σ. To interpret the Euler characteristic of
O (D)intermsofintersectiontheory,wemustchangecomputationsinvolvingthe
X
toric irrelevant ideal into computations involving the Stanley-Reisner ideal. For a
polynomial ring R = K[x ,...,x ] endowed with the fine (also called Zd) grading
1 d
6 HALSCHENCK
deg(x )=e ∈ZdandsquarefreemonomialidealM,thefollowingresultofMusta¸tˇa
i i
[[16], Corollary 3.1] provides the bridge:
(7) ToriR(M∨,K)m ≃ExtR|m|−i(R/M,R)−m if m∈{0,1}d, else 0.
Letting Z ={0,1}d, applying Musta¸tˇa’s result yields:
toriS(S/B(Σ),K)D′ = toriS(S/B(Σ),K)m
mX∈Z,
φ(m)=D′
(8)
|m|−i+1
= extS (S/IΣ,S)−m
mX∈Z,
φ(m)=D′
Lemma 3.7. For a complete fan Σ⊆N ⊗ZR≃Rn with |Σ(1)|=d,
(1) Extj(S/I ,S)=0 for all j 6=d−n.
S Σ
(2) In the Zd grading, Extd−n(S/I ,S)≃S/I (1).
S Σ Σ
Proof. From Definition 3.2, I is the Stanley-Reisner ideal of the simplicial sphere
Σ
P , which is Gorenstein by Corollary II.5.2 of [22]. Since dimP =n−1,
Σ Σ
codim(I )=(d−1)−(n−1)=d−n.
Σ
Everythingfollows fromthis, savethat S/I is shifted by 1. The Gorensteinprop-
Σ
erty means the minimal free resolution of S/I is of the form
Σ
k k
0−→S(−α)∂−d→−n S(−β )∂d−−→n−1 ···−→ S(−γ )−[I→Σ] S −→S/I −→0,
j j Σ
Mj=1 Mj=1
where ∂ is (up to signs)the transpose ofthe matrix ofminimal generators[I ].
d−n Σ
To show that the shift in Extd−n is 1, we use a result of Hochster. For a complex
∆ and weight α, let ∆| ={σ ∈∆|σ ⊆α}. Equating the multidegree 1 with the
α
full simplex on all vertices of ∆, Hochster’s formula (5.12 of [15]) yields
TordS−n(S/IΣ,K)1 =Hn−1(Σ|1,K).
Since Σ|1 ≃PΣ ≃Sn−1, the result follows. e (cid:3)
Example 3.8. The Stanley-Reisner ring for the fan Σ of Example 3.4 has a Z4
graded minimal free resolution
 −x2x4  S(−1,0,−1,0)
0−→S(−1,−1,−1,−1)−−−x−1−x−3−−→ ⊕ −h−−x−1x−−3−−x−2−x−4−→i S −→S/IΣ.
S(0,−1,0,−1)
Thus, Ext2(S/I ,S) ≃ S(1,1,1,1)/I . The simplicial complex P consists of
Σ Σ Σ
vertices [1],[2],[3],[4] and edges [12],[23],[34],[41] and is homotopic to S1. Since
the multidegrees are all smaller than 1 in the pointwise order, Σ|1 =PΣ, so
K=H1(S1,K)=H1(Σ|1,K)=Tor2S(S/IΣ,K)1,
showing the shift α inethe last stepeof the free resolution of S/I is S(−1).
Σ
EULER CHARACTERISTIC OF COHERENT SHEAVES ON SIMPLICIAL TORICS 7
4. Proof of Equation 1
We now prove Equation 1. By Equation 5,
n+1
χ(O (D))=s − (−1)iexti(S/B(Σ)[l],S(D)) .
X D S 0
Xi=0
n+1
Let γ(m)=sl·φ(m)+D and E= (−1)iextiS(S/B(Σ)[l],S(D))0. It suffices to show
iP=0
E =sD+ (−1)|m|−d+n+1dimK(S/IΣ)1−m·γ(m).
mX∈Z\0
First, observe that
d
E = (−1)i toriS(S/B(Σ),K)m ·γ(m).
Xi=0 D′∈XCl(XΣ)(cid:16) mX∈Z, (cid:17)
φ(m)=D′
d
(9) = (−1)i toriS(S/B(Σ),K)m·γ(m).
Xi=0 mX∈Z
d
=sD+ (−1)i toriS(S/B(Σ),K)m·γ(m).
Xi=1 mX∈Z\0
The first line follows from Lemma 3.1, the second line is simply a rearrangement,
and the third line follows from the observation that
sD =tor0S(S/B(Σ),K)0·γ(0).
For i≥0,
TorS(B(Σ),K)≃TorS (S/B(Σ),K),
i i+1
so using Equation 7 we may rewrite the last line of Equation 9 as
d−1
(10) sD+ (−1)i+1 extS|m|−i(S/IΣ,S)−m·γ(m).
Xi=0 mX∈Z\0
By Lemma 3.7, Ext|m|−i(S/I ,S) is nonzero iff |m|−i=d−n, and
S Σ
Extd−n(S/I ,S)≃S/I (1).
S Σ Σ
Since the only nonzero terms in Equation 10 occur for i =|m|−d+n we rewrite
Equation 10 as
=sD+ (−1)|m|−d+n+1extdS−n(S/IΣ,S)−m·γ(m)
mX∈Z\0
(11)
=sD+ (−1)|m|−d+n+1dimK(S/IΣ)1−m·γ(m)
mX∈Z\0
This shows that
E =sD+ (−1)|m|−d+n+1dimK(S/IΣ)1−m·γ(m),
mX∈Z\0
and Equation 1 follows. 2
8 HALSCHENCK
Example 4.1. Consider the divisor D =3D −5D onthe HirzebruchsurfaceH
3 4 2
from Figure 1. Since the support function for D is not convex, D is not nef. Thus,
computing χ(O (D)) involves more than a simple global section computation. A
H2
direct calculation with Riemann-Roch for surfaces shows that
χ(O (D))=4.
H2
Using the methods of §9.4 of [4], it can be shown that h0(D)=0, h1(D)=2, and
h2(D)=6. Now we illustrate how to apply Equation 1. Let
1 −2 1 0
φ=
(cid:18)0 1 0 1(cid:19)
so that the Class group is given by
Z4 −φ→Z2 ≃Cl(H )−→0.
2
The Eisenbud-Musta¸tˇa-Stillman bound of Equation4 is l=80, but a careful anal-
ysis (see Example 3.6 of [6]) shows that in this case taking l=4 is sufficient. Then
for example with m=(0,1,0,1) we have φ(m)=(−2,2) so since D =(3,−5),
S4·φ(m)+D =S(−5,3) =H0(OH2(−5,3)),
and the dimension of this space is two. However,
(S/IΣ)1−(0,1,0,1) =(S/IΣ)(1,0,1,0) =0,
since x x ∈I . Acheck showsthat allterms inthe summationvanish, savewhen
1 3 Σ
m∈{(1,1,0,1),(0,1,1,1),(1,1,1,1)}
For the first two values, φ(m)=(−1,2), and we compute
S4·φ(m)+D =S(−1,3) =H0(OH2(−1,3)),
whichhas dimensiontwelve. Since 1−mis either(0,0,1,0)or(1,0,0,0),forthese
two values of m,
dimK(S/IΣ)1−m =1
Since |m|−d+n=1,these twoweightscontribute (−1)·2·12=−24to the Euler
characteristic. For the remaining weight m = (1,1,1,1), the Stanley-Reisner ring
is one dimensional in degree 1−m=(0,0,0,0), and φ(1,1,1,1)=(0,2) and
S4·φ(m)+D =S(3,3) =H0(OH2(3,3)),
which has dimension 28. Since |m|−d+n=2 the contribution is positive, thus
χ(O (3D −5D ))=−24+28=4.
H2 3 4
Problem As noted in the introduction, this work began as an attempt to find a
toric proof of Hirzebruch-Riemann-Roch using Equation 1; it would be interesting
to find such a proof. A proof of Equation 1 also follows from results of Maclagan-
Smith [14], I thank Greg Smith for noting this.
Acknowledgements Computations were performed using Macaulay2 [8] by
Graysonand Stillman, and NormalToricVarieties[21] by Greg Smith.
EULER CHARACTERISTIC OF COHERENT SHEAVES ON SIMPLICIAL TORICS 9
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Schenck: MathematicsDepartment,University ofIllinois, Urbana,IL 61801,USA
E-mail address: [email protected]