Torsion of dierentials on toric varieties
Klaus Altmann
Institut fur reine Mathematik, Humboldt-Universitat zu Berlin
Ziegelstr. 13a, D-10099 Berlin, Germany.
E-mail: [email protected]
Abstract
We introduce an invariant for semigroups with cancellation property. When
the semigroup equals the set of lattice points in a rational, polyhedral cone,
then this invariant describes the torsion of the dierential sheaf on the associated toric variety.
Finally, as an example, we present the case of two-dimensional cones (corresponding to two-dimensional cyclic quotient singularities).
1 An invariant for semigroups
(1.1)
Let S be a commutative semigroup with 0 and cancellation property (i.e.
a + s = b + s implies a = b for a; b; s 2 S ). In particular, S can be embedded into a
group, and the notion a for a 2 S makes sense. Assume that (inside this group)
S \ ( S ) = f0g; then via
a b :() a b 2 S ;
S turns also into a partially ordered set.
For each ` 2 S we will dene a certain abelian group T` . Their direct sum T :=
`2S T` or the T`'s itself should be considered an invariant of the original semigroup.
If S is nitely generated, then so will the T`'s. In case S equals the set of lattice
points of some rational, polyhedral cone (i.e. generated by a nite set of rays through
rational points) in IRn, we will show that T CI coincides with the torsion of the
dierential sheaf Y of the ane toric variety Y = Spec CI [S ] (CI [S ] denotes the
semigroup algebra of S over CI ).
1
This paper was written at M.I.T. and supported by a DAAD-fellowship.
1
(1.2)
For any subset P S we denote by L(P ) the abelian group of integral
relations between elements of P , i.e.
L(P ) := ker ZZ P ! spanZZ P S S :
For each q 2 L(P ) we dene its support as
supp q := fs 2 P j qs 6= 0g ;
it is a nite subset of P . Then, every q 2 L(P ) is uniquely representable as a
dierence q = q q with q ; q 2 IN P and supp q \ supp q = ;. In particular,
+
+
X
s2S
qs s =
+
+
X
s2S
qs s 2 spanIN P S ;
and we denote this element by q (the \bar-value" of q).
Denition: For every ` 2 S let S` := fs 2 S j s `g. Then, we dene
.
.
T` := L(S`) hq 2 L(S`) j q `i = hq 2 L(S ) j supp q `i hq 2 L(S ) j q `i :
(hq 2 L(S`) j q `i means the subgroup generated by those q's, and \supp q `" is
just an abbreviation for \s ` for all s 2 supp q".)
(1.3)
Proposition: Let E S be a subset that generates the semigroup S .
Denoting E` := E \ S` = fs 2 E j s `g, we obtain for ` 2 S
.
.
T` = L(E`) hq 2 L(E`) j q `i = hq 2 L(E ) j supp q `i hq 2 L(E ) j q `i :
Proof: We regard the canonical map
.
.
' : L(E` ) hq 2 L(E` ) j q `i ! L(S`) hq 2 L(S`) j q `i :
' is surjective: Suppose a relation q 2 L(S`) involves s ; . . . ; sN 2 S` (i.e. supp q =
fs ; . . . ; sN g). Each sv can be represented as a sum of elements from E providv
v
ing
relation q(sv ) with
P a special
P q (s ) =v s `. Then, if q equals the relation
v
[ v qv s = 0], we obtain
v qv q (s ) 2 L(E` ).
P q
v
(By the way, q~ := q
v qv q (s ) can be regarded as similar to q , but each occurrence of sv is replaced by an appropriate sum of elements from E . In particular, the
bar-value of q~ is less or equal than that of q.)
P
' is injective: Let q 2 L(E`) such that '(q) = 0, i.e. it splits into a sum q = i qi
with qi 2 L(S`); qi `. Treating each of the summands as we did q previously, we
obtain
X
qi = q~i + qvi q(sv ) with q~i 2 L(E`); q~i `; and sv 2 S`:
1
1
v
2
Hence, up to relations from hq 2 L(E`) j q `i, we may assume that
q=
X
sv 2S` nE`
gv q(sv ) 2 L(S` ) ZZ S
`
for some integer coecients gv . However, since q is actually contained in L(E` ) ZZ E , evaluating this equation at each of the sv 2 S` n E` yields always gv = 0. 2
`
Corollary: If S is a nitely generated semigroup, then the T` are nitely generated
abelian groups.
(1.4)
We conclude this general section with a trivial, but important lemma.
Lemma: Let ` 2 S , and dene S 0 S as the semigroup generated by S`. The
relation \S 0 " associated to S 0 is not the restriction of that of S ; in general, \S0 "
is stricter than \S ". However,
(i) (S 0)` = S`, and
(ii) the abelian groups T` coincide for S and S 0.
Proof: For (i) assume that s 2 S fullls s S `. By denition, there is an t 2 S
such that s + t = `. In particular, s; t 2 S` S 0, and we obtain s S0 `.
The same trick works for (ii): If q 2 L(S`) is a relation with q S `, then there
2
exists a t 2 S such that q + t = `. In particular, t 2 S` S 0, hence q S0 `.
Hence, to compute T`, we may always change the semigroup and assume that E =
E`.
2 Rational, polyhedral cones
(2.1)
For this section we assume that S ZZ n equals the set of lattice points
of some rational, polyhedral cone in IRn. In particular, it denes an ane toric
variety Y := Spec CI [S ]. (For general facts about toric varieties see [Da] or [Od]).
Denition: Let Y = Spec B be an arbitrary ane scheme over the complex num-
bers, i.e. let B be an arbitrary (commutative) CI -algebra. Then, the B -module of
Kahler dierentials Y together with the canonical dierential d : B ! Y are
dened by the following universal property:
(i) d is additive, kills the constants, and it fullls the rule
1
1
d(fg) = f d(g) + g d(f ) for f; g 2 B:
3
(ii) For each B -module Q all maps d0 : B ! Q meeting the properties similar to
(i) factorize through d by a uniquely determined B -linear map Y ! Q.
Indeed, the Kahler dierentials Y do exist, and they are uniquely determined by
the previous denition. For a proof of these facts and basic properties (for instance
the two fundamental exact sequences) see Chapter 10 of [Ma].
1
1
Theorem: In the toric situation the CI [S ]-module of Kahler dierentials Y and its
torsion submodule tors (
Y ) are ZZ n-graded. For ` 2 S we have
tors (
Y )(`) = T` ZZ CI ;
and tors (
Y ) vanishes in the remaining degrees. Moreover, if s 2 S , then the canonical map T` ! T` s describes the multiplication with xs 2 CI [S ].
1
1
1
1
+
(2.2)
Proof: Step 1: Let E = fs ; . . . ; sw g be a generating set of the
P semigroup
w
S ; in particular, we have a surjective map : IN
! S ; a 7! i ai si. Let
m := f(a; b) j a; b 2 IN w ; (a) = (b) in S g.
0
+1
+1
Then, Y can be regarded as the closed subset of CI w dened by the ideal I =
(z a zb j (a; b) 2 m) CI [z ; . . . ; zw ]. We obtain the standard exact sequence (cf.
Theorem 58 in [Ma])
+1
0
d
I=I !
CI
2
1
w
+1
CI z0;
[
;z
... w ]
inducing
CI [S ] ! Y ! 0 ;
1
CI [S ]m ! CI [S ]w ! Y ! 0
via composing d with the canonical surjection CI [S ]m ! I=I .
+1
1
2
The maps in the latter exact sequence could be described as follows: Denote by
feab g and fei g the standard bases of CI [S ]m and CI [S ]w , respectively; and for an
s 2 S denote by xs 2 CI [S ] the corresponding element in the semigroup algebra, e.g.
xs zi (mod I ). Then,Pthe image of eab in I=I is the equation z a zb , and this
maps onto d(za z b ) = wi (ai bi ) x a s d zi 2 CI +1 CI [S ]. In particular,
+1
i
2
( )
=0
(eab ) =
w
X
i=0
i
1
(ai bi) x a
( )
si
w
ei :
(At the rst glance, might not always map into CI [S ]w . However, if (a) si 2= S ,
then this would imply ai = bi = 0.)
Finally, ei 2 CI [S ]w maps onto d xs 2 Y by the second map.
Step 2: The CI -linear map CI [S ] ! ZZ n ZZ CI [ZZ n] ; xs 7! s xs is a CI -derivation: It
kills the constants, and for s; t 2 S we have xs(t xt ) + xt (s xs) = (s + t) xs t.
Hence, by denition of the Kahler dierentials, we obtain a CI [S ]-linear map
Y ! ZZ n ZZ CI [ZZ n] ; d xs 7! s xs :
+1
i
+1
1
+
1
4
On the other hand, ZZ n ZZ CI [ZZ n ] can be identied with the module of Kahler
dierentials on the torus (CI )n = Spec CI [ZZ n], and the previous map corresponds
to the restriction of dierentials from Y onto the open subset (CI )n Y . Since
CI is just a localisation of Y , we obtain that this map has exactly tors (
Y ) as
its kernel.
Putting this fact together with the information from the rst step, we obtain the
sequence of CI [S ]-modules
1
(
1
)n
()
1
! CI [S ]w
CI [S ]m
! ZZ n ZZ CI [ZZ n]
7
!
si xs
+1
ei
i
and, moreover, tors (
Y ) = ker =im .
Step 3: Dening deg(eab ) := (a) = (b) 2 S for CI [S ]m , deg(ei ) := si 2 S for
CI [S ]w , and deg(s xt ) := t 2 ZZ n for ZZ n ZZ CI [ZZ n], the sequence () turns out to
be ZZ n-graded. In particular, to calculate ker =im , we can deal with each degree
` 2 S separately (degrees from ZZ n n S do not appear).
P
Claim: T` ZZ CI ! (ker =im )` ; q 7! s 2E qi x` s ei is an isomorphism of
CI -vector spaces.
P
Proof: For q 2 L(E` ), the sum s 2E qi x` s ei denes an element of (CI [S ]w )` .
Applying , we obtain
1
+1
i
i
i
+1
i
(
P
P
` s ei ) =
` s (si xs )
s 2E qi x
Ps 2E qi xi
= ( s 2E qi s ) x`
= 0:
i
i
i
i
i
i
P
Obviously, each element of (ker )` can be obtained that way: If s 2E qi (x) ei 2
CI [S ]w is of degree `, we obtain qi(x) = qi x` s (qi 2 CI with qi = 0Punless ` si).
Moreover, we have just seen that kills this element if and only if s 2E qi si = 0,
i.e. it comes from a relation q 2 L(E` ).
On the other hand, q 2 L(E`) with q ` can be written as q = a b (a; b 2
IN w ; (a) = (b) `). Then,
i
i
+1
i
+1
X
si 2E
(ai bi) x`
si
ei = x`
(a) (eab ) ;
2
and those elements generate (im )`.
(2.3)
Remark: We never used the fact that Y = Spec CI [S ] is a normal variety. Hence, Theorem (2.1) remains true, if S ZZ n is just supposed to be a nitely
generated semigroup with S \ ( S ) = f0g. In particular, over IR, S generates a rational, polyhedral cone IRn, but there may exist nitely many points of \ ZZ n
not belonging to S .
5
Example: If S IN , then Y is a curve singularity. For instance, S := IN n f1g is
generated by the integers 2 and 3, and Y CI is the cusp given by the equation
2
y x = 0.
Let us compute the torsion of Y in this case: With E = fs := 2; s := 3g we have
the only relation q := [3s 2s = 0]. Since q 2 S , on the one hand, but q = 6, on
the other hand, we obtain T = T = ZZ q, i.e. tors (
Y ) is a one-dimensional vector
space concentrated in the singular point.
2
3
1
0
0
1
5
5
1
1
3 Two-dimensional, cyclic quotient singularities
(3.1)
Two-dimensional, cyclic quotient singularities coincide with the twodimensional, ane toric varieties; they were rst investigated by Riemenschneider,
[Ri]. We want to compute our invariant for this special case.
Let S be the set of lattice points in the two-dimensional cone spanned by some
primitive vectors s ; sw 2 ZZ . Then, following x1.6 in [Od], E = fs ; . . . ; sw g is
built from all lattice points of the compact part of the boundary of conv (S n f0g);
in particular, @ (conv (S n f0g)) consists of w primitive edges (containing no interior
lattice point). Every pair of adjacent elements si ; si (i = 1; . . . ; w) provides a
ZZ -basis for the lattice ZZ , and there are relations
0
2
0
1
2
si + si = ai si (ai 2 IN; ai 2; i = 1; . . . ; w 1) :
1
+1
sw
r
This is a rough draft of S ; in fact it
cannot exactly be shaped as shown in
the gure.
r
E
E
E
r
s
4
E
s
E
E
E
3
r
J
r
s
J
a
2
a
a
a
a
s
1
r
h
hh
h
h
hh
s
0
r
hh
Remark: (cf. Lemma 1.20 in [Od]) The coecients ai can be obtained from the
continued fraction
n =a
n q
a
1
2
6
1
a3
1
...
with n := det(s ; sw ) and q 2 f1; . . . ; n 1g such that nj(qs + sw ).
0
0
(3.2)
Since ai 2 (i = 1; . . . ; w 1), the sets E` E are \connected" (i.e.
they are shaped as E` = fsp; sp ; . . . ; sp k g for some p; k) for every ` 2 S . Hence,
to compute T`, we may assume that E` = fs ; . . . ; sw g (cf. Lemma (1.4)).
+1
+
0
Lemma: Let ` 2 S such that E` = fs ; . . . ; sw g. Then,
(i) ` s + sw , and
0
0
(ii) T` = 0 unless ` = s0 + sw .
Proof: (i) The vectors s and sw form a IR-basis of IR , and, for an s 2 ZZ , the
condition \s 2 S " is equivalent to having non-negative coordinates only. Hence, the
inequalities ` s ; sw imply that both coordinates of ` are not smaller than 1, i.e.
0
2
2
0
` s sw has non-negative coordinates.
(ii) Let ` > s + sw be given; we will show that T` = 0. We distinguish two cases:
Case 1: ` (s + sw ) s (or similarly ` (s + sw ) sw ).
Then, since si s + sw , we know that s + si 2s + sw ` for i = 2; . . . ; w.
Summing up the equations sj + sj = aj sj (1 j i 1) provides relations qi
0
0
0
0
0
0
0
1
(q i )
s
0
+ si
= (s
1
0
+1
+ si
1
i 1
X
)+
j =1
(aj 2) sj
( i = 2; . . . ; w )
having exactly s + si ` as their bar-values. In particular, they yield zero in T`.
On the other hand, if we are given an arbitrary relation q 2 L(E ), then we can
use qw ; . . . ; q to eliminate sw ; . . . ; s step by step from the support of q (without
changing its value in T`). Since s and s are linearly independent, q has to be trivial
then.
Case 2: Not Case 1; in particular,
` (s + sw ) si for some i = 1; . . . ; w 1.
Pw
w
Then, we have ` (s + s ) = j gj sj with non-negative integers gj and, moreover, gi 1. (There are no summands involving s or sw , because this would t in
the rst case.)
Again, we startPwith an arbitrary q 2 L(E ). First, we use the relation s + sw =
(s + sw ) + jw (aj 2) sj (bar-value s + sw `) to eliminate s (if i = 1) or
sw (if i 2) from supp q.
If w = 2, then we are already done. Otherwise, we use the similar relations expressing si + sw (if i = 1) or s + si (if i 2) by the generators in between to eliminate
sw or s , respectively. The result is a relation q with supp q fs ; . . . ; sw g.
Finally, we know that
0
2
2
0
1
0
1
=1
0
0
0
1
1
1
=1
0
0
0
0
1
` = (s + sw ) +
0
w 1
X
j =1
gj sj = (s + sw ) +
1
7
1
w 1
X
j =1
(gj + aj 2) sj :
1
Hence, the fact T` = 0 for the cone spanned by s and sw (induction by w) tells
us that L(s ; . . . ; sw ) is spanned by relations with bar-value not greater than `.
In particular, our q can be reduced to zero.
2
1
1
(3.3)
1
1
Lemma: Assume ` = s + sw (including E` = fs ; . . . ; sw g). Then,
8
for w 2
< 0
w
=
for a = . . . = aw = 2
T` = ZZ
0
[(
:
0
1) 2]
1
ZZ
1
otherwise :
(The second case (a = . . . = aw = 2) means that the points s ; . . . ; sw are sitting
on an ane line; the corresponding cyclic quotient singularity equals the cone over
the rational normal curve of degree w.)
1
0
1
Proof: The case w 2 is obvious. Assuming a = . . . = aw = 2, the entire col1
1
lection of relations with bar-value not greater than ` is given by the [w=2] equations
s + sw = s + sw = . . . = s w= + sw
0
1
1
[
2]
w=2] ;
[
and we obtain (w 1) [w=2] = [(w 1)=2] for the rank of T`.
Finally, if w 3 and not a = . . . = aw = 2, then
1
`=s
0
+ sw
=s
1
1
+ sw
1
+
w 1
X
(ai 2) si >hs1 ;s 1i s + sw
w
i=1
1
1
shows (cf. Lemma (3.2)) that L(s ; . . . ; sw ) hq 2 L(E ) j q `i. On the other
hand, the only relations involving s or sw and having a bar-value ` are those
representing s + sw as linear combination
ofPs ; . . . ; sw .
Pw
(Indeed, if for instancePs + i gi si = wi hi si P
is such a relation (gi; hi w
i
w
0; gi hi = 0), then s + i gi s ` = s + s implies i gi si sw . Hence gw = 0
or 1, and gi = 0 for i 6= w. Moreover, since s is not representable by other generators, this implies that s + sw forms the left hand side of the relation - and the right
hand side has to be built from s ; . . . ; sw then.)
2
1
1
0
0
1
0
0
=0
=0
1
=1
0
0
0
1
1
(3.4)
We are gathering our results and obtain the following description of the
invariants T` for a two-dimensional cyclic quotient singularity:
Theorem: Let S and a ; . . . ; aw as in (3.1); dene a := aw := 3.
(1) Let sp; sp k 2 E (0 p < p + k w) be elements such that
(i) k 3, and
1
1
0
+
(ii) at least one of the numbers ap ; . . . ; ap+k is greater than two.
8
Then, ` := sp + sp+k uniquely determines p and k, and we have
T` =
ZZ
ZZ
k
[(
=
1) 2]
for ap+1 = . . . = ap+k 1 = 2
otherwise :
The abelian group T` vanishes in the remaining degrees.
(2) For T = `T` we obtain T = ZZ (w
1)(
w
=
2) 2
. In particular,
w 1 :
dim (tors (
Y )) =
2
1
(w + 1 equals the embedding dimension of the cyclic quotient singularity Y ).
Proof: For (1) assume that we are given some ` 2 S . Then, E` is shaped as
E` = fsp; . . . ; sp kg, and by Lemma (3.2) we know that T` = 0 unless ` = sp + sp k .
+
+
sp
+ sp+k ,
On the other hand, if ` =
then Lemma (3.3) tells us about T` . The
only thing being left is asking the other way around: What is the condition for an
` := sp + sp k to yield exactly E` = fsp; . . . ; sp k g?
Obviously, E` does always contain fsp; . . . ; sp k g, and we show that it is exactly the
condition ap = . . . = ap k = 2 saying that both sets are not equal:
Assume E` = fsp i; . . . ; sp; . . . ; sp k ; . . . ; sp k j g (w.l.o.g. i j 0, i 6= 0), then we
obtain a chain of inequalities
+
+
+
+
+
sp + sp k = ` sp i + sp
+
k j
+ +
+ +
sp
i+1 + sp+k+j
1
. . . sp
i+j + sp+k :
If i > j , this would imply that two dierent elements of E (sp and sp i j ) would
be comparable in S . Hence, i = j , and all signs in the previous chain turn into
equalities implying ap i = . . . = ap = . . . = ap k = . . . = ap k i = 2.
The reversed direction is easy; the equalities ap = . . . = ap k = 2 imply ` =
sp + sp k , hence sp ; sp k 2 E`.
+
+1
1
+ +1
1
+
+ +
1
+
+ +1
To prove the second part of the theorem, we have to count dimensions. Assume
that the compact part of the boundary @ (conv (S n f0g)) consists of P
m edges, each
containing wi 1 (i = 1; . . . ; m) interior lattice points. In particular, mi wi = w.
Then, we have
(a) w possibilities of choosing two dierent points sp and sp k from E ;
(b) 2w 1 of those pairs with 1 k 2;
(c) w possibilities of choosing two dierent points sp and sp k from the i-th
edge;
(d) 2wi 1 of those pairs with 1 k 2.
=1
+1
2
+
i +1
2
+
9
Hence, we obtain
w+1
2
(2w 1)
m
X
i=1
m
wi + 1 + X
(2wi 1) = w 1
2
2
i
=1
m
X
i=1
wi 1
2
possibilities of choosing pairs sp; sp k 2 E with k 3 and such that at least one of
the numbers ap ; . . . ; ap k is greater than two. Those pairs yield T` = ZZ .
+
+1
+
1
On the other hand, let fsq ; . . . ; sq w g form the i-th edge. Then, its only pairs
(sp; sp k ) meeting the assumption that at least one of the numbers ap; . . . ; ap k is
greater than two are (sq ; sq k ) and (sq w k ; sq w ) with 1 k wi. Each of them
providing a contribution of ZZ k = (which is automatically zero if k = 1; 2), we
obtain for the entire i-th edge
+ i
+
+
+ i
[(
2
wX
i 1
k=3
+
+ i
1) 2]
k 1 + 1 wi 1 = wi 1
2
2
2
2
dimensions for T .
Acknowledgement: I would like to thank the referee for many useful hints and
suggestions.
References
[Da] Danilov, V.I.: The Geometry of Toric Varieties. Russian Math. Surveys 33/2
(1978), 97-154.
[Ma] Matsumura, H.: Commutative Algebra. W.A.Benjamin, New York 1970.
[Ri] Riemenschneider, O.: Deformationen von Quotientensingularitaten (nach
zyklischen Gruppen). Math. Ann. 209 (1974), 211-248.
[Od] Oda, T.: Convex bodies and algebraic geometry. Ergebnisse der Mathematik
und ihrer Grenzgebiete (3/15), Springer-Verlag, 1988.
10

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