Summary: Torsion of differentials on toric varieties
Klaus Altmann \Lambda
Institut f¨ur reine Mathematik, HumboldtUniversit¨at zu Berlin
Ziegelstr. 13a, D10099 Berlin, Germany.
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 differential sheaf on the asso
ciated toric variety.
Finally, as an example, we present the case of twodimensional cones (corre
sponding to twodimensional 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 \Gammaa for a 2 S makes sense. Assume that (inside this group)
S `` (\GammaS) = f0g; then via
a – b :() a \Gamma b 2 S ;
S turns also into a partially ordered set.
For each ` 2 S we will define a certain abelian group T ` . Their direct sum T :=