A 
-module on a smooth complex algebraic
variety 
,
in the left-module convention, is a sheaf of modules over
the sheaf 
of algebraic differential operators, quasicoherent over
the structure sheaf 
. The sheaf 
is generated by multiplication by regular functions and
differentiation by vector fields. Their action on
a local section 
satisfies the Leibniz rule
 |
For example, 
is a left 
-module
by its natural differentiation action. On affine space,
this theory is equivalent to modules over the Weyl algebra
(Ginzburg 1998).
Twisted 
-modules instead use a sheaf
of twisted differential operators, locally resembling 
but glued with additional twisting data. Complex exponents
of a line bundle can specify such a twist without
requiring an actual fractional tensor power of the line
bundle.
-Modules." 1998.