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.

See also

Differential Operator, Quantum Geometric Langlands Correspondence, Sheaf, Weyl Algebra

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References

Ginzburg, V. "Lectures on -Modules." 1998. https://math.berkeley.edu/~nadler/ginzburg.dmodules.pdf.

Cite this as:

Weisstein, Eric W. "D-Module." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/D-Module.html

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