nLab rank of a coherent sheaf

Idea

As the concept of coherent sheaf is a generalization of that of (complex, algebraic vector bundle) vector bundle, the concept of rank of a coherent sheaf is the corresponding generalization of that of rank of a vector bundle.

Definition

e.g. (Huybrechts-Lehn 96, def. 1.2.2)

Related concepts

• degree of a coherent sheaf

• slope of a coherent sheaf

• stable coherent sheaf, Bridgeland stability condition

References

This appeared in the Grothendieck’s FGA in order to treat the moduli problems leading to Quot schemes. See Nitsure’s contribution at FGA explained for a modern remake.

• Daniel Huybrechts, Manfred Lehn, The Geometry of the Moduli Spaces of Sheaves, 1996 (pdf)