Monad (homological algebra)

Not to be confused with Monad (category theory).

In homological algebra, a monad is a 3-term complex

A → B → C

of objects in some abelian category whose middle term B is projective, whose first map A → B is injective, and whose second map B → C is surjective. Equivalently, a monad is a projective object together with a 3-step filtration B ⊃ ker(B → C) ⊃ im(A → B). In practice A, B, and C are often vector bundles over some space, and there are several minor extra conditions that some authors add to the definition. Monads were introduced by Horrocks. ^[1] .

See also

• ADHM construction

References

1. Horrocks, G. (October 1964). "Vector Bundles on the Punctured Spectrum of a Local Ring". Proceedings of the London Mathematical Society. s3-14 (4): 689–713. doi:10.1112/plms/s3-14.4.689.

Further reading

• Barth, Wolf; Hulek, Klaus (1978), "Monads and moduli of vector bundles", Manuscripta Mathematica, 25 (4): 323–347, doi:10.1007/BF01168047, ISSN   0025-2611, MR   0509589, Zbl   0395.14007