Abstract
This chapter deals with Abelian categories, Grothendieck topologies, presheaves and sheaves.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
But although infinite products and coproducts do exist, they are not equal: The direct sum of a family of Abelian groups \(\{ A_{i} \}_{i \in \ensuremath {{\mathbb{N}}}}\) is the subset of the direct product A 1×A 2×…×A n ×⋯ consisting of all tuples such that only a finite number of coordinates are different from the zero element in the respective A i ’s.
- 2.
A relation ρ on a set B generates an equivalence relation ∼ by putting a∼b if either a=b, or there is a sequence a∼a 1∼a 2∼…∼a n =b, or a sequence b=b 1∼b 2∼…∼b m =a.
- 3.
Hence addition and multiplication with an element in R may be defined on the set of equivalence classes by performing the operations on elements representing the classes and taking the resulting classes. The details are left to the reader.
- 4.
Note that if we turn the contravariant functor into a covariant one as
, then the two kinds of limits are interchanged. - 5.
One readily verifies that direct image f ∗ of a presheaf commutes with forming the associated sheaf.
, then the two kinds of limits are interchanged.References
Bucur, I., Deleanu, A.: Categories and Functors. Interscience Publication. Wiley, New York (1968)
Grothendieck, A.: Éléments de géométrie algébrique. i–iv. Publ. Math. I.H.E.S., 4, 8, 11, 17, 20, 24, 28, 32
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Holme, A. (2012). Abelian Categories. In: A Royal Road to Algebraic Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19225-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-19225-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-19224-1
Online ISBN: 978-3-642-19225-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)