-
Recent Posts
Archives
- March 2023
- January 2023
- May 2020
- April 2020
- March 2020
- June 2018
- July 2016
- June 2016
- May 2016
- March 2015
- February 2015
- January 2015
- December 2014
- December 2013
- November 2013
- July 2013
- June 2013
- May 2013
- March 2013
- February 2013
- January 2013
- December 2012
- November 2012
- October 2012
- September 2012
- August 2012
- April 2012
- March 2012
- February 2012
- January 2012
- December 2011
- November 2011
- October 2011
Categories
Meta
Pages
Tag Archives: right-exact
Commutative Algebra 52
Direct Limits of Rings Let be a directed system of rings. Regard them as a directed system of abelian groups (i.e. ℤ-modules) and take the direct limit A. Proposition 1. The abelian group A has a natural structure of a … Continue reading
Posted in Advanced Algebra
Tagged adjoint functors, coinduced modules, colimits, duals, left-exact, limits, right-exact, tensor products
Leave a comment
Commutative Algebra 51
Limits Are Left-Exact By example 6 and proposition 2 in the previous article, one is inclined to conclude that taking the colimit in is a right-exact functor, but there is a rather huge issue here: the functors are between and … Continue reading
Posted in Advanced Algebra
Tagged colimits, directed limits, directed sets, left-exact, limits, right-exact
2 Comments
Commutative Algebra 50
Adjoint Functors Adjoint functors are a general construct often used for describing universal properties (among other things). Take two categories and . Definition. Covariant functors and are said to be adjoint if we have isomorphisms which are natural in A and … Continue reading
Posted in Advanced Algebra
Tagged adjoint functors, category theory, colimits, hom functor, left-exact, limits, right-exact, tensor products, universal properties
2 Comments
Commutative Algebra 29
Distributivity Finally, tensor product is distributive over arbitrary direct sums. Proposition 1. Given any family of modules , we have: Proof Take the map which takes . Note that this is well-defined: since only finitely many are non-zero, only finitely … Continue reading
Posted in Advanced Algebra
Tagged hom functor, induced modules, localization, right-exact, tensor products, yoneda lemma
2 Comments
Commutative Algebra 26
Left-Exact Functors We saw (in theorem 1 here) that the localization functor is exact, which gave us a whole slew of nice properties, including preservation of submodules, quotient modules, finite intersection/sum, etc. However, exactness is often too much to ask … Continue reading
Posted in Advanced Algebra
Tagged additive functors, hom functor, induced modules, left-exact, right-exact
Leave a comment
Tensor Product over Noncommutative Rings
Following the earlier article on tensor products of vector spaces, we will now look at tensor products of modules over a ring R, not necessarily commutative. It turns out we have to distinguish between left and right modules now. Indeed recall … Continue reading
Posted in Notes
Tagged bimodules, hom functor, left-exact, modules, right-exact, tensor products
Leave a comment
Mathematics and Such 