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Tag Archives: rightexact
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. ℤmodules) 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, leftexact, limits, rightexact, tensor products
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Commutative Algebra 51
Limits Are LeftExact By example 6 and proposition 2 in the previous article, one is inclined to conclude that taking the colimit in is a rightexact 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, leftexact, limits, rightexact
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, leftexact, limits, rightexact, 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 welldefined: since only finitely many are nonzero, only finitely … Continue reading
Posted in Advanced Algebra
Tagged hom functor, induced modules, localization, rightexact, tensor products, yoneda lemma
2 Comments
Commutative Algebra 26
LeftExact 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, leftexact, rightexact
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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, leftexact, modules, rightexact, tensor products
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