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Tag Archives: exact sequences
Commutative Algebra 25
Arbitrary Collection of Modules Finally, we consider the case where we have potentially infinitely many modules. Proposition 1. For a collection of A-modules , we have Proof First claim: we will show that the LHS satisfies the universal property for … Continue reading
Posted in Advanced Algebra
Tagged direct products, direct sums, exact sequences, local properties, local rings, localization
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Commutative Algebra 21
Exact Sequences When studying homomorphisms of modules over a fixed ring, we often encounter sequences like this: where each is a homomorphism of modules. This sequence may terminate (on either end) or it may continue indefinitely. Note on indices: usually … Continue reading
Posted in Advanced Algebra
Tagged additive functors, exact sequences, functors, modules, short exact sequences
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Exact Sequences and the Grothendieck Group
As before, all rings are not commutative in general. Definition. An exact sequence of R-modules is a collection of R-modules and a sequence of R-module homomorphisms: such that for all i. Examples 1. The sequence is exact if and only if f … Continue reading
Posted in Notes
Tagged composition series, exact sequences, grothendieck group, modules, short exact sequences
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