Primer año de posgradoAlgebraPlanificado
Modules
Modules over rings, submodules and quotients, exact sequences, free, projective and injective modules, tensor products, and the structure theorem for finitely generated modules over a principal ideal domain.
math.algebra.modules
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Se basa en
Requisitos previos obligatorios, hasta el fondo: 28
Aprenderás a
- Work with module homomorphisms, kernels, cokernels and exact sequences.
- Prove the structure theorem for finitely generated modules over a PID, and deduce the classification of finitely generated abelian groups and the Jordan normal form.
- Compute tensor products of modules and describe their effect on exact sequences.
Definición
- A module over a ring R is an abelian group with an action of R satisfying the distributive and associative laws.
- A sequence A → B → C is exact at B if the image of the first map equals the kernel of the second.
- A module P is projective if Hom(P, −) carries surjections to surjections.
Teorema
- A finitely generated module over a PID R is isomorphic to a direct sum R/(d₁) ⊕ ⋯ ⊕ R/(dₖ) with d₁ | d₂ | ⋯ | dₖ.
- Free modules are projective, and over a PID every submodule of a free module is free.
- M ⊗_R − is right exact but in general not exact.
Errores frecuentes
- Expecting modules to behave like vector spaces: ℤ/2ℤ is a ℤ-module with no basis.
- Assuming tensoring preserves injectivity: multiplication by 2 on ℤ is injective, but becomes zero after tensoring with ℤ/2ℤ.
Conduce a
Descubrimientos opcionales
- Adjunctions The tensor–hom adjunction Hom(M ⊗ N, P) ≅ Hom(M, Hom(N, P)), natural in all three variables, is a basic example of an adjunction.
Itinerarios de aprendizaje
Lecturas complementarias
- David S. Dummit and Richard M. Foote, Abstract Algebra
- Serge Lang, Algebra