Lecture Notes
Categories and functors
- Lecture 0 (January 11, 2021).
Definition of categories, injective and surjective morphisms, functors, and examples.
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- Lecture 1 (January 13, 2021).
Faithful and full functors, Natural transformations, Equivalence of categories.
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- Lecture 2 (January 15, 2021).
Proof of Theorem 1.6, Adjoint functors and examples.
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- Lecture 3 (January 20, 2021).
Adjoint functors continued: unit and counit of adjunction.
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- Lecture 4 (January 22, 2021).
Yoneda embedding theorem. Representable functors.
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- Lecture 5 (January 25, 2021).
Representable functors cntd. Direct sums and direct products.
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- Lecture 6 (January 27, 2021).
Direct and inverse limits. Examples of direct limits.
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- Lecture 7 (January 29, 2021).
Right-directedness for direct limits. Examples of inverse limits.
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Homological algebra
- Lecture 8 (February 1, 2021).
Additive categories. Kernel and cokernel of a morphism.
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- Lecture 9 (February 3, 2021).
Abelian categories. Additive functors.
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- Lecture 10 (February 5, 2021).
Left and right exact functors. Hom functors are left exact.
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- Lecture 11 (February 10, 2021).
Definition of tensor product.
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- Lecture 12 (February 12, 2021).
Tensor functor. Tensor-Hom adjointness.
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- Lecture 13 (February 15, 2021).
Preview of coming attractions. Derived functors.
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- Lecture 14 (February 17, 2021).
Category of complexes. Cohomology functors. Homotopic morphisms.
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- Lecture 15 (February 19, 2021).
Long exact sequence in cohomology. Snake Lemma.
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- Lecture 16 .
Injective and projective objects. Several equivalent properties.
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- Lecture 17 .
Injective/projective resolutions. Uniqueness.
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- Lecture 18 .
Baer's criterion. Enough injectives in R-mod.
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- Lecture 19 .
Ext and Tor functors.
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- Lecture 20 .
Flat modules.
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- Lecture 21 .
Flatness is a local property.
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- Lecture 22 .
Two ways of computing Ext/Tor are the same.
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- Lecture 23 .
Spectral sequences - quick glance.
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Galois theory
- Lecture 24 .
Galois theory: introduction. Field extensions.
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- Lecture 25 .
Algebraic extensions.
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- Lecture 26 .
Splitting extensions: existence and uniqueness.
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- Lecture 27 .
Symmetric polynomials.
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- Lecture 28 .
Fundamental theorem of algebra. Galois group - definition.
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- Lecture 29 .
Dedekind's independence of characters theorem. Artin's theorem.
Separable and normal extensions.
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- Lecture 30 .
Separable/normal extns cntd. Perfect fields and p-radical extensions.
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- Lecture 31 .
Fundamental theorem of Galois theory.
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- Lecture 32 .
Topological groups. Krull topology on (infinite) Galois groups.
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- Lecture 33 .
Galois group of the closure of a finite field.
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- Lecture 34 .
Noether's equations. Galois cohomology.
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- Lecture 35 .
Norm, trace, Hilbert's 90-th problem and cyclic extensions.
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- Lecture 36 .
Kummer's theorem.
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- Lecture 37 .
Primitive element and Normal basis theorems.
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