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Postmodern Algebra
Sam Macdonald
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You can't use 'macro parameter character #' in math mode
You can't use 'macro parameter character #' in math mode
Front Matter
Colophon
Acknowledgements
I
Homological Algebra
1
The Category of Chain Complexes
1.1
Chain Complexes and Short Exact Sequences
Homology
1.2
The Category of Chain Complexes
Establishing
Ch
(
R
)
The Homology Functor
Homotopy
Complexes of Complexes
1.3
Split Exact Sequences
Definition and the Splitting Lemma
When Do Short Exact Sequences Split?
1.4
Long Exact Sequences
The Snake Lemma
The Long Exact Sequence in Homology
2
R
-Mod
2.1
Hom
There’s No Place Like
Hom
Additive Functors
Exact Functors
Hom
is Left Exact
2.2
Tensor Products
Biadditive Maps and First Properties
Elements in Tensor Products
Bimodules and When Tensor Products are Modules
The Tensor Product Functor
Tensor is Right Exact
2.3
Localization
2.4
Hom-Tensor Adjunction
3
Projectives and Injectives
3.1
Projective Modules
Definition, Free Modules are Projective
Direct Summands
3.2
Injective Modules
Definition and Baer’s Criterion
Divisible Modules
Every Module Embeds into an Injective Module
3.3
Flat Modules
Definition
Torsion Submodules
3.4
Commutative Local Rings
4
Resolutions
4.1
Projective Resolutions
Definition, Existence, Examples
A Brief Introduction to Graded Modules
Uniqueness of Minimal Projective Resolutions
Syzygys and Betti Numbers
4.2
Injective Resolutions
5
Derived Functors
5.1
The General Construction
Making Sure Derived Functors Make Sense
Forcing Exactness
5.2
A First Look at
Ext
and
Tor
Double Complexes
Balancing
Tor
and
Ext
5.3
Computing Ext and Tor
The Constructions
Examples
5.4
Other Derived Functors
Group Homology
Local Cohomology
6
Abelian Categories
6.1
What’s an Abelian Category?
Monomorphisms and Epimorphisms
Additive Categories
Kernels and Cokernels
Definition of an Abelian Category
6.2
Complexes and Homology
Complexes in Abelian Categories
Homology in Abelian Categories
6.3
Functors
Exact Functors: Revisited
The Yoneda Lemma: Revisited
The Diagram Chasers: Revisited
6.4
Projectives and Injectives
Projectives, Injective, and Resolutions: Revisited
Split Exact Sequences: Revisited
Unique Resolutions and the Horseshoe Lemma: Revisited
6.5
Derived Functors
Derived Functors: Revisited
7
Spectral Sequences
7.1
Intro to Spectral Sequences
What is a Spectral Sequence?
Graded and Bigraded Modules
Filtrations
7.2
Convergence
7.3
Filtered Complexes
7.4
Double Complexes
II
Commutative Algebra
8
Finiteness Conditions
8.1
Generated Modules
8.2
Algebras
8.3
Algebra and Module Finite
8.4
Noetherian Rings
Noetherian Rings
Noetherian Modules
Hilbert’s Basis Theorem
8.5
Invariant Rings: Application I
9
Graded Rings
9.1
Graded Rings
Graded Rings
Graded Modules and Homomorphisms
9.2
Finiteness Conditions for Graded Algebras
9.3
Invariant Rings: Application II
10
Local Rings
10.1
Local Rings
10.2
Localization
10.3
NAK
11
Prime Ideals
11.1
Prime and Maximal Ideals
11.2
The Spectrum of a Ring
11.3
Operations
Colons and Annihilators
Prime Avoidance
12
Decomposing Ideals
12.1
Minimal Primes and Support
12.2
Associated Primes
12.3
Primary Decomposition
12.4
The Krull Intersection Theorem
13
Integral Extensions
13.1
Integral Extensions
13.2
Over, Up, and Down
14
Dimension Theory
14.1
Dimension and Height
14.2
Noether Normalizations
14.3
Height and Number of Generators
14.4
Systems of Parameters
III
Commutative Algebra II
15
Domains II
15.1
Artinian Rings
15.2
Discrete Valuation Rings and Dedekind Domains
Discrete Valuation Rings
Dedekind Domains
Fractional Ideals
16
Depth and Grade
16.1
Depth and Grade
16.2
The Koszul Complex
16.3
Macaulay2
17
Cohen-Macaulay Rings
17.1
IV
Local Cohomology
18
Injective Modules
18.1
Essential Extensions and Injective Hulls
18.2
Injective Modules in Noetherian Rings
18.3
Mathis Duality
19
Local Cohomology
19.1
Definitions of Local Cohomology
19.2
The Čech Complex
19.3
Properties and Vanishing Theorems
20
The Canonical Module
20.1
Local Duality
20.2
Gorenstein Rings
Gorenstein
Worksheet on Gorenstein rings
20.3
Canonical Modules
20.4
Graded Local Duality and Regularity
Backmatter
A
Foundational Knowledge
A.1
Ring Theory
A.2
Module Theory
A.3
Topology
B
A Lifetime Supply of Category Theory
B.1
Categories
Definition and First Examples
Diagrams and Morphisms
Opposite Categories and Subcategories
B.2
Functors
Functor? Damn Near Killed ’er!
Properties of Functors
The
Hom
Functors
B.3
Natural Transformations
Natural Transformations
The Yoneda Lemma
C
Macaulay2
C.1
Getting Started
C.2
Asking Macaulay2 for Help
C.3
Basic Commands
Colophon
🔗
Chapter
10
Local Rings
10.1
Local Rings
10.2
Localization
10.3
NAK
