Algebraic structures like groups, rings, modules, and vector spaces form the foundation of homological algebra. These structures have specific properties and operations that define their behavior and relationships.
Homomorphisms are crucial tools for studying these structures, preserving their essential properties when mapping between them. Understanding kernels, images, and exact sequences helps analyze the relationships and connections between different algebraic structures.
Algebraic Structures
Groups and Rings
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consists of a set G together with a binary operation ∗ satisfying the following axioms:
Closure: For all a,b∈G, a∗b∈G
Associativity: For all a,b,c∈G, (a∗b)∗c=a∗(b∗c)
: There exists an element e∈G such that for all a∈G, a∗e=e∗a=a
Inverse element: For each a∈G, there exists an element a−1∈G such that a∗a−1=a−1∗a=e
extends the concept of a group by adding a second binary operation, usually denoted as +, and requiring the set to also have the structure of an abelian group under this operation
A ring (R,+,∗) satisfies the following axioms:
(R,+) is an abelian group with identity element 0
(R,∗) is a monoid with identity element 1
Distributivity: For all a,b,c∈R, a∗(b+c)=(a∗b)+(a∗c) and (a+b)∗c=(a∗c)+(b∗c)
Modules and Vector Spaces
generalizes the notion of a by allowing the scalars to be from a ring instead of a field
A left R-module consists of an abelian group (M,+) and a ring (R,+,∗) together with an operation ⋅:R×M→M (called ) such that for all r,s∈R and x,y∈M:
(r+s)⋅x=r⋅x+s⋅x
(r∗s)⋅x=r⋅(s⋅x)
1⋅x=x, where 1 is the multiplicative identity in R
Vector space is a special case of a module where the ring R is a field F
A vector space over a field F is an abelian group (V,+) together with a scalar operation ⋅:F×V→V satisfying the following axioms for all a,b∈F and u,v∈V:
(a+b)⋅v=a⋅v+b⋅v
a⋅(u+v)=a⋅u+a⋅v
(a∗b)⋅v=a⋅(b⋅v)
1⋅v=v, where 1 is the multiplicative identity in F
Homomorphisms and Related Concepts
Homomorphisms and Their Properties
is a structure-preserving map between two algebraic structures of the same type (e.g., groups, rings, modules, vector spaces)
A φ:G→H satisfies φ(a∗b)=φ(a)∗φ(b) for all a,b∈G
A ψ:R→S satisfies ψ(a+b)=ψ(a)+ψ(b) and ψ(a∗b)=ψ(a)∗ψ(b) for all a,b∈R
of a homomorphism φ:G→H is the set of elements in G that map to the identity element in H, denoted as ker(φ)={g∈G:φ(g)=eH}
The kernel is always a normal of G
of a homomorphism φ:G→H is the set of elements in H that are mapped to by some element in G, denoted as Im(φ)={φ(g):g∈G}
The image is always a subgroup of H
Exact Sequences
is a sequence of homomorphisms between algebraic structures (e.g., groups, rings, modules, vector spaces) such that the image of each homomorphism is equal to the kernel of the next homomorphism
A is an exact sequence of the form 0→AfBgC→0, where 0 denotes the trivial group, ring, module, or vector space
In a short exact sequence, f is (one-to-one), g is (onto), and Im(f)=ker(g)
Splitting lemma states that a short exact sequence 0→AfBgC→0 splits (i.e., B≅A⊕C) if and only if there exists a homomorphism h:C→B such that g∘h=idC