1.2 Basic algebraic structures and homomorphisms

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

• Group consists of a set $G$ together with a binary operation $*$ satisfying the following axioms:
  • Closure: For all $a, b \in G$, $a * b \in G$
  • Associativity: For all $a, b, c \in G$, $(a * b) * c = a * (b * c)$
  • Identity element: There exists an element $e \in G$ such that for all $a \in G$, $a * e = e * a = a$
  • Inverse element: For each $a \in G$, there exists an element $a^{-1} \in G$ such that $a * a^{-1} = a^{-1} * a = e$
• Ring extends the concept of a group by adding a second binary operation, usually denoted as addition $+$, 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 $a, b, c \in R$, $a * (b + c) = (a * b) + (a * c)$ and $(a + b) * c = (a * c) + (b * c)$

Modules and Vector Spaces

• Module generalizes the notion of a vector space 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 $\cdot : R \times M \to M$ (called scalar multiplication) such that for all $r, s \in R$ and $x, y \in M$:
    • $(r + s) \cdot x = r \cdot x + s \cdot x$
    • $(r * s) \cdot x = r \cdot (s \cdot x)$
    • $1 \cdot x = x$, where $1$ is the multiplicative identity in $R$
• Vector space is a special case of a module where the ring $R$ is a field $\mathbb{F}$
  • A vector space over a field $\mathbb{F}$ is an abelian group $(V, +)$ together with a scalar multiplication operation $\cdot : \mathbb{F} \times V \to V$ satisfying the following axioms for all $a, b \in \mathbb{F}$ and $u, v \in V$:
    • $(a + b) \cdot v = a \cdot v + b \cdot v$
    • $a \cdot (u + v) = a \cdot u + a \cdot v$
    • $(a * b) \cdot v = a \cdot (b \cdot v)$
    • $1 \cdot v = v$, where $1$ is the multiplicative identity in $\mathbb{F}$

Homomorphisms and Related Concepts

Homomorphisms and Their Properties

• Homomorphism is a structure-preserving map between two algebraic structures of the same type (e.g., groups, rings, modules, vector spaces)
  • A group homomorphism $\varphi : G \to H$ satisfies $\varphi(a * b) = \varphi(a) * \varphi(b)$ for all $a, b \in G$
  • A ring homomorphism $\psi : R \to S$ satisfies $\psi(a + b) = \psi(a) + \psi(b)$ and $\psi(a * b) = \psi(a) * \psi(b)$ for all $a, b \in R$
• Kernel of a homomorphism $\varphi : G \to H$ is the set of elements in $G$ that map to the identity element in $H$, denoted as $\ker(\varphi) = \{g \in G : \varphi(g) = e_H\}$
  • The kernel is always a normal subgroup of $G$
• Image of a homomorphism $\varphi : G \to H$ is the set of elements in $H$ that are mapped to by some element in $G$, denoted as $\operatorname{Im}(\varphi) = \{\varphi(g) : g \in G\}$
  • The image is always a subgroup of $H$

Exact Sequences

• Exact sequence 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 short exact sequence is an exact sequence of the form $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$, where $0$ denotes the trivial group, ring, module, or vector space
    • In a short exact sequence, $f$ is injective (one-to-one), $g$ is surjective (onto), and $\operatorname{Im}(f) = \ker(g)$
  • Splitting lemma states that a short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ splits (i.e., $B \cong A \oplus C$) if and only if there exists a homomorphism $h : C \to B$ such that $g \circ h = \operatorname{id}_C$