6.1 Projective and injective resolutions

Projective and injective resolutions are key tools in homological algebra. They help us understand modules by breaking them down into simpler pieces, allowing us to compute important algebraic invariants and derived functors.

These resolutions are built using special types of modules with nice lifting properties. They form the backbone of many calculations in homological algebra, connecting abstract concepts to concrete computations we can perform.

Projective and Injective Modules

Properties and Definitions of Projective and Injective Modules

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• Projective module is a module $P$ such that for any surjective homomorphism $f: M \to N$ and any homomorphism $g: P \to N$, there exists a homomorphism $h: P \to M$ with $f \circ h = g$
• Injective module is a module $I$ such that for any injective homomorphism $f: M \to N$ and any homomorphism $g: M \to I$, there exists a homomorphism $h: N \to I$ with $h \circ f = g$
• Projective modules are direct summands of free modules ($P \oplus Q \cong F$ for some module $Q$ and free module $F$)
• Injective modules are direct summands of divisible modules ($I \oplus J \cong D$ for some module $J$ and divisible module $D$)

Resolutions using Projective and Injective Modules

• Projective resolution of a module $M$ is an exact sequence $\cdots \to P_1 \to P_0 \to M \to 0$ where each $P_i$ is a projective module
  • Used to compute derived functors of covariant functors ($\operatorname{Tor}_n(M,N) = H_n(M \otimes_R P_\bullet)$ where $P_\bullet$ is a projective resolution of $N$)
• Injective resolution of a module $M$ is an exact sequence $0 \to M \to I^0 \to I^1 \to \cdots$ where each $I^i$ is an injective module
  • Used to compute derived functors of contravariant functors ($\operatorname{Ext}^n(M,N) = H^n(\operatorname{Hom}_R(M, I^\bullet))$ where $I^\bullet$ is an injective resolution of $N$)
• Every module has a projective resolution and an injective resolution
  • Constructed using enough projectives/injectives property ($\operatorname{Hom}_R(P,-)$ is exact for projective $P$, $\operatorname{Hom}_R(-,I)$ is exact for injective $I$)

Complexes and Exact Sequences

Definitions and Properties of Complexes

• Chain complex is a sequence of modules and homomorphisms $\cdots \to M_{n+1} \xrightarrow{d_{n+1}} M_n \xrightarrow{d_n} M_{n-1} \to \cdots$ such that $d_n \circ d_{n+1} = 0$ for all $n$
  • Cochain complex has arrows going in the opposite direction ($M^n \xrightarrow{d^n} M^{n+1}$)
• Homology of a chain complex at $M_n$ is defined as $H_n(M) = \ker d_n / \operatorname{im} d_{n+1}$
  • Cohomology of a cochain complex at $M^n$ is defined as $H^n(M) = \ker d^n / \operatorname{im} d^{n-1}$
• Acyclic complex is a complex with trivial homology ($H_n(M) = 0$ for all $n$)
  • Exact sequence is an acyclic complex ($0 \to M' \to M \to M'' \to 0$ is exact iff $\operatorname{im} f = \ker g$ at each stage)

Minimal Resolutions and Applications

• Minimal resolution is a projective or injective resolution with the smallest possible terms
  • Minimality condition: $d_n(P_n) \subseteq \mathfrak{m}P_{n-1}$ for all $n$ where $\mathfrak{m}$ is the maximal ideal of the base ring
• Minimal resolutions are unique up to isomorphism
  • Used to define invariants of modules (Betti numbers, Bass numbers)
• Projective dimension of $M$ is the length of its minimal projective resolution ($\operatorname{pd} M = \sup\{n \mid P_n \neq 0\}$)
  • Injective dimension of $M$ is the length of its minimal injective resolution ($\operatorname{id} M = \sup\{n \mid I^n \neq 0\}$)