5.4 Existence theorems and applications

Projective and injective resolutions are powerful tools in homological algebra. They allow us to study modules through sequences of simpler objects, giving us insight into their structure and properties.

These resolutions are crucial for computing derived functors like Ext and Tor. They help us measure how far a module is from being projective or injective, and provide a way to understand the global structure of rings.

Projective and Injective Resolutions

Existence of Projective and Injective Modules

• Every module $M$ over a ring $R$ has a projective resolution
  • Construct a projective resolution by taking a free resolution and replacing each free module with a projective module
• Every module $M$ over a ring $R$ has an injective resolution
  • Construct an injective resolution by embedding $M$ into an injective module and repeating the process
• A ring $R$ has enough projectives if every $R$-module is a quotient of a projective module
  • Equivalent to every $R$-module having a projective resolution
• A ring $R$ has enough injectives if every $R$-module can be embedded into an injective module
  • Equivalent to every $R$-module having an injective resolution

Properties of Projective and Injective Resolutions

• Projective resolutions are unique up to homotopy equivalence
  • Two projective resolutions of the same module are chain homotopic
• Injective resolutions are unique up to homotopy equivalence
  • Two injective resolutions of the same module are chain homotopic
• Projective resolutions can be used to compute derived functors of right exact functors
  • Example: $\operatorname{Ext}^n_R(M,N)$ is the $n$-th homology of $\operatorname{Hom}_R(P_\bullet, N)$, where $P_\bullet$ is a projective resolution of $M$
• Injective resolutions can be used to compute derived functors of left exact functors
  • Example: $\operatorname{Ext}^n_R(M,N)$ is the $n$-th cohomology of $\operatorname{Hom}_R(M, I^\bullet)$, where $I^\bullet$ is an injective resolution of $N$

Fundamental Homological Lemmas and Theorems

Horseshoe Lemma

• Given a short exact sequence of chain complexes, there exists a short exact sequence of the corresponding homology groups
  • Constructs a long exact sequence in homology from a short exact sequence of chain complexes
• Provides a tool for computing homology groups of chain complexes
  • Example: If $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$ is a short exact sequence of chain complexes, then there is a long exact sequence $\cdots \to H_n(A_\bullet) \to H_n(B_\bullet) \to H_n(C_\bullet) \to H_{n-1}(A_\bullet) \to \cdots$

Comparison Theorem

• Given a morphism of chain complexes that induces isomorphisms on homology, the chain complexes are chain homotopic
  • Useful for comparing resolutions and proving uniqueness up to homotopy
• Allows for the construction of chain maps between resolutions
  • Example: If $P_\bullet$ and $Q_\bullet$ are projective resolutions of a module $M$, then there exists a chain map $f_\bullet: P_\bullet \to Q_\bullet$ lifting the identity on $M$, and $f_\bullet$ is a homotopy equivalence

Derived Functors

Ext and Tor Functors

• $\operatorname{Ext}^n_R(M,N)$ is the $n$-th derived functor of $\operatorname{Hom}_R(M,-)$ applied to $N$
  • Measures the failure of the functor $\operatorname{Hom}_R(M,-)$ to be exact
• $\operatorname{Tor}_n^R(M,N)$ is the $n$-th derived functor of $- \otimes_R N$ applied to $M$
  • Measures the failure of the functor $- \otimes_R N$ to be exact
• $\operatorname{Ext}^n_R(M,N)$ can be computed using a projective resolution of $M$ or an injective resolution of $N$
  • $\operatorname{Ext}^n_R(M,N) = H^n(\operatorname{Hom}_R(P_\bullet, N))$, where $P_\bullet$ is a projective resolution of $M$
  • $\operatorname{Ext}^n_R(M,N) = H^n(\operatorname{Hom}_R(M, I^\bullet))$, where $I^\bullet$ is an injective resolution of $N$
• $\operatorname{Tor}_n^R(M,N)$ can be computed using a projective resolution of $M$ or a flat resolution of $N$
  • $\operatorname{Tor}_n^R(M,N) = H_n(P_\bullet \otimes_R N)$, where $P_\bullet$ is a projective resolution of $M$
  • $\operatorname{Tor}_n^R(M,N) = H_n(M \otimes_R F_\bullet)$, where $F_\bullet$ is a flat resolution of $N$

Global Dimension

• The global dimension of a ring $R$ is the supremum of the projective dimensions of all $R$-modules
  • Measures how far the ring is from being semisimple
• A ring has finite global dimension if and only if every module has a finite projective resolution
  • Example: Semisimple rings have global dimension 0, as every module is projective
• The global dimension of a ring is equal to the supremum of the injective dimensions of all $R$-modules
  • Consequence of the balance of $\operatorname{Ext}$ functor: $\operatorname{Ext}^n_R(M,N) \cong \operatorname{Ext}^n_R(N,M)$ for all $n > \max(\operatorname{pd} M, \operatorname{id} N)$