7.2 Definition and properties of Ext functor

The Ext functor is a powerful tool in homological algebra that measures how far the Hom functor is from being exact. It's derived from the Hom functor and provides insights into module structures and relationships.

Ext groups are computed using injective resolutions and play a crucial role in long exact sequences. They exhibit properties like dimension shifting and balance, connecting different modules and allowing for complex algebraic computations.

Definition and Basic Properties

Ext Functor and its Derivation

• Ext functor is a bifunctor that measures the deviation of the Hom functor from being exact
• Defined as the right derived functor of the Hom functor
  • Derived functors are a way to extend a functor that is not exact to a family of functors that measure the degree to which the original functor fails to be exact
  • The right derived functors are obtained by applying the original functor to an injective resolution of the second argument and taking homology
• Notation: $\operatorname{Ext}^n_R(A,B)$ denotes the $n$-th Ext group of $A$ and $B$ over the ring $R$

Properties of the Hom Functor

• Hom functor $\operatorname{Hom}_R(A,-)$ is a left exact functor for any $R$-module $A$
  • Preserves exact sequences of the form $0 \to B \to C \to D$
  • Does not necessarily preserve exactness at later terms in the sequence
• Hom functor is contravariant in the first argument and covariant in the second argument
  • $\operatorname{Hom}_R(-,B)$ is contravariant
  • $\operatorname{Hom}_R(A,-)$ is covariant

Resolutions and Long Exact Sequences

Injective and Projective Resolutions

• Injective resolution of an $R$-module $A$ is an exact sequence of the form:
  • $0 \to A \to I^0 \to I^1 \to \cdots$ where each $I^i$ is an injective $R$-module
  • Used to compute right derived functors ($\operatorname{Ext}$)
• Projective resolution of an $R$-module $A$ is an exact sequence of the form:
  • $\cdots \to P_1 \to P_0 \to A \to 0$ where each $P_i$ is a projective $R$-module
  • Used to compute left derived functors ($\operatorname{Tor}$)
• Existence of resolutions:
  • Every module over a ring with enough injectives admits an injective resolution
  • Every module over a ring with enough projectives admits a projective resolution

Long Exact Sequences

• Fundamental tool in homological algebra to study relationships between Ext groups
• For a short exact sequence of $R$-modules $0 \to A \to B \to C \to 0$, there is a long exact sequence:
  • $0 \to \operatorname{Hom}_R(X,A) \to \operatorname{Hom}_R(X,B) \to \operatorname{Hom}_R(X,C) \to \operatorname{Ext}^1_R(X,A) \to \operatorname{Ext}^1_R(X,B) \to \cdots$
• Allows the computation of Ext groups by breaking them into shorter exact sequences
• Connects Ext groups of different modules in a functorial way

Advanced Properties and Theorems

Dimension Shifting and Balance

• Dimension shifting: relationship between Ext groups of different dimensions
  • For $R$-modules $A$ and $B$ and an integer $n$, there is an isomorphism:
    • $\operatorname{Ext}^{n+1}_R(A,B) \cong \operatorname{Ext}^n_R(A,C)$ where $C$ is the cokernel of an injective hull of $B$
  • Allows computation of higher Ext groups from lower ones
• Balance: symmetry between left and right Ext functors
  • For $R$-modules $A$ and $B$, there is an isomorphism:
    • $\operatorname{Ext}^n_R(A,B) \cong \operatorname{Ext}^n_{R^{op}}(B,A)$ where $R^{op}$ is the opposite ring of $R$
  • Reflects the duality between injective and projective modules

Universal Coefficient Theorem

• Relates Ext groups to homology and cohomology groups
• For a ring $R$, an $R$-module $A$, and a chain complex $C$ of projective $R$-modules, there is a short exact sequence:
  • $0 \to \operatorname{Ext}^1_R(H_{n-1}(C),A) \to H^n(C;A) \to \operatorname{Hom}_R(H_n(C),A) \to 0$
• Allows computation of cohomology groups using Ext groups and homology groups
• Particularly useful in algebraic topology and algebraic geometry