The is a powerful tool in homological algebra that measures how far the 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
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Ext functor is a bifunctor that measures the deviation of the Hom functor from being exact
Defined as the right 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 of the second argument and taking homology
Notation: ExtRn(A,B) denotes the n-th Ext group of A and B over the ring R
Properties of the Hom Functor
Hom functor HomR(A,−) is a left exact functor for any R-module A
Preserves exact sequences of the form 0→B→C→D
Does not necessarily preserve at later terms in the sequence
Hom functor is contravariant in the first argument and covariant in the second argument
HomR(−,B) is contravariant
HomR(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→A→I0→I1→⋯ where each Ii is an injective R-module
Used to compute right derived functors (Ext)
Projective resolution of an R-module A is an exact sequence of the form:
⋯→P1→P0→A→0 where each Pi is a projective R-module
Used to compute left derived functors (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