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
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:
ExtRn+1(A,B)≅ExtRn(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:
ExtRn(A,B)≅ExtRopn(B,A) where Rop is the opposite ring of R
Reflects the duality between injective and projective modules
Universal Coefficient Theorem
Relates Ext groups to homology and 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→ExtR1(Hn−1(C),A)→Hn(C;A)→HomR(Hn(C),A)→0
Allows computation of cohomology groups using Ext groups and homology groups
Particularly useful in algebraic topology and algebraic geometry
Key Terms to Review (18)
Additivity: Additivity refers to the property of certain functors in homological algebra where the functor preserves direct sums. This means that if you apply an additive functor to a direct sum of objects, it is equivalent to taking the direct sum of the functor applied to each individual object. This property is crucial for understanding how derived functors behave, especially when examining left and right derived functors and their applications.
Classifying Spaces: Classifying spaces are topological spaces that serve as a universal space for a given category of bundles, helping to classify fiber bundles up to isomorphism. They play a crucial role in algebraic topology, particularly in the study of vector bundles and principal bundles, by allowing mathematicians to translate geometric problems into algebraic terms.
Cohomology: Cohomology is a mathematical concept that assigns algebraic invariants to topological spaces and chain complexes, capturing information about their structure and relationships. It provides a dual perspective to homology, focusing on the study of cochains and cocycles, which can reveal properties of spaces that homology alone might miss. This tool is essential in various areas of mathematics, connecting geometry, algebra, and topology.
Derived functor: A derived functor is a concept in homological algebra that extends the idea of a functor by associating it with a sequence of objects (called derived objects) which capture the 'homological information' of the original functor. This allows one to study properties of functors that are not preserved under direct application, particularly in the context of resolving modules and understanding exact sequences.
Exactness: Exactness in homological algebra refers to a property of sequences of objects and morphisms that captures the idea of preserving structure in a way that connects the input and output accurately. It ensures that the image of one morphism equals the kernel of the next, providing a precise relationship among the objects involved and facilitating the understanding of how algebraic structures behave under various operations.
Ext functor: The Ext functor is a fundamental tool in homological algebra that measures the extent to which a module fails to be projective, providing a way to study extensions of modules. This concept connects deeply to various areas such as the historical development of homological algebra, category theory, and the computations involved in related functors like Tor.
Ext^n: The functor $$\text{Ext}^n$$ is a key concept in homological algebra, representing the derived functors of the Hom functor. It captures the information about extensions of modules, particularly concerning how modules can be built from others and how they can fit into exact sequences. The notation $$\text{Ext}^n(A, B)$$ describes the set of equivalence classes of extensions of the module $$A$$ by the module $$B$$, where $$n$$ indicates the degree of the extension.
Henri Cartan: Henri Cartan was a prominent French mathematician known for his significant contributions to algebraic topology and homological algebra, particularly in developing the theory of sheaves and derived functors. His work laid essential groundwork for later developments in category theory and cohomology, impacting various mathematical areas including group cohomology and spectral sequences.
Hom: The term 'Hom' refers to a functor that represents the set of morphisms between two objects in a category. It serves as a fundamental building block in category theory, providing insights into the relationships between objects, particularly when studying their structure and properties. This concept is essential in understanding how different algebraic structures can interact with one another through mappings or functions.
Homological Dimension: Homological dimension refers to a measure of the complexity of modules in terms of their projective or injective resolutions. It indicates how many steps are needed to resolve a module using projective or injective modules, helping to classify modules based on their structure and behavior. This concept is crucial when studying the Ext functor, as it directly relates to the calculation of Ext groups and the properties of modules over rings.
Injective Module: An injective module is a type of module that has the property that any homomorphism from a submodule can be extended to the entire module. This means that if you have a short exact sequence where one of the modules is injective, it allows for certain extensions and lifting properties that are crucial in homological algebra. The concept connects deeply with projective modules and plays a significant role in constructing projective and injective resolutions, understanding exact sequences, and utilizing the Ext functor effectively.
Long exact sequence: A long exact sequence is a sequence of abelian groups or modules connected by homomorphisms, which satisfies exactness at every point, indicating that the image of each homomorphism equals the kernel of the next. This concept is crucial in understanding the behavior of homology and cohomology theories, allowing one to relate different algebraic structures through their exact sequences and facilitating computations in various contexts.
Projective Module: A projective module is a type of module that satisfies a lifting property with respect to homomorphisms, meaning that for any surjective homomorphism, any module homomorphism from the projective module can be lifted to the original module. This concept is crucial for understanding direct sums and the behavior of modules under exact sequences, particularly how projective modules can be used to construct resolutions and relate to the Ext functor.
Resolution: In homological algebra, a resolution is a sequence of modules and morphisms that allows for the study of the properties of modules via projective or injective approximations. This tool is essential for analyzing Ext and Tor functors, as it provides a framework to compute these derived functors and their applications in various contexts, such as in finding solutions to algebraic problems.
Samuel Eilenberg: Samuel Eilenberg was a prominent mathematician known for his groundbreaking contributions to the field of homological algebra, particularly in the development of derived functors and the Ext functor. His work laid the foundation for many essential concepts in modern algebra, influencing various mathematical areas including topology and category theory.
Short Exact Sequence: A short exact sequence is a sequence of algebraic structures and homomorphisms where the image of one morphism equals the kernel of the next, typically represented as 0 → A → B → C → 0. This definition serves as a foundation for understanding various concepts in algebra, particularly in relation to how structures relate to one another through homomorphisms and their exactness properties.
Tensor Product: The tensor product is a construction that combines two algebraic structures, such as vector spaces or modules, into a new one that captures the essence of their interactions. It plays a crucial role in various mathematical areas by allowing the formation of bilinear maps and enabling the representation of more complex relationships between these structures. In addition to its foundational importance in algebra, the tensor product serves as a building block for other concepts, such as functoriality and derived functors.
Tor Functor: The Tor functor is a derived functor that measures the extent to which a sequence fails to be exact when applied to modules. It plays a vital role in homological algebra, connecting algebraic properties of modules with topological invariants, and it helps in understanding the relationships between different algebraic structures.