Glossary

6.3 Universal properties of derived functors

3 min read•august 7, 2024

Derived functors are powerful tools in homological algebra, measuring how far a functor is from being exact. They help us understand complex structures by breaking them down into simpler parts we can analyze.

Universal properties of derived functors give us a way to characterize these tools uniquely. This approach lets us work with derived functors abstractly, without worrying about the specific constructions used to define them.

Universal Properties and Functors

Universal Properties and Natural Transformations

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  • A universal property defines an object in terms of its relationships with other objects, rather than by its internal structure
  • Universal properties are used to characterize objects and morphisms in category theory
  • provide a way to compare functors between categories
    • A natural transformation η:F→G\eta: F \to G is a family of morphisms ηX:F(X)→G(X)\eta_X: F(X) \to G(X) for each object XX in the source category, such that for every morphism f:X→Yf: X \to Y, the diagram commutes: G(f)∘ηX=ηY∘F(f)G(f) \circ \eta_X = \eta_Y \circ F(f)
  • Universal properties and natural transformations play a crucial role in the study of derived functors and homological algebra

Delta and Cohomological Functors

  • A is a sequence of functors Tn:A→BT^n: \mathcal{A} \to \mathcal{B} between abelian categories, along with connecting homomorphisms δn:Tn(A′)→Tn+1(A)\delta^n: T^n(A') \to T^{n+1}(A) for each short exact sequence 0→A→A′→A′′→00 \to A \to A' \to A'' \to 0 in A\mathcal{A}, satisfying certain axioms
    • The connecting homomorphisms form a : ⋯→Tn(A)→Tn(A′)→Tn(A′′)→δnTn+1(A)→⋯\cdots \to T^n(A) \to T^n(A') \to T^n(A'') \xrightarrow{\delta^n} T^{n+1}(A) \to \cdots
  • A is a contravariant delta functor, i.e., a delta functor with Tn(f):Tn(B)→Tn(A)T^n(f): T^n(B) \to T^n(A) for each morphism f:A→Bf: A \to B in A\mathcal{A}
  • Examples of cohomological functors include the functors ExtRn(A,−)\text{Ext}^n_R(A, -) and the cohomology functors Hn(X;−)H^n(X; -) in algebraic topology

Derived Functors and Exact Sequences

Effacement and Derived Functors

  • The states that for any additive functor F:A→BF: \mathcal{A} \to \mathcal{B} between abelian categories and any object AA in A\mathcal{A}, there exists an epimorphism P→AP \to A with PP projective such that F(P)→F(A)F(P) \to F(A) is an epimorphism
  • The effacement theorem is used to construct derived functors by taking projective of objects
  • Given a left exact functor F:A→BF: \mathcal{A} \to \mathcal{B} and an object AA in A\mathcal{A}, the right derived functors RnF(A)R^nF(A) are defined as the cohomology of the complex F(P∙)F(P_\bullet), where P∙→AP_\bullet \to A is a projective resolution of AA
    • The derived functors measure the failure of FF to be exact

Long Exact Sequences and Connecting Homomorphisms

  • A long exact sequence is a sequence of homomorphisms between abelian groups or modules ⋯→An−1→fn−1An→fnAn+1→⋯\cdots \to A_{n-1} \xrightarrow{f_{n-1}} A_n \xrightarrow{f_n} A_{n+1} \to \cdots such that the image of each homomorphism is equal to the kernel of the next: Im(fn−1)=Ker(fn)\text{Im}(f_{n-1}) = \text{Ker}(f_n)
  • Long exact sequences often arise from short exact sequences of chain complexes by taking homology or cohomology
  • The in a long exact sequence is a homomorphism δn:Hn(C′′)→Hn−1(C)\delta_n: H_n(C'') \to H_{n-1}(C) induced by the short exact sequence of chain complexes 0→C→C′→C′′→00 \to C \to C' \to C'' \to 0
    • The connecting homomorphism measures the obstruction to lifting cycles in C′′C'' to cycles in C′C'
  • Examples of long exact sequences include the long exact sequence in homology associated with a short exact sequence of chain complexes and the long exact sequence of a pair in singular homology

Key Terms to Review (23)

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.
Cohomological Functor: A cohomological functor is a type of functor that assigns to each object in a category a cohomology group, which encapsulates important algebraic and topological information about the object. These functors arise naturally in homological algebra and provide a way to systematically study derived functors, ultimately allowing us to derive deeper insights into the structure of mathematical objects.
Connecting Homomorphism: A connecting homomorphism is a morphism that arises in the context of exact sequences, specifically serving as a bridge between different chain complexes. It captures the relationship between the homology of the chain complexes and helps facilitate the transfer of algebraic information across these complexes. This concept plays a crucial role in linking different algebraic structures, especially when analyzing how they interact under sequences and derived functors.
Delta Functor: A delta functor is a construction in homological algebra that captures the relationship between a category and its derived category, often denoted as \(\Delta\). It serves to systematically encode the necessary data for constructing derived functors, allowing one to represent derived functors as functors that arise from this delta structure. This concept plays a crucial role in defining universal properties that characterize the behavior of derived functors.
Effacement Theorem: The effacement theorem is a fundamental result in homological algebra that describes how certain derived functors can be represented in terms of projective resolutions. This theorem connects derived functors with the structure of exact sequences, highlighting that under certain conditions, the derived functors can vanish when taking higher derived functors from a projective resolution. It emphasizes the significance of universal properties in understanding how these functors behave.
Eilenberg-Steenrod Axioms: The Eilenberg-Steenrod axioms are a set of properties that define the category of singular homology in algebraic topology. They serve as a foundation for homological algebra by establishing how functors behave with respect to topological spaces, ensuring consistency and a systematic approach to deriving properties of topological invariants. These axioms connect to various concepts in topology and homological algebra, providing essential tools for understanding cellular homology and derived functors.
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: In homological algebra, the term 'ext' refers to a functor that provides a way to measure the extent to which a module fails to be projective. Specifically, 'Ext' is used to define derived functors of the Hom functor, which captures the idea of extensions of modules. This concept is essential for understanding the relationships between modules and their extensions, particularly in the context of left and right derived functors as well as universal properties that these derived functors satisfy.
Functoriality: Functoriality is a fundamental concept in category theory that emphasizes the idea of structure-preserving mappings between categories. It reflects how mathematical structures behave under various transformations, allowing for consistent relationships between different objects and morphisms in those structures. This property is essential for connecting various aspects of homological algebra, including the manipulation of sequences and the study of functors like Tor and derived functors.
Grothendieck's Six Operations: Grothendieck's six operations are a framework in algebraic geometry and derived categories that consists of six functors: direct image, inverse image, proper direct image, proper inverse image, exceptional functor, and dualizing functor. These operations are fundamental for understanding how sheaves and cohomology behave under various geometric transformations, providing a powerful tool for studying the relationships between different spaces and their properties.
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.
Injective Dimension: Injective dimension is a measure of how far a module is from being injective, specifically defined as the shortest length of an injective resolution of that module. It plays a vital role in homological algebra, connecting the notions of injective modules, projective resolutions, and derived functors, while also influencing computations involving Tor and Ext.
Jean-Pierre Serre: Jean-Pierre Serre is a prominent French mathematician known for his work in algebraic geometry, topology, and number theory. His contributions have had a significant impact on various areas of mathematics, particularly through the development of cohomological methods and the theory of derived categories.
Left derived functor: A left derived functor is a construction in homological algebra that extends a given functor defined on a category to the derived category, providing a way to measure the failure of exactness. It is created by applying a sequence of projective resolutions to an object and then applying the original functor to these resolutions. This process allows one to capture important topological and algebraic properties of the objects involved, revealing deeper connections in their structures.
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.
Natural Transformations: Natural transformations are a way of relating functors, which are mappings between categories, in a coherent manner. They provide a framework to compare different functors that operate on the same category, preserving the structure and relationships within those categories. In the context of derived functors, natural transformations allow for the connection between different functors that arise in homological algebra, highlighting their universal properties and how they interact with specific sequences or constructions.
Projective Dimension: Projective dimension is a homological invariant that measures the complexity of a module by determining the length of the shortest projective resolution of that module. It connects deeply with various concepts in homological algebra, such as resolutions, derived functors, and cohomology theories, showcasing how modules relate to projective modules and their role in understanding other homological properties.
Resolutions: In homological algebra, resolutions are sequences of modules and morphisms that provide a way to approximate or represent a given module in terms of simpler modules. This concept is crucial for understanding derived functors, as resolutions help in computing these functors through the process of taking derived functors of the exact sequences formed by these resolutions.
Right Derived Functor: A right derived functor is a construction in homological algebra that extends a functor from a category of modules to a derived category, capturing information about the behavior of the functor when applied to exact sequences. It is defined using projective resolutions, allowing us to measure how far a functor is from being exact by quantifying the failure to preserve exactness in sequences. This concept highlights the importance of universal properties by associating these functors with higher-level homological invariants.
Spectral Sequences: Spectral sequences are powerful computational tools in homological algebra that provide a method for calculating homology and cohomology groups through a sequence of approximations. They allow mathematicians to systematically derive information about complex algebraic structures by filtering through layers of associated chain complexes, leading to eventual convergence towards desired invariants.
Tor: Tor is a functor that measures the failure of flatness between two modules in a category, particularly in the context of homological algebra. It arises as a left derived functor of the Hom functor, which means it captures important information about how well a module behaves with respect to another module when we take exact sequences into account. This concept is essential for understanding derived functors and their universal properties in various algebraic structures.
Universal Coefficient Theorem: The Universal Coefficient Theorem is a fundamental result in algebraic topology and homological algebra that relates the homology or cohomology groups of a topological space to its singular homology or cohomology groups with coefficients in a module. It establishes a way to compute these groups when changing from integer coefficients to coefficients in any abelian group, bridging the gap between homology and cohomology.
Universal Property of Limits: The universal property of limits is a foundational concept in category theory that defines how limits can be uniquely characterized by their universal mapping properties. It establishes that for any cone over a diagram, there exists a unique morphism from the cone's apex to the limit, making limits a way to 'capture' all the information of a diagram in a single object. This property plays a crucial role in understanding derived functors, as it links the concept of limits with their ability to create new mathematical structures through functorial operations.
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