Glossary

5.4 Existence theorems and applications

3 min read•august 7, 2024

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 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

Top images from around the web for Existence of Projective and Injective Modules

  • A Characterisation of Morita Algebras in Terms of Covers | SpringerLink View original

    Is this image relevant?

  • Category:Homological algebra - Wikimedia Commons View original

    Is this image relevant?

  • A Characterisation of Morita Algebras in Terms of Covers | SpringerLink View original

    Is this image relevant?

1 of 3

Top images from around the web for Existence of Projective and Injective Modules

  • A Characterisation of Morita Algebras in Terms of Covers | SpringerLink View original

    Is this image relevant?

  • Category:Homological algebra - Wikimedia Commons View original

    Is this image relevant?

  • A Characterisation of Morita Algebras in Terms of Covers | SpringerLink View original

    Is this image relevant?

1 of 3
  • Every module MM over a ring RR has a projective resolution
    • Construct a projective resolution by taking a free resolution and replacing each free module with a projective module
  • Every module MM over a ring RR has an injective resolution
    • Construct an injective resolution by embedding MM into an injective module and repeating the process
  • A ring RR has enough projectives if every RR-module is a quotient of a projective module
    • Equivalent to every RR-module having a projective resolution
  • A ring RR has enough injectives if every RR-module can be embedded into an injective module
    • Equivalent to every RR-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: Ext⁡Rn(M,N)\operatorname{Ext}^n_R(M,N) is the nn-th homology of Hom⁡R(P∙,N)\operatorname{Hom}_R(P_\bullet, N), where P∙P_\bullet is a projective resolution of MM
  • Injective resolutions can be used to compute derived functors of left exact functors
    • Example: Ext⁡Rn(M,N)\operatorname{Ext}^n_R(M,N) is the nn-th cohomology of Hom⁡R(M,I∙)\operatorname{Hom}_R(M, I^\bullet), where I∙I^\bullet is an injective resolution of NN

Fundamental Homological Lemmas and Theorems

Horseshoe Lemma

  • Given a 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→A∙→B∙→C∙→00 \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 ⋯→Hn(A∙)→Hn(B∙)→Hn(C∙)→Hn−1(A∙)→⋯\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∙P_\bullet and Q∙Q_\bullet are projective resolutions of a module MM, then there exists a chain map f∙:P∙→Q∙f_\bullet: P_\bullet \to Q_\bullet lifting the identity on MM, and f∙f_\bullet is a homotopy equivalence

Derived Functors

Ext and Tor Functors

  • Ext⁡Rn(M,N)\operatorname{Ext}^n_R(M,N) is the nn-th derived functor of Hom⁡R(M,−)\operatorname{Hom}_R(M,-) applied to NN
    • Measures the failure of the functor Hom⁡R(M,−)\operatorname{Hom}_R(M,-) to be exact
  • Tor⁡nR(M,N)\operatorname{Tor}_n^R(M,N) is the nn-th derived functor of −⊗RN- \otimes_R N applied to MM
    • Measures the failure of the functor −⊗RN- \otimes_R N to be exact
  • Ext⁡Rn(M,N)\operatorname{Ext}^n_R(M,N) can be computed using a projective resolution of MM or an injective resolution of NN
    • Ext⁡Rn(M,N)=Hn(Hom⁡R(P∙,N))\operatorname{Ext}^n_R(M,N) = H^n(\operatorname{Hom}_R(P_\bullet, N)), where P∙P_\bullet is a projective resolution of MM
    • Ext⁡Rn(M,N)=Hn(Hom⁡R(M,I∙))\operatorname{Ext}^n_R(M,N) = H^n(\operatorname{Hom}_R(M, I^\bullet)), where I∙I^\bullet is an injective resolution of NN
  • Tor⁡nR(M,N)\operatorname{Tor}_n^R(M,N) can be computed using a projective resolution of MM or a flat resolution of NN
    • Tor⁡nR(M,N)=Hn(P∙⊗RN)\operatorname{Tor}_n^R(M,N) = H_n(P_\bullet \otimes_R N), where P∙P_\bullet is a projective resolution of MM
    • Tor⁡nR(M,N)=Hn(M⊗RF∙)\operatorname{Tor}_n^R(M,N) = H_n(M \otimes_R F_\bullet), where F∙F_\bullet is a flat resolution of NN

Global Dimension

  • The global dimension of a ring RR is the supremum of the projective dimensions of all RR-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 RR-modules
    • Consequence of the balance of Ext⁡\operatorname{Ext} functor: Ext⁡Rn(M,N)≅Ext⁡Rn(N,M)\operatorname{Ext}^n_R(M,N) \cong \operatorname{Ext}^n_R(N,M) for all n>max⁡(pd⁡M,id⁡N)n > \max(\operatorname{pd} M, \operatorname{id} N)

Key Terms to Review (18)

Abelian Categories: Abelian categories are a type of mathematical structure that generalizes several important concepts in algebra, characterized by having all morphisms, kernels, and cokernels that behave nicely, allowing for the existence of limits and colimits. They provide a framework in which one can work with exact sequences and homological properties, making them essential for understanding complex structures in homological algebra. The richness of their structure allows for a more generalized approach to concepts like exactness and isomorphism, which are crucial in the motivation behind homological methods and their various applications.
Applications to representation theory: Applications to representation theory explore how algebraic structures can be represented through linear transformations on vector spaces, connecting abstract algebra to concrete examples in linear algebra. This interplay allows mathematicians to understand and classify various algebraic entities by analyzing their representations, leading to deeper insights in areas such as group theory and module theory. In essence, these applications provide tools to study symmetries and transformations within mathematical structures.
Applications to Sheaf Theory: Applications to sheaf theory refer to the practical use of sheaf concepts in various mathematical fields, particularly algebraic geometry and topology, to solve problems involving local-global relationships. This involves utilizing the properties of sheaves, such as gluing data and local sections, to understand complex structures and morphisms within different mathematical contexts. This connection is particularly evident in existence theorems, where the ability to construct or deduce sheaves leads to significant results regarding the representation of objects and their properties.
Daniel Quillen: Daniel Quillen was an influential mathematician known for his groundbreaking work in homological algebra and category theory. He made significant contributions to the understanding of derived categories and the formulation of the concept of model categories, which have had a lasting impact on the field. His ideas transformed how mathematicians approach homological methods, particularly in relation to algebraic topology and abstract algebra.
Derived Functors: Derived functors are a way to extend the concept of functors in category theory to measure how much a given functor fails to be exact. They provide a systematic way to derive additional information from a functor by analyzing its relationship with exact sequences and chain complexes. Derived functors are particularly useful in homological algebra as they connect various algebraic structures, allowing us to study properties like the existence of certain modules and their relationships.
Equivalence of Categories: Equivalence of categories is a concept in category theory that describes when two categories are, in a certain sense, structurally the same. This notion is not just about having the same objects and morphisms but instead emphasizes the existence of functors between the two categories that create a correspondence preserving the relationships between their objects and morphisms. Essentially, if two categories are equivalent, they can be considered interchangeable for purposes such as studying properties of mathematical structures.
Exact Categories: Exact categories are a generalization of abelian categories where one can define a notion of exactness using distinguished triangles. They allow for the study of homological properties in a more flexible context, bridging the gap between various algebraic structures and categorical frameworks. This concept enables the formulation of existence theorems that address the conditions under which certain objects or morphisms can be found or constructed.
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.
Existence of injective resolutions: The existence of injective resolutions refers to the property that every module can be embedded into an injective module, which can then be extended to an injective resolution. This concept is crucial in homological algebra as it facilitates the study of modules through their resolutions, especially in relation to derived functors and Ext groups.
Existence of Projective Resolutions: The existence of projective resolutions refers to the fact that every module has a projective resolution, which is a type of exact sequence where all the modules involved are projective. This concept is essential in homological algebra as it allows for the construction of derived functors, which are used to measure how far a given functor is from being exact. The existence of such resolutions highlights the significance of projective modules in the study of module categories and the computation of Ext and Tor functors.
Ext Groups: Ext groups, denoted as $$\text{Ext}^n(A, B)$$, are mathematical constructions in homological algebra that measure the extent to which a module $A$ fails to be projective relative to another module $B$. They serve as a generalization of the concept of extensions and classify the equivalence classes of short exact sequences that can be formed with these modules. The presence of Ext groups provides insight into the structure and properties of modules, particularly in terms of their homomorphisms and extensions.
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.
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.
Homotopy category: The homotopy category is a mathematical framework that allows for the study of topological spaces and algebraic structures up to homotopy equivalence, providing a way to simplify and categorize complex constructions. It effectively captures the essential features of spaces and maps by identifying them when they can be continuously transformed into each other, thus emphasizing their topological properties rather than their precise form. This concept is particularly useful in various areas of mathematics, including the formulation of existence theorems and applications, as well as in the study of Hochschild and cyclic homology.
Inductive Limits: Inductive limits, also known as direct limits, refer to a way of constructing a new object from a directed system of objects and morphisms in category theory. This process allows us to combine a sequence of objects and maps into a single object that captures the essence of the entire system, making it essential for discussing properties and existence theorems in algebraic structures.
Projective limits: Projective limits are a way of defining an object in a category as a limit of a directed system, often used in the context of sequences or families of objects and morphisms. This concept captures the idea of taking an inverse limit across various stages or levels, essentially piecing together information from smaller or simpler objects to construct a more complex one. Projective limits have significant implications in various mathematical contexts, particularly in constructing objects that reflect the properties of their components.
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.
Tor Groups: Tor groups, often denoted as $$\text{Tor}$$, are derived functors that measure the extent to which a functor fails to be exact. They arise in homological algebra when considering the tensor product of modules, specifically identifying obstructions to exactness in the sequence of modules. The significance of Tor groups extends to various existence theorems, providing insight into the structure of modules and their relationships.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide