Cohomology theories

Cohomology theories are mathematical frameworks used in algebraic topology, geometry, and related fields to study topological spaces and their properties. They serve as tools for assigning algebraic invariants to topological spaces, allowing for deeper insights into their structure. Cohomology theories capture essential features such as connectivity, holes, and other topological characteristics. ### Key Concepts in Cohomology Theories 1.

Alexander–Spanier cohomology is a cohomology theory used in algebraic topology that serves to study topological spaces. It extends the notion of singular cohomology, providing a way to compute topological invariants of spaces whether or not they are nice enough to have a smooth structure. It was introduced by John W. Alexander and Paul Spanier. ### Definition and Basic Ideas 1.

André–Quillen cohomology is a concept in algebraic geometry and homological algebra that provides a way to study deformations of algebraic structures, particularly in the context of algebraic varieties and schemes. It was introduced by the mathematicians Michèle André and Daniel Quillen in the context of their work on deformation theory.

BRST quantization is a formalism used in the field of quantum field theory to handle systems with gauge symmetries. It is named after the physicists Bonora, Reisz, Sirlin, and Tyutin, who contributed to its development. BRST stands for Becchi-Rouet-Stora-Tyutin, referring to the key researchers who formulated the method. The motivation for BRST quantization arises from the challenges associated with quantizing gauge theories.

Bivariant theory is a concept in algebraic topology and homotopy theory that studies the relationships between different homological or homotopical invariants using a bivariant framework. It essentially generalizes classical invariant theory (like cohomology and homology) to consider pairs of spaces or pairs of morphisms, allowing for a more nuanced and flexible understanding of how different spaces can interact.

Bredon cohomology is a type of cohomology theory that is particularly useful in the context of spaces with group actions. It was introduced by Glen Bredon in the 1960s and is designed to study topological spaces with an additional structure of a group action, often leading to insights in equivariant topology.

Brown–Peterson cohomology is a homology theory in algebraic topology that is particularly focused on stable homotopy and complex cobordism. Introduced by Ronald Brown and F. P. Peterson in the context of stable homotopy theory, it serves as a tool for studying the cohomological properties of topological spaces, especially with respect to the stable homotopy category.

In the context of mathematics, particularly in the study of Lie groups and Lie algebras, a **Cartan pair** refers to a specific structure that arises in the theory of semisimple Lie algebras.

Chromatic homotopy theory is a branch of algebraic topology that studies stable homotopy groups of spheres and related phenomena through the lens of chromatic filtration. It originated from attempts to better understand the relationship between stable homotopy theory and complex-oriented cohomology theories, particularly in the context of the stable homotopy category.

Coherent sheaf cohomology is a concept in algebraic geometry and sheaf theory, dealing with the study of coherent sheaves on algebraic varieties. Coherent sheaves are a generalization of vector bundles and are important because they allow for the treatment of sections and their relationships in a more general setting.

Cohomology is a fundamental concept in algebraic topology and other fields of mathematics that studies the properties of spaces through algebraic invariants. It provides a way to associate a sequence of abelian groups or vector spaces to a topological space, which can help in understanding its structure and features.

Cohomology of a stack is a concept that extends the idea of cohomology from algebraic topology and algebraic geometry to the realm of stacks, which are sophisticated objects that generalize schemes and sheaves. Stacks allow one to systematically handle problems involving moduli spaces, particularly when there are nontrivial automorphisms or when the objects involved have "geometric" or "categorical" structures.

Cohomology with compact support is a concept in algebraic topology and differential geometry that generalizes the notion of cohomology by focusing on those cochains that vanish outside of compact sets. This has important implications for the study of properties of spaces when dealing with functions or forms that are localized in compact subsets.

Crystalline cohomology is a cohomology theory in algebraic geometry and arithmetic geometry that is particularly useful for studying schemes over fields of characteristic \( p \). Developed primarily by Pierre Deligne in the 1970s, it is related to several important concepts in both algebraic geometry and number theory.

De Rham cohomology is a mathematical concept from the field of differential geometry and algebraic topology that studies the topology of smooth manifolds using differential forms. It provides a bridge between analysis and topology by utilizing the properties of differential forms and their relationships through the exterior derivative. ### Key Concepts 1. **Differentiable Manifolds**: A differentiable manifold is a topological space that is locally similar to Euclidean space and has a well-defined notion of differentiability.

Deligne cohomology is a cohomology theory that generalizes the classical notions of singular cohomology by incorporating additional structures, specifically those related to sheaf theory and algebraic geometry. It was introduced by Pierre Deligne in the context of his work on the Weil conjectures and arithmetic geometry.

Dolbeault cohomology is a mathematical concept that arises in the field of complex differential geometry and algebraic geometry. It provides a way to study the properties of complex manifolds by using differential forms. In essence, Dolbeault cohomology is a specific kind of cohomology theory that is particularly suited to complex manifolds. While ordinary cohomology deals with real-valued differential forms, Dolbeault cohomology focuses specifically on complex-valued differential forms.

Elliptic cohomology is a branch of algebraic topology that generalizes classical cohomology theories using the framework of elliptic curves and modular forms. It is an advanced topic that blends ideas from algebraic geometry, number theory, and homotopy theory. ### Key Features 1.

The term "Factor system" can refer to various concepts depending on the context, including mathematics, economics, and systems theory. Here are a few interpretations: 1. **Mathematics**: In mathematics, a factor system typically refers to a collection of factors that can be used to break down numbers or algebraic expressions into their constituent parts. For example, in number theory, factorization involves expressing a number as a product of its prime numbers.

Galois cohomology is a branch of mathematics that studies objects known as "cohomology groups" in the context of Galois theory, which is a part of algebra concerned with the symmetries of polynomial equations. To understand Galois cohomology, we start with a few key ideas: 1. **Galois Groups**: A Galois group is a group associated with a field extension, representing the symmetries of the roots of polynomials.

Gelfand–Fuks cohomology is a concept in the field of mathematics that arises from the study of infinite-dimensional Lie algebras and their representations. It provides a powerful tool for analyzing and understanding the structure of these algebras, particularly in the context of the theory of differential operators and the geometry of manifolds. The cohomology theory was developed by Israel Gelfand and Sergei Fuks in the 1960s.

Group cohomology is a mathematical tool used in algebraic topology, group theory, and various other areas of mathematics. It provides a way to study the properties of groups using cohomological methods, which are analogous to those used in homology theory but focus on the algebraic structure associated with groups.

The Hodge–de Rham spectral sequence is a mathematical tool used in algebraic topology and differential geometry, specifically in the context of studying the relationships between differential forms on a smooth manifold and the topology of that manifold. This spectral sequence arises from the filtration provided by the Hodge decomposition theorem in conjunction with the de Rham complex of differential forms. ### Overview 1.

Infinitesimal cohomology is a concept from the field of algebraic geometry and is particularly associated with the study of formal schemes and deformation theory. It provides a way to study the local behavior of schemes using a "cohomological" approach that incorporates infinitesimal neighborhoods. In more detailed terms, infinitesimal cohomology typically arises in contexts involving the study of deformations of algebraic objects.

Koszul cohomology is a concept from algebraic topology and homological algebra that arises in the context of differential graded algebras and the study of the algebraic invariants associated with topological spaces or algebraic varieties. It is named after Jean-Pierre Serre and Jean Koszul, who developed the foundational ideas related to this cohomology theory.

Kähler differentials are a concept from algebraic geometry and commutative algebra. They arise in the context of the study of a ring \( R \) and its associated differentials with respect to a base field or a base ring. Specifically, Kähler differentials provide a way to study the infinitesimal behavior of functions and their properties on schemes.

Lie algebra cohomology is a mathematical concept that arises in the study of Lie algebras, which are algebraic structures used extensively in mathematics and physics to describe symmetries and conservation laws. Cohomology, in this context, refers to a homological algebra framework that helps in analyzing the structure and properties of Lie algebras.

Cohomology theories are mathematical frameworks used in algebraic topology, algebraic geometry, and other areas to study the properties of topological spaces and algebraic structures. Here’s a list of notable cohomology theories, each with unique properties and applications: 1. **Singular Cohomology**: The most fundamental cohomology theory for topological spaces, using singular simplices. It is defined for any topological space and provides multiplicative structures.

Local cohomology is a concept in algebraic geometry and commutative algebra that extends the notion of ordinary cohomology to study the local behavior of a module over a ring, particularly with respect to a specified ideal. It is particularly useful for understanding the properties of sheaves and modules around points in a space or in relation to certain subvarieties.

Monsky–Washnitzer cohomology is a type of cohomology theory developed in the context of the study of schemes, particularly over fields of positive characteristic. It is named after mathematicians Paul Monsky and Michiel Washnitzer, who introduced the concept in 1970s. This cohomology theory is specifically designed to work with algebraic varieties defined over fields of characteristic \( p > 0 \) and offers a way to analyze their geometric and topological properties.

Motivic cohomology is a concept in algebraic geometry and topology that generalizes classical cohomology theories to the framework of algebraic varieties. It is particularly influential in the study of algebraic cycles, motives, and the relationship between algebraic geometry and topology. ### Background Motivic cohomology was introduced in the context of the theory of motives, which aims to unify various cohomological approaches to algebraic varieties.

Nonabelian cohomology is a branch of mathematics that studies the cohomological properties of nonabelian structures, particularly in the context of group theory and algebraic geometry. It generalizes classical cohomology theories to contexts where the groups involved do not necessarily obey the commutative property, hence the term "nonabelian.

P-adic cohomology is a branch of mathematics that studies the properties of algebraic varieties and schemes over p-adic fields using cohomological methods. It is particularly important in number theory, algebraic geometry, and arithmetic geometry, as it provides tools to understand the relationships between algebraic structures and their properties over p-adic numbers.

In the context of cohomology, a pullback is a construction that allows you to take a cohomology class on a target space and "pull it back" to a cohomology class on a domain space via a continuous map. This is particularly common in algebraic topology and differential geometry. ### Formal Definition Let \( f: X \to Y \) be a continuous map between two topological spaces \( X \) and \( Y \).

Quantum cohomology is a branch of mathematics that combines concepts from algebraic geometry, symplectic geometry, and quantum physics. It arises in the study of certain moduli spaces and has applications in various fields, including string theory, mathematical physics, and enumerative geometry. At a high level, quantum cohomology seeks to extend classical cohomology theories, particularly for projective varieties, to incorporate quantum effects, which can be thought of as counting curves under certain conditions.

Sheaf cohomology is a fundamental concept in algebraic geometry and topology that provides a way to study the properties of sheaves on topological spaces or schemes. It serves as a powerful tool for capturing global sections of sheaves and understanding their finer structures. ### Key Concepts 1.

Spencer cohomology is a mathematical framework used in the study of differential operators and the cohomology of various algebraic and geometric structures. It is a cohomology theory primarily associated with the analysis of differential equations, particularly in the context of differential forms and sheaf theory on smooth manifolds.

Weil cohomology theory is a set of tools and concepts in algebraic geometry and number theory developed by André Weil to study the properties of algebraic varieties over fields, particularly over finite fields and more generally over local fields. It was introduced as a way to provide a cohomology theory that would capture essential topological and algebraic features of varieties and is particularly characterized by its application to counting points on varieties over finite fields.

Witt vector cohomology is a tool in algebraic geometry and number theory that utilizes Witt vectors to study the cohomological properties of schemes in the context of p-adic cohomology theories. Witt vectors are a generalization of the notion of numbers in a ring, particularly for fields of characteristic \( p \), and they allow the construction of an effective cohomology theory that preserves useful algebraic properties. ### Key Concepts 1.

Étale cohomology is a cohomological theory in algebraic geometry that provides a means to study the properties of algebraic varieties over fields, particularly in the context of fields that are not algebraically closed. It was developed in the mid-20th century, notably by Alexander Grothendieck, and is part of the broader framework of schemes in modern algebraic geometry.

Čech cohomology is a mathematical tool used in algebraic topology to study the properties of topological spaces. Named after the Czech mathematician Eduard Čech, this cohomology theory is particularly useful for analyzing spaces that may not be well-behaved in a classical sense.