Homological algebra

Homological algebra is a branch of mathematics that studies algebraic structures and their relationships using concepts and methods from homology and cohomology. It originated from the study of algebraic topology but has since become a central area in various fields of mathematics, including algebra, geometry, and category theory.

Spectral sequences are a powerful mathematical tool used primarily in algebraic topology, homological algebra, and algebraic geometry. They provide a systematic method for computing homology groups, cohomology groups, or other related invariants of topological spaces or algebraic objects. ### Definition and Construction A spectral sequence consists of a sequence of pages (or terms), each represented as a collection of abelian groups or modules, along with differentials that relate these groups across the pages.

The Bockstein spectral sequence is a mathematical tool in the field of homological algebra and algebraic topology, particularly in the study of spectral sequences. It arises in the context of computing homology and cohomology groups with coefficients in a group or ring, especially when the coefficients can be viewed as a module over a more complex ring.

The Chromatic Spectral Sequence is a tool in stable homotopy theory, particularly in the study of stable homotopy groups of spheres and related objects. It is mainly concerned with the chromatic filtration, which categorizes stable homotopy groups based on their interactions with complex oriented theories, such as complex cobordism and various versions of K-theory.

The EHP spectral sequence is a tool in homotopy theory and stable homotopy theory, particularly involving the study of the stable homotopy groups of spheres. It is named after the mathematicians Eilenberg, Henriques, and Priddy—hence EHP. The EHP spectral sequence arises from the framework of stable homotopy types and is associated with the "suspension" of spaces and the mapping spaces between them.

The term "Exact couple" can refer to different concepts depending on the context. 1. **Mathematics**: In mathematics, particularly in the field of algebra and topology, an "exact couple" refers to a specific type of diagram used in homological algebra that combines two different chain complexes. Exact couples are used to construct spectral sequences, which are tools that allow mathematicians to compute homology groups by breaking complex problems into simpler parts.

The Five-Term Exact Sequence is a concept in algebraic topology and homological algebra, particularly in the context of derived functors and spectral sequences. It often arises in the study of homology and cohomology theories. In general, an exact sequence is a sequence of algebraic objects (like groups, modules, or vector spaces) linked by homomorphisms where the image of one homomorphism equals the kernel of the next.

The Frölicher spectral sequence is a tool in the field of differential geometry, particularly useful in the study of differentiable manifolds and their associated sheaf-theoretic or cohomological structures. It provides a way to compute the sheaf cohomology associated with the global sections of a sheaf of differential forms on a smooth manifold.

The Grothendieck spectral sequence is a powerful tool in algebraic geometry and homological algebra, providing a method for computing the derived functors of a functor that is defined in terms of a different functor. It is commonly used in the context of sheaf cohomology. The context in which the Grothendieck spectral sequence typically arises is in the cohomology of sheaves on a topological space (often a variety or scheme).

The May spectral sequence is a mathematical tool used in algebraic topology, particularly in the study of stable homotopy theory and the homotopy theory of spectra. Named after M. M. May, it is particularly useful for computing homotopy groups of spectra and understanding stable homotopy categories. The May spectral sequence arises in the context of a type of cohomology theory called stable cohomology.

A spectral sequence is a mathematical tool used in algebraic topology, homological algebra, and related fields to compute homology or cohomology groups that may be difficult to compute directly. It provides a method to systematically approximate these groups through a sequence of pages (typically indexed by integers) and associated differentials.

The Čech-to-derived functor spectral sequence is a tool in homological algebra and sheaf theory that relates Čech cohomology to derived functors, particularly sheaf cohomology. This spectral sequence emerges in contexts where one is interested in understanding the relationship between local properties, codified by Čech cohomology, and global properties captured by derived functors like the derived functors of sheaf cohomology. ### Overview of the Components Involved 1.

An acyclic model generally refers to a system or structure that does not contain cycles. In various contexts, this term can have different meanings, but it is commonly used in the fields of computer science, mathematics, and data structures. Here are a few specific contexts in which an acyclic model might be referenced: 1. **Graph Theory**: In graph theory, an acyclic graph is a graph that does not contain any cycles.

An acyclic object is typically a term used in the context of data structures, graphs, and programming. An acyclic structure does not contain cycles, meaning there are no paths that loop back to an earlier point in the structure. Here are some contexts where the term might be applied: 1. **Graphs**: An acyclic graph is a directed or undirected graph that has no cycles.

Banach algebra cohomology is a branch of functional analysis and abstract algebra that studies Banach algebras using the techniques of cohomology. It provides a way to investigate the structure of Banach algebras and their representations through the lens of cohomological methods, which originated in algebraic topology. ### Basic Concepts: 1. **Banach Algebras**: A Banach algebra \( A \) is a complete normed algebra over the field of complex or real numbers.

The Cartan–Eilenberg resolution is a method in homological algebra that provides a way to resolve certain algebraic structures (such as modules or complexes) using projective or injective resolutions. It is particularly useful in the context of derived functors and in studying the homological properties of chain complexes.

Chiral homology is a mathematical concept that arises in the field of homotopy theory, particularly in the study of algebraic topology and homological algebra. It is a special type of homology theory that aims to capture certain geometric and algebraic properties of topological spaces or algebraic structures that are sensitive to orientation or chirality (i.e., handedness).

Cohomology of algebras is a mathematical framework that extends the concepts of cohomology from topological spaces and differential geometry to algebraic structures, such as associative algebras, Lie algebras, and more generally, algebraic structures equipped with an action. At its core, cohomology assigns algebraic invariants (usually in the form of cohomology groups or cohomology spaces) to an algebraic object, which can provide insights into its structure and properties.

Cyclic homology is a concept in mathematics, specifically in the field of algebraic topology and homological algebra. It generalizes the idea of homology theories and is closely related to the study of algebraic structures known as "differential graded algebras" (DGAs). Cyclic homology was introduced by the mathematician Jean Leray in the context of algebraic topology and further developed by others, particularly by Alain Connes in the 1980s.

In the context of category theory, a **Delta-functor** (or simply a **delta functor**) typically refers to a specific type of functor that is associated with a structure of some kind that behaves like a "difference" operator, often in relation to algebraic constructs or topological spaces. The term is not universally standardized, and its exact meaning can vary based on the context or specific area of mathematics being discussed.

The Eilenberg–Ganea theorem is a fundamental result in algebraic topology, specifically in the theory of topological spaces and homotopy theory. Named after mathematicians Samuel Eilenberg and Tadeusz Ganea, the theorem concerns the relationship between the fundamental group of a space and its higher homotopy groups.

The Eilenberg–Zilber theorem is a result in algebraic topology and homological algebra that provides a way to compute the singular homology of a product of two topological spaces. Specifically, the theorem addresses the relationship between the singular chains on the product of two spaces and the singular chains on the spaces themselves.

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Factorization homology is a concept from the field of algebraic topology and homotopy theory, particularly in the context of homology theories that are associated with topological spaces and manifolds. It is a way of deriving a homology theory from the structure of a space by using "factorization" properties of differentials forms or more general coefficients.

Filtered algebra generally refers to structures in algebra where there is a filtration, which is a systematic way to impose a structure or hierarchy on the elements of the algebra. More formally, a filtered algebra is an algebra equipped with a filtration, which is an ascending chain of subalgebras indexed by a directed set. This can be useful in various branches of mathematics, particularly in homological algebra, algebraic topology, and representation theory. ### Definitions and Properties 1.

The Five Lemma is a result in the field of homological algebra, particularly in the context of derived categories and spectral sequences. It provides a criterion for when a five-term exact sequence of chain complexes splits. This lemma is commonly used in the study of abelian categories and the derived functor theory.

The gonality of an algebraic curve is a fundamental invariant that measures the complexity of the curve in terms of the degree of the simplest map to the projective line \(\mathbb{P}^1\).

Grothendieck's Tôhoku paper, formally titled "Éléments de géométrie algébrique I: La théorie des schémas," is a seminal work published in 1960 in the journal "Tohoku Mathematical Journal." This paper is one of the foundational texts in the field of algebraic geometry and represents a major step forward in the development of the theory of schemes.

Hochschild homology is an important concept in algebraic topology and homological algebra, often used to study algebraic structures, particularly associative algebras. It was introduced by Gerhard Hochschild in the 1940s. ### Definition and Construction Hochschild homology is typically defined for a unital associative algebra \( A \) over a field \( k \) (or more generally, over a commutative ring).

A Hodge structure is a concept in algebraic geometry and differential geometry that is used to study the relationships between algebraic and topological properties of complex manifolds. It provides a bridge between the geometric structure of a manifold and its algebraic properties. A Hodge structure on a vector space \( V \) over the complex numbers can be described as a decomposition of the space into subspaces that reflect the complex geometry of the underlying manifold.

Homological dimension is a concept from homological algebra that measures the "size" or "complexity" of an object in terms of its projective or injective resolutions. It provides a way to classify objects in terms of their relationships with projective and injective modules, often in the context of modules over rings or sheaves over topological spaces.

The Horseshoe Lemma is a result in topology and functional analysis, specifically in the context of the study of topological vector spaces. It is particularly important in the field of functional analysis, where it has applications in various areas including the theory of nonlinear operators and differential equations. The lemma generally states that under certain conditions, a continuous linear operator defined on a Banach space can be approximated by finite-dimensional spaces in a specific way.

Hyperhomology is a concept in algebraic topology and homological algebra that generalizes the notion of homology. It is typically used in the context of derived categories and can be thought of as a way to derive information from a more complicated algebraic structure, often involving sheaves or simplicial sets. In essence, hyperhomology provides a way to compute homological invariants associated with a diagram of objects in a category.

The Koszul complex is a construction in algebraic topology and commutative algebra that arises in the study of modules over a ring and their syzygies. It is particularly useful for understanding the structure of modules over a polynomial ring and in computing homological properties.

The Künneth theorem is an important result in algebraic topology that relates the homology groups of a product of two topological spaces to the homology groups of the individual spaces. It is particularly useful in the computation of homology groups for spaces that can be expressed as products of simpler spaces.

Homological algebra is a branch of mathematics that studies homology in a variety of contexts, primarily within the field of algebra. Its applications span across various areas of mathematics, including algebraic topology, algebraic geometry, and representation theory. Below is a list of some key topics and concepts within homological algebra: 1. **Chain Complexes**: Structures consisting of a sequence of abelian groups (or modules) connected by boundary maps.

In homological algebra, a mapping cone is a construction that allows us to define a new complex from a given morphism of chain complexes. It plays a significant role in various contexts, such as in the study of derived categories and in the formulation of the Long Exact Sequence in homology.

Matrix factorization in algebra refers to the process of decomposing a matrix into a product of two or more matrices. This can reveal underlying structures in the data represented by the original matrix, simplify computations, and enable various applications in fields such as statistics, machine learning, and computer graphics. ### Types of Matrix Factorization 1.

Mixed Hodge modules are a sophisticated mathematical concept that arises in the intersection of algebraic geometry, Hodge theory, and the theory of perverse sheaves. They were introduced by the mathematician Mitsuhiro Shintani in a 2000 paper and are part of the framework for understanding the properties of complex algebraic varieties and their singularities.

A Mixed Hodge structure is a mathematical concept in the field of algebraic geometry and Hodge theory, which provides a framework for understanding the topology of complex algebraic varieties and their connections with complex geometry. This notion generalizes the classical Hodge structure, which arises in the study of smooth projective varieties.

The Nine Lemma is a result in algebraic topology, specifically in the study of homotopy theory. It deals with examining the relationships between spaces that can be constructed from homotopy types and the existence of certain maps between them. The lemma provides a way to relate complications in a certain diagram of spaces and maps to giving conditions under which some homotopy properties hold true.

A perverse sheaf is a concept from algebraic geometry and sheaf theory, particularly in the context of the theory of derived categories and the study of singularities. It is a specific kind of sheaf that has been equipped with additional structure that allows for a refined understanding of the topology of spaces, particularly within the framework of non-abelian derived categories.

Secondary calculus and cohomological physics are advanced topics that emerge from the field of mathematics and theoretical physics. Here is an overview of each: ### Secondary Calculus Secondary calculus, also referred to in some literature as "secondary calculus of variations" or "higher-order calculus," is an extension of classical calculus that deals with variations of functionals, especially in the context of higher derivatives and secondary derivatives. In classical calculus of variations, one typically solves problems involving the optimization of functional (e.g.

Semiorthogonal decomposition is a concept in mathematics, particularly in the fields of functional analysis and category theory. It refers to a method of breaking down a complex structure into simpler components that satisfy certain orthogonality conditions. In a more specific context, particularly in algebraic geometry and derived categories, semiorthogonal decomposition allows the decomposition of a category—typically a derived category of coherent sheaves—into simpler subcategories that have a well-defined relationship with each other.

The Short Five Lemma is a tool in algebraic topology, specifically in the context of homological algebra and the theory of derived functors. It deals with the properties of cohomology in the setting of a commutative diagram of chain complexes and can be used to derive relationships between cohomology groups of various objects.

The Snake Lemma is a fundamental result in homological algebra, particularly in the study of abelian categories and exact sequences. It describes a way to construct a long exact sequence of homology groups from a commutative diagram involving two short exact sequences.

The Splitting Lemma is a foundational concept in the field of algebraic topology and homological algebra. It generally pertains to the behavior of certain sequences or diagrams in category theory, specifically focusing on the properties of morphisms (or maps) in a category.

Tate cohomology is a concept in algebraic topology and algebraic geometry that generalizes classical cohomology theories. It is particularly useful in the context of studying topological spaces that have group actions or in the context of number theory and arithmetic geometry.

A triangulated category is a particular type of category that arises in the context of homological algebra and derived categories. It provides a framework to study homological properties by relating them to geometric intuition through triangles, similar to how one uses exact sequences in abelian categories.

The Universal Coefficient Theorem is a fundamental result in algebraic topology, particularly in the context of homology and cohomology theories. It provides a way to relate the homology of a topological space to its homology with coefficients in an arbitrary abelian group.