is a powerful tool in noncommutative geometry, capturing key information about algebras and categories. It measures deformations, extensions, and structural properties, providing insights into the algebraic and geometric nature of noncommutative spaces.
The Hochschild forms the foundation for this theory. Through its cohomology groups, cup product, and Gerstenhaber bracket, it reveals a rich algebraic structure that connects to various areas of mathematics and physics.
Definition of Hochschild cohomology
Hochschild cohomology is a cohomology theory for associative algebras that captures important structural information about the algebra
It is a fundamental tool in noncommutative geometry, providing a way to study the and extensions of algebras
Hochschild cohomology for algebras
Top images from around the web for Hochschild cohomology for algebras
L∞-algebras and their cohomology | Emergent Scientist View original
Is this image relevant?
From the Potential to the First Hochschild Cohomology Group of a Cluster Tilted Algebra ... View original
Is this image relevant?
L∞-algebras and their cohomology | Emergent Scientist View original
Is this image relevant?
L∞-algebras and their cohomology | Emergent Scientist View original
Is this image relevant?
From the Potential to the First Hochschild Cohomology Group of a Cluster Tilted Algebra ... View original
Is this image relevant?
1 of 3
Top images from around the web for Hochschild cohomology for algebras
L∞-algebras and their cohomology | Emergent Scientist View original
Is this image relevant?
From the Potential to the First Hochschild Cohomology Group of a Cluster Tilted Algebra ... View original
Is this image relevant?
L∞-algebras and their cohomology | Emergent Scientist View original
Is this image relevant?
L∞-algebras and their cohomology | Emergent Scientist View original
Is this image relevant?
From the Potential to the First Hochschild Cohomology Group of a Cluster Tilted Algebra ... View original
Is this image relevant?
1 of 3
For an associative algebra A over a field k, the Hochschild cohomology HH∗(A) is defined as the cohomology of a certain cochain complex constructed from A
The n-th Hochschild cohomology group HHn(A) consists of the n-cocycles modulo the n-coboundaries in this cochain complex
Hochschild cohomology groups are vector spaces over k that encode information about the algebra A (derivations, extensions, deformations)
Hochschild cohomology for categories
Hochschild cohomology can also be defined for categories, generalizing the notion for algebras
For a small category C, the Hochschild cohomology HH∗(C) is defined using a cochain complex constructed from the morphisms in C
This cohomology theory captures information about the category C (natural transformations, extensions, deformations)
Hochschild cohomology for categories is a key tool in the study of noncommutative spaces and their deformation theory
Hochschild cochain complex
The Hochschild cohomology of an algebra A is defined as the cohomology of the Hochschild cochain complex C∗(A)
This cochain complex encodes the algebraic structure of A and is the main object of study in Hochschild cohomology
Construction of cochain complex
The n-th cochain group Cn(A) consists of k-linear maps A⊗n→A (n-cochains)
The coboundary maps δn:Cn(A)→Cn+1(A) are defined using the algebra structure of A
For f∈Cn(A) and a1,…,an+1∈A, δn(f)(a1,…,an+1) is given by a certain formula involving f and the multiplication in A
The Hochschild cochain complex is the sequence of cochain groups and coboundary maps: ⋯→Cn−1(A)δn−1Cn(A)δnCn+1(A)→⋯
Differential in cochain complex
The coboundary maps δn in the Hochschild cochain complex satisfy δn+1∘δn=0, making C∗(A) a cochain complex
The condition δn+1∘δn=0 ensures that the image of δn−1 (n-coboundaries) is contained in the kernel of δn (n-cocycles)
The Hochschild cohomology groups HHn(A) are defined as the quotient of the n-cocycles by the n-coboundaries: HHn(A)=ker(δn)/im(δn−1)
Cup product in Hochschild cohomology
The Hochschild cohomology HH∗(A) has a natural product structure called the cup product, making it a graded algebra
The cup product combines cochains to produce higher-degree cochains and induces a well-defined product on cohomology classes
Definition of cup product
For cochains f∈Cn(A) and g∈Cm(A), their cup product f∪g∈Cn+m(A) is defined by a certain formula involving the multiplication in A
(f∪g)(a1,…,an+m)=f(a1,…,an)⋅g(an+1,…,an+m)
The cup product is compatible with the coboundary maps, meaning δ(f∪g)=δ(f)∪g+(−1)nf∪δ(g)
This compatibility ensures that the cup product descends to a well-defined product on Hochschild cohomology: HHn(A)⊗HHm(A)→HHn+m(A)
Associativity of cup product
The cup product on Hochschild cohomology is associative: (f∪g)∪h=f∪(g∪h) for cochains f,g,h
This associativity follows from the associativity of the multiplication in the algebra A
As a result, the Hochschild cohomology HH∗(A) is a graded associative algebra under the cup product
The associativity of the cup product is a crucial property in understanding the algebraic structure of Hochschild cohomology
Gerstenhaber algebra structure
In addition to the cup product, the Hochschild cohomology HH∗(A) has a graded Lie bracket called the Gerstenhaber bracket
Together, the cup product and Gerstenhaber bracket make HH∗(A) a Gerstenhaber algebra, a rich algebraic structure central to deformation theory
Gerstenhaber bracket on Hochschild cohomology
The Gerstenhaber bracket is a bilinear map [−,−]:HHn(A)⊗HHm(A)→HHn+m−1(A) defined on cohomology classes
For cochains f∈Cn(A) and g∈Cm(A), their Gerstenhaber bracket [f,g] is defined by a certain formula involving the composition of cochains
The Gerstenhaber bracket satisfies graded antisymmetry and the graded Jacobi identity, making HH∗(A) a graded Lie algebra
Compatibility of cup product and bracket
The cup product and Gerstenhaber bracket on Hochschild cohomology are compatible, satisfying the graded Poisson identity:
[f∪g,h]=[f,h]∪g+(−1)(n−1)mf∪[g,h] for f∈HHn(A),g∈HHm(A),h∈HHk(A)
This compatibility makes HH∗(A) a Gerstenhaber algebra, a graded version of a Poisson algebra
The Gerstenhaber algebra structure on Hochschild cohomology plays a fundamental role in the deformation theory of algebras and categories
Hochschild cohomology vs cyclic homology
Hochschild cohomology is closely related to another important invariant called
While Hochschild cohomology captures deformations and extensions of algebras, cyclic homology is more closely tied to the trace and noncommutative differential forms
Connes' periodicity exact sequence
There is a long exact sequence, called Connes' periodicity exact sequence, relating Hochschild and cyclic homology:
⋯→HHn(A)IHCn(A)SHCn−2(A)BHHn+1(A)→⋯
The maps I,S,B in the sequence are certain natural transformations between the Hochschild and cyclic complexes
This exact sequence provides a powerful tool for computing cyclic homology groups in terms of Hochschild cohomology groups
Cyclic vs Hochschild cohomology groups
The cyclic homology groups HCn(A) are often easier to compute than Hochschild cohomology groups due to their periodic nature
In many cases, the cyclic homology groups exhibit a periodicity (e.g., HCn(A)≅HCn+2(A)), while Hochschild cohomology groups can be more varied
Studying the relationship between Hochschild and cyclic homology via Connes' periodicity sequence is a key aspect of noncommutative geometry
Both theories provide complementary perspectives on the structure and geometry of noncommutative spaces
Applications of Hochschild cohomology
Hochschild cohomology has numerous applications in various areas of mathematics, including algebra, geometry, and mathematical physics
It serves as a fundamental tool for studying deformations, extensions, and the structure of algebras and categories
Deformation theory of algebras
The Hochschild cohomology groups HH2(A) and HH3(A) play a central role in the deformation theory of algebras
Elements of HH2(A) correspond to first-order deformations of the algebra A, while elements of HH3(A) control the obstructions to extending these deformations
Studying the Hochschild cohomology of an algebra provides insight into its rigidity and the space of its possible deformations
Classifying extensions of algebras
Hochschild cohomology can be used to classify extensions of algebras
An extension of an algebra A by an A- M is determined by a Hochschild 2-cocycle in HH2(A,M)
The group HH2(A,M) classifies the equivalence classes of extensions of A by M, with the trivial extension corresponding to the zero element
Hochschild cohomology in algebraic geometry
Hochschild cohomology appears naturally in algebraic geometry, particularly in the study of coherent sheaves and their deformations
For a smooth algebraic variety X, the Hochschild cohomology of the structure sheaf OX is related to the tangent and cotangent bundles of X
Hochschild cohomology provides a way to study infinitesimal deformations of algebraic varieties and their categories of coherent sheaves
It is a key ingredient in the formulation of deformation quantization and the study of noncommutative algebraic geometry
Computations of Hochschild cohomology
Computing Hochschild cohomology groups can be a challenging task, but several important results and techniques are available
In some cases, Hochschild cohomology groups can be computed explicitly, while in others, they can be related to more familiar invariants
Hochschild-Kostant-Rosenberg theorem
The Hochschild-Kostant-Rosenberg (HKR) theorem is a fundamental result that computes the Hochschild cohomology of a smooth commutative algebra
For a smooth commutative algebra A over a field of characteristic zero, the HKR theorem states that HHn(A)≅ΩA/kn, where ΩA/kn is the module of Kähler differentials
This theorem establishes a deep connection between Hochschild cohomology and differential forms in the commutative setting
Hochschild cohomology of group algebras
For a group algebra k[G] of a finite group G over a field k, the Hochschild cohomology can be computed using group cohomology
There is an isomorphism HHn(k[G])≅Hn(G,k[G]), where Hn(G,k[G]) is the group cohomology of G with coefficients in the group algebra k[G]
This result allows for the computation of Hochschild cohomology of group algebras using the tools and techniques of group cohomology
Hochschild cohomology of quiver algebras
Quiver algebras, which arise from directed graphs (quivers), provide an important class of algebras with rich Hochschild cohomology
For a quiver algebra A, the Hochschild cohomology can often be computed using the combinatorics of the underlying quiver
In some cases, the Hochschild cohomology of a quiver algebra can be described in terms of the path algebra and the ideal of relations defining the algebra
Techniques from and homological algebra are often employed in the study of Hochschild cohomology of quiver algebras
Key Terms to Review (18)
Adams Spectral Sequence: The Adams spectral sequence is a powerful tool in homological algebra and algebraic topology that allows for the computation of stable homotopy groups of spheres. It connects the algebraic invariants of a space, like cohomology, to its topological features, facilitating a deeper understanding of the relationships between different spaces. By using this sequence, mathematicians can systematically derive information about these groups through a series of approximations and filtrations.
B. m. keller: B. M. Keller is a mathematician known for his contributions to noncommutative geometry and Hochschild cohomology. His work often focuses on the relationship between algebraic structures and topological spaces, shedding light on how these areas intersect within the broader context of algebraic geometry and operator algebras.
Bimodule: A bimodule is a mathematical structure that serves as a module for two different rings simultaneously, allowing for interaction between them. This concept is crucial in noncommutative algebra, particularly as it facilitates the study of representations and dualities of algebraic structures. Bimodules provide a way to connect different algebraic systems and enable the exploration of their properties in a unified manner.
Chain Complex: A chain complex is a sequence of abelian groups or modules connected by boundary homomorphisms, where the composition of two consecutive maps is zero. This structure captures the idea of 'chains' of elements that can be combined and manipulated, serving as a foundational tool in algebraic topology and homological algebra. In the context of Hochschild cohomology, chain complexes are used to study the properties of algebras and their modules, allowing for computations that reveal deeper insights about the algebraic structures involved.
Cochain Complex: A cochain complex is a sequence of abelian groups or modules connected by homomorphisms that capture the algebraic structure of topological spaces. This concept is fundamental in homological algebra and serves as a backbone for understanding various cohomology theories, including Hochschild cohomology, which specifically deals with algebras and their modules.
Cyclic homology: Cyclic homology is a branch of algebraic topology and homological algebra that extends classical homology theories to incorporate a cyclic symmetry. It provides tools for studying algebras, particularly noncommutative algebras, by capturing their structure in a way that reflects both algebraic and geometric properties. This concept is deeply linked to higher K-theory, cohomological theories, and the study of invariants associated with algebraic structures.
Deformation Theory: Deformation theory is a mathematical framework that studies the ways in which a given geometric object can be 'deformed' into another, allowing for an understanding of the structure and properties of the object under continuous transformations. It connects closely to various areas, such as algebraic geometry and topology, providing insights into how structures can change while preserving certain features. This concept is crucial for analyzing the stability of structures, exploring representations, and understanding cohomological dimensions in various contexts.
Derived Category: A derived category is a mathematical construction that provides a framework to study complexes of objects, particularly in the context of homological algebra. It allows for the manipulation and categorization of objects and morphisms while considering their relationships through homology. This approach captures more information than traditional categories, making it essential in areas such as algebraic geometry, representation theory, and particularly Hochschild cohomology.
Gerhard Hochschild: Gerhard Hochschild was a prominent mathematician known for his work in algebra, particularly in the areas of representation theory and cohomology. His contributions to Hochschild cohomology have had a significant impact on the study of algebraic structures, influencing various fields including noncommutative geometry and category theory.
Hochschild Cohomology: Hochschild cohomology is a mathematical framework that studies the properties of algebraic structures, particularly associative algebras, by examining their derived functors. It plays a crucial role in understanding the structure of algebras and their modules, providing insights into deformation theory and representation theory. This concept connects to cyclic cohomology through its formulation and applications in noncommutative geometry, as both theories investigate similar algebraic phenomena but from different perspectives.
Hochschild Homology: Hochschild homology is an important concept in algebraic topology and noncommutative geometry that measures the homological properties of algebras and modules. It provides a way to study the structure of algebras through chains of their bimodules and has deep connections to Hochschild cohomology, which focuses on cohomological aspects. This duality between homology and cohomology forms a critical part of understanding the algebraic invariants in various mathematical contexts.
Hochschild–Kostant–Rosenberg Theorem: The Hochschild–Kostant–Rosenberg Theorem is a fundamental result in algebraic geometry and noncommutative geometry that establishes an isomorphism between Hochschild cohomology and certain algebraic structures associated with differential forms. This theorem bridges the gap between algebraic and geometric interpretations by linking the cohomological properties of algebras to the geometry of their associated schemes, thereby enhancing our understanding of deformation theory and representation theory.
Hochschild–Serre spectral sequence: The Hochschild–Serre spectral sequence is a powerful tool in algebraic topology and homological algebra that provides a way to compute the Hochschild cohomology of a module or an algebraic structure through a filtration process. It connects the cohomology of groups with that of their subgroups, revealing deep relationships between their structures and allowing for the transfer of information across different levels.
K-theory: K-theory is a branch of mathematics that studies vector bundles and their generalizations through the use of algebraic topology and homological algebra. It provides a framework for understanding the structure of these bundles, allowing for the classification of topological spaces and algebras, which has deep implications in various mathematical fields, including geometry and number theory.
Localization: Localization is a mathematical process that focuses on studying objects in a neighborhood around a point or in relation to a specific subset. This approach allows for the examination of properties and structures that may behave differently in local contexts compared to global ones. In various branches of mathematics, particularly in K-theory and Hochschild cohomology, localization plays a vital role in understanding how certain structures can be simplified or understood more deeply by focusing on these specific regions.
Noncommutative algebra: Noncommutative algebra is a branch of mathematics that studies algebras where the multiplication operation does not necessarily satisfy the commutative property, meaning that for some elements $a$ and $b$, it holds that $ab \neq ba$. This area includes various structures like matrix algebras and operator algebras, which are crucial for understanding complex mathematical frameworks, including those used in quantum mechanics and functional analysis. Noncommutative algebra is also integral to the development of theories such as noncommutative geometry, which broadens the scope of traditional geometry by incorporating these algebraic structures.
Representation Theory: Representation theory is the study of how algebraic structures, such as groups and algebras, can be realized as linear transformations of vector spaces. This branch of mathematics connects abstract algebra with linear algebra and has significant applications in various areas, including physics and geometry.
Spectral Sequence: A spectral sequence is a mathematical tool that helps in computing homology and cohomology groups through a filtration process. It organizes data in a way that allows for systematic calculations, revealing deeper structures in algebraic topology and related fields. The construction involves a sequence of pages, each consisting of a collection of abelian groups and differential maps connecting them, which can ultimately converge to the desired cohomological information.