Dihedral And Reflexive Homology Theory Of Polynomial Algebra

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2014

In the world of chain complexes En-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic En-homology of an En-algebra computes the homology of an n-fold algebraic delooping. The aim of this paper is to construct two spectral sequences for calculating these homology groups and to treat some concrete classes of examples such as Hochschild cochains, graded polynomial algebras and chains on iterated loop spaces. In characteristic zero we gain an identification of the summands in Pirashvili's Hodge decomposition of higher order Hochschild homology in terms of derived functors of indecomposables of Gerstenhaber algebras and as the homology of exterior and symmetric powers of derived Kähler differentials.

Information Sciences Letters, 2022

In this research, the basic definitions of an operad, graded algebra, and A ∞-module are introduced. The A ∞-algebras and their (co)homology are studied to obtain the relations between the cyclic and dihedral (co)homology. The L ∞-algebras are discussed, and the relations of the isomorphism between primitive and indecomposable elements in the L ∞-algebras are presented. We demonstrate the relation between cyclic and dihedral (co)homology of L ∞-algebras. Finally, the Mayer-Vietoris sequence of L ∞-algebras is investigated.

Journal of Pure and Applied Algebra, 1998

In this paper we study spectral sequences converging to the Hochschild and cyclic (co) homologies of a tilly group-graded algebra, and their invariance under Morita and derived equivalences induced by graded bimodules.

Scientific African, 2021

In this research, the basic definitions of an operad, graded algebra, and A ∞-module are introduced. The A ∞-algebras and their (co)homology are studied to obtain the relations between the cyclic and dihedral (co)homology. The L ∞-algebras are discussed, and the relations of the isomorphism between primitive and indecomposable elements in the L ∞-algebras are presented. We demonstrate the relation between cyclic and dihedral (co)homology of L ∞-algebras. Finally, the Mayer-Vietoris sequence of L ∞-algebras is investigated.

2016

In the world of chain complexes En-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic En-homology of an En-algebra computes the homology of an n-fold algebraic delooping. The aim of this paper is to construct two spectral sequences for calculating these homology groups and to treat some concrete classes of examples such as Hochschild cochains, graded polynomial algebras and chains on iterated loop spaces. In characteristic zero we gain an identification of the summands in Pirashvili's Hodge decomposition of higher order Hochschild homology in terms of derived functors of indecomposables of Gerstenhaber algebras and as the homology of exterior and symmetric powers of derived Kähler differentials.

Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using derived functors and the symmetric bar construction of Fiedorowicz. The symmetric homology of group rings is related to stable homotopy theory. Two chain complexes are constructed that compute symmetric homology, as well as two spectral sequences. In the setup of the second spectral sequence, a complex isomorphic to the suspension of the cycle-free chessboard complex of Vrecica and Zivaljevic appears. Homology operations are defined on the symmetric homology groups over Z/p, p a prime. Finally, an explicit partial resolution is presented, permitting the computation of the zeroth and first symmetric homology groups of finite-dimensional algebras.

Transactions of The American Mathematical Society - TRANS AMER MATH SOC, 1986

We present a new free resolution for k as an G-module, where G is an associative augmented algebra over a field k. The resolution reflects the combinatorial properties of G. Introduction. Let k be a field and let G be an associative augmented fc-algebra. For many purposes one wishes to have a projective resolution of k as a G-module. The bar resolution is always easy to define, but it is often too large to use in practice. At the other extreme, minimal resolutions may exist, but they are often hard to write down in a way that is amenable to calculations. The main theorem of this paper presents a compromise resolution. Though rarely minimal, it is small enough to offer some bounds but explicit enough to facilitate calculations. As it relies heavily upon combinatorial constructions, it is best suited for analyzing otherwise tricky algebras given via generators and relations. Since several results we get as consequences of the main theorem have been obtained before through other means, this paper may be viewed as generalizing and unifying several seemingly unrelated ideas. In particular, we are generalizing Priddy's results on Koszul algebras , extending homology computations by Govorov [9] and Backelin [3], and complementing Bergman's methods regarding the diamond lemma . Three results may be of interest. The homology of the modp Steenrod algebra is given in terms of the homology of a new chain complex smaller than the A-algebra in Theorem 3.5. Formula ( ) offers an efficient algorithm for the determination of Hubert series, and Theorem 4.2 asserts the existence of new bounds on the torsion groups of commutative graded rings. and the main theorem. Throughout this paper, k denotes any field and G is an associative fc-algebra with unity. The field k embeds in G via n: k ^-> G and we suppose that G has an augmentation, i.e., a fc-algebra map £ : G -> fc for which n is a right inverse. S denotes a set of generators for G as a fc-algebra and k(S) is the free associative fc-algebra with unity on S. There is a canonical surjection /: k(S) -> G once S is chosen, and the augmentation e is determined once we know e(x) for each x G S. In particular, this means that k{S) may be augmented such that / becomes a map of augmented algebras. To S we associate a function e: S -> Z+ called a grading. In the absence of a more compelling choice we often take e to be grading by length, i.e., e(x) = 1 for

2002

Using the long exact sequence of nonabelian derived functors, an eight term exact sequence of Lie algebra homology with Λ/qΛ coefficients is obtained, where Λ is a ground ring and q is a nonnegative integer. Hopf formulas for the second and third homology of a Lie algebra are proved. The condition for the existence and the description of the universal q-central relative extension of a Lie epimorphism in terms of relative homologies are given.

Journal of Mathematical Sciences, 2007

In the survey, we deal with the following situation. Let A be a graded algebra or a differential graded algebra. Adjoining a set x of free (in any sense) indeterminates, we make a new differential graded algebra A x by setting the differential values d : x → A on x. In the general case, such a construction is called the Shafarevich complex. Beginning with classical examples like the bar-complex, Koszul complex, and Tate resolution, we discuss noncommutative (and sometimes even nonassociative) versions of these notions. The comparison with the Koszul complex leads to noncommutative regular sequences and complete intersections; Tate's process of killing cycles gives noncommutative DG resolutions and minimal models. The applications include the Golod-Shafarevich theorem, growth measures for graded algebras, characterizations of algebras of low homological dimension, and a homological description of Gröbner bases. The same constructions for categories of algebras with identities (like Lie or Jordan algebras) allow one to give a homological description of extensions and deformations of PI-algebras.

Mathematics and Statistics, 2023

A scheme is a type of mathematical construction that extends the concept of algebraic variety in a number of ways, including accounting for multiplicities and being defined over any commutative ring. In this article, we study some properties of the cyclic and dihedral homology theory in schemes. We study the long exact sequence of cyclic homology of scheme and prove some results. So, we introduce and study Morita-equivalence in cyclic homology of schemes and proof the main relation between trace map and inclusion map. Our goal is to explain product structures on cyclic homology groups ℍ𝒞∗(𝒳/𝒮). Especially, we show ℍ𝒞∗(𝒳/𝒮)=⨂ 𝑛∈ℤℍ𝒞𝑛(𝒳/𝒮) of algebra. We give the relationsbetween dihedral homology (ℍ𝒟(𝒰)) and cyclic homology(ℍ𝒞(𝒰)) of schemes, therefore: ℍ𝒞𝑛(𝒰)=−ℍ𝒟𝑛+1(𝒰)⨁ +ℍ𝒟𝑛(𝒰). We explain the tracemap andinclusion map of cyclic homology for scheme algebra which takes form: 𝑖𝑛𝑐:ℍ𝒞𝑛(𝒰,𝒱)→ℍ𝒞𝑛(𝑀𝑚(𝒰),𝑀𝑚(𝒱)) and 𝑡𝓇:ℍ𝒞𝑛(𝑀𝑚(𝒰),𝑀𝑚(𝒱))→ℍ𝒞𝑛( 𝒰,𝒱 ). For the shuffle map 𝑠ℎ:𝒰⨂𝒱→𝒰×𝒱, we obtain the long exact sequence of cyclic homology for scheme: ⋯→ℍ𝒞𝑛(𝒰×𝒱)𝑖→⨁𝑟+𝑠=𝑛ℍ𝒞𝑟(𝒰)⨂ℍ𝒞𝑠 𝐵(𝒱)𝑆⨂−1−⨂𝑆→ ⨁𝑝+𝑞=𝑛−2ℍ𝒞𝑝(𝒰)⨂ℍ𝒞𝑞(𝒱)𝜕→ℍ𝒞𝑛−1(𝒰×𝒱)→⋯. We give the long exact sequence of dihedral homology for scheme: ⋯→ℍ𝒞𝑛(𝒰×𝒱)𝑖→⨁𝑟+𝑠=𝑛ℍ𝒞𝑟(𝒰)⨂ℍ𝒞𝑠 𝐵(𝒱)𝑆⨁−1−⨂𝑆→ ⨁𝑝+𝑞=𝑛−2ℍ𝒞𝑝(𝒰)⨂ℍ𝒞𝑞(𝒱)𝜕→ℍ𝒞𝑛−1(𝒰×𝒱)→⋯ . For any three 𝒰,𝒱 and 𝒲 algebra, we write the next long exact sequence as a commutative diagram: ⋯→ℍ𝒞𝑛(𝒰∗)𝑓∗→ℍ𝒞𝑛(𝒱∗)𝑔∗→ℍ𝒞𝑛(𝒲∗)𝛿→ℍ𝒞𝑛−1(𝒰∗)𝑓∗→ℍ𝒞𝑛−1(𝒱∗)𝑔∗→ℍ𝒞𝑛−1(𝒲∗)→⋯. For all 𝒰,𝒱 and 𝒲 schemes, we give the long exact sequence of dihedral homology as: ⋯→ℍ𝒟𝑛(𝒰∗)𝑓∗→ℍ𝒟𝑛(𝒱∗)𝑔∗→ℍ𝒟𝑛(𝒲∗)𝛿→ℍ𝒟𝑛−1(𝒰∗)𝑓∗→ℍ𝒟𝑛−1(𝒱∗)𝑔∗→ℍ𝒟𝑛−1(𝒲∗)→⋯.