First and Second Module Cohomology Groups for Non Commutative Semigroup Algebras

The aim of this paper is to report some results on the cohomology groups \(Ext_{{A^e}}^n\) and the Hochschild cohomology groups H n (A),where A is an algebra given by arbitrary quiver (not necessarily a finite quiver) and relations. In this case, the two cohomologies no longer coincide. In particular, we prove that \(Ext_{{A^e}}^1(A,A) = 0\) if and only if Q is a tree, where A = kQ with Q an arbitrary quiver; and that H 1 (A) = 0 if and only if Q is a finite tree, where A = kQ/I is a monomial algebra. This generalizes the corresponding results of Happel, Bardzell and Marcos to the infinite-dimensional case.

The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let Λt be the Yoneda algebra of a reconstruction algebra of type A1 over a field k. In this paper, a minimal projective bimodule resolution of Λt is constructed, and the k-dimensions of all Hochschild homology and cohomology groups of Λt are calculated explicitly.

Sorin Popa [Po 1] [Po 2] has proved several results on the existence and properties of hyperfinite subfactors R of a type II1 factor M with the relative commutant R0 \M of R in M equal to C1. These theorems have been used in various cohomology calculations [CES, CS, PS, CPSS], as averaging over an amenable subgroup that generates the hyperfinite subfactor is a major step in showing that the continuous and completely bounded Hochschild cohomology groups are equal. It has seemed reasonable that Popa's results could be extended from factors to general type II1 von Neumann algebras by direct integral theory [KR, Chapter 4]. However, we do not know of such an attempt. Direct integral theory can be used directly to prove cohomology is zero and deduce results like [CPSS, Theorems 5.4 and 5.5] however these theorems on the continuous Hochschild cohomology for a von Neumann algebra with Cartan subalgebras are deduced from the theorems in this paper. This paper provides direct proofs of Popa's main two results in [Po 1] by modifying his proofs using an interpolation type result for projections in a maximal abelian selfadjoint sub-algebra (masa) of the type II1 algebra. This introduction contains a more detailed description of how our results extend Popa's, the basic definitions, and a brief reference to their use in the calculation of Hochschild cohomology groups in von Neumann algebras. Though averaging plays an important role in calculating the Hochschild cohomology of von Neumann algebras for all types I ; II1; II1; III of von Neumann algebras (see [Ri]), the results proved here are only used in the type II1 situation. The reason is that the type I 's are already trivially hyperfinite, and the type II1 and III von Neumann algebras may be handled by their stability under tensoring with B H†. This tensor factor B H† of M in the II1 and III cases gives a suitable hyperfinite algebra over which to average. Popa [Po 1] restricts his attention to type II II1 and II1† factors. MATH. SCAND. 85 (1999), 105^120

Let \Lambda be a two-sided Noetherian Gorenstein k -algebra, for k a field. We introduce Tate–Hochschild homology and cohomology groups for \Lambda , which are defined for all degrees, non-negative as well as negative, and which agree with the usual Hochschild homology and cohomology groups for all degrees larger than the injective dimension of \Lambda . We prove certain duality theorems relating the Tate–Hochschild (co)homology groups in positive degree to those in negative degree, in the case where \Lambda is a Frobenius algebra. We explicitly compute all Tate–Hochschild (co)homology groups for certain classes of Frobenius algebras, namely, certain quantum complete intersections.

Macaulay is a system for computing in algebraic geometry and cummutative algebra; it is capable of a variety of computations which are tedious or impossible to perform by hand. The primitive types in the system are polynomials, matrices, ideals, polynomial rings, modules, maps between rings, and complexes of modules. The system performs algebraic manipulation on objects of these types. The possible manipulations include the computation of standard (Gröbner) bases, modules of syzygies, finite free resolutions, Hilbert polynomials and functions. Using these basic operations, a variety of derived operations are possible, such as projections, ideal intersections, and the computation of coherent sheaf cohomology groups. The algorithm used for constructing standard (Gröbner) bases and syzygies is described in ([Buc76], [Zac78], [Sch80], [Bay82], [BaSt86a]).

In 2002, an introductory workshop was held at the Mathematical Sciences Research Institute in Berkeley to survey some of the many directions of the commutative algebra field. Six principal speakers each gave three lectures, accompanied by a help session, describing the interaction of commutative algebra with other areas of mathematics for a broad audience of graduate students and researchers. This book is based on those lectures, together with papers from contributing researchers. David Benson and Srikanth Iyengar present an introduction to the uses and concepts of commutative algebra in the cohomology of groups. Mark Haiman considers the commutative algebra of n points in the plane. Ezra Miller presents an introduction to the Hilbert scheme of points to complement Professor Haiman's paper. Further contributors include David Eisenbud and Jessica Sidman; Melvin Hochster; Graham Leuschke; Rob Lazarsfeld and Manuel Blickle; Bernard Teissier; and Ana Bravo.

The Lie algebra structure of the first Hochschild cohomology group for monomial algebras

The aim of this article is to present a detailed algebraic computation of the Hochschild and cyclic homology groups of the Yang–Mills algebras YM( n ) ( n ∈ ℕ ≧2 ) defined by A. Connes and M. Dubois-Violette in [8], continuing thus the study of these algebras that we have initiated in [17]. The computation involves the use of a spectral sequence associated to the natural filtration on the universal enveloping algebra YM( n ) provided by a Lie ideal 𝔱𝔶𝔪( n ) in 𝔶𝔪( n ) which is free as Lie algebra. As a corollary, we describe the Lie structure of the first Hochschild cohomology group.

Let A be a basic connected finite dimensional algebra over an algebraically closed field, with ordinary quiver without oriented cycles. Given a presentation of A by quiver and admissible relations, Assem and de la Peña have constructed an embedding of the space of additive characters of the fundamental group of the presentation into the first Hochschild cohomology group of A. We compare the embeddings given by the different presentations of A. In some situations, we characterise the images of these embeddings in terms of (maximal) diagonalizable subalgebras of the first Hochschild cohomology group (endowed with its Lie algebra structure).

By using a representation of a Lie algebra on the second Hochschild cohomology group, we construct an obstruction class to extensibility of derivations and a short exact sequence of Wells type for an abelian extension of an associative algebra.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Let \( \tilde \Lambda \) be the ℤ2-Galois covering of the Grassmann algebra Λ over a field k of characteristic not equal to 2. In this paper, the dimensional formulae of Hochschild homology and cohomology groups of \( \tilde \Lambda \) are calculated explicitly. And the cyclic homology of \( \tilde \Lambda \) can also be calculated when the underlying field is of characteristic zero. As a result, we prove that there is an isomorphism from ⊕i≥1HHi(\( \tilde \Lambda \) ) to ⊕i≥1HHi(Λ).

A semigroup S is called left (right) weakly commutative if, for every a, b ∈ S, there exist x ∈ S and a positive integer n such that (ab) n = bx ((ab) n = xa). A semigroup which is both left and right weakly commutative is called a weakly commutative semigroup. In this chapter we deal with left weakly commutative, right weakly commutative and weakly commutative semigroups. It is shown that a semigroup is a semilattice of left archimedean (right archimedean, t-archimedean) semigroups if and only if it is right weakly commutative (left weakly commutative, weakly commutative). It is proved that a weakly commutative 0-simple semigroup is a group with a zero adjoined. Moreover, a semigroup is weakly commutative archimedean and contains an idempotent element if and only if it is an ideal extension of a group by a nil semigroup. We get, as a consequence, that a semigroup is weakly commutative and regular if and only if it is a Clifford semigroup. We show that a right (left) weakly commutative semigroup is embeddable into a group if and only if it is cancellative. At the end of the chapter we deal with the least weakly separative congruence on weakly commutative semigroups. It is proved that if 5 is a left weakly commutative semigroup then σ defined by a σ b if and only if ab n = bn+1 and ba n = an+1 for a positive integer n is a weakly separative congruence on 5. Similarly, if S is a right weakly commutative semigroup then r defined by a τ b if and only if b n a = b n+1 and a n b = an+1 for some positive integer n is a weakly separative congruence on S. Moreover, π = σ ∩ τ is the least weakly separative congruence on a weakly commutative semigroup.

For a simplicial complex X on {1,2, ..., n} we define enriched homology and cohomology modules. They are graded modules over k[x_1, ..., x_n] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein* complexes X, and also orientable manifolds in terms of the enriched modules. We introduce the notion of girth for simplicial complexes and make a conjecture relating the girth to invariants of the simplicial complex. We also put strong vanishing conditions on the enriched homology modules and describe the simplicial complexes we then get. They are block designs and include Steiner systems S(c,d,n) and cyclic polytopes of even dimension.

The Hochschild cohomology of an associative algebra records information about the deformations of that algebra, and hence the first step toward understanding its deformations is an examination of the Hochschild cohomology. In this dissertation, we use techniques from homological algebra, invariant theory, and combinatorics to analyze the Hochschild cohomology of skew group algebras arising from finite cyclic groups acting on polynomial rings over fields of arbitrary characteristic. These algebras are the natural semidirect product of the group ring with the polynomial ring. Many families of algebras arise as deformations of skew group algebras, such as symplectic reflection algebras and rational Cherednik algebras. We give an explicit description of the Hochschild cohomology governing graded deformations of skew group algebras for cyclic groups acting on polynomial rings. For skew group algebras, a description of the Hochschild cohomology is known in the nonmodular setting (i.e., when the characteristic of the field and the order of the group are coprime). However, in the modular setting (i.e., when the characteristic of the field divides the order of the group), much less is known, as techniques commonly used in the nonmodular setting are not available.