Yoneda Product Research Articles

Higher Yoneda product structures and Iwasawa algebras modulo $p$

Higher Yoneda product structures and Iwasawa algebras modulo $p$

Ext-Algebras of Graded Skew Extensions

In this paper, we study the Ext-algebras of graded skew extensions. For a connected graded algebra $A$ and a graded automorphism $\sigma$, we analyze the Yoneda product of the Ext-algebra of graded skew extension $A[z;\sigma]$, and prove this Ext-algebra is an $R$-smash product of the Ext-algebra of $A$ and the one of polynomial algebra $k[z]$ as an associative algebra.

Spectral sequences for the cohomology rings of a smash product

Stefan and Guichardet have provided Lyndon–Hochschild–Serre type spectral sequences which converge to the Hochschild cohomology and Ext groups of a smash product. We show that these spectral sequences carry natural multiplicative structures, and that these multiplicative structures can be used to calculate the cup product on Hochschild cohomology and the Yoneda product on an Ext algebra.

Extension groups of tautological sheaves on Hilbert schemes

We give formulas for the extension groups between tautological sheaves and more generally between tautological objects twisted by natural line bundles on the Hilbert scheme of points on a smooth quasi-projective surface. As a consequence we observe that a tautological object can never be a spherical or P n \mathbb {P}^n -object. We also provide a description of the Yoneda products.

Hochschild cohomology ring of the generalized quaternion algebras

We will give an efficient bimodule projective resolution of the generalized quaternion Z algebra Γ. As a main result, we will determine the ring structure of the Hochschild cohomology HH*(Γ) by calculating the Yoneda products using this resolution.

The Hochschild cohomology ring of preprojective algebras of type [formula omitted

We compute the Hochschild cohomology of the preprojective algebra of generalized Dynkin type Ln over any commutative ring in which 2 is invertible. In particular, we describe the bigraded ring structure of the Hochschild cohomology ring under the Yoneda product by giving an explicit presentation by homogeneous generators and relations.

Hochschild cohomology ring of a maximal order of the quaternion algebra

We give an efficient bimodule projective resolution of a maximal Z-order Λ of the ordinary quaternion algebra over Q, and therefore we determine the ring structure of the Hochschild cohomology of Λ by calculating the Yoneda products using this resolution.

Hochschild cohomology rings of [formula omitted]-Koszul algebras

In this paper we determine the multiplicative structure of Hochschild cohomology rings of d -Koszul algebras with the Yoneda products based on an explicit construction of a chain map Δ : P → P ⊗ A P lifting the identity with P a minimal projective bimodule resolution of a d -Koszul algebra A .

Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras

A main contribution of this paper is the explicit construction of comparison morphisms between the standard bar resolution and Bardzell's minimal resolution for truncated quiver algebras over arbitrary fields (TQA's). As a direct application we describe explicitly the Yoneda product and derive several results on the structure of the cohomology ring of TQA's over a field of characteristic zero. For instance, we show that the product of odd degree cohomology classes is always zero. We prove that TQA's associated with quivers with no cycles or with neither sinks nor sources have trivial cohomology rings. On the other side we exhibit a fundamental example of a TQA with nontrivial cohomology ring. Finally, for truncated polynomial algebras in one variable, we construct explicit cohomology classes in the bar resolution and give a full description of their cohomology ring.

Hochschild Cohomology Ring of an Order of a Simple Component of the Rational Group Ring of the Generalized Quaternion Group

We will give an efficient bimodule projective resolution of an order Γ, where Γ is an order of a simple component of the rational group ring ℚ Q 2 r of the generalized quaternion 2-group Q 2 r of order 2 r+2. Moreover, we will determine the ring structure of the Hochschild cohomology HH*(Γ) by calculating the Yoneda products using this bimodule projective resolution.

Module Structure of an Injective Resolution

Let A be the ring obtained by localizing the polynomial ring κ[X, Y, Z, W] over a field κ at the maximal ideal (X, Y, Z, W) and modulo the ideal (XW − YZ). Let 𝔭 be the ideal of A generated by X and Y. We study the module structure of a minimal injective resolution of A/𝔭 in detail using local cohomology. Applications include the description of , where M is a module constructed by Dutta, Hochster and McLaughlin, and the Yoneda product of .

A generalized Yoneda algebra of an algebra associated with a cyclic quiver

Let A=KΓ/(Xk), where KΓ is the path algebra of a cyclic quiver Γ over a field K,X is the sum of all arrows of Γ and k is a positive integer. In this paper, we describe the ring structure of the generalized Yoneda algebra ⊕i≥0ExtAi(A/Jl,A/Jl) of A with multiplication given by the Yoneda product, where J denotes the Jacobson radical of A and l is a positive integer with l≤k.

A periodic projective bimodule resolution of an algebra associated with a cyclic quiver and a separable algebra, and the Hochschild cohomology ring

Let Δ be a separable algebra over a commutative ring R and f(x) a monic polynomial over the center of Δ. We deal with the R-algebra Λ=ΔΓ/(f(Xs)), where ΔΓ is the path algebra of the cyclic quiver Γ with s vertices and s arrows, and X is the sum of all arrows. We show that Λ has a periodic projective bimodule resolution of period 2. Moreover, by using the resolution, we describe the structure of the Hochschild cohomology ring of Λ by means of the Yoneda product.

On Hochschild cohomology rings of Fibonacci algebras

Let Λ be a Fibonacci algebra over a field k. The multiplication of Hochschild cohomology ring of Λ induced by the Yoneda product is described explicitly. As a consequence, the multiplicative structure of Hochschild cohomology ring of Λ is proved to be trivial.

Cohomology Theories of Hopf Bimodules and Cup-Product

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by M. Gerstenhaber and S. D. Schack, and by C. Ospel. We prove, when A is finite-dimensional, that they are all equal to the Ext functor on the module category of an associative algebra associated to A, described by C. Cibils and M. Rosso. We also give an expression for a cup-product in the cohomology defined by C. Ospel, and prove that it corresponds to the Yoneda product of extensions.

Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product

We prove that the category of Hopf bimodules over any Hopf algebra has enough injectives, which enables us to extend some results on the unification of Hopf bimodule cohomologies of [R. Taillefer, PhD thesis, 2001; arXiv preprint math.QA/0005019] to the infinite dimensional case. We also prove that the cup-product defined on these cohomologies is graded-commutative. Unlike the algebra case (see [S. Schwede, J. Reine Angew. Math. 498 (1998) 153–172]), these methods do not give a non-trivial Gerstenhaber algebra structure on the cohomology we consider. We also comment that the other approach to finding such a structure that we know of (see [M. Farinati, A. Solotar, arXiv preprint math.KT/0207243]) also gives a trivial Gerstenhaber algebra structure.

The work of Jan-Erik Roos on the cohomology of commutative rings

This is an attempt to present one aspect of the work of Jan-Erik Roos. A glance at the list of publications reveals three clearly deÞned periods in his life as an algebraist. During the Þrst one he studied abelian categories, obtaining fundamental results on derived functors of inverse limits. They are contained in (3), (5)-(9), (11)-(17), (19)-(21). In the second period he focused on the homological theory of non-commutative rings, producing methods and results of lasting interest, among them a truly classic theorem—the determination of the global dimension of Weyl algebras. The papers (4), (18), (22)-(26), and (31) (treating related questions from commutative ring theory) contain the results of that period. Bjork (Bj) has given an overview in the context of contemporary and subsequent research. The work discussed here starts in the mid-1970s, when Jan-Erik turned to homo- logical problems on Þnitely generated modules over commutative noetherian local (or graded) rings. He has produced fascinating results on the structure of free res- olutions of modules of inÞnite projective dimension, and has investigated deep and mysterious links between homological properties of commutative rings and topo- logical spaces. His study of numerical invariants encoded in Poincare series, and of algebraic invariants determined by Yoneda products and by homology products, brings an unusual degree of integration between these components. This highly original and technically dicult work also brings to mind other qual- ities, such as elegance and optimism. A quick look at the many rings appearing on the following pages shows that there is nothing contrived about his 'examples': they are deÞned by the kind of simple expressions in few variables that one scrib- bles on a piece of paper to have something 'concrete' to play with. Appearances notwithstanding, some of these rings have been craftily constructed to posess a desired property. Others have been found by sifting, with the determination of a gold prospector, through computations of (literally!) thousands of examples. The purpose in this survey is to provide a guide to some of Jan-Erik's Þnds. A dierent perspective of work completed by 1985 can be found in the article of Anick and Halperin (AH). Connection with topology, which were discussed early on by Lemaire (Le) in a Bourbaki talk, and recently by Hess (He) in historical context,

Minimal projective resolutions

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the Ext \operatorname {Ext} -algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the “no loop” conjecture.

Cohomology theories of Hopf bimodules and cup-product

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by Gerstenhaber and Schack, and by Ospel. We prove, when all the spaces involved are finite dimensional, that they are all equal to the Ext functor on the module category of an associative algebra X associated to A, as described by Cibils and Rosso. We also give an expression for a cup-product in the cohomology defined by Ospel, and prove that it corresponds to the Yoneda product of extensions.

On the hochschild cohomology of truncated cycle algebras

The purpose of this paper is to study the Hochschild cohomology ring H *(⋀)of algebras of the form ⋀ = kZe /JN , where Ze is an oriented cycle with e vertices and J is the ideal generated by the arrows, N≥2. We provide a new description of the Yoneda product in H *(⋀). and prove that this is a finitely generated infinite dimensional ring. In addition we show that algebras of the form ⋀ = kZe /JN are not derived equivalent unless they are isomorphic.