Yoneda Product Research Articles

On Hochschild Cohomology of Preprojective Algebras, II

We study the Hochschild cohomology of a finite-dimensional preprojective algebra; this is periodic by a result of A. Schofield. We determine the ring structure of the Hochschild cohomology ring given by the Yoneda product. As a result we obtain an explicit presentation by generators and relations.

Homology of maximal orders in central simple algebras

and moreover that HH2(O/R) = 0. These two facts greatly facilitate the computation of the cyclic homology HC∗(O/R) ([3]). This paper presents non-commutative analogues of the main results of [3]. In particular, L is to be replaced by a central simple algebra D over K, while O is taken to be a maximal R-order in D. Of course, O is not, in general, uniquely defined by R and D, but it turns out that the homology of O/R is independent of the choice of maximal order. We prove that (0.1) remains valid in the non-commutative context but that the odd Hochschild homology groups vanish. The first section is devoted to generalities on Hochschild homology. We do not claim to prove any new result but only to establish notation and give self-contained proofs of various facts for which we could not find convenient references. The main results, the periodicity theorem and the vanishing theorem, are proved in §2 when R is a complete discrete valuation ring with perfect residue field. We construct an element in Hochschild cohomology such that Yoneda product gives the desired periodicity. The calculations needed to compute the low-dimensional Hochschild homology groups are laborious, but the resulting formulae are quite simple. The globalization is carried out in §3 and is completely standard. The vanishing of odd Hochschild homology guarantees that the Loday-Quillen spectral sequence degenerates at E, so it is easy to compute cyclic homology, in both the local and the global case, up to extension. This is explained in §4, where the extension problem is partially solved by an application of the universal coefficient theorem.

Boundedness and periodicity of modules over QF rings

be a minimal projective resolution. M is called bounded if the lengths of the kernels Ker di have a common upper bound, and M is called periodic if some Ker di is isomorphic to M. In [ 1, p. 7761, Alperin posed the following question: Is every bounded KG-module periodic? Alperin answered this question in the affirmative in case K is algebraic over its prime field [l, Theorem 11. Without any restriction on K, Eisenbud settled Alperin’s question in [6, Theorem 9.21. In Section 1 of the present paper, we look at an arbitrary QF ring R instead of a group algebra, and we give a condition which forces a bounded R-module M to be periodic. Before we will formulate our result, we make some preliminary remarks. Let M and X be modules over a ring R. With the Yoneda product for multiplication, the abelian group ExtX(M, M) = Oi,,, Extk(M, M) becomes a graded ring which we call the extension ring of M, and the abelian group Ext,*(M, X) becomes a graded right ExtR(M, M)-module. We recall some classical examples of extension rings. Let G be a finite group, and let K be a field on which G acts trivially. Then the cohomology ring H*(G, K) is the extension ring Ext,*(K, K). More generally, let M be a finitely generated KG-module. Then the ring Hom,(M, M) is a G-ring, and the cohomology ring H*(G, Hom,(M, M)) (with cup product) is canonically isomorphic to the extension ring ExtX(M, M) (with Yoneda product). These rings have been intensively studied; see Carlson [3] for details and for further references. Other well-known exampes of extension rings arise in the homological algebra of local rings, namely the extension

On the homological structures associated with certain artin algebras

If R is a ring, there are several graded algebras which are associated with R. If M is any R-module, then Exti(M, M) = @ C,“=O Ext”,(M, M) is a graded algebra, where the product is the Yoneda product. If Jz! c RMod is any collection of R-modules, then 0 CM6 .& CNt ,A( Ext;?;(M, N) is also a graded ring, where the product is the Yoneda product, where it is defined, and zero elsewhere. In this paper, we shall give a description of these structures for certain classes of modules over rings satisfying certain conditions. In particular, in this paper, all rings are assumed to be finite-dimensional algebras over some fixed algebraically closed field F, and all modules are left modules. If R is such a ring, yR (or just 9, if the context ensures no ambiguity) E RMod will be a collection containing one representative of each isomorphism class of simple R-modules. We shall compute the structure of Exts(S, S) for SEY and of Ext(R)= @CsG,Y’CTtTfExtr?;(S, T) when R is pro-uniserial, injec-uniserial, or Nakayama.

Commutative rings, algebraic topology, graded lie algebras and the work of Jan-Erik Roos

The classic book, Homological Algebra, by Cartan and Eilenberg begins in the following way: ‘During the last decade the methods of algebraic topology have invaded extensive- ly the domain of pure algebra . . . The invasion has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras and associative algebras . . . We present here a single cohomology (and also a homology) theory which embodies all three.’ A few lines later the authors mention Hilbert’s syzygy theorem as an application; this is one instance of the usefulness of these methods when applied to the study of commutative rings. A central role in Homological Algebra is played by the functors Tor and Ext. Ap- plied to a ring homomorphism R + k (k a field) they give a graded k-vector space Tor$(k, k) whose graded dual Ei = Ext,*(k, k) is, equipped with the Yoneda product, a graded algebra. With the introduction of differential homological algebra by Eilenberg and Moore [14] the functor Tor was extended to differential graded algebras (DGA’s). They showed, in particular, that for the cochain algebra C* of a pointed l-connected CW complex of finite type: Torc*(k,

A note on Yoneda product

A note on Yoneda product

ON A HOMOLOGICAL CHARACTERIZATION OF A CERTAIN CLASS OF LOCAL RINGS

It is shown that the nondegeneracy of the Yoneda product ( is either a noetherian module or a complex of finite projective dimension, and is the residue field) characterizes the regularity of the ring , whereas the isomorphism characterizes the fact that is Gorenstein.Bibliography: 8 titles.

Yoneda Products in the Cartan-Eilenberg Change of Rings Spectral Sequence With Applications to BP ∗ (BO(n))

Yoneda Products in the Cartan-Eilenberg Change of Rings Spectral Sequence With Applications to BP ∗ (BO(n))

Yoneda products in the Cartan-Eilenberg change of rings spectral sequence with applications to 𝐵𝑃_{∗}(𝐵𝑂(𝑛))

Yoneda product structure is defined on a Cartan-Eilenberg change of rings spectral sequence. Application is made to a factorization theorem for the E 2 {E_2} -term of the Adams spectral sequence for Brown-Peterson homology of the classifying spaces B O ( n ) BO(n) .

Two conjectures in the homology of local rings

This is a conjecture of the author [6]. Analogous results have been proved by Evens [2] for finite groups and conjectured by Venkov for nilpotent algebras. We show that Conjecture II is true in several classes of local rings for which Conjecture I has been proved. In the process, another class of rings is added to the list for which Conjecture I is true. Tora(k, k) has the structure of a commutative Hopf algebra whose dual Hopf algebra is isomorphic to Ext,(k, k) with the Yoneda product [7]. Our approach in each case will be to show that the primitive elements of TorR(k, k) form a finite dimensional vector space over k, so that dually, Ext,(k, k) is finitely generated. For the convenience of the reader, a description of the Hopf algebra structure on TorR(k, k) is given in Section 1. Also included is a proof of the above mentioned duality. Then Conjecture II is verified for local complete intersections in Section 2 and for a class of local rings first studied by Golod

Pairings and Products in the Homotopy Spectral Sequence

Smash and composition pairings, as well as Whitehead products are constructed in the unstable Adams spectral sequence; and these pairings and products are described homologically on the ${E_2}$. level. In the special case of the Massey-Peterson spectral sequence, the composition action is given homologically by the Yoneda product, while the Whitehead product vanishes. It is also shown that the unstable Adams spectral sequence over the rationals, with its Whitehead products, is given by the primitive elements in the rational cobar spectral sequence.

Pairings and products in the homotopy spectral sequence

Smash and composition pairings, as well as Whitehead products are constructed in the unstable Adams spectral sequence; and these pairings and products are described homologically on the E 2 {E_2} . level. In the special case of the Massey-Peterson spectral sequence, the composition action is given homologically by the Yoneda product, while the Whitehead product vanishes. It is also shown that the unstable Adams spectral sequence over the rationals, with its Whitehead products, is given by the primitive elements in the rational cobar spectral sequence.