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On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra. (English) Zbl 0595.16020

Algebra, algebraic topology and their interactions, Proc. Conf., Stockholm 1983, Lect. Notes Math. 1183, 291-338 (1986).
[For the entire collection see Zbl 0577.00005.]
Let k be a field and R a ring with a ring epimorphism \(R\to k\). k is a module over R and \(E=Ext_ R(k,k)\) is a graded algebra. The purpose of this paper is to examine in two cases how well the subalgebra generated by \(E^ 1\) approximates E.
In the first case, R is an augmented algebra over k, generally noncommutative. As algebras, \(Ext_ R(k,k)=H(T(I)^*,d^*)\) where \(T(I)^*\) is the dual of the tensor algebra on I, the augmentation ideal. After reviewing S. Priddy’s results for R two-homogeneous [Trans. Am. Math. Soc. 152, 39-60 (1970; Zbl 0261.18016)], the author demonstrates, on the added condition that R be a Koszul algebra, the equivalence of seven statements, including that \(Ext^ 1_ R(k,k)\) generates \(Ext_ R(k,k).\)
In the second case, (R,m) is a local Noetherian commutative ring with maximal ideal m, \(k=R/m\) is the residue field. If k has a minimal resolution \(Y=U\otimes F\) where U is a Serre R-algebra and \(dF\subset m^ 2Y\), then \(Hom_ R(U,k)=[Ext^ 1_ R(k,k)],\) the subalgebra generated by \(Ext^ 1\). If \(m^ 4=0\), \(Ext_ R(k,k)\simeq [Ext^ 1_ R(k,k)]\otimes T(V),\) where T(V) is the Hopf-sub-algebra on an appropriate graded vector space V.
For both cases, there are results on the Poincaré series \(P_ R\).
Reviewer: R.M.Najar

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
16W50 Graded rings and modules (associative rings and algebras)
16S20 Centralizing and normalizing extensions