The behaviour of \(\text{Ext}_ A(R,A)\) through a morphism
The analogies between local algebra and rational homotopy theory [see \textit{L. Avramov} and \textit{S. Halperin} in: Algebra, algebraic topology and their interactions, Proc. Conf., Stockholm 1983, Lect. Notes Math. 1183, 1-27 (1986; Zbl 0588.13010)] gave rise to the extensive study of the invariant \(\text{Ext}_A(k,A)\) when \(A\) is an augmented differential graded algebra over a field \(k\). For a morphism \(A\to B\) of augmented differential graded algebras (with augmentations \(A\to k\), \(B\to l\)), the author studies a morphism \[ \phi : \text{Ext}_A(k,A)\otimes_k\text{Ext}_F(l,F)\to \text{Ext}_B(l,B) \] in which \(F\) is the homotopy fibre of the morphism \(A\to B\). The result by \textit{L. L. Avramov, H. B. Foxby} and \textit{J. Lescot} [Trans. Am. Math. Soc. 335, No. 2, 497-523 (1993; Zbl 0770.13007)] is generalized as follows: The morphism \(\phi\) is an isomorphism provided that the homology of the fibre \(F\) is bounded. Finally, some interesting topological consequences of this result are presented.
scientific article; zbMATH DE number 1038255
scientific article