The behaviour of \(\text{Ext}_ A(R,A)\) through a morphism. (English) Zbl 0880.13008
The analogies between local algebra and rational homotopy theory [see L. Avramov and 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 L. L. Avramov, H. B. Foxby and 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.
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 L. L. Avramov, H. B. Foxby and 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.
Reviewer: M.Golasiński (Toruń)
MSC:
| 13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |
| 18G15 | Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) |
Keywords:
homotopy fibre; KS-resolution; spectral sequence; Yoneda product; Ext; augmented differential graded algebraReferences:
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