What question did this study set out to answer?

The aim is to explore the interaction between homology structures in fibration contexts using coalgebra theory.

December 1, 1965

Homology and fibrations I Coalgebras, cotensor product and its derived functors

Key Points

The aim is to explore the interaction between homology structures in fibration contexts using coalgebra theory.
Defined coalgebras over a commutative ring and their graded comodules.
Introduced the cotensor product of comodules and derived functors related to injective resolutions.
Developed algebraic structures and established connections to geometric scenarios using commuting diagrams.
The derived functor Cotor A(A, B) was introduced, resulting in spectral sequences.
Under specific flatness conditions, Cotor R(A)(H(A), H(B)) is isomorphic to the term E~(A, A, B).
These findings integrate algebraic constructs within geometric frameworks, facilitating deeper insights.

Abstract

The s tudy of the relations between the homology structure of the base space, the total space and the fiber of a fibration offers ample opportunity for application of homological algebra. This series of papers develops some of this algebra and derives its relations with the geometric situations. In this paper the basic notion is that of a (graded differential) coalgebra A over a commutative ring K. Left and right (graded differential) A-comodules are defined as well as a cotensor product A DAB of a right A-comodule A and a left A-comodule B. Using a suitable relative notion of an injective resolution the derived functor Cotor A (A, B) (which is a graded K-module) is defined. This functor is the target of a spectral sequence Er (A, A, B), d r and under some flatness conditions (which are always satisfied if K is a field) the term E~ (A, A, B) is isomorphic with Cotor R (A) (H (A), H (B) ). This algebraic apparatus is developed in w w 2-10. The contact with geometry is established in the following way. We consider a commuting diagram E' 9~E

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