Cartan–Eilenberg resolution

Cartan–Eilenberg resolution is a type of double complex used in Homological algebra to resolve a complex of objects by injective or projective objects in such a way that total complexes compute derived functors and induce spectral sequences. It provides a functorial and systematic method to replace a bounded-below or bounded-above complex by a bicomplex whose rows and columns are acyclic in prescribed senses, facilitating computations in contexts ranging from Sheaf theory to Group cohomology and Algebraic topology.

Definition and Construction

A Cartan–Eilenberg resolution of a complex C in an abelian category equipped with enough injectives or enough projectives is a first quadrant double complex I^{*,*} together with a morphism of complexes C → Tot(I^{*,*}) such that each column I^{p,*} is an injective (or projective) resolution of C^p and the horizontal differentials induce chain maps between those resolutions compatible with the structure of C. The construction uses successive choices of injective hulls or projective covers in the spirit of the methods of Henri Cartan and Samuel Eilenberg, similarly to techniques found in the work of Jean-Pierre Serre, Koszul, and Godeaux. When constructed in categories like modules over a ring R or sheaves on a scheme X, the resolution interacts with classical tools such as Čech cohomology, Godement resolution, Bar resolution, and Koszul complex.

Properties and Existence

Cartan–Eilenberg resolutions exist under hypotheses comparable to those guaranteeing the existence of ordinary injective or projective resolutions, for instance in categories with enough injectives as in models used by Alexander Grothendieck, Jean-Louis Verdier, and Pierre Deligne. Key properties include functoriality up to homotopy, compatibility with truncation functors studied by Serre and Grothendieck, and the fact that the total complex Tot(I^{*,*}) is quasi-isomorphic to the original complex C, mirroring results of Daniel Quillen and Michael Atiyah in derived contexts. The double complex admits two canonical filtrations (row and column) which yield convergent first quadrant spectral sequences analogous to those introduced by Jean Leray and used extensively by Germain Henri Hess and John Milnor.

Relationship to Derived Functors and Spectral Sequences

The Cartan–Eilenberg resolution is instrumental in defining and computing derived functors such as Ext and Tor in derived categories associated with Grothendieck abelian category frameworks developed by Grothendieck and Verdier. Applying a left or right exact functor F to a Cartan–Eilenberg resolution yields a bicomplex whose total homology computes the right or left derived functors R^iF or L_iF, recovering classical spectral sequences like the Grothendieck spectral sequence of Jean Leray and the Hochschild–Serre spectral sequence used by Claude Chevalley and Gerhard Hochschild. The two spectral sequences arising from the row and column filtrations correspond to iterated derived functor computations, connecting to machinery developed by Henri Cartan, Samuel Eilenberg, Spencer Bloch, and Joseph Bernstein in representation-theoretic and arithmetic applications.

Examples and Computations

Typical examples include taking C to be a single module concentrated in degree zero over a ring R, recovering the classical injective or projective resolutions that appear in Noetherian ring settings studied by Emmy Noether and David Hilbert. For complexes of sheaves on a scheme S as in the work of Alexander Grothendieck and Jean-Pierre Serre, Cartan–Eilenberg resolutions refine the Godement resolution and connect to hypercohomology computations used by Pierre Deligne and Luc Illusie. In group cohomology, resolving a cochain complex of Eilenberg–MacLane space-valued modules reproduces the Hochschild–Serre and Lyndon spectral sequences familiar from the works of Hermann Weyl and John Tate. Explicit computations for Koszul-type complexes or bar resolutions demonstrate the passage from bicomplex homology to Ext and Tor, echoing examples treated by Koszul, Cartan, and Eilenberg.

Historical Context and Attribution

The construction and systematic use of Cartan–Eilenberg resolutions appear in the mid-20th century development of homological algebra spearheaded by Henri Cartan and Samuel Eilenberg in their foundational treatments that paralleled contemporaneous advances by Alexander Grothendieck, Jean-Pierre Serre, Maurice Auslander, and Daniel Quillen. Their methods built on earlier resolutions used in Algebraic topology by figures such as Henri Poincaré and Eduard Čech and were integrated into the modern language of derived categories advanced by Jean-Louis Verdier and Grothendieck during the formulation of the six operations formalism in algebraic geometry.

Category:Homological algebra