Holmstrom Generalized sheaf cohomology

Generalized sheaf cohomology

Brown:

Abstract: Cohomology groups $E^{pq}_2 = H^p(X, \pi_{-q} E ) \implies H^{p+q}(X, E) chunk78422660categorychunk --- ## Generalized sheaf cohomology [MathSciNet](http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&H^q(X,E)$ are defined, where $X$ is a topological space, and $E$ is sheaf on $X$ with values in Kan’s category of spectra. These groups generalize the ordinary cohomology groups of $X$ with coefficients in an abelian sheaf, as well as the generlized cohomology of $X$ in the usual sense. The groups are defined by means of the “homotopical algebra” of Quillen applied to suitable categories of sheaves. The study of the homotopy category of sheaves of spectra requires an abstract homotopy theory more general than Quillen’s, and this is developed in Part 1 of the paper. Finally, the basic cohomological properties are proved, including a spectral sequence (in generalized cohomology theory) which generalizes the Atiyah-Hirzebruch spectral sequence (in generalized cohomology theory) and the “locak to global” spectral sequence (in sheaf cohomology theory).

Generalized sheaf cohomology

The spectral sequence is