The Cohomological Dimension of Certain Grothendieck Topologies

1966 ◽  
Vol 83 (3) ◽  
pp. 572 ◽  
Author(s):  
Stephen S. Shatz
Keyword(s):  
Cohomological Dimension ◽  
Grothendieck Topologies

scholarly journals Coarse cohomology with twisted coefficients

2020 ◽  
Vol 70 (6) ◽  
pp. 1413-1444
Author(s):  
Elisa Hartmann
Keyword(s):  
Sheaf Cohomology ◽  
Grothendieck Topology ◽  
Coarse Structure ◽  
Relative Cohomology ◽  
Number Of Ends ◽  
Constant Coefficients ◽  
Coarse Space ◽  
Grothendieck Topologies

AbstractTo a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

Grothendieck Topologies

Categories ◽  
1972 ◽  
pp. 291-319
Author(s):  
Horst Schubert
Keyword(s):  
Grothendieck Topologies

scholarly journals On the Dimension of Modules and Algebras, I

1955 ◽  
Vol 8 ◽  
pp. 49-57 ◽  
Author(s):  
Samuel Eilenberg ◽  
Masatoshi Ikeda ◽  
Tadasi Nakayama
Keyword(s):  
Alternative Treatment ◽  
Cohomological Dimension ◽  
Subsequent Paper

In [5], Ikeda-Nagao-Nakayama gave a characterization of algebras of cohomological dimension ≦n In a subsequent paper [4] Eilenberg gave an alternative treatment of the same question. The present paper is devoted to the discussion of a number of questions suggested by the results of [4] and [5]. Among others it is shown that the conditions employed in stating the main results in [4] and [5] are equivalent, so that the main results of these two papers are in accord. Further, the cohomological dimension of a residue-algebra is studied in terms of that of the original algebra and the (module-) dimension of the associated ideal. The terminology and notation employed here are that of [3].

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