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Tyler Lawson
Tyler Lawson
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With respect to the updated question: yes, there are other cohomology theories. For example, if $H$ is ordinary sheaf cohomology, then we can define a new sheaf cohomology theory $K$ by $K^q({\cal F}) = H^q({\cal F}) \times H^{q+1}({\cal F})$$K^q({\cal F}) = H^q({\cal F}) \times H^{q-1}({\cal F})$. (This is a special instance of a hypercohomology construction which is genuinely important in some areas.)

With respect to the original question: A. Grothendieck, "Sur quelques points d’algèbre homologique" (the Tohoku paper); Cartan-Eilenberg, "Homological algebra".

With respect to the updated question: yes, there are other cohomology theories. For example, if $H$ is ordinary sheaf cohomology, then we can define a new sheaf cohomology theory $K$ by $K^q({\cal F}) = H^q({\cal F}) \times H^{q+1}({\cal F})$. (This is a special instance of a hypercohomology construction which is genuinely important in some areas.)

With respect to the original question: A. Grothendieck, "Sur quelques points d’algèbre homologique" (the Tohoku paper); Cartan-Eilenberg, "Homological algebra".

With respect to the updated question: yes, there are other cohomology theories. For example, if $H$ is ordinary sheaf cohomology, then we can define a new sheaf cohomology theory $K$ by $K^q({\cal F}) = H^q({\cal F}) \times H^{q-1}({\cal F})$. (This is a special instance of a hypercohomology construction which is genuinely important in some areas.)

With respect to the original question: A. Grothendieck, "Sur quelques points d’algèbre homologique" (the Tohoku paper); Cartan-Eilenberg, "Homological algebra".

Source Link
Tyler Lawson
Tyler Lawson
  • 52.6k
  • 9
  • 187
  • 251

With respect to the updated question: yes, there are other cohomology theories. For example, if $H$ is ordinary sheaf cohomology, then we can define a new sheaf cohomology theory $K$ by $K^q({\cal F}) = H^q({\cal F}) \times H^{q+1}({\cal F})$. (This is a special instance of a hypercohomology construction which is genuinely important in some areas.)

With respect to the original question: A. Grothendieck, "Sur quelques points d’algèbre homologique" (the Tohoku paper); Cartan-Eilenberg, "Homological algebra".