Skip to main content
MathOverflow

Return to Answer

added 2 characters in body
Source Link
Donu Arapura
Donu Arapura
  • 35.2k
  • 2
  • 94
  • 160

To be honest, the question is a bit borderline for MO, but I'll make a few comments anyway. I would argue that there is nothing inherently geometric about the definition of (universal) delta functors, but the end result is certainly very useful in geometry, and algebraic geometry in particular. Prior to Grothendieck's Tohoku paper, sheaf cohomology was defined as Cech cohomology. This is very nice and concrete, and works well in most cases, but it does have pathologies. Grothendieck, in Tohoku, realized that the right definition for sheaf cohomology was via derived functors invented a few years before by Cartan-Eilenberg. But he added many insights of his own.

To be specific, one would like a short exact sequence of sheaves to give a long exact sequence of cohomologies, i.e. the global sections functor should extend to a $\delta$-functor. But perhaps there are many ways to do this, or no ways at all-- so pick the one that dominates all the others, which would be the universal $\delta$-functor, provided it exists. Grothendieck went on to show that it did s exist exist (for sheaf cohomology), and that it also coincided with the right derived functor for global sections.

Hope that helps.

To be honest, the question is a bit borderline for MO, but I'll make a few comments anyway. I would argue that there is nothing inherently geometric about the definition of (universal) delta functors, but the end result is certainly very useful in geometry, and algebraic geometry in particular. Prior to Grothendieck's Tohoku paper, sheaf cohomology was defined as Cech cohomology. This is very nice and concrete, and works well in most cases, but it does have pathologies. Grothendieck, in Tohoku, realized that the right definition for sheaf cohomology was via derived functors invented a few years before by Cartan-Eilenberg. But he added many insights of his own.

To be specific, one would like a short exact sequence of sheaves to give a long exact sequence of cohomologies, i.e. the global sections functor should extend to a $\delta$-functor. But perhaps there are many ways to do this, or no ways at -- so pick the one that dominates all the others, which would be the universal $\delta$-functor, provided it exists. Grothendieck went on to show that it did s exist (for sheaf cohomology), and that it also coincided with the right derived functor for global sections.

Hope that helps.

To be honest, the question is a bit borderline for MO, but I'll make a few comments anyway. I would argue that there is nothing inherently geometric about the definition of (universal) delta functors, but the end result is certainly very useful in geometry, and algebraic geometry in particular. Prior to Grothendieck's Tohoku paper, sheaf cohomology was defined as Cech cohomology. This is very nice and concrete, and works well in most cases, but it does have pathologies. Grothendieck, in Tohoku, realized that the right definition for sheaf cohomology was via derived functors invented a few years before by Cartan-Eilenberg. But he added many insights of his own.

To be specific, one would like a short exact sequence of sheaves to give a long exact sequence of cohomologies, i.e. the global sections functor should extend to a $\delta$-functor. But perhaps there are many ways to do this, or no ways at all-- so pick the one that dominates all the others, which would be the universal $\delta$-functor, provided it exists. Grothendieck went on to show that it did exist (for sheaf cohomology), and that it also coincided with the right derived functor for global sections.

Hope that helps.

Source Link
Donu Arapura
Donu Arapura
  • 35.2k
  • 2
  • 94
  • 160

To be honest, the question is a bit borderline for MO, but I'll make a few comments anyway. I would argue that there is nothing inherently geometric about the definition of (universal) delta functors, but the end result is certainly very useful in geometry, and algebraic geometry in particular. Prior to Grothendieck's Tohoku paper, sheaf cohomology was defined as Cech cohomology. This is very nice and concrete, and works well in most cases, but it does have pathologies. Grothendieck, in Tohoku, realized that the right definition for sheaf cohomology was via derived functors invented a few years before by Cartan-Eilenberg. But he added many insights of his own.

To be specific, one would like a short exact sequence of sheaves to give a long exact sequence of cohomologies, i.e. the global sections functor should extend to a $\delta$-functor. But perhaps there are many ways to do this, or no ways at -- so pick the one that dominates all the others, which would be the universal $\delta$-functor, provided it exists. Grothendieck went on to show that it did s exist (for sheaf cohomology), and that it also coincided with the right derived functor for global sections.

Hope that helps.