"Derived functor"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| 3번째 줄: | 3번째 줄: | ||
* extend a left invariant functor to get a derived functor | * extend a left invariant functor to get a derived functor | ||
* then we get a cohomology theory | * then we get a cohomology theory | ||
| − | * e.g. sheaf cohomology of a topological space X with coefficients in a sheaf | + | * e.g. sheaf cohomology of a topological space X with coefficients in a sheaf <math>\mathcal F</math> = the right derived functor of the global section functor |
==left invariant functors== | ==left invariant functors== | ||
===global section functor=== | ===global section functor=== | ||
| − | * a functor from sheaves on | + | * a functor from sheaves on <math>X</math> to abelian groups defined by |
| − | + | :<math> | |
\mathcal F \mapsto H^{0}(X, \mathcal F) | \mathcal F \mapsto H^{0}(X, \mathcal F) | ||
| − | + | </math> | |
===invariants=== | ===invariants=== | ||
| − | * | + | * <math>G</math> : group |
| − | * from modules of | + | * from modules of <math>G</math> to abelian groups |
| − | + | :<math> | |
M\mapsto M^{G} | M\mapsto M^{G} | ||
| − | + | </math> | |
2020년 11월 13일 (금) 21:21 판
introduction
- basic tool to define cohomology theory
- extend a left invariant functor to get a derived functor
- then we get a cohomology theory
- e.g. sheaf cohomology of a topological space X with coefficients in a sheaf \(\mathcal F\) = the right derived functor of the global section functor
left invariant functors
global section functor
- a functor from sheaves on \(X\) to abelian groups defined by
\[ \mathcal F \mapsto H^{0}(X, \mathcal F) \]
invariants
- \(G\) : group
- from modules of \(G\) to abelian groups
\[ M\mapsto M^{G} \]