Grothendieck topology: Difference between revisions
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The notion of a ''Grothendieck topology'' or ''site'' | {{subpages}} | ||
The notion of a '''Grothendieck topology''' or '''site'''' captures the essential properties necessary for constructing a robust theory of cohomology of sheaves. The theory of Grothendieck topologies was developed by Alexander Grothendieck and Michael Artin. | |||
==Definition== | ==Definition== | ||
A ''Grothendieck topology'' <math>T</math> consists of | |||
#A category, denoted <math>cat(T)</math> | |||
#A set of coverings <math>\{U_i\to U\}</math>, denoted <math>cov(T)</math>, such that | |||
##<math>\{id:U\mapsto U\}\in cov(T)</math> for each object <math>U</math> of <math>cat(T)</math> | |||
##If <math>\{U_i\to U\}\in cov(T)</math>, and <math>V\to U</math> is any morphism in <math>cat(T)</math>, then the canonical morphisms of the fiber products determine a covering <math>\{U_i\times_U V\to V\}\in cov(T)</math> | |||
##If <math>\{U_i\to U\}\in cov(T)</math> and <math>\{V_{i,j}\to U_i\}\in cov(T)</math>, then <math>\{V_{i,j}\to U_i\to U\}\in cov(T)</math> | |||
==Examples== | ==Examples== | ||
#A standard topological space <math>X</math> becomes a category <math>op(X)</math> when you regard the open subsets of <math>X</math> as objects, and morphisms are inclusions. An open covering of open subsets <math>U</math> clearly verify the axioms above for coverings in a site. Notice that a [[presheaf]] of rings is just a contravariant functor from the category <math>op(X)</math> into the category of rings. | #A standard topological space <math>X</math> becomes a category <math>op(X)</math> when you regard the open subsets of <math>X</math> as objects, and morphisms are inclusions. An open covering of open subsets <math>U</math> clearly verify the axioms above for coverings in a site. Notice that a [[presheaf]] of rings is just a contravariant functor from the category <math>op(X)</math> into the category of rings. | ||
#'''The Small Étale Site''' Let <math>S</math> be a scheme. Then the category of étale schemes over <math>S</math> (i.e., <math>S</math>-schemes <math>X</math> over <math>S</math> whose structural morphisms are étale) | #'''The Small Étale Site''' Let <math>S</math> be a scheme. Then the [[Étale morphism|category of étale schemes]] over <math>S</math> (i.e., <math>S</math>-schemes <math>X</math> over <math>S</math> whose structural morphisms are étale) becomes a site if we require that coverings are jointly surjective; that is, | ||
==Sheaves on Sites== | |||
[[Category: | In analogy with the situation for topological spaces, a presheaf may be defined as a contravariant functor | ||
such that for all coverings <math>\{U_i\to U\}\in cov(T)</math>, the diagram | |||
<math>0\to F(U)\to \prod F(U_i)\to \prod F(U_i\times_U U_j)</math> | |||
is exact.[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 06:00, 24 August 2024
The notion of a Grothendieck topology or site' captures the essential properties necessary for constructing a robust theory of cohomology of sheaves. The theory of Grothendieck topologies was developed by Alexander Grothendieck and Michael Artin.
Definition
A Grothendieck topology consists of
- A category, denoted
- A set of coverings , denoted , such that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{id:U\mapsto U\}\in cov(T)} for each object Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cat(T)}
- If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{U_i\to U\}\in cov(T)} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\to U} is any morphism in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cat(T)} , then the canonical morphisms of the fiber products determine a covering Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{U_i\times_U V\to V\}\in cov(T)}
- If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{U_i\to U\}\in cov(T)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{V_{i,j}\to U_i\}\in cov(T)} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{V_{i,j}\to U_i\to U\}\in cov(T)}
Examples
- A standard topological space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} becomes a category Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle op(X)} when you regard the open subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} as objects, and morphisms are inclusions. An open covering of open subsets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} clearly verify the axioms above for coverings in a site. Notice that a presheaf of rings is just a contravariant functor from the category Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle op(X)} into the category of rings.
- The Small Étale Site Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} be a scheme. Then the category of étale schemes over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} (i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} -schemes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} whose structural morphisms are étale) becomes a site if we require that coverings are jointly surjective; that is,
Sheaves on Sites
In analogy with the situation for topological spaces, a presheaf may be defined as a contravariant functor such that for all coverings Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{U_i\to U\}\in cov(T)} , the diagram Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\to F(U)\to \prod F(U_i)\to \prod F(U_i\times_U U_j)} is exact.