Infinite direct sum of sheaf of modules is same as sheaf formed by infinite direct sum of modules?
This is a statement made in Mumford, Oda Algebraic Geometry II.(Chpt 1, Sec 2. Prop 2.7) http://www.dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf
If $M_a$ is any collection of $R$ modules and denote $X=Spec(R)$, then $overline{oplus_aM_a}=oplus_aoverline{M_a}$ where $overline{M}$ denotes formation of sheaf of $O_X$ modules.
The proof goes along by using $Spec(R_f)$ affine which enforces quasi-compact. Then both sides of equation are canonically identified.(It is not hard to check this equality.) Then extension to a unique isomorphic sheaf by $mathcal{B}$-sheaf extension.
$textbf{Q:}$ Isn't this is sort of saying infinite direct sum of quasi-coherent sheaves of modules is quasi-coherent over affine scheme? Or am I misunderstanding the statement.
$textbf{Q':}$ What is the example of infinite direct sum of sheaves of free module over affine scheme fails to be a sheaf?
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 22:26
user45765
2,6222722
add a comment |
This is a statement made in Mumford, Oda Algebraic Geometry II.(Chpt 1, Sec 2. Prop 2.7) http://www.dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf
If $M_a$ is any collection of $R$ modules and denote $X=Spec(R)$, then $overline{oplus_aM_a}=oplus_aoverline{M_a}$ where $overline{M}$ denotes formation of sheaf of $O_X$ modules.
The proof goes along by using $Spec(R_f)$ affine which enforces quasi-compact. Then both sides of equation are canonically identified.(It is not hard to check this equality.) Then extension to a unique isomorphic sheaf by $mathcal{B}$-sheaf extension.
$textbf{Q:}$ Isn't this is sort of saying infinite direct sum of quasi-coherent sheaves of modules is quasi-coherent over affine scheme? Or am I misunderstanding the statement.
$textbf{Q':}$ What is the example of infinite direct sum of sheaves of free module over affine scheme fails to be a sheaf?
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 22:26
user45765
2,6222722
add a comment |
This is a statement made in Mumford, Oda Algebraic Geometry II.(Chpt 1, Sec 2. Prop 2.7) http://www.dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf
If $M_a$ is any collection of $R$ modules and denote $X=Spec(R)$, then $overline{oplus_aM_a}=oplus_aoverline{M_a}$ where $overline{M}$ denotes formation of sheaf of $O_X$ modules.
The proof goes along by using $Spec(R_f)$ affine which enforces quasi-compact. Then both sides of equation are canonically identified.(It is not hard to check this equality.) Then extension to a unique isomorphic sheaf by $mathcal{B}$-sheaf extension.
$textbf{Q:}$ Isn't this is sort of saying infinite direct sum of quasi-coherent sheaves of modules is quasi-coherent over affine scheme? Or am I misunderstanding the statement.
$textbf{Q':}$ What is the example of infinite direct sum of sheaves of free module over affine scheme fails to be a sheaf?
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 22:26
user45765
2,6222722
This is a statement made in Mumford, Oda Algebraic Geometry II.(Chpt 1, Sec 2. Prop 2.7) http://www.dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf
If $M_a$ is any collection of $R$ modules and denote $X=Spec(R)$, then $overline{oplus_aM_a}=oplus_aoverline{M_a}$ where $overline{M}$ denotes formation of sheaf of $O_X$ modules.
The proof goes along by using $Spec(R_f)$ affine which enforces quasi-compact. Then both sides of equation are canonically identified.(It is not hard to check this equality.) Then extension to a unique isomorphic sheaf by $mathcal{B}$-sheaf extension.
$textbf{Q:}$ Isn't this is sort of saying infinite direct sum of quasi-coherent sheaves of modules is quasi-coherent over affine scheme? Or am I misunderstanding the statement.
$textbf{Q':}$ What is the example of infinite direct sum of sheaves of free module over affine scheme fails to be a sheaf?
abstract-algebra algebraic-geometry commutative-algebra
abstract-algebra algebraic-geometry commutative-algebra
asked Jan 3 at 22:26
user45765
2,6222722
asked Jan 3 at 22:26
user45765
2,6222722
asked Jan 3 at 22:26
user45765
2,6222722
asked Jan 3 at 22:26
user45765
2,6222722
asked Jan 3 at 22:26
user45765
2,6222722
2,6222722
add a comment |
add a comment |
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