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Definition of line bundle associated to a Cartier Divisor.

Properties of quotient sheavesEffective Cartier divisorsDefinition of Cartier divisorsQuestions about the definition of a Cartier Divisor from Liu pg 256Cartier Divisor corresponds on $X$ to $(U_i,f_i)$Do I correctly understand the constructions involved in definition of Cartier divisor?Sheaf associated to a Cartier divisorRepresentation of an effective Cartier divisor: is there an imprecision in Liu's book?Using definition of Cartier divisors on $mathbb{P}^1$.Constructing an invertible sheaf from a Cartier divisor?

1

$begingroup$

Let $D$ be a Cartier Divisor on a scheme $X$ represented by ${(U_i, f_i)}$ where $U_i$ is an open cover on $X$, $f_i in Gamma(U_i, mathcal{K}^*)$ and such that $f_i/f_j in Gamma(U_i cap U_j, mathcal{O}^*)$.

We define the sheaf associated to $D$ denoted by $mathcal{L}(D)$ to be the sub $mathcal{O}_X$-module of $mathcal{K}$ generated by $f_i^{-1}$ on $U_i$.

Why take the inverse here?

It seems that $D$ as described already is a line bundle via the map $mathcal{O}_X vert_{U_i} to mathcal{D} vert_{U_i}$ defined as $1 mapsto f_i$.

algebraic-geometry

share|cite|improve this question

asked Mar 12 at 21:37

JadwigaJadwiga

2,07811025

$endgroup$

add a comment | 

1

$begingroup$

Let $D$ be a Cartier Divisor on a scheme $X$ represented by ${(U_i, f_i)}$ where $U_i$ is an open cover on $X$, $f_i in Gamma(U_i, mathcal{K}^*)$ and such that $f_i/f_j in Gamma(U_i cap U_j, mathcal{O}^*)$.

We define the sheaf associated to $D$ denoted by $mathcal{L}(D)$ to be the sub $mathcal{O}_X$-module of $mathcal{K}$ generated by $f_i^{-1}$ on $U_i$.

Why take the inverse here?

It seems that $D$ as described already is a line bundle via the map $mathcal{O}_X vert_{U_i} to mathcal{D} vert_{U_i}$ defined as $1 mapsto f_i$.

algebraic-geometry

share|cite|improve this question

asked Mar 12 at 21:37

JadwigaJadwiga

2,07811025

$endgroup$

add a comment | 

$begingroup$

Let $D$ be a Cartier Divisor on a scheme $X$ represented by ${(U_i, f_i)}$ where $U_i$ is an open cover on $X$, $f_i in Gamma(U_i, mathcal{K}^*)$ and such that $f_i/f_j in Gamma(U_i cap U_j, mathcal{O}^*)$.

We define the sheaf associated to $D$ denoted by $mathcal{L}(D)$ to be the sub $mathcal{O}_X$-module of $mathcal{K}$ generated by $f_i^{-1}$ on $U_i$.

Why take the inverse here?

It seems that $D$ as described already is a line bundle via the map $mathcal{O}_X vert_{U_i} to mathcal{D} vert_{U_i}$ defined as $1 mapsto f_i$.

algebraic-geometry

share|cite|improve this question

asked Mar 12 at 21:37

JadwigaJadwiga

2,07811025

$endgroup$

share|cite|improve this question

asked Mar 12 at 21:37

JadwigaJadwiga

2,07811025

share|cite|improve this question

asked Mar 12 at 21:37

JadwigaJadwiga

2,07811025

asked Mar 12 at 21:37

JadwigaJadwiga

2,07811025

asked Mar 12 at 21:37

JadwigaJadwiga

2,07811025