2 TeX

Let X $X$ be an arbitrary scheme. A quasi coherent sheaf \cal F $\cal F$ is said to be injective if Hom_{ $Hom_{ O_X}(-, \cal F) F)$ is exact. We can also regard a quasi coherent sheaf \cal G $\cal G$ on X $X$ such that for all open subset U $U$ of X, \cal G(U) $X$, $\cal G(U)$ is an injective \cal O_X-module$\cal O_X$-module. So we can ask a question that

1)Is

1) Is there any relation between these tow two kind of sheaves?

2)Which

2) Which conditions on X $X$ (or on\cal Fon $\cal F$) are needed to regard the first kink kind of these sheaves (\cal F$\cal F$) equivalent to the second one?

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# componentwise injective quasi coherent sheaves

Let X be an arbitrary scheme. A quasi coherent sheaf \cal F is said to be injective if Hom_{ O_X}(-, \cal F) is exact. We can also regard a quasi coherent sheaf \cal G on X such that for all open subset U of X, \cal G(U) is an injective \cal O_X-module. So we can ask a question that

1)Is there any relation between these tow kind of sheaves?

2)Which conditions on X (or on\cal F) are needed to regard the first kink of these sheaves (\cal F) equivalent to the second one?