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As explained in this question:

Classifying Quasi-coherent Sheaves on Projective Schemes,

given a quasi-coherent sheaf $\mathcal{F}$ on $\mathbb{P}^n$, we have an isomorphism $\Gamma_*(\mathcal{F})^\tilde{}\cong \mathcal{F}$ while given a graded $A[x_0,...,x_n]$ module $M$, we don't always have $M \cong \Gamma_*(M^\tilde{}).$ Torsion module will be a counterexample.

In general, we have an equivalence of category between

(category of quasi-coherent sheaves on $\mathbb{P}^n$)
$\cong$ (category of graded $A[x_0,...,x_n]$ modules M)/(category of torsion modules)

where (category of torsion modules) is a Serre subcategory.

I am not so familiar with categorical quotients. My question is how to translate this categorical result into down-to-earth language? In particular, given two quasi-coherent sheaves $M_1\tilde{}$ and $M_2\tilde{}$ what is a necessary and sufficient condition for $M_1$ and $M_2$ such that $M_1\tilde{} \cong M_2\tilde{}$?

KReiser
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Mathstudent
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1 Answers1

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If there is a morphism $f: M \rightarrow N $ then a necessary and sufficient condition for $f$ to be an isomorphism is kernel $f$ and cokernel $f$ are both torsion modules.

Mathstudent
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