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The Stacks project

Remark 31.12.9. Let X be an integral locally Noetherian scheme. Thanks to Lemma 31.12.8 we know that the reflexive hull \mathcal{F}^{**} of a coherent \mathcal{O}_ X-module is coherent reflexive. Consider the category \mathcal{C} of coherent reflexive \mathcal{O}_ X-modules. Taking reflexive hulls gives a left adjoint to the inclusion functor \mathcal{C} \to \textit{Coh}(\mathcal{O}_ X). Observe that \mathcal{C} is an additive category with kernels and cokernels. Namely, given \varphi : \mathcal{F} \to \mathcal{G} in \mathcal{C}, the usual kernel \mathop{\mathrm{Ker}}(\varphi ) is reflexive (Lemma 31.12.7) and the reflexive hull \mathop{\mathrm{Coker}}(\varphi )^{**} of the usual cokernel is the cokernel in \mathcal{C}. Moreover \mathcal{C} inherits a tensor product

\mathcal{F} \otimes _\mathcal {C} \mathcal{G} = (\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G})^{**}

which is associative and symmetric. There is an internal Hom in the sense that for any three objects \mathcal{F}, \mathcal{G}, \mathcal{H} of \mathcal{C} we have the identity

\mathop{\mathrm{Hom}}\nolimits _\mathcal {C}(\mathcal{F} \otimes _\mathcal {C} \mathcal{G}, \mathcal{H}) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {C}(\mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{H}))

see Modules, Lemma 17.22.1. In \mathcal{C} every object \mathcal{F} has a dual object \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{O}_ X). Without further conditions on X it can happen that

\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \not\cong \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{O}_ X) \otimes _\mathcal {C} \mathcal{G} \quad \text{and}\quad \mathcal{F} \otimes _\mathcal {C} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{O}_ X) \not\cong \mathcal{O}_ X

for \mathcal{F}, \mathcal{G} of rank 1 in \mathcal{C}. To make an example let X = \mathop{\mathrm{Spec}}(R) where R is as in More on Algebra, Example 15.23.17 and let \mathcal{F}, \mathcal{G} be the modules corresponding to M. Computation omitted.


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