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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