Let $\mathcal{C}$ be a cocomplete linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of a dualizable object $V \in \mathcal{C}$ is the composite

$1 \xrightarrow{\mathrm{coev}} V \otimes \check{V} \cong \check{V} \otimes V \xrightarrow{\mathrm{ev}} 1.$

For example, if $\mathcal{C}=\mathsf{Qcoh}(X)$ for some scheme $X$, then the rank of a locally free sheaf $V$ coincides with the usual locally constant function which is called the rank. Of course, $V=0$ when the rank of $V$ is zero.

Question. In general, if the rank of a dualizable object $V$ is zero, do we have $V=0$? If not, what is a simple counterexample $(\mathcal{C},V)$?

For example this is true when $\mathcal{C}$ Tannakian (Deligne, Catégories tannakiennes, Lemme 7.3), and more generally when $\mathcal{C}$ is weakly Tannakian (see Schäppi's paper arXiv:1312.6358). But this property seems to be so intuitive that it "should" hold in more general situations.

Let $\mathcal{C}$ be a cocomplete $\mathbb{Q}$-linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of a dualizable object $V \in \mathcal{C}$ is the composite

$1 \xrightarrow{\mathrm{coev}} V \otimes \check{V} \cong \check{V} \otimes V \xrightarrow{\mathrm{ev}} 1.$

For example, if $\mathcal{C}=\mathsf{Qcoh}(X)$ for some scheme $X$, then the rank of a locally free sheaf $V$ coincides with the usual locally constant function which is called the rank, multiplied with $1$ in $\Gamma(X,\mathcal{O}_X)$. Of course, $V=0$ when the rank of $V$ is zero.

Question. In general, if the rank of a dualizable object $V$ is zero, do we have $V=0$? If not, what is a simple counterexample $(\mathcal{C},V)$?

For example this is true when $\mathcal{C}$ Tannakian (Deligne, Catégories tannakiennes, Lemme 7.3), and more generally when $\mathcal{C}$ is weakly Tannakian (see Schäppi's paper arXiv:1312.6358). But this property seems to be so intuitive that it "should" hold in more general situations.

Let $\mathcal{C}$ be a cocomplete linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of a dualizable object $V \in \mathcal{C}$ is the composite

$1 \xrightarrow{\mathrm{coev}} V \otimes \check{V} \cong \check{V} \otimes V \xrightarrow{\mathrm{ev}} 1.$

For example, if $\mathcal{C}=\mathsf{Qcoh}(X)$ for some scheme $X$, then the rank of a locally free sheaf $V$ coincides with the usual locally constant function which is called the rank. Of course, $V=0$ whenthe rank of $V$ is zero.

Question. In general, if the rank of a dualizable object $V$ is zero, do we have $V=0$?

For example this is true when $\mathcal{C}$ Tannakian (Deligne, Catégories tannakiennes, Lemme 7.3), and more generally when $\mathcal{C}$ is weakly Tannakian (see Schäppi's paper arXiv:1312.6358). But this property seems to be so intuitive that it "should" hold in more general situations.

Let $\mathcal{C}$ be a cocomplete linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of a dualizable object $V \in \mathcal{C}$ is the composite

$1 \xrightarrow{\mathrm{coev}} V \otimes \check{V} \cong \check{V} \otimes V \xrightarrow{\mathrm{ev}} 1.$

For example, if $\mathcal{C}=\mathsf{Qcoh}(X)$ for some scheme $X$, then the rank of a locally free sheaf $V$ coincides with the usual locally constant function which is called the rank. Of course, $V=0$ when the rank of $V$ is zero.

Question. In general, if the rank of a dualizable object $V$ is zero, do we have $V=0$? If not, what is a simple counterexample $(\mathcal{C},V)$?

For example this is true when $\mathcal{C}$ Tannakian (Deligne, Catégories tannakiennes, Lemme 7.3), and more generally when $\mathcal{C}$ is weakly Tannakian (see Schäppi's paper arXiv:1312.6358). But this property seems to be so intuitive that it "should" hold in more general situations.