The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

17.23 Rank and determinant

Let $(X, \mathcal{O}_ X)$ be a ringed space. Consider the category $\mathcal{C}$ of finite locally free $\mathcal{O}_ X$-modules. This is an exact category (see Injectives, Remark 19.9.6) whose admissible epimorphisms are surjections and whose admissible monomorphisms are kernels of surjections. Moreover, there is a set of isomorphism classes of objects of $\mathcal{C}$ (proof omitted). Thus we can form the Grothendieck $K$-group $K(\mathcal{C})$, which is often denoted $K_0^{naive}(X)$. Explicitly, in this case $K_0^{naive}(X)$ is the abelian group generated by $[\mathcal{E}]$ for $\mathcal{E}$ a finite locally free $\mathcal{O}_ X$-module, subject to the relations

\[ [\mathcal{E}'] = [\mathcal{E}] + [\mathcal{E}''] \]

whenever there is a short exact sequence $0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{E}'' \to 0$ of finite locally free $\mathcal{O}_ X$-modules.

Ranks. Given a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$, the rank is a locally constant function

\[ r = r_\mathcal {E} : X \longrightarrow \mathbf{Z}_{\geq 0},\quad x \longmapsto \text{rank}_{\mathcal{O}_{X, x}} \mathcal{E}_ x \]

This makes sense as $\mathcal{E}_ x \cong \mathcal{O}_{X, x}^{\oplus r(x)}$ and this uniquely determines $r(x)$. By definition of locally free modules the function $r$ is locally constant. If $0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{E}'' \to 0$ is a short exact sequence of finite locally free $\mathcal{O}_ X$-modules, then $r_\mathcal {E} = r_{\mathcal{E}'} + r_{\mathcal{E}''}$, Thus the rank defines a homomorphism

\[ K_0^{naive}(X) \longrightarrow \text{Map}_{cont}(X, \mathbf{Z}),\quad [\mathcal{E}] \longmapsto r_\mathcal {E} \]

Determinants. Given a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ we obtain a disjoint union decomposition

\[ X = X_0 \amalg X_1 \amalg X_2 \amalg \ldots \]

with $X_ i$ open and closed, such that $\mathcal{E}$ is finite locally free of rank $i$ on $X_ i$ (this is exactly the same as saying the $r_\mathcal {E}$ is locally constant). In this case we define $\det (\mathcal{E})$ as the invertible sheaf on $X$ which is equal to $\wedge ^ i(\mathcal{E}|_{X_ i})$ on $X_ i$ for all $i \geq 0$. Since the decomposition above is disjoint, there are no glueing conditions to check. By Lemma 17.23.1 below this defines a homomorphism

\[ \det : K_0^{naive}(X) \longrightarrow \mathop{\mathrm{Pic}}\nolimits (X),\quad [\mathcal{E}] \longmapsto \det (\mathcal{E}) \]

of abelian groups.

Lemma 17.23.1. Let $X$ be a ringed space. Let $0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{E}'' \to 0$ be a short exact sequence of finite locally free $\mathcal{O}_ X$-modules, Then there is a canonical isomorphism

\[ \det (\mathcal{E}') \otimes _{\mathcal{O}_ X}\det (\mathcal{E}'') \longrightarrow \det (\mathcal{E}) \]

of $\mathcal{O}_ X$-modules.

Proof. We can decompose $X$ into disjoint open and closed subsets such that both $\mathcal{E}'$ and $\mathcal{E}''$ have constant rank on them. Thus we reduce to the case where $\mathcal{E}'$ and $\mathcal{E}''$ have constant rank, say $r'$ and $r''$. In this situation we define

\[ \wedge ^{r'}(\mathcal{E}') \otimes _{\mathcal{O}_ X} \wedge ^{r''}(\mathcal{E}'') \longrightarrow \wedge ^{r' + r''}(\mathcal{E}) \]

as follows. Given local sections $s'_1, \ldots , s'_{r'}$ of $\mathcal{E}'$ and local sections $s''_1, \ldots , s''_{r''}$ of $\mathcal{E}''$ we map

\[ s'_1 \wedge \ldots \wedge s'_{r'} \otimes s''_1 \wedge \ldots \wedge s''_{r''} \quad \text{to}\quad s'_1 \wedge \ldots \wedge s'_{r'} \wedge \tilde s''_1 \wedge \ldots \wedge \tilde s''_{r''} \]

where $\tilde s''_ i$ is a local lift of the section $s''_ i$ to a section of $\mathcal{E}$. We omit the details. $\square$

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