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Tag 03DG

18.18. Intrinsic properties of modules

Let $\mathcal{P}$ be a property of sheaves of modules on ringed topoi. We say $\mathcal{P}$ is an intrinsic property if we have $\mathcal{P}(\mathcal{F}) \Leftrightarrow \mathcal{P}(f^*\mathcal{F})$ whenever $(f, f^\sharp) : (\mathop{\textit{Sh}}\nolimits(\mathcal{C}'), \mathcal{O}') \to (\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O})$ is an equivalence of ringed topoi. For example, the property of being free is intrinsic. Indeed, the free $\mathcal{O}$-module on the set $I$ is characterized by the property that $$ \mathop{\rm Mor}\nolimits_{\textit{Mod}(\mathcal{O})}( \bigoplus\nolimits_{i \in I} \mathcal{O}, \mathcal{F}) = \prod\nolimits_{i \in I} \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(\mathcal{C})}(\{*\}, \mathcal{F}) $$ for a variable $\mathcal{F}$ in $\textit{Mod}(\mathcal{O})$. Alternatively, we can also use Lemma 18.17.2 to see that being free is intrinsic. In fact, each of the properties defined in Definition 18.17.1 is intrinsic for the same reason. How will we go about defining other intrinsic properties of $\mathcal{O}$-modules?

The upshot of Lemma 18.7.2 is the following: Suppose you want to define an intrinsic property $\mathcal{P}$ of an $\mathcal{O}$-module on a topos. Then you can proceed as follows:

  1. Given any site $\mathcal{C}$, any sheaf of rings $\mathcal{O}$ on $\mathcal{C}$ and any $\mathcal{O}$-module $\mathcal{F}$ define the corresponding property $\mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F})$.
  2. For any pair of sites $\mathcal{C}$, $\mathcal{C}'$, any special cocontinuous functor $u : \mathcal{C} \to \mathcal{C}'$, any sheaf of rings $\mathcal{O}$ on $\mathcal{C}$ any $\mathcal{O}$-module $\mathcal{F}$, show that $$ \mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F}) \Leftrightarrow \mathcal{P}(\mathcal{C}', g_*\mathcal{O}, g_*\mathcal{F}) $$ where $g : \mathop{\textit{Sh}}\nolimits(\mathcal{C}) \to \mathop{\textit{Sh}}\nolimits(\mathcal{C}')$ is the equivalence of topoi associated to $u$.

In this case, given any ringed topos $(\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O})$ and any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ we simply say that $\mathcal{F}$ has property $\mathcal{P}$ if $\mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F})$ is true. And Lemma 18.7.2 combined with (2) above guarantees that this is well defined.

Moreover, the same Lemma 18.7.2 also guarantees that if in addition

  1. (3)    For any morphism of ringed sites $(f, f^\sharp) : (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D})$ such that $f$ is given by a functor $u : \mathcal{D} \to \mathcal{C}$ satisfying the assumptions of Sites, Proposition 7.14.6, and any $\mathcal{O}_\mathcal{D}$-module $\mathcal{G}$ we have $$ \mathcal{P}(\mathcal{D}, \mathcal{O}_\mathcal{D}, \mathcal{F}) \Rightarrow \mathcal{P}(\mathcal{C}, \mathcal{O}_\mathcal{C}, f^*\mathcal{F}) $$

then it is true that $\mathcal{P}$ is preserved under pullback of modules w.r.t. arbitrary morphisms of ringed topoi.

We will use this method in the following sections to see that: locally free, locally generated by sections, locally generated by $r$ sections, finite type, finite presentation, quasi-coherent, and coherent are intrinsic properties of modules.

Perhaps a more satisfying method would be to find an intrinsic definition of these notions, rather than the laborious process sketched here. On the other hand, in many geometric situations where we want to apply these definitions we are given a definite ringed site, and a definite sheaf of modules, and it is nice to have a definition already adapted to this language.

    The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 1875–1982 (see updates for more information).

    \section{Intrinsic properties of modules}
    \label{section-intrinsic}
    
    \noindent
    Let $\mathcal{P}$ be a property of sheaves of modules on ringed topoi.
    We say $\mathcal{P}$ is an {\it intrinsic property} if we have
    $\mathcal{P}(\mathcal{F}) \Leftrightarrow \mathcal{P}(f^*\mathcal{F})$
    whenever $(f, f^\sharp) :
    (\Sh(\mathcal{C}'), \mathcal{O}')
    \to
    (\Sh(\mathcal{C}), \mathcal{O})$
    is an equivalence of ringed topoi.
    For example, the property of being free is intrinsic. Indeed, the free
    $\mathcal{O}$-module on the set $I$ is characterized by the property
    that
    $$
    \Mor_{\textit{Mod}(\mathcal{O})}(
    \bigoplus\nolimits_{i \in I} \mathcal{O},
    \mathcal{F})
    =
    \prod\nolimits_{i \in I} \Mor_{\Sh(\mathcal{C})}(\{*\},
    \mathcal{F})
    $$
    for a variable $\mathcal{F}$ in $\textit{Mod}(\mathcal{O})$.
    Alternatively, we can also use Lemma \ref{lemma-global-pullback}
    to see that being free is intrinsic. In fact, each of the properties
    defined in Definition \ref{definition-global} is intrinsic for the
    same reason.
    How will we go about defining other intrinsic properties of
    $\mathcal{O}$-modules?
    
    \medskip\noindent
    The upshot of
    Lemma \ref{lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}
    is the following: Suppose you want to define
    an intrinsic property $\mathcal{P}$ of an $\mathcal{O}$-module on a
    topos. Then you can proceed as follows:
    \begin{enumerate}
    \item Given any site $\mathcal{C}$, any sheaf of rings $\mathcal{O}$
    on $\mathcal{C}$ and any $\mathcal{O}$-module $\mathcal{F}$
    define the corresponding
    property $\mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F})$.
    \item For any pair of sites $\mathcal{C}$, $\mathcal{C}'$, any
    special cocontinuous functor $u : \mathcal{C} \to \mathcal{C}'$,
    any sheaf of rings $\mathcal{O}$ on $\mathcal{C}$ any
    $\mathcal{O}$-module $\mathcal{F}$, show that
    $$
    \mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F})
    \Leftrightarrow
    \mathcal{P}(\mathcal{C}', g_*\mathcal{O}, g_*\mathcal{F})
    $$
    where $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$
    is the equivalence of topoi associated to $u$.
    \end{enumerate}
    In this case, given any ringed topos $(\Sh(\mathcal{C}), \mathcal{O})$
    and any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ we simply say that
    $\mathcal{F}$ has property $\mathcal{P}$ if
    $\mathcal{P}(\mathcal{C}, \mathcal{O}, \mathcal{F})$ is true.
    And Lemma \ref{lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}
    combined with (2) above guarantees that this is well defined.
    
    \medskip\noindent
    Moreover, the same
    Lemma \ref{lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}
    also guarantees that if in addition
    \begin{enumerate}
    \item[(3)] For any morphism of ringed sites
    $(f, f^\sharp) :
    (\mathcal{C}, \mathcal{O}_\mathcal{C})
    \to
    (\mathcal{D}, \mathcal{O}_\mathcal{D})$
    such that $f$ is given by a functor
    $u : \mathcal{D} \to \mathcal{C}$ satisfying the
    assumptions of Sites, Proposition \ref{sites-proposition-get-morphism},
    and any $\mathcal{O}_\mathcal{D}$-module $\mathcal{G}$
    we have
    $$
    \mathcal{P}(\mathcal{D}, \mathcal{O}_\mathcal{D}, \mathcal{F})
    \Rightarrow
    \mathcal{P}(\mathcal{C}, \mathcal{O}_\mathcal{C}, f^*\mathcal{F})
    $$
    \end{enumerate}
    then it is true that $\mathcal{P}$ is preserved under pullback
    of modules w.r.t.\ arbitrary morphisms of ringed topoi.
    
    \medskip\noindent
    We will use this method in the following sections to see
    that:
    locally free,
    locally generated by sections,
    locally generated by $r$ sections,
    finite type,
    finite presentation,
    quasi-coherent, and
    coherent
    are intrinsic properties of modules.
    
    \medskip\noindent
    Perhaps a more satisfying method would be to find an intrinsic definition
    of these notions, rather than the laborious process sketched here.
    On the other hand, in many geometric situations where we want to apply
    these definitions we are given a definite ringed site, and a definite
    sheaf of modules, and it is nice to have a definition already adapted to
    this language.

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