The Stacks project

Lemma 37.33.1. Let $f : X \to S$ be a flat, proper morphism of finite presentation. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. For a morphism $g : T \to S$ consider the base change diagram

\[ \xymatrix{ X_ T \ar[d]_ p \ar[r]_ q & X \ar[d]^ f \\ T \ar[r]^ g & S } \]

Assume $\mathcal{O}_ T \to p_*\mathcal{O}_{X_ T}$ is an isomorphism for all $g : T \to S$. Then there exists an immersion $j : Z \to S$ of finite presentation such that a morphism $g : T \to S$ factors through $Z$ if and only if there exists a finite locally free $\mathcal{O}_ T$-module $\mathcal{N}$ with $p^*\mathcal{N} \cong q^*\mathcal{E}$.

Proof. Observe that the fibres $X_ s$ of $f$ are connected by our assumption that $H^0(X_ s, \mathcal{O}_{X_ s}) = \kappa (s)$. Thus the rank of $\mathcal{E}$ is constant on the fibres. Since $f$ is open (Morphisms, Lemma 29.25.10) and closed we conclude that there is a decomposition $S = \coprod S_ r$ of $S$ into open and closed subschemes such that $\mathcal{E}$ has constant rank $r$ on the inverse image of $S_ r$. Thus we may assume $\mathcal{E}$ has constant rank $r$. We will denote $\mathcal{E}^\vee = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{E}, \mathcal{O}_ X)$ the dual rank $r$ module.

By cohomology and base change (more precisely by Derived Categories of Schemes, Lemma 36.30.4) we see that $E = Rf_*\mathcal{E}$ is a perfect object of the derived category of $S$ and that its formation commutes with arbitrary change of base. Similarly for $E' = Rf_*\mathcal{E}^\vee $. Since there is never any cohomology in degrees $< 0$, we see that $E$ and $E'$ have (locally) tor-amplitude in $[0, b]$ for some $b$. Observe that for any $g : T \to S$ we have $p_*(q^*\mathcal{E}) = H^0(Lg^*E)$ and $p_*(q^*\mathcal{E}^\vee ) = H^0(Lg^*E')$. Let $j : Z \to S$ and $j' : Z' \to S$ be immersions of finite presentation constructed in Derived Categories of Schemes, Lemma 36.31.4 for $E$ and $E'$ with $a = 0$ and $r = r$; these are roughly speaking characterized by the property that $H^0(Lj^*E)$ and $H^0((j')^*E')$ are finite locally free modules compatible with pullback.

Let $g : T \to S$ be a morphism. If there exists an $\mathcal{N}$ as in the lemma, then, using the projection formula Cohomology, Lemma 20.54.2, we see that the modules

\[ p_*(q^*\mathcal{L}) \cong p_*(p^*\mathcal{N}) \cong \mathcal{N} \otimes _{\mathcal{O}_ T} p_*\mathcal{O}_{X_ T} \cong \mathcal{N}\quad \text{and similarly }\quad p_*(q^*\mathcal{E}^\vee ) \cong \mathcal{N}^\vee \]

are finite locally free modules of rank $r$ and remain so after any further base change $T' \to T$. Hence in this case $T \to S$ factors through $j$ and through $j'$. Thus we may replace $S$ by $Z \times _ S Z'$ and assume that $f_*\mathcal{E}$ and $f_*\mathcal{E}^\vee $ are finite locally free $\mathcal{O}_ S$-modules of rank $r$ whose formation commutes with arbitrary change of base (small detail omitted).

In this situation if $g : T \to S$ be a morphism and there exists an $\mathcal{N}$ as in the lemma, then the map (cup product in degree $0$)

\[ p_*(q^*\mathcal{E}) \otimes _{\mathcal{O}_ T} p_*(q^*\mathcal{E}^\vee ) \longrightarrow \mathcal{O}_ T \]

is a perfect pairing. Conversely, if this cup product map is a perfect pairing, then we see that locally on $T$ we may choose a basis of sections $\sigma _1, \ldots , \sigma _ r$ in $p_*(q^*\mathcal{E})$ and $\tau _1, \ldots , \tau _ r$ in $p_*(q^*\mathcal{E}^\vee )$ whose products satisfy $\sigma _ i \tau _ j = \delta _{ij}$. Thinking of $\sigma _ i$ as a section of $q^*\mathcal{E}$ on $X_ T$ and $\tau _ j$ as a section of $q^*\mathcal{E}^\vee $ on $X_ T$, we conclude that

\[ \sigma _1, \ldots , \sigma _ r : \mathcal{O}_{X_ T}^{\oplus r} \longrightarrow q^*\mathcal{E} \]

is an isomorphism with inverse given by

\[ \tau _1, \ldots , \tau _ r : q^*\mathcal{E} \longrightarrow \mathcal{O}_{X_ T}^{\oplus r} \]

In other words, we see that $p^*p_*q^*\mathcal{E} \cong q^*\mathcal{E}$. But the condition that the cupproduct is nondegenerate picks out a retrocompact open subscheme (namely, the locus where a suitable determinant is nonzero) and the proof is complete. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BDP. Beware of the difference between the letter 'O' and the digit '0'.