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 (1)
Comment #10082 by ZL on