Let $k$ be an algebraically closed field, $T$ a $k$-scheme (can assume connected) and $X$ a projective variety over $k$. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k T$ flat over $T$. Assume further that the dual $\mathcal{H}om_{\mathcal{O}_{X_T}}(\mathcal{F},\mathcal{O}_{X_T}) $ is flat over $T$, where $X_T:=X \times_k T$. Suppose for some point $t \in T$, the natural morphism $$\mathcal{H}om_{\mathcal{O}_{X_T}}(\mathcal{F},\mathcal{O}_{X_T}) \otimes k(t) \to \mathcal{H}om_{\mathcal{O}_{X_t}}(\mathcal{F}_t,\mathcal{O}_{X_t})$$ is an isomorphism (or surjection), where $k(t)$ is the residue field of $\mathcal{O}_{T,t}$, $X_t:=X \times_k t$ and $\mathcal{F}_t$ is the restriction of $\mathcal{F}$ to $X_t$. Then, does there exist an open neighbourhood $U$ of $t$ in $T$ such that for all $u \in U$, the induced morphism:$$\mathcal{H}om_{\mathcal{O}_{X_T}}(\mathcal{F},\mathcal{O}_{X_T}) \otimes k(u) \to \mathcal{H}om_{\mathcal{O}_{X_u}}(\mathcal{F}_u,\mathcal{O}_{X_u})$$ is an isomorphism (or surjection)?

EDIT: An equivalent (slightly vague) formulation of this question is, whether the sheaves $\operatorname{Ext}^i(\mathcal{F},m_u)$ is upper-semicontinuous in some sense, where $m_u$ is the maximal ideal $\mathcal{O}_{X_T,u}$?

  • $\begingroup$ Check out Bănică, Putinar, Schumacher, "Variation der globalen Ext in Deformationen kompakter komplexer Räume", Math. Ann. 250 (1980), in particular Satz 2. $\endgroup$ – pgraf Feb 6 '16 at 13:36

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