# Restriction of Ext sheaves on closed subschemes

Let $$f:X\rightarrow C$$ be a morphism, where $$C$$ is a smooth curve. For $$t\in C$$ let $$i_t:X_t = f^{-1}(t)\rightarrow X$$ be the inclusion of the fiber of $$f$$ over $$t$$, and let $$\mathcal{F}$$ a coherent sheaf on $$X$$ that is flat over $$C$$.

Does there exist an isomorphism $$i_t^{*}\mathcal{E}xt^1(\mathcal{F},\mathcal{O}_X)\cong \mathcal{E}xt^1(i_t^{*}\mathcal{F},\mathcal{O}_{X_t})$$ ?

• Take $f =Id_C$ and $\mathcal{F} = \mathcal{O}_t$. The RHS vanishes whereas $\mathcal{E}xt^1(\mathcal{O}_t, \mathcal{O}_X) \simeq \mathcal{O}_t$. So there's no isomorphsm in general. – HYL Jan 13 '19 at 5:16
• We are assuming that $\mathcal{F}$ is a coherent sheaf on $X$ that is flat over $C$. – gxg Jan 13 '19 at 11:27
• The identity map is flat and the structure sheaf is flat, but if you want a more 'complicated' example how about $\mathcal{F} = \mathcal{O}_X$ and X = Proj(E) where E is a rank 2 vb over a smooth non-rational curve. – meh Jun 12 '19 at 20:05

• It is seems that in order to use Lemma 15.87.2 the flatness of the morphism is fundamental. In my case the inclusion $i_t:X_t\rightarrow X$ is not flat. – gxg Jan 12 '19 at 22:02