Exercise 111.48.1. Let $k$ be a field. Let $X = \mathbf{P}^3_ k$. Let $L \subset X$ and $P \subset X$ be a line and a plane, viewed as closed subschemes cut out by $1$, resp., $2$ linear equations. Compute the dimensions of for all $i$. Make sure to do both the case where $L$ is contained in $P$ and the case where $L$ is not contained in $P$.
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{O}_ L, \mathcal{O}_ P) \]
Exercise 111.48.2. Let $k$ be a field. Let $X = \mathbf{P}^ n_ k$. Let $Z \subset X$ be a closed $k$-rational point viewed as a closed subscheme. For example the point with homogeneous coordinates $(1 : 0 : \ldots : 0)$. Compute the dimensions of for all $i$.
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{O}_ Z, \mathcal{O}_ Z) \]
Exercise 111.48.3. Let $X$ be a ringed space. Define cup-product maps for $\mathcal{O}_ X$-modules $\mathcal{F}, \mathcal{G}, \mathcal{H}$. (Hint: this is a super general thing.)
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{G}, \mathcal{H}) \times \mathop{\mathrm{Ext}}\nolimits ^ j_ X(\mathcal{F}, \mathcal{G}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^{i + j}_ X(\mathcal{F}, \mathcal{H}) \]
Exercise 111.48.4. Let $X$ be a ringed space. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module with dual $\mathcal{E}^\vee = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{O}_ X)$. Prove the following statements
$\mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ i_{\mathcal{O}_ X}( \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{E}, \mathcal{G}) = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ i_{\mathcal{O}_ X}( \mathcal{F}, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \mathcal{G}) = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ i_{\mathcal{O}_ X}( \mathcal{F}, \mathcal{G}) \otimes _{\mathcal{O}_ X} \mathcal{E}^\vee $, and
$\mathop{\mathrm{Ext}}\nolimits ^ i_ X( \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{E}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{F}, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \mathcal{G})$.
Here $\mathcal{F}$ and $\mathcal{G}$ are $\mathcal{O}_ X$-modules. Conclude that
Exercise 111.48.5. Let $X$ be a ringed space. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. Construct a trace map for all $i$. Generalize to a trace map for any $\mathcal{O}_ X$-module $\mathcal{F}$.
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{E}, \mathcal{E}) \to H^ i(X, \mathcal{O}_ X) \]
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{E}, \mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{F}) \to H^ i(X, \mathcal{F}) \]
Exercise 111.48.6. Let $k$ be a field. Let $X = \mathbf{P}^ d_ k$. Set $\omega _{X/k} = \mathcal{O}_ X(-d - 1)$. Prove that for finite locally free modules $\mathcal{E}$, $\mathcal{F}$ the cup product on Ext combined with the trace map on Ext produces a nondegenerate pairing. Hint: you can either reprove duality in this setting or you can reduce to cohomology of sheaves and apply the Serre duality theorem as proved in the lectures.
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{E}, \mathcal{F} \otimes _{\mathcal{O}_ X} \omega _{X/k}) \times \mathop{\mathrm{Ext}}\nolimits ^{d - i}_ X(\mathcal{F}, \mathcal{E}) \to \mathop{\mathrm{Ext}}\nolimits _ X^ d(\mathcal{F}, \mathcal{F} \otimes _{\mathcal{O}_ X} \omega _{X/k}) \to H^ d(X, \omega _{X/k}) = k \]
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