The Stacks project

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 $P$.

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 $i$.

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.)

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

1. $\mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ i_{\mathcal{O}_ X}( \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{E}, \mathcal{G}) = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ i_{\mathcal{O}_ X}( \mathcal{F}, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \mathcal{G}) = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ i_{\mathcal{O}_ X}( \mathcal{F}, \mathcal{G}) \otimes _{\mathcal{O}_ X} \mathcal{E}^\vee$, and

2. $\mathop{\mathrm{Ext}}\nolimits ^ i_ X( \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{E}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{F}, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \mathcal{G})$.

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 $\mathcal{F}$.

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.