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Let $\mathcal{P}$ be the category of perverse sheaves on $\mathbb{P}^1$ over the field $\mathbb{C}$, where strata are the point $Z = {0}$, and its complement $U$. Let $i: Z \rightarrow \mathbb{P}^1$ and $j: U \rightarrow \mathbb{P}^1$ be the inclusions. Then I have found that the simple objects in the category $\mathcal{P}$ are $\underline{\mathbb{C}}_{\mathbb{P}^1}[1]$, and $i_{*}\underline{\mathbb{C}}_Z$.

  • Why does the object $j_{*} \underline{\mathbb{C}}_U[1] $ lie in $\mathcal{P}$?
  • What are ext-groups between the two simple objects in $\mathcal{P}$?
  • What are the projective covers of the two simple objects in $\mathcal{P}$?
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