Equivariant Constructible Perverse Foundations Examples

The group ${\mathbb{C}}^{\times }$ acts trivially on $\mathrm{pt}$, and the frame space is $U_N={\mathbb{C}}^N\setminus \{0\}$; the quotient $U_N\mathrm{/}{\mathbb{C}}^{\times }$ is the space of complex lines in ${\mathbb{C}}^N$, i.e. ${\mathbb{P}}^{N-1}$. Under the resolution triple, the constant equivariant object $k$ corresponds to the pair $(k,k_{{\mathbb{P}}^{N-1}})$ because the pullback of $k$ along $U_N\to \mathrm{pt}$ is the constant sheaf $k_{U_N}$, which descends along the ${\mathbb{C}}^{\times }$-bundle to $k_{{\mathbb{P}}^{N-1}}$. So $Q_N(k)=k_{X_N}=k_{{\mathbb{P}}^{N-1}}$.

The finite cells. Use the filtration ${\mathbb{P}}^0\subset {\mathbb{P}}^1\subset \cdots \subset {\mathbb{P}}^{N-1}$ by coordinate subspaces. For $j\ge 1$, the map from the closed unit ball in ${\mathbb{C}}^j$ to ${\mathbb{P}}^j$ given by $v\mapsto [v:\sqrt{1-\parallel v{\parallel }^2}]$ sends its boundary to ${\mathbb{P}}^{j-1}$ by $v\mapsto [v:0]$, and is a homeomorphism on the interior to the affine chart complementary to ${\mathbb{P}}^{j-1}$; its inverse chart coordinate is $z\mapsto z\mathrm{/}\sqrt{1+\parallel z{\parallel }^2}$. The adjunction space formed by attaching this ball to ${\mathbb{P}}^{j-1}$ maps continuously and bijectively to ${\mathbb{P}}^j$. It is compact and the target is Hausdorff by [F3], so this map is a homeomorphism. Induction gives a finite CW structure with one cell in each degree $0,2,\dots ,2(N-1)$ and no odd cells. Its cellular cochain differentials vanish; [F4] consequently gives singular cohomology $k$ in those even degrees and zero otherwise. Projective space is metrizable (embed a line as its orthogonal projection matrix) and its smooth ball charts give contractible-open bases; [F7] identifies this computation with constant-sheaf cohomology and hence with $\mathrm{Hom}(k,k[m])$.

Fix $m\ge 0$. Choose $N$ with $r_N=2(N-1)\ge m$, so that the comparison range of [F2] applies; then ${\mathrm{Ext}}_G^m(k,k)\cong {\mathrm{Hom}}_{{\mathbb{P}}^{N-1}}(k,k[m])\cong H^m({\mathbb{P}}^{N-1};k)$ by [F7] and step 1.2, which is $k$ for even $m$ and $0$ for odd $m$. For $m<0$ choose any $N$ with $2(N-1)\ge \mathrm{\mid }m\mathrm{\mid }$; the same comparison applies and $H^m({\mathbb{P}}^{N-1};k)=0$ because a CW complex has no negative cohomology. Hence ${\mathrm{Ext}}_G^m(k,k)\cong k$ for even $m\ge 0$ and vanishes otherwise, which is the displayed computation.

The computation of step 2.1 is degreewise finite: each group is either $k$ or $0$. Infinitely many even degrees are nonzero, namely all $m=2j$ with $j\ge 0$, so the total direct sum ${⨁}_m{\mathrm{Ext}}_G^m(k,k)$ contains a copy of $k$ in infinitely many degrees and is infinite-dimensional.

On the ordinary point, [F6] gives ${\mathrm{Hom}}_{D^b(\mathrm{pt})}(k,k[2])=0$, while the equivariant group ${\mathrm{Ext}}_G^2(k,k)$ is $k$ by step 2.1. The restriction functor to the ordinary point would have to induce the comparison map ${\mathrm{Ext}}_G^2(k,k)\to {\mathrm{Hom}}_{\mathrm{pt}}(k,k[2])$ on these Hom groups, and this map cannot be injective because its target is zero and its source is nonzero; hence restriction is not derived fully faithful. This proves all three assertions. AC is present only as recorded in [A1].