Equivariant Borel–Moore Correspondence Convolution and Constructible Ext — Examples

The two sources and the objects. The map $f_1:\mathrm{p}\mathrm{t}\to \mathrm{p}\mathrm{t}$ is proper with $d_1=0$ and $L_1=f_{1!}{k}_{\mathrm{p}\mathrm{t}}[0]=k$, and the map $f_2:{\mathbb{P}}^1\to \mathrm{p}\mathrm{t}$ is proper with $d_2=1$ and $L_2=f_{2!}{k}_{{\mathbb{P}}^1}[1]=R\mathrm{\Gamma }({\mathbb{P}}^1,k)[1]$, the last equality using $f_{2\ast }=f_{2!}$ for proper $f_2$. The fiber products are $Z_{12}=Z_{21}=\mathrm{p}\mathrm{t}{\times }_{\mathrm{p}\mathrm{t}}{\mathbb{P}}^1={\mathbb{P}}^1$.

The cohomology of the projective line. By [F3] the Schubert cells give ${\mathbb{P}}^1$ one $0$-cell and one $2$-cell, so its constant-coefficient cellular cochain complex is $k\oplus k[-2]$. By [F4] it computes singular cohomology, and [F5] identifies this with constant-sheaf cohomology: $H^0=H^2=k$ and the other groups vanish. A complex of vector spaces with these cohomology groups is quasi-isomorphic to $k\oplus k[-2]$: choose cocycles representing the two generators to obtain a cochain map from $k\oplus k[-2]$ to the complex inducing these cohomology isomorphisms. Thus $R\mathrm{\Gamma }({\mathbb{P}}^1,k)\simeq k\oplus k[-2]$.

The forward corner. For complexes of $k$-vector spaces one has $\mathrm{Hom}(k,C[n])=H^n(C)$, so ${\mathrm{Ext}}^r(L_1,L_2)=\mathrm{Hom}(k,R\mathrm{\Gamma }({\mathbb{P}}^1,k)[1+r])=H^{1+r}({\mathbb{P}}^1,k)$, which by step 1.2 is $k$ for $r=-1$ and for $r=1$ and is zero otherwise; in particular both nonzero groups are one-dimensional.

The reverse corner. Similarly ${\mathrm{Ext}}^r(L_2,L_1)=\mathrm{Hom}(R\mathrm{\Gamma }({\mathbb{P}}^1,k)[1],k[r])=\mathrm{Hom}(R\mathrm{\Gamma }({\mathbb{P}}^1,k),k[r-1])=\mathrm{Hom}(k\oplus k[-2],k[r-1])=\mathrm{Hom}(k,k[r-1])\oplus \mathrm{Hom}(k,k[r+1])$, which is $k$ exactly when $r=1$ or $r=-1$ and is zero otherwise; so both corners have the same two nonzero degrees.

The degree check. By [F1] the isomorphism ${\mathrm{\Phi }}_{12}^r$ identifies ${\mathrm{Ext}}^r(L_1,L_2)$ with $H_{d_1+d_2-r}^{\mathrm{B}\mathrm{M}}({\mathbb{P}}^1;k)=H_{1-r}^{\mathrm{B}\mathrm{M}}({\mathbb{P}}^1;k)$: the degree $r=-1$ corresponds to Borel–Moore degree $2$ and $r=1$ to Borel–Moore degree $0$, and the reverse corner $(2,1)$ gives the same two degrees. These values match the computation: by [F2] and step 1.2, $H_j^{\mathrm{B}\mathrm{M}}({\mathbb{P}}^1;k)={\mathrm{Hom}}_{D_c^b({\mathbb{P}}^1;k)}(k_{{\mathbb{P}}^1},k_{{\mathbb{P}}^1}[2-j])=H^{2-j}({\mathbb{P}}^1;k)$ is $k$ for $j=2$ and for $j=0$. The single-source shifts fail this comparison in both corners: $2{\dim }_{\mathbb{C}}{X}_1-r=-r$ is $1$ for $r=-1$ and $-1$ for $r=1$, and $2{\dim }_{\mathbb{C}}{X}_2-r=2-r$ is $3$ for $r=-1$ and $1$ for $r=1$, while the required degrees are $2$ and $0$. Hence the summed $d_1+d_2$ normalization is the one compatible with the two unequal source dimensions.

The geometry and the cohomology of the projective line. The fiber products are $Z_{12}=Z_{21}={\mathbb{P}}^1$. By [F3] its Schubert CW structure has one $0$-cell and one $2$-cell, so the cellular cochain complex is $k\oplus k[-2]$. By [F4] it computes singular cohomology, and [F6] identifies this with constant-sheaf cohomology. Choosing cocycles representing the two generators gives a quasi-isomorphism $k\oplus k[-2]\to R\mathrm{\Gamma }({\mathbb{P}}^1,k)$. Hence $L_2\simeq k[1]\oplus k[-1]$ and $L_1=k$.

The two classes and their composites. The unit $\alpha :k\to R\mathrm{\Gamma }({\mathbb{P}}^1,k)$ includes the degree-zero summand, generating ${\mathrm{Ext}}^{-1}(L_1,L_2)=k$. Evaluation at a point projects to that summand; its shift $\beta :L_2\to k[1]$ generates ${\mathrm{Ext}}^1(L_2,L_1)=k$, since $\mathrm{Hom}(k[-2],k)=0$. Therefore $\beta [-1]\circ \alpha ={\mathrm{id}}_k$. The reverse composite $\alpha [1]\circ \beta$ is also nonzero: it is the projector of $L_2=k[1]\oplus k[-1]$ onto $k[1]$, and is neither zero nor ${\mathrm{id}}_{L_2}$.

The convolution check. Apply [F2] with $(i,j,k)=(1,2,1)$ and degrees $r=-1$, $s=1$ to the composable pair $\alpha :L_1\to L_2[r]$ and $\beta :L_2\to L_1[s]$: since $\beta [r]\circ \alpha =\beta [-1]\circ \alpha ={\mathrm{id}}_k$ by step 2.1, the product theorem gives ${\mathrm{\Phi }}_{11}^0({\mathrm{id}}_k)={\mathrm{\Phi }}_{12}^{-1}(\alpha )\ast {\mathrm{\Phi }}_{21}^1(\beta )$, so $\alpha \ast \beta ={\mathrm{id}}_k$ in the source-first convention, and the right-hand side is the convolution product of [F1]. The corner degrees are $d_1+d_2-r=2$ for $\alpha$ and $d_2+d_1-s=0$ for $\beta$.

The geometric product. The adjunction chain of [F1] sends $\alpha$ to ${\mathrm{id}}_{k_{{\mathbb{P}}^1}}$, hence to the positive fundamental class $[{\mathbb{P}}^1]$. Write $i:\mathrm{p}\mathrm{t}\hookrightarrow {\mathbb{P}}^1$ for the evaluation point. Evaluation is the global-sections image of $k_{{\mathbb{P}}^1}\to i_{\ast }k$, and its adjoint under $R\mathrm{\Gamma }⊣a^{!}$, $a:{\mathbb{P}}^1\to \mathrm{p}\mathrm{t}$, is the composite $k_{{\mathbb{P}}^1}\to i_{\ast }k=i_{\ast }{i}^{!}{\omega }_{{\mathbb{P}}^1}\to {\omega }_{{\mathbb{P}}^1}$, with the last arrow the proper trace. Thus ${\mathrm{\Phi }}_{21}^1(\beta )=i_{\ast }[\mathrm{p}\mathrm{t}]$ with coefficient $+1$. Geometrically ${\mathbb{P}}^1\times \{i\}$ meets the middle diagonal of ${\mathbb{P}}^1\times {\mathbb{P}}^1$ transversely at $(i,i)$; the map from its tangent line to the complex normal line of the diagonal is an isomorphism, preserving the real orientation. The positive Thom cap of [F5] therefore gives that point with multiplicity $+1$, and proper pushforward to $Z_{11}=\mathrm{p}\mathrm{t}$ gives $[\mathrm{p}\mathrm{t}]$. This computes $[{\mathbb{P}}^1]\ast [\mathrm{p}\mathrm{t}]=[\mathrm{p}\mathrm{t}]$, with middle degree drop $2$ and no permutation sign, agreeing with step 3.1. It verifies a transverse instance of the calculus, not its general nontransverse case. AC is inherited through [A1].