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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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A point and projective-line check of source-first convolution and opposite composition

Example

Assume the Axiom of Choice as inherited from the cited constructible and cellular suppliers. In the point/projective-line geometry with $E = p t$ , $X_{1} = p t$ , $X_{2} = P^{1}$ , $d_{1} = 0$ , $d_{2} = 1$ and $L_{1} = k$ , $L_{2} = R Γ (P^{1}, k) [1]$ in the setting of Equivariant Borel–Moore homology and oriented correspondence convolution, let $α \in {Ext}^{- 1} (L_{1}, L_{2})$ be the unit $k \to R Γ (P^{1}, k)$ and let $β \in {Ext}^{1} (L_{2}, L_{1})$ be point evaluation $R Γ (P^{1}, k) [1] \to k [1]$ , that is, evaluation $H^{0} (P^{1}, k) \to k$ shifted by $1$ and extended by zero on $H^{2}$ . Then $α * β = β [- 1] \circ α = {id}_{k}$ in the source-first convention of Equivariant Borel–Moore homology and oriented correspondence convolution, and under the correspondence isomorphism of Proper-pushforward Ext is equivariant Borel–Moore homology of the fiber product one has $[P^{1}] * [p t] = [p t]$ : the refined diagonal intersection of the fundamental class of $P^{1}$ with the point class is the point class, and proper pushforward to the point corner gives its generator.

Facts & Assumptions

Given: the Axiom of Choice; the trivial group $G = 1$ ; the base $E = p t$ ; the sources $X_{1} = p t$ and $X_{2} = P^{1}$ with $d_{1} = 0$ , $d_{2} = 1$ ; the proper maps $f_{1} : p t \to p t$ and $f_{2} : P^{1} \to p t$ ; the objects $L_{1} = k$ and $L_{2} = R Γ (P^{1}, k) [1]$ ; the unit $α : k \to R Γ (P^{1}, k)$ and the evaluation $β : R Γ (P^{1}, k) [1] \to k [1]$ ; and the conventions of Equivariant Borel–Moore homology and oriented correspondence convolution.

[F1]

Proper-pushforward Ext is equivariant Borel–Moore homology of the fiber product supplies natural isomorphisms $Φ_{i j}^{r} : Hom (L_{i}, L_{j} [r]) \tilde{\to} H_{d_{i} + d_{j} - r}^{B M, G} (Z_{i j}; k)$ for all $r$ and $(i, j)$ , and Equivariant Borel–Moore homology and oriented correspondence convolution fixes the source-first product $a * b = h_{*} Δ_{X_{j}}^{!} (a ⊠ b)$ with degree drop $2 d_{j}$ , the corner degrees and the normalized diagonal class.

[F2]

Verdier-duality identification intertwines Yoneda product and oriented convolution supplies $Φ_{i k}^{r + s} (β [r] \circ α) = Φ_{i j}^{r} (α) * Φ_{j k}^{s} (β)$ for composable morphisms, with the identity mapped to $[Δ_{X_{i}}]$ .

[F3]

The projective line $P^{1}$ is smooth of complex dimension $1$ (Complex projective space and its holomorphic charts) and proper over $Spec C$ (Finite-dimensional projective space is proper over every base); under $P^{1} = {Gr}_{1} (C^{2})$ the Schubert cells of Stiefel spaces, Grassmannians, and tautological bundles, Schubert cells in real and complex Grassmannians and Schubert cells give the stable Grassmannian CW structure give a finite CW structure with one cell in real dimension $0$ and one in real dimension $2$ .

[F4]

With no cells in adjacent dimensions every cellular differential vanishes and $H_{n} ≅ C_{n}^{c e l l}$ (A CW complex with no cells in adjacent dimensions has zero cellular boundary), and cellular cochains compute cohomology with local coefficients (Cellular cochains compute cohomology with local coefficients).

[F5]

Complex-oriented diagonal Thom classes realize exceptional exchange on supports fixes the positive complex orientation and the normalization of the diagonal Thom intersection: the positive generator is the one evaluating to $+ 1$ on positively oriented normal fibers, and the refined operation is independent of transversality.

[F6]

Under AC, constant-sheaf cohomology on a metric space with a contractible-open neighborhood basis equals singular cohomology (Locally contractible metric spaces have singular constant sheaf cohomology). The projective-line charts and its Hermitian-projector embedding in [F3] supply those hypotheses.

[A1]

The Axiom of Choice is inherited through the cited constructible and cellular suppliers; the explicit finite computations below are choice-free (The Axiom of Choice).

Verification

technique · direct

1.1F3F4F6algebra

The geometry and the cohomology of the projective line. The fiber products are $Z_{12} = Z_{21} = P^{1}$ . 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 $k \oplus k [- 2] \to R Γ (P^{1}, k)$ . Hence $L_{2} ≃ k [1] \oplus k [- 1]$ and $L_{1} = k$ .

2.1step 1.1algebra

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

3.1F1F2step 1.1step 2.1

The convolution check. Apply [F2] with $(i, j, k) = (1, 2, 1)$ and degrees $r = - 1$ , $s = 1$ to the composable pair $α : L_{1} \to L_{2} [r]$ and $β : L_{2} \to L_{1} [s]$ : since $β [r] \circ α = β [- 1] \circ α = {id}_{k}$ by step 2.1, the product theorem gives $Φ_{11}^{0} ({id}_{k}) = Φ_{12}^{- 1} (α) * Φ_{21}^{1} (β)$ , so $α * β = {id}_{k}$ 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 $α$ and $d_{2} + d_{1} - s = 0$ for $β$ .

4.1A1F1F2F5step 2.1step 3.1∎

The geometric product. The adjunction chain of [F1] sends $α$ to ${id}_{k_{P^{1}}}$ , hence to the positive fundamental class $[P^{1}]$ . Write $i : p t ↪ P^{1}$ for the evaluation point. Evaluation is the global-sections image of $k_{P^{1}} \to i_{*} k$ , and its adjoint under $R Γ ⊣ a^{!}$ , $a : P^{1} \to p t$ , is the composite $k_{P^{1}} \to i_{*} k = i_{*} i^{!} ω_{P^{1}} \to ω_{P^{1}}$ , with the last arrow the proper trace. Thus $Φ_{21}^{1} (β) = i_{*} [p t]$ with coefficient $+ 1$ . Geometrically $P^{1} \times {i}$ meets the middle diagonal of $P^{1} \times P^{1}$ 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 $Z_{11} = p t$ gives $[p t]$ . This computes $[P^{1}] * [p t] = [p t]$ , 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].

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