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Unequal-dimensional proper pushforwards carry the summed-dimension Ext/Borel–Moore degree

Example

Assume the Axiom of Choice as inherited from the cited constructible and cellular suppliers. For the trivial group $G$ , $E = p t$ , $X_{1} = p t$ and $X_{2} = P^{1}$ with $d_{1} = 0$ , $d_{2} = 1$ , put $L_{1} = k$ and $L_{2} = R Γ (P^{1}, k) [1]$ in the setting of Equivariant Borel–Moore homology and oriented correspondence convolution. Then ${Ext}^{r} (L_{1}, L_{2})$ is $k$ for $r = - 1, 1$ and zero otherwise, and the corresponding summed-dimension Borel–Moore degrees $d_{1} + d_{2} - r$ are $2$ and $0$ ; the reverse corner ${Ext}^{r} (L_{2}, L_{1})$ has the same two nonzero degrees. The unaugmented shifts $2 \dim_{C} X_{1} - r = - r$ and $2 \dim_{C} X_{2} - r = 2 - r$ give the wrong degrees in at least one corner, so the summed $d_{1} + d_{2}$ normalization of Proper-pushforward Ext is equivariant Borel–Moore homology of the fiber product is the one compatible with the two unequal source dimensions.

Facts & Assumptions

Given: the Axiom of Choice; the trivial group $G = 1$ ; the base $E = p t$ ; the sources $X_{1} = p t$ with $d_{1} = 0$ and $X_{2} = P^{1}$ with $d_{2} = 1$ ; the proper maps $f_{1} : p t \to p t$ and $f_{2} : P^{1} \to p t$ ; the objects $L_{1} = f_{1!} k_{p t} [0]$ and $L_{2} = f_{2!} k_{P^{1}} [1]$ ; and the normalization of Equivariant Borel–Moore homology and oriented correspondence convolution.

[F1]

Assume AC. For the equivariant correspondence data of Equivariant Borel–Moore homology and oriented correspondence convolution, Proper-pushforward Ext is equivariant Borel–Moore homology of the fiber product supplies natural isomorphisms $Φ_{i j}^{r} : {Hom}_{D_{G, c}^{b} (E; k)} (L_{i}, L_{j} [r]) \tilde{\to} H_{d_{i} + d_{j} - r}^{B M, G} (Z_{i j}; k)$ for every integer $r$ and every pair $(i, j)$ , compatible with the Borel approximation.

[F2]

In Equivariant Borel–Moore homology and oriented correspondence convolution, $H_{j}^{B M, G} (X; k) = {Hom}_{D_{G, c}^{b} (X; k)} (k_{X}, ω_{X}^{G} [- j])$ , and ordinary Borel–Moore homology is $H_{j}^{B M} (X; k) = Hom (k_{X}, ω_{X} [- j])$ , which for smooth complex $X$ of pure dimension $d$ , where $ω_{X} = k_{X} [2 d]$ , equals $H^{2 d - j} (X; k)$ . For the trivial group the nerve of Equivariant constructible derived category of complex varieties is constant at $X$ . Its Cartesian-cohomology subcategory is equivalent to the ordinary bounded constructible category by the identity model with $r_{N} = \infty$ and $Q_{N} = i d$ of General linear Stiefel approximations stabilize bounded constructible Hom; the Cartesian condition is still imposed on simplicial sheaves.

[F3]

The complex projective line $P^{1}$ is a smooth complex manifold of complex dimension $1$ (Complex projective space and its holomorphic charts), and its structure morphism $P_{C}^{1} \to Spec C$ is proper (Finite-dimensional projective space is proper over every base with $S = Spec C$ and $n = 1$ ). Under the standard identification $P^{1} = {Gr}_{1} (C^{2})$ , 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 exactly one cell in real dimension $0$ and one in real dimension $2$ .

[F4]

If a CW complex has no cells in adjacent dimensions, every cellular differential vanishes and $H_{n} (X; G) ≅ C_{n}^{c e l l} (X; G)$ (A CW complex with no cells in adjacent dimensions has zero cellular boundary); for a CW pair and local system the cellular cochain complex computes cohomology, $H^{n} (C_{c e l l}^{*} (X, A; L)) ≅ H^{n} (X, A; L)$ (Cellular cochains compute cohomology with local coefficients).

[F5]

Under AC, a metric space with a contractible-open neighborhood basis has constant-sheaf cohomology naturally equal to its singular cohomology (Locally contractible metric spaces have singular constant sheaf cohomology). The projective-line charts of [F3] give such a basis; its rank-one Hermitian-projector embedding gives a metric.

[A1]

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

Verification

technique · direct

1.1F2F3given

The two sources and the objects. The map $f_{1} : p t \to p t$ is proper with $d_{1} = 0$ and $L_{1} = f_{1!} k_{p t} [0] = k$ , and the map $f_{2} : P^{1} \to p t$ is proper with $d_{2} = 1$ and $L_{2} = f_{2!} k_{P^{1}} [1] = R Γ (P^{1}, k) [1]$ , the last equality using $f_{2 *} = f_{2!}$ for proper $f_{2}$ . The fiber products are $Z_{12} = Z_{21} = p t \times_{p t} P^{1} = P^{1}$ .

1.2F3F4F5algebra

The cohomology of the projective line. By [F3] the Schubert cells give $P^{1}$ 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 $R Γ (P^{1}, k) ≃ k \oplus k [- 2]$ .

2.1step 1.2algebra

The forward corner. For complexes of $k$ -vector spaces one has $Hom (k, C [n]) = H^{n} (C)$ , so ${Ext}^{r} (L_{1}, L_{2}) = Hom (k, R Γ (P^{1}, k) [1 + r]) = H^{1 + r} (P^{1}, k)$ , 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.

3.1step 1.2step 2.1algebra

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

4.1A1F1F2step 1.2step 3.1∎

The degree check. By [F1] the isomorphism $Φ_{12}^{r}$ identifies ${Ext}^{r} (L_{1}, L_{2})$ with $H_{d_{1} + d_{2} - r}^{B M} (P^{1}; k) = H_{1 - r}^{B M} (P^{1}; k)$ : 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, $H_{j}^{B M} (P^{1}; k) = {Hom}_{D_{c}^{b} (P^{1}; k)} (k_{P^{1}}, k_{P^{1}} [2 - j]) = H^{2 - j} (P^{1}; k)$ is $k$ for $j = 2$ and for $j = 0$ . The single-source shifts fail this comparison in both corners: $2 \dim_{C} X_{1} - r = - r$ is $1$ for $r = - 1$ and $- 1$ for $r = 1$ , and $2 \dim_{C} X_{2} - r = 2 - r$ 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.

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