ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

How statement and proof provenance work

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

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Kunneth as a two-column spectral sequence over a PID

Example

Assume AC. For bounded-below free PID complexes $C, D$ , put $T_{n} = ⨁_{i + j = n} H_{i} C \otimes H_{j} D$ and $U_{n} = ⨁_{i + j = n - 1} {Tor}_{1} (H_{i} C, H_{j} D)$ . Künneth has columns $E_{0, q}^{2} = T_{q}$ and $E_{1, q}^{2} = U_{q + 1}$ and gives $0 \to T_{n} \to H_{n} (C \otimes D) \to U_{n} \to 0$ .

Over $Z$ take $C_{1} = Z a \oplus Z b$ , $C_{0} = Z c$ , $d a = 2 c, d b = 0$ , and $D_{1} = Z x$ , $D_{0} = Z y$ , $d x = 2 y$ , zero elsewhere. Its page has $Z / 2$ at $(0, 0), (0, 1), (1, 0)$ , zero elsewhere, and the target has $H_{0} = Z / 2$ , $H_{1} = (Z / 2)^{2}$ , zero otherwise.

Facts & Assumptions

Given: The free PID hypotheses, with the homological tensor sign $d (u \otimes v) = d u \otimes v + (- 1)^{∣ u ∣} u \otimes d v$ .

[F1]

The PID two-column Künneth collapse has the displayed tensor inclusion and Tor quotient, and admits nonnatural splittings under AC (PID Kunneth is a two-column collapse).

Verification

1.1

F1 gives zero in every column $p > 1$ , because free presentations have free kernels under AC. A later differential changes $p$ by $- r$ for $r \geq 2$ , so no two surviving columns can be joined. Hence $E^{2} = E^{\infty}$ . The increasing target filtration is $F_{- 1} = 0$ , $F_{0} = T_{n}$ and $F_{1} = H_{n}$ , with $F_{1} / F_{0} = U_{n}$ . Its maps are the cross product and Tor quotient of F1. AC is used in free-submodule/projectivity and section choices, and a splitting is additional to this natural exact sequence.

2.1

In the numerical case $H_{0} C = Z / 2$ , $H_{1} C = Z$ , $H_{0} D = Z / 2$ and $H_{1} D = 0$ . Tensoring the rank-one resolution $Z \overset{2}{\to} Z$ of $Z / 2$ with $Z / 2$ gives a zero differential, so its Tor-zero and Tor-one groups are both $Z / 2$ . The one-term resolution of $Z$ gives tensor $Z / 2$ and zero positive Tor. Thus the three listed page positions are precisely the surviving ones.

F1step 1.1

3.1

Directly the degree-one tensor differential sends $α (a \otimes y) + β (b \otimes y) + γ (c \otimes x)$ to $2 (α + γ) c \otimes y$ . Its kernel is generated by $u = b \otimes y$ and $t = a \otimes y - c \otimes x$ . Degree-two differentials send $a \otimes x$ to $- 2 t$ and $b \otimes x$ to $- 2 u$ , an injective map onto $2 Z u \oplus 2 Z t$ . Thus $H_{2} = 0$ , $H_{1} = (Z / 2) [u] \oplus (Z / 2) [t]$ , and $H_{0} = Z (c \otimes y) / 2 = Z / 2$ . In degree one the tensor subobject is $⟨ [u] ⟩$ , and the Tor quotient is generated by the image of $[t]$ , in agreement with F1.

F1step 2.1

4.1

Sending the quotient generator to $[t]$ gives a section. The automorphism $a \mapsto a + b$ fixes both end terms but sends $[t] \mapsto [t] + [u]$ and fixes $[u]$ , so neither lift of the quotient generator is invariant. This verifies nonnaturality in the actual computed extension. Degree zero has one quotient and filtration $F_{- 1} = 0, F_{0} = H_{0}$ ; all other degrees except one are zero. These finite filtrations solve reconstruction and convergence for the example. The matrices and their kernels require no additional choice.

F1step 1.1step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Weibel, 5.6.4 (standard reference, not scraped)