Koszul complexes as derived intersections and splitting obstructions

Definitions and mathematical statement

Paul Balmer asks whether tensor products, exterior powers and cones reconstruct the Koszul complex $K(P,s)$ from the two-term complex $P\xrightarrow{s}A$, and whether this construction extends to tensor triangulated categories. For a finite projective module $P$ of constant rank $n$, the Koszul complex has terms $\bigwedge^iP$ and differential \[ d_s(p_1\wedge\cdots\wedge p_i)=\sum_{j=1}^i(-1)^{j-1}s(p_j)p_1\wedge\cdots\widehat{p_j}\cdots\wedge p_i. \] The source's remark replaces $P\to A$ by $X\to\mathbf1$ in a tensor triangulated category and seeks a class $c(X)\in GW(\mathcal K)$ obstructing a summand $\mathbf1$. Defining such a Grothendieck--Witt group needs duality and coherence data beyond an unspecified triangulated category. Madhav Nori's original remark asserts that such a construction exists, without giving it.

The attached unreviewed aid sets $S=\operatorname{Sym}_A(P)$ and lets $A_s,A_0$ be $S$-algebras via $s$ and zero. It reports \[ A_s\otimes_S^{\mathbf L}A_0\simeq K(P,s). \] It proposes the analogous free-commutative-algebra/bar construction in a presentable symmetric monoidal stable infinity-category, subject to the needed colimit compatibility. This is additional enhancement data; bare tensor triangulated axioms do not provide a canonical functorial bar construction. The spectral free commutative algebra must also be distinguished from the ordinary algebraic symmetric algebra when comparing to the finite exterior-power complex.

The aid reports determinant-twisted, $n$-shifted symmetric self-duality, attributed to Balmer--Gille, and a supported Grothendieck--Witt/Witt class on the zero locus of $s$. It claims that $K(P,s)$ is zero in the derived category exactly when the chosen map $s:P\to A$ splits, equivalently is surjective. This detects the chosen map, not whether some other map splits $P$. Under regular noetherian and $1/2$ hypotheses it further reports that forgetting support yields an ${\mathbb{A}}^1$-invariant class independent of $s$, whose nonvanishing obstructs any free rank-one summand of $P$. For $A={\mathbb{R}}[x]$, $P=A$, $s=x$, it reports a nonzero supported Witt class with zero absolute Witt image. Thus object vanishing, supported quadratic obstruction and absolute section-independent obstruction are distinct levels of information. Exact duality signs, twists, support and group conventions are not supplied.

Short English statement

Construct Koszul objects from a map to the unit and identify the data needed for a tensor-triangular extension. The source distinguishes testing a chosen splitting map from an absolute quadratic obstruction to any trivial summand.

Sources

• Paul Balmer, Question 11 and accompanying remark, and Madhav Nori's existence remark in AIM Projective modules and ${\mathbb{A}}^1$-homotopy theory: https://aimath.org/pastworkshops/projectiveA1problems.pdf. Canonical location: aim-algebraic-geometry-notes.json, notes[247]. The record is labeled exact despite severe extraction damage; supplied verification explicitly repairs "cones" and category terminology. It does not reproduce the full repaired statement. No PDF was consulted.
• The attached aid attributes Koszul self-duality to Balmer--Gille without a complete citation and proposes the obstruction synthesis with low novelty confidence. It is machine generated and not peer reviewed. The original literature field is empty.
• Local registry: data/unsolvedmath/problems.json, permanent catalogue entry, exact source code catalogue entry.