Singular free-summand descent and a cuspidal top-Chern counterexample

Definitions and mathematical statement

The reconstructed question, attributed to Anand Sawant, asks: when are the preceding Questions 1--9 on projective modules and ${\mathbb{A}}^1$-homotopy well-defined for singular varieties, where have they been considered, and what is their status? Those nine questions are not included in this record, so a complete source-specific status survey cannot be formulated from it. Different extensions of vector bundles, Chern classes, Chow groups and motivic classification can yield different singular versions.

The attached aid studies a free-summand part of that question. For a commutative noetherian ring $A$ and finite projective $A$-module $P$, write \[ \operatorname{Um}(P^\vee)=\{\varphi\in\operatorname{Hom}_A(P,A):\varphi(P)=A\}. \] Such a functional splits and is equivalent to a decomposition $P\simeq A\oplus Q$. The aid reports that for a nilpotent ideal $J\subset A$, existence of such a free line summand is equivalent for $P$ and $P/JP$. This is existence invariance, not a bijection of all individual lifted functionals.

For reduced noetherian $A$ with finite normalization $B$, let $I=\{a\in A:aB\subseteq A\}$ be the conductor, $C=A/I$, $D=B/I$. The conductor square gives $A=B\times_D C$. With $P_T=P\otimes_A T$, the aid reports the exact compatibility description \[ \operatorname{Um}(P^\vee)\simeq \operatorname{Um}(P_B^\vee) \mathbin{\times}_{\operatorname{Um}(P_D^\vee)} \operatorname{Um}(P_C^\vee). \] Thus a free summand downstairs requires compatible unimodular functionals, not merely separate summands on the normalization and conductor subscheme.

For $A=k[t^2,t^3]$, it uses $B=k[t]$, conductor $I=(t^2,t^3)$, $C=k$, $D=k[t]/(t^2)$. Gluing trivial line bundles by $1+at\in D^\times$ gives the aid's line bundle $L_a$, nontrivial for each $a\ne0$. It reports \[ \mathrm{CH}_0(\operatorname{Spec}A)=0, \qquad c_1(L_a)\cap[\operatorname{Spec}A]=0, \] in Fulton's homological Chow theory, although $L_a$ has no free line summand, since a rank-one module with such a summand would be free. This supplies the aid's odd-dimensional, dimension-one counterexample to a direct singular extension of the preceding top-Chern detection question using that homological class. It does not refute every operational, motivic or other possible singular Chern-class formulation.

Short English statement

Ask which earlier projective-module and ${\mathbb{A}}^1$-homotopy questions extend to singular varieties and what is known about them. The attached aid treats free-summand descent and reports a cuspidal line-bundle example showing that vanishing of a Fulton homological top Chern class need not detect a free summand on a singular curve.

Sources

• Local UnsolvedMath registry, permanent ID $20000247$, exact public code catalogue entry; canonical location aim-algebraic-geometry-notes.json, notes[246].
• AIM workshop "Projective modules and $A^1$-homotopy theory," Question 10, attributed in the supplied reconstruction to Anand Sawant; source URL https://aimath.org/pastworkshops/projectiveA1problems.pdf. The supplied verification says the PDF was inspected in its own run; this drafting step did not consult it. No year is supplied; the literature field is empty. The OCR tail "4 SARANG SANE" has unresolved page/header or attribution status.
• Attached research attempt 1, "Singular free-summand descent and a cuspidal top-Chern counterexample," referring to Fulton homological Chow groups without full bibliographic citations.