Gts local sheaf divisor annihilation

In the category of zero-dimensional coherent sheaves on $X=\mathrm{Tot}(K_{{\mathbb{P}}^3})$, with positive lengths explicitly admitted, a skyscraper $k(p)$ on the pullback of a hyperplane belongs to both framing concentration opens specified below. Multiplication by the divisor's defining section annihilates its nonzero framing. Thus concentration and admission alone do not extend the section-multiplication embedding argument to this category. The example does not satisfy the torsion-free hypothesis used in the source proof of Lemma 6.13 and is not a counterexample to that theorem.

Bojko, Wall-crossing for Calabi–Yau fourfolds: framework, tools, and applications, Assumptions 5.1 and 5.10; Definition 5.3; §6.2; §6.3; Assumption 6.9; Remark 6.10; Example 6.11; Theorem 6.12; Lemma 6.13; Equation (6.14); §7.2.

Bojko, Kuhn, Liu, and Thimm, Wall-crossing for equivariant DT4 invariants, §2.2.8, §3.2, Assumption 4.2.4, and Theorem 1.2.1; used only under their stated hypotheses.

Let $Y={\mathbb{P}}_{\mathbb{C}}^3$ and $X=\mathrm{Tot}(K_Y)$ with projection $\pi :X\to Y$. Choose a hyperplane $H\subset Y$, set $D={\pi }^{-1}(H)$, and let $s$ be the pulled-back section defining this effective Cartier divisor. Choose a closed point $p\in D$ on the zero section. Let $\mathcal{A}={\mathrm{Coh}}_0(X)$, the abelian category of coherent sheaves with finite support, and use the class map $[F]\mapsto \mathrm{\ell }(F)\in \mathrm{\Gamma }=\mathbb{Z}$, where $\mathrm{\ell }(F)$ is the sum of its local module lengths. Set $C(\mathcal{A})=E(\mathcal{A})={\mathbb{Z}}_{>0}$ and $\varphi (F)=1$ for every nonzero $F$. All objects are semistable, and the HN property holds with one nonzero factor. No restriction to torsion-free sheaves is imposed in this diagnostic datum.

Take the two original framing indices $j=1,2$ with divisors $D_j=jD$ and functors ${\mathrm{F}\mathrm{r}}_j(F)=H^0(X,F(jD))$. Put ${\mathcal{A}}_1={\mathcal{A}}_2=\mathcal{A}$ and take ${\mathcal{U}}_j$ to be the entire length-$n$ locus for each fixed $n>0$. Finite-support sheaves have no higher cohomology; tensoring by a line preserves length, so these framing functors are exact of rank $n$. For flat families with support finite over the parameter base, their degree-zero direct images are rank-$n$ vector bundles and commute with base change. These are the geometric concentration data used here; no identification with a separately chosen source enhancement is required. In particular the residue sheaf $k(p)={\mathcal{O}}_{X,p}\mathrm{/}{\mathfrak{m}}_p$ has length one, has admitted class $1\in E(\mathcal{A})$, and lies in ${\mathcal{U}}_1\cap {\mathcal{U}}_2$.

lem-gts-original-k-direct-sum-framing-oriented-gysin, lem-gts-original-k-nondegenerate-change-of-framing name the held construction interfaces discussed by this diagnostic. Their conclusions are not assumed or used to establish the calculations below; this example does not prove those interfaces.