Phase and cohomology bounds from semistable skyscrapers

At every stalk $R$ of $S$, the residue-field Koszul resolution and balanced tensor product compute ${\mathrm{Tor}}_i^R(k,M)=0$ for $i>\dim R$, for every finite $M$. The projective-dimension criterion in F3 therefore gives projective dimension at most $\dim R\le n$. Finite module resolutions consequently have Tor amplitude $[-n,0]$. Bounded coherent complexes are perfect locally: resolve their finitely many cohomology modules and use their truncation triangles to totalize finite free complexes. For a sheaf on projective $S$, global negative-twist surjections from F4 and the preceding stalk bound make the $n$th syzygy locally free (a finite presentation whose stalk is free is free on a neighborhood, by lifting a basis and applying Nakayama to kernel and cokernel). Thus global finite locally free resolutions also exist, and bounded coherent complexes are in the triangulated closure of vector bundles.

For perfect complexes $A,B$, finite total Hom complexes and the trace of F2 give the functorial duality $\mathrm{Hom}(A,B[i])\cong \mathrm{Hom}(B,A\otimes {\omega }_S[n-i]{)}^{\ast }\mathrm{.}$ Here the pairing is evaluation followed by the trace, with the total-complex signs. To verify perfectness, first take vector bundles $A,B$, where the assertion is precisely F2 applied to $A^{\vee }\otimes B$; it is then preserved by shifts, finite sums and cones in either argument, by the long exact Hom sequences and their duals. Step 1.1 extends it to all bounded coherent complexes. This derives the complex duality used here rather than attributing coherent-complex duality directly to a vector-bundle statement.

Let $E\in \mathcal{P}(\theta )$ with $0<\theta \le 1$. For $i<0$, $\mathrm{Hom}(E,k(x)[i])=0$ since $\theta >1+i$. For $i>n$, step 2.1 identifies this group with the dual of $\mathrm{Hom}(k(x),E[n-i]\otimes {\omega }_S)$, which is isomorphic to $\mathrm{Hom}(k(x),E[n-i])$ after choosing an identification of the one-dimensional fiber of ${\omega }_S$ at $x$ with $\mathbb{C}$. The latter vanishes because $1>\theta +n-i$. This uses semistability of $k(x)$, with no stability assumption and no invariance of $\sigma$ under tensoring by ${\omega }_S$.

Localize $E$ at a closed point $x$. Its perfect stalk has a bounded finite free complex representative. Whenever a differential matrix has a unit entry, elementary row and column operations split off a two-term contractible free summand; repeating leaves a bounded minimal complex with all differentials in the maximal ideal. Applying ${\mathrm{Hom}}_R(-,k(x))$ then gives zero differential, so its terms in degree $-i$ are dual to ${\mathrm{Ext}}_R^i(E_x,k(x))$. Global derived Hom into $k(x)$ is this local Hom: a finite vector-bundle representative has a Hom complex supported at $x$, whose higher sheaf cohomology vanishes. Step 3.1 makes every term outside degrees $[-n,0]$ zero. Hence $H^q(E{)}_x=0$ outside those degrees. Every nonzero coherent sheaf on a finite-type complex scheme has a closed point in its support, so $E$ has cohomology only in $[-n,0]$. Taking the finite HN filtration extends this inclusion from semistables to $\mathcal{P}((0,1])$.

A semistable object of phase $>1$, normalized by an integer shift into phase $(0,1]$, is by step 4.1 in standard cohomological degrees at most $-1$. Thus its Hom to any sheaf is zero. A semistable object of phase $\le -n$ normalizes by a shift of at most $-n-1$ and lies in standard cohomological degrees at least $1$, so its Hom from any sheaf is zero. The same holds for finite extensions. Apply the top and bottom HN truncation triangles of a sheaf: if a top factor had phase $>1$, its nonzero map to the sheaf would contradict the first vanishing; if a bottom factor had phase $\le -n$, the nonzero map from the sheaf to that factor would contradict the second. Equivalently these are the orthogonal descriptions of the truncation subcategories furnished by a slicing. Therefore every sheaf lies in $\mathcal{P}((-n,1])$.

Step 4.1 gives $E$ cohomological amplitude $[-n,0]$. The sheaf $F$ has Tor amplitude $[-n,0]$ by step 1.1. Tensoring their finite locally free complexes, or using the finite Tor spectral sequence of their standard cohomology filtrations, gives $E{\otimes }^{\mathbf{L}}F$ cohomological amplitude $[-2n,0]$. Its standard truncation filtration therefore has factors $H^{-j}(E{\otimes }^{\mathbf{L}}F)[j]$ for $0\le j\le 2n$. After shift $r$, step 5.1 puts each factor in $\mathcal{P}((r+j-n,r+j+1])$, contained in $\mathcal{P}((r-n,r+2n+1])$. For $r=n+1$ the lower endpoint is $1$, strictly excluded. Extensions preserve that strict lower bound. This proves all assertions, including $n=0$ and zero objects.