Perfect complexes on schemes and their support

PDF p. 16: Perfect complexes with support

Assume the Axiom of Choice and fix suitable universes. Let $X$ be any scheme. Let $\mathcal{D}(X)$ be the stable derived infinity-category obtained from cochain complexes of Zariski ${\mathcal{O}}_X$-modules by inverting quasi-isomorphisms. A quasi-isomorphism induces an isomorphism on every cohomology sheaf. Its homotopy category is the ordinary unbounded derived category $D({\mathcal{O}}_X)$.

An object $E$ is perfect if every point has an open neighbourhood on which $E$ is equivalent to a bounded complex of finite locally free modules. Bounded means nonzero in only finitely many degrees; finite locally free means locally a finite direct sum of copies of the structure sheaf. Bounds may vary between neighbourhoods. Write $\mathrm{Perf}(X)$ for the full enhanced subcategory on these objects, retaining all ambient mapping spaces and their composition coherences.

The support of $E$ is $\mathrm{Supp}(E)={\bigcup }_{n\in \mathbb{Z}}\{x\in X:H^n(E{)}_x\mathrm{\ne }0\}$. For any subset $T\subseteq X$, define ${\mathrm{Perf}}_T(X)$ as the full subcategory of perfect objects with support contained in $T$. If $T$ is closed and $j:X\setminus T\hookrightarrow X$ is the open immersion, this is equivalently the full subcategory on objects with $j^{\ast }E\simeq 0$; thus these complexes are acyclic on the open complement. For arbitrary $T$, the condition is interpreted stalkwise.

These full subcategories are stable and idempotent complete. Their finite limits and colimits are inherited from $\mathcal{D}(X)$; their exact triangles are the images in the homotopy category of ambient cofiber sequences $E\to F\to G\to E[1]$. For cochains, $E[1{]}^n=E^{n+1}$ with differential $-d$.

For a commutative unital ring $A$, degreewise associated sheaf gives an exact equivalence $\mathrm{Perf}(A)\simeq \mathrm{Perf}(\mathrm{Spec}A)$, where $\mathrm{Perf}(A)$ is the full subcategory of $\mathcal{D}(A)$ on objects represented by bounded complexes of finitely generated projective $A$-modules.

Local perfection implies quasi-coherent cohomology proves that the local condition is independent of representatives and gives quasi-coherent cohomology. Restriction, stalks and cohomological support proves invariance of support and the open-complement equivalence. Stable supported perfect subcategories and retracts proves the stable and idempotent-complete subcategory assertions. Enhanced affine perfect equivalence proves the enhanced affine comparison for the actual associated-sheaf functor, and Perfect complexes over a ring and its graded version supplies the ring convention. No global bound or global vector-bundle resolution is imposed.