Nonconnective K theory of perfect complexes

PDF p. 3: Nonconnective algebraic K theory: enhanced prerequisite interface

Assume the Axiom of Choice. A qcqs scheme is a quasi-compact quasi-separated scheme. For such X, realize the full sheaf-derived stable idempotent-complete category Perf(X) of Perfect complexes on schemes and their support by the full perfect fiber widehat Perf(X) in the single localized K-flat family of One coherent K-flat family and its supplied perfect fibers. Each fiber is identified with the supplied Perf(X) by the actual exact inclusion-induced equivalence. The supplier has specified no prior scheme-indexed tensor/pullback diagram; here its coherent interface is, by definition, the one produced by that common family. Its tensor and pullbacks are those of Coherent derived tensor and pullback on perfect complexes. By Essential smallness of perfect and supported categories these fibers are essentially small. Extend 𝕂 once to the equivalent domain of essentially small stable idempotent-complete infinity-categories, rather than selecting independent skeletons for separate maps. Let 𝕂 be the nonconnective functor of External nonconnective lax symmetric monoidal K-theory, and define

$K(X)=\mathbb{K}(\mathrm{Perf}(X)),K_i(X)={\pi }_{i}K(X)(i\in \mathbb{Z})\mathrm{.}$

All groups use the indexing of Stable spectra and exact triangles; negative i are allowed and are not set to zero by a connective truncation. In the displayed formulas Perf and its supported categories mean these common-family realizations of the supplied enhanced categories. Equivalent coherent small presentations give naturally equivalent spectra and diagrams.

For every subset T⊆X define

$K_T(X)=\mathbb{K}({\mathrm{Perf}}_T(X)),K_{i,T}(X)={\pi }_i{K}_T(X)(i\in \mathbb{Z})\mathrm{.}$

The supported category consists of perfect complexes whose cohomology stalks vanish off T. For closed T this is equivalently acyclicity on the open complement. Its exact inclusion in Perf(X) supplies $K_T$(X)→K(X). Here $K_T$(X) is defined by that category, without presupposing a fiber identification with K(X)→K(X∖T).

For every morphism f:X→Y of qcqs schemes, exact derived pullback gives f*:K(Y)→K(X). On pairs it gives f*:$K_T$(Y)→$K_{f^{-1}T}$(X). These are coherent contravariant operations; on affine finite-projective representatives they are derived extension of scalars. No flatness, regularity or finite Tor-dimension assumption on f is imposed.

Derived tensor and the lax symmetric monoidal 𝕂 give the coherent E∞ multiplication K(X)∧K(X)→K(X), with unit induced by $O_X$. Thus K(X) is a commutative ring spectrum. Each $K_T$(X) is a module over K(X), and there are coherent support pairings $K_T$(X)∧$K_S$(X)→$K_{T\cap S}$(X). They induce $K_{i,T}$(X)⊗$K_{j,S}$(X)→$K_{i+j,T\cap S}$(X) for all integers i,j. Pullback respects the products and module actions. For proper T no distinguished ring unit on $K_T$(X) is asserted; its specified structure is the ambient module and supported pairings.

1.1 Use the common-family model of [F1] and apply [F3] to its whole coherent diagram and to each full support subfamily. This defines the displayed spectra, inclusion maps and all-integer groups independently of equivalent small presentations, as proved in [F4]. [F1, F3, F4]

2.1 Apply [F3] to the exact pullbacks and coherent separately exact pairings in [F2]. The functorial, ring, module and support-product assertions are precisely those of [F4]; smash of shifted sphere maps gives the degree i+j formula. The unit belongs to the ambient Perf(X), so it does not create a unit in a proper support category. This proves the interface. [F2, F3, F4, step 1.1] ∎

A degree-zero comparison with a Grothendieck group, comparison with a particular classical nonconnective model, and a scheme localization theorem identifying a support category as the fiber are separate results with their own hypotheses. No such theorem is inferred from this definition. No identification with coherent-sheaf G-theory on a singular scheme is made.