Derived moduli prestack of perfect complexes

Fix a characteristic-zero field $k$ and the derived-affine model in the derived-scheme supplier. For a derived $k$-scheme $X$ and a connective derived $k$-algebra $A$, put $X_A=X\times \mathrm{Spec}(A)$. Define $\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}(X)(A)$ to be the maximal $\mathrm{\infty }$-groupoid whose objects are perfect quasi-coherent complexes on $X_A$ and whose morphisms are equivalences in the derived category. A perfect complex is, locally on $X_A$, quasi-isomorphic to a bounded complex of finite-rank locally free modules. On a derived affine $\mathrm{Spec}(A)$, a perfect module means a dualizable object of $\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}(\mathrm{Spec}(A))$, with evaluation and coevaluation satisfying the triangle identities. No equivalence between these two presentations beyond the stated local models is asserted here.

The derived moduli prestack $\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}(X)$ is the functor $A\mapsto \mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}(X)(A)$, with pullback along every derived algebra map. Its universal object on $X\times \mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}(X)$, when the stack is represented, is the tautological perfect complex. An $A$-point $x$ of $\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}(X)$ is therefore a family $E_x$ on $X_A$, not only a closed point of the ordinary truncation.

This definition does not assert that Perf(X) is globally a derived Artin stack, locally of finite presentation, or equipped with a shifted form. Those are separate representability and geometricity results. It also does not assert the tangent-complex formula for the moduli stack.

Derived Artin stack and cotangent complex

Derived schemes and the cotangent complex of a morphism

Small dg categories, perfect modules, and Morita deformation