structures in a cohesive (∞,1)-topos
derived smooth geometry
In the context of dg-geometry an ∞-stack is called perfect if its (∞,1)-category of quasicoherent ∞-stacks (of modules over the structure sheaf ) is generated from compact objects/dualizable objects: modules that are locally perfect chain complexes.
Let be a field of characteristic 0. Let be the Lawvere theory of commutative associative algebras over . When this is regarded as an (∞,1)-algebraic theory, the --algebras are modeled (by the monoidal Dold-Kan correspondence equivalently) by the
model structure on dg-algebras (over , in non-positive degree, with positively graded differential).
of the opposite (∞,1)-category of these -algebras.
where on the right we take the -category of -modules over the -algebra , regarded as an unbounded dg-algebra.
We have then
This appears as (Ben-ZviFrancisNadler, section 3.1).
For a ∞-stack , the -category is the full sub--category of consisting of those modules that are prefect over every affine .
This appears as (Ben-ZviFrancisNadler, definition 3.1).
A ∞-stack is called a perfect stack if
it has affine diagonal ;
and is the (∞,1)-category of ind-objects
of the full sub-(∞,1)-category of perfect complexes of modules on .
A morphism is said to be perfect morphism if its fibers over affines are perfect.
A stable (∞,1)-category is compactly generated if it has a small set of compact object that are generators in the sense that if for we have that is equivalent to the zero morphism, then is the zero object.
For a ∞-stack with affine diagonal, the following are equivalent:
of the -algebras of quasicoherent ∞-stacks to perfect $-stacks constitutes a geometric ∞-function theory: this assignment commutes with (∞,1)-pullbacks and admits a ggood pull-push theory of integral transforms on sheaves.
Perfect stacks cover a broad array of spaces of interest, with notable exceptions being the (constant ∞-stack on a) classifying space of a topological group such as the circle or the classifying spaces of most algebraic groups in non-zero characteristic. This is because if is perfect, then the global sections functor must preserve colimits, which fails when the global sections of the structure sheaf is ‘too large’, as in the previous cases.
But the following are examples of perfect -stacks
quasi-compact derived schemes with affine diagonal;
the total space of a quasi-projective morphism over a perfect base;
a quasi-projective derived scheme;
The concept of a perfect stack in the context of dg-geometry is considered in