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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).
The higher geometry/derived geometry over formal duals of these algebras is sometimes called dg-geometry: a general space in this context is given by an ∞-stack over a full sub-(∞,1)-site
of the opposite (∞,1)-category of these -algebras.
The (∞,2)-presheaf of quasicoherent ∞-stacks is
given by
where on the right we take the -category of -modules over the -algebra , regarded as an unbounded dg-algebra.
By the co-Yoneda lemma we may express every as an (∞,1)-colimit of representables
We have then
This appears as (Ben-ZviFrancisNadler, section 3.1).
(Ben-ZviFrancisNadler, section 3.1).
Let . An -module is a perfect module if it lies in the smallest sub-(∞,1)-category of containing and closed under finite (∞,1)-colimits and retracts.
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:
is perfect
is
compactly generated,
and its compact and dualizable objects coincide.
The assigmnent
of the -algebras of quasicoherent ∞-stacks to perfect $-stacks constitutes a [[geometric ∞-function theory]]: this assignment commutes with [[(∞,1)-pullback]]s 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 group]]s in non-zero [[characteristic]]. This is because if is perfect, then the [[global section]]s 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 scheme]]s with affine diagonal;
the total space of a quasi-projective morphism over a perfect base;
a quasi-projective [[derived scheme]];
the quotient of a quasi-projective [[derived scheme]] by a linear action of an affine group (for of [[characteristic]] 0).
The concept of a perfect stack in the context of [[dg-geometry]] is considered in
[[!redirects perfect infinity-stacks]] [[!redirects perfect ∞-stack]] [[!redirects perfect ∞-stacks]]
Last revised on December 14, 2010 at 00:48:27. See the history of this page for a list of all contributions to it.