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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 zero. 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 in Ben-Zvi, Francis & Nadler, section 3.1.
(Ben-Zvi, Francis & Nadler, 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 in Ben-Zvi, Francis & Nadler, 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 objects that are generators in the sense that if for we have that is equivalent to the zero morphism, then is the zero object.
For an ∞-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)-pullbacks and admits a good 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 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
Last revised on March 5, 2024 at 03:44:29. See the history of this page for a list of all contributions to it.