Perfect complexes on algebraic stacks

Hall, Jack; Rydh, David

doi:10.1112/s0010437x17007394

Cited by 61 publications

(132 citation statements)

References 54 publications

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“…The following example summarizes some results from [, § 2] that we will use in this article. Example Let

j : U \to X

be a quasi‐compact, quasi‐separated and representable (more generally, concentrated) morphism of algebraic stacks.…”

Section: Subcategories Of Triangulated Categoriesmentioning

confidence: 99%

“…Every compact object of

{sans-serifD}_{qc} false(X false)

is a perfect complex, and for schemes, algebraic spaces, and

double-struckQ

‐stacks with affine stabilizers the converse holds. This is all discussed in detail in .…”

Section: Subcategories Of Triangulated Categoriesmentioning

confidence: 99%

“…We have that

t \in {⟨ C ⟩}^{⊥} \Leftrightarrow Hom false(c, t false) = 0, \forall c \in false⟨ scriptC false⟩ \Leftrightarrow Hom false(c, t false) = 0, \forall c \in C .

Since

c

is compact, it follows that

{false⟨ scriptC false⟩}^{⊥}

is localizing. The claims (2) and (3) is Thomason's Localization Theorem (see [, Theorem 2.1] or [, Theorem 3.12]). For (4), we note that

q : T \to T / ⟨ C ⟩

has a right adjoint (Bousfield localization) that preserves coproducts (smashing localization).…”

Section: Subcategories Of Triangulated Categoriesmentioning

confidence: 99%

“…Let

X

be a quasi‐compact and quasi‐separated algebraic stack. We let

{sans-serifD}_{qc} false(X false)

denote its unbounded derived category of quasi‐coherent sheaves and let

{sans-serifD}_{qc} {(X)}^{c}

denote the thick subcategory of compact objects of

{sans-serifD}_{qc} false(X false)

. If

X

is a scheme or algebraic space, then there is an equality

{sans-serifD}_{qc} {(X)}^{c} = Perf false(X false)

.…”

Section: Introductionmentioning

confidence: 99%

“…When

X

has finite stabilizers, Theorem was first proven by the first author [, Theorem 1.2]. Recall that quasi‐compact algebraic stacks with quasi‐finite and separated diagonal satisfy the Thomason condition [, Theorem A]. Such stacks are concentrated precisely when they are tame , for example, of characteristic zero.…”

Section: Introductionmentioning

confidence: 99%

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