the stable derived category of a noetherian scheme

By Henning Krause. Compositio Mathematica, 2004

381 citations recorded by the source corpus.

for a noetherian scheme, we introduce its unbounded stable derived category. this leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. some applications are included, for instance an analogue of maximal cohen–macaulay approximations, a construction of tate cohomology, and an extension of the classical grothendieck duality. in addition, the relevance of the stable derived category in modular representation theory is indicated.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/eddfb152b75ea3039fed4d41ab7b0d4d8023c354

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Citations in the corpus

Models for singularity categories (2012, background): “Note that by [16] the conditions are indeed satisfied for the sequence Kac(I(A )) → K(I(A )) → D(A ) if A is a locally Noetherian Grothendieck category such that D(A ) is compactly generated.”
Singular support of coherent sheaves and the geometric Langlands conjecture (2012, background): “” (As a minor detail, both [Orl1] and [Kra] work with Noetherian classical schemes.”
Existence of Gorenstein projective resolutions and Tate cohomology (2007, background): “Some other injective cousins of the results of this paper are given in [18].”
Formal completions and idempotent completions of triangulated categories of singularities (2009, background): “Krause [10] that the category D(cohX) can be considered as the category of compact object in the homotopy category of injective quasi-coherent sheaves H(InjX) for a Noetherian scheme X.”
The stable module category of a general ring (2014, background): “ of absolutely clean modules is covering for any ring R. When Ris left Noetherian, Krause constructs a recollement involving K(Inj), the stable derived category S(R), and the derived category D(R) in [Kra05]. Half of this recollement arises because S(R) is a Bousﬁeld localization of K(Inj). The othe”
Cluster-tilted algebras are Gorenstein and stably Calabi–Yau (2005, background): “It is a triangulated category and an instance of a stable derived category in the sense of [40].”
Hereditary abelian model categories (2015, methodology): “The fact that K(Inj) is compactly generated whenever R is (left) noetherian, or just a locally noetherian Grothendieck category, has been known since [43].”
Coherent analogues of matrix factorizations and relative singularity categories (2011, background): “The construction of the functor DbSing(Z) −→ D st Sing(Z) in [19] is given in terms of the Verdier localization functor Q : D(Z–qcoh) −→ D(Z–qcoh) by the triangulated subcategory D Sing(Z) ⊂ D (Z–qcoh) and its adjoint functors on both”
A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, methodology): “ch categories do exist. Since Db(coh GX) admits an enhancement to a dg-category, we could use the derived category of dg-modules over the enhancement. Or, a more geometric construction due to Krause, [Kra05], uses the homotopy category of injective complexes of quasi-coherent sheaves, in the non-equ”
Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence (2009, background): “Parts of this paper have been written when I was visiting the Institut des Hautes Études Scientifiques, which I wish to thank for the excellent working conditions.”
Subcategories of singularity categories via tensor actions (2011, methodology): “The particular category which will concern us is the stable derived category of Krause [31], namely”
Colocalizing subcategories and cosupport (2010, background): “As W is compact in KðInj kGÞ, see [18], Lemma 2.”
Representations of Algebras and Related Topics (2011, background): “plexes. This is a triangulated category in which the triangles come from the mapping cone construction. This Module categories for ﬁnite group algebras 13 category is investigated in detail in Krause [24], Benson and Krause [13]. Let K acInj(kG) be the full subcategory of KInj(kG) whose objects are ”
On triangulated orbit categories (2005, background): “These include the category of projectivem odulesovera preprojectivealgebra ofgeneralized D ynkin typein thesenseof Happel-Preiser-Ringel[29],whosetriangulated structuregoesback to Auslander-Reiten’s work [6],[44],[7].”