Finite homological dimension and a derived equivalence

Sanders, William; Sane, Sarang

doi:10.1090/tran/6882

Cited by 3 publications

(8 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is proved locally and follows from the proof in the affine case (e.g. [11] has an explicit proof ).…”

Section: The Foxby Morphismmentioning

confidence: 78%

“…In the case, when X = Spec(A) is affine and Cohen-Macaulay, Sane and Sanders [11] established these equivalences. Methods in [11] relies on a construction of a map K • −→ M • of complexes of modules from some direct sum of (exact) Koszul complexes K • to a given complex M • of modules, so that the nonzero homology of K • surjects onto the corresponding homology of M • .…”

Section: Introductionmentioning

confidence: 95%

“…Methods in [11] relies on a construction of a map K • −→ M • of complexes of modules from some direct sum of (exact) Koszul complexes K • to a given complex M • of modules, so that the nonzero homology of K • surjects onto the corresponding homology of M • . The construction was originally due to H.-B.…”

Section: Introductionmentioning

confidence: 99%

“…Foxby [3] that appeared in a preprint. By now, several other expositions and versions of the same is available in the literature [4,15,10,11]. To meet the goals of this paper, a similar morphism of complexes of coherent sheaves on quasi-projective schemes X was constructed (2.3).…”

Section: Introductionmentioning

confidence: 99%

“…To meet the goals of this paper, a similar morphism of complexes of coherent sheaves on quasi-projective schemes X was constructed (2.3). Other than that, the proof of the main theorem adapts the inductive arguments in [11], which involves further technicalities and finesse in this non-affine situation. Overall, these methods emanate from the methodologies developed in [7][8][9] in the context of Witt theories, by constructing cones, with smaller width, of morphisms of complexes.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Foxby-morphism and derived equivalences

Mandal

2015

Journal of Algebra

View full text Add to dashboard Cite

“…This is proved locally and follows from the proof in the affine case (e.g. [11] has an explicit proof ).…”

Section: The Foxby Morphismmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Foxby-morphism and derived equivalences

Mandal

2015

Journal of Algebra

View full text Add to dashboard Cite

Witt, GW, K-theory of quasi-projective schemes

Mandal

2017

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

In this article we continue our investigation of the Derived Equivalences over noetherian quasi-projective schemes X, over affine schemes Spec (A). For integers k ≥ 0, let CM k (X) denote the category of coherent O X -modules F , with locally free dimension dim V (X) (F ) = k = grade(F ). We prove that there is a zig-zag equivalence D b CM k (X) → D k (V (X)) of the derived categories. It follows that there is a sequence of zig-zag maps K CM k+1 (X)of the Ktheory spectra that is a homotopy fibration. In fact, this is analogous to the fibrations of the G-theory spaces of Quillen (see proof of [Q, Theorem 5.4]). We also establish similar homotopy fibrations of GW-spectra and GW -bispectra. * Partially supported by a General Research Grant (no 2301857) from U. of Kansas localization theorems, applicable to non-regular schemes. Further, while developments in Grothendieck-Witt theory (GW -theory) and Witt theory followed the foot prints of Ktheory ([S3, B1]), due to lack of any natural duality on Coh(X), the situation in these two areas appear even less complete. When, X is non-regular, the category M(X) of coherent sheaves with finite V (X)-dimension differs from Coh(X). There appears to be a gap in the literature of K-theory, GW -theory, and Witt theory, with respect to the place of the category M(X). One can speculate, whether this lack of completeness is attributable to this gap? The goal of this one and the related articles is to work on this gap and attempt to establish the said literature on non-regular schemes at the same pedestal as that of regular schemes. For quasi-projective schemes over noetherian affine schemes, this goal is accomplished to up some degree of satisfaction. The special place of the full subcategory CM k (X) ⊂ M(X) would also be clear subsequently, where for integers k ≥ 0, CM k (X) will denote the full subcategory of objects F in M(X), with dim V (X) (F ) = grade (F ) = k.With respect to a certain facets of these three areas, namely Algebraic K-theory, Grothendieck-Witt (GW ) theory and Witt theory, a common thread between them is their invariance properties with respect equivalences of the associated Derived categories. We review some of the results on such invariances. For example, recall the theorem of Thomanson-Trobaugh ([TT, Theorem 1.9.8]): suppose A → B is a functor of complicial exact categories with weak equivalences. Assume that the associated functor of the triangulated categories T A → T B is an equivalence. Then, the induced map K(A) → K(B) of the K-theory spaces is a homotopy equivalence (see [S1, 3.2.24]). The non-connective version of this theorem was given by Schlichting ([S2, Theorem 9], also see [S1, 3.2.29]): which states that under the relaxed hypothesis that, if T A → T B is an equivalence up to factors, then it induces a homotopy equivalence K(A) → K(B) of the K-theory spectra. While K-theory is defined for complicial exact categories with weak equivalences, Schlichting defined Grotherndieck Witt (GW ) spectra and bispectra ([S3], also see §A) of pointed dg categorie...

show abstract