Geometricity for derived categories of algebraic stacks

Bergh, Daniel; Lunts, Valery A.; Schnürer, Olaf M.

doi:10.1007/s00029-016-0280-8

Cited by 28 publications

(54 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We formulate our results in terms of general (homotopy) limits of dg-categories, under appropriate compatibility assumptions on the structure functors. We apply this theory to obtain new semi-orthogonal decompositions of geometric interest, complementing our results in [27] and extending results by other authors [17,3].…”

Section: Introductionsupporting

confidence: 71%

“…That is, we need to prove that if ( A 2 , u)) ≃ 0. This however follows immediately by the calculation of the Hom-complexes in C given by (3). Indeed, since we have inclusions…”

Section: Preordered Semi-orthogonal Decompositionsmentioning

confidence: 79%

“…With different methods, and in the setting of classical triangulated categories, related results have been obtained in the recent paper [4]. We remark that the concept of preordered semi-orthogonal decomposition was also considered in [3]. We will follow closely the account of psod-s contained in the previous paper of the authors [27].…”

Section: Preordered Semi-orthogonal Decompositionsmentioning

confidence: 90%

“…which recovers the semi-orthogonal factors given by [3,Theorem 4.9]. It will be useful to explain how the semi-orthogonal factors A k χ are constructed in the simple normal crossing case, and some of their basic properties.…”

Section: 1mentioning

confidence: 96%

“…Before giving a proof of Proposition 4.1 we will make a few preliminary considerations. If D ⊂ X is a simple normal crossing divisor, Proposition 4.1 is a reformulation of Theorem 4.9 from [3]. In order to translate back to the statement of [3,Theorem 4.9], it is sufficient to note that S k decomposes as the disjoint union of strata of codimension k. Indeed, in the simple normal crossing setting, all strata are already normal.…”

Section: 1mentioning

confidence: 99%

See 4 more Smart Citations

Gluing semi-orthogonal decompositions

2020

View full text Add to dashboard Cite

We introduce preordered semi-orthogonal decompositions (psod-s) of dg-categories. We show that homotopy limits of dg-categories equipped with compatible psod-s carry a natural psod. This gives a way to glue semi-orthogonal decompositions along faithfully-flat covers, extending some results of [4]. As applications we will:(1) construct semi-orthogonal decompositions for root stacks of log pairs (X, D) where D is a (not necessarily simple) normal crossing divisors, generalizing results from [17] and [3], (2) compute the Kummer flat K-theory of general log pairs (X, D), generalizing earlier results of Hagihara and Nizio l in the simple normal crossing case [15], [23].

show abstract

Section: Introductionsupporting

confidence: 71%

“…That is, we need to prove that if ( A 2 , u)) ≃ 0. This however follows immediately by the calculation of the Hom-complexes in C given by (3). Indeed, since we have inclusions…”

Section: Preordered Semi-orthogonal Decompositionsmentioning

confidence: 79%

Section: Preordered Semi-orthogonal Decompositionsmentioning

confidence: 90%

Section: 1mentioning

confidence: 96%

Section: 1mentioning

confidence: 99%

See 3 more Smart Citations

Gluing semi-orthogonal decompositions

2020

View full text Add to dashboard Cite

show abstract

An embedding problem for finite local torsors over twisted curves

Otabe

2021

Mathematische Nachrichten

View full text Add to dashboard Cite

In his previous paper, the author proposed as a problem a purely inseparable analogue of the Abhyankar conjecture for affine curves in positive characteristic and gave a partial answer to it, which includes a complete answer for finite local nilpotent group schemes. In the present paper, motivated by the Abhyankar conjectures with restricted ramifications due to Harbater and Pop, we study a refined version of the analogous problem, based on a recent work on tamely ramified torsors due to Biswas–Borne, which is formulated in terms of root stacks. We study an embedding problem to conclude that the refined analogue is true in the solvable case.

show abstract

The circle quantum group and the infinite root stack of a curve

Sala

Schiffmann

2019

Sel. Math. New Ser.

View full text Add to dashboard Cite

In the present paper, we give a definition of the quantum group U υ (sl(S 1 )) of the circle S 1 := R/Z, and its fundamental representation. Such a definition is motivated by a realization of a quantum group U υ (sl(S 1 Q )) associated to the rational circle S 1 Q := Q/Z as a direct limit of U υ ( sl(n))'s, where the order is given by divisibility of positive integers. The quantum group U υ (sl(S 1 Q )) arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack X ∞ over a fixed smooth projective curve X defined over a finite field. Via this Hall algebra approach, we are able to realize geometrically the fundamental and the tensor representations, and a family of symmetric tensor representations, depending on the genus g X , of U υ (sl(S 1 Q )). Moreover, we show that U υ ( sl(+∞)) and U υ ( sl(∞)) are subalgebras of U υ (sl(S 1 Q )). As proved by T. Kuwagaki in the appendix, the quantum group U υ (sl(S 1 )) naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle S 1 . 30 4.5. Hecke operators on line bundles and the fundamental representation of U υ ( gl(n)) 34 4.6. Tensor and symmetric tensor representations of U υ ( gl(n)) 35 5. Relations between different Hall algebras 35 6. Hall algebra of the infinite root stack over a curve 36 6.1. Preliminaries on Hall algebras 36 6.2. Hecke algebra and U υ (sl(S 1 Q )) 37 6.3. Shuffle algebra presentation of U > ∞ 44 6.4. Hecke operators on line bundles and the fundamental representation of U υ (sl(S 1 Q )) 47 6.5. Tensor and symmetric tensor representations of U υ (sl(S 1 Q )) 47 6.6. Representation of the completed Hecke algebra 47 7. Comparisons with U υ sl(+∞)) and U υ sl(∞)) 48 Appendix A. Some results from [Lin14] 50 Appendix B. U υ (sl(S 1 )) from mirror symmetry 57 Acknowledgments 57 B.1. Mirror symmetry for P 1 ∞ 57 B.2. U υ (sl(S 1 )) from the mirror side 58 B.3. Fundamental representation 60 B.4. The composition Hall algebra of S 1 60 B.5. Other Dynkin types 60 References 62 1.1. Definition of U υ (sl(S 1 )). Let us introduce some notation. We call interval of S 1 a half-open interval J = [a, b[⊆ S 1 . We say that an interval J is rational if J ⊆ S 1 Q . We say that an interval J (resp. rational interval J) is strict if J = S 1 (resp. J = S 1 Q ). For an interval J, we denote by χ J its

show abstract

Geometricity for derived categories of algebraic stacks

Cited by 28 publications

References 45 publications

Gluing semi-orthogonal decompositions

Gluing semi-orthogonal decompositions

An embedding problem for finite local torsors over twisted curves

The circle quantum group and the infinite root stack of a curve

Contact Info

Product

Resources

About