Serre duality for non-commutative ${\mathbb {P}}^{1}$-bundles

Nyman, Adam

doi:10.1090/s0002-9947-04-03523-8

Cited by 21 publications

(29 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark. The analogous result was shown (again with the assumption that the base schemes are isomorphic) for general smooth projective base in [33]. The semiorthogonal decomposition yields a significant streamlining of the argument, however.…”

Section: Remarksupporting

confidence: 59%

The birational geometry of noncommutative surfaces

Rains

2019

Preprint

View full text Add to dashboard Cite

We show that any commutative rationally ruled surface with a choice of anticanonical curve admits a 1-parameter family of noncommutative deformations parametrized by the Jacobian of the anticanonical curve, and show that many standard facts from commutative geometry (blowups commute, Quot schemes are projective, etc.) carry over. The key new tool in studying these deformations is a relatively simple description of their derived categories and the relevant t-structures; this also allows us to establish nontrivial derived equivalences for deformations of elliptic surfaces. We also establish that the category of line bundles (suitably defined) on such a surface has a faithful representation in which the morphisms are difference or differential operators, and thus find that difference/differential equations can be viewed as sheaves on such surfaces. In particular, we find that many moduli spaces of sheaves on such surfaces have natural interpretations as moduli spaces of equations with (partially) specified singularities, and in particular find that the "isomonodromy" interpretation of discrete Painlevé equations and their generalizations has a natural geometric interpretation (twisting sheaves by line bundles).

show abstract

Section: Remarksupporting

confidence: 59%

The birational geometry of noncommutative surfaces

Rains

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…By [15, Theorem 1.2] and Lemma 2.6, it suffices to show that φ = a µ( * ψ) −1 makes the diagram commute. This fact follows from [11,Corollary 6.7].…”

Section: )mentioning

confidence: 75%

“…η ′ * K * →K whose unlabeled maps are canonical. The diagram commutes by the functoriality of (−) * and by [11,Corollary 6.3], and the compositions of the horizontal maps equal the map ǫ defined in the statement of Lemma 2.8. On the one hand, if we replace φ * by a µ(ψ * * ) −1 , the outer circuit of (2-5) commutes by [11,Corollary 6.7].…”

Section: )mentioning

confidence: 99%

“…The diagram commutes by the functoriality of (−) * and by [11,Corollary 6.3], and the compositions of the horizontal maps equal the map ǫ defined in the statement of Lemma 2.8. On the one hand, if we replace φ * by a µ(ψ * * ) −1 , the outer circuit of (2-5) commutes by [11,Corollary 6.7]. On the other hand, by [15, Theorem 1.2], and Lemma 2.6, this choice of φ * is unique making the outer circuit of (2-5) commute.…”

Section: )mentioning

confidence: 99%

“…Homological preliminaries. We recall [11,Section 2.2] that (e k A) ≥k+n is defined to be the sum of K − K-bimodules ⊕ i≥0 e k A k+n+i . We define A ≥n := ⊕ k (e k A) ≥k+n with its natural A − A-bimodule structure and we let A 0 denote the…”

Section: Vector Bundles Over P(v )mentioning

confidence: 99%

See 2 more Smart Citations

The geometry of arithmetic noncommutative projective lines

Nyman¹

2014

Journal of Algebra

View full text Add to dashboard Cite

Abstract. Let k be a perfect field and let K/k be a finite extension of fields. An arithmetic noncommutative projective line is a noncommutative space of the form ProjS K (V ), where V be a k-central two-sided vector space over K of rank two and S K (V ) is the noncommutative symmetric algebra generated by V over K defined by M. Van den Bergh [26]. We study the geometry of these spaces. More precisely, we prove they are integral, we classify vector bundles over them, we classify them up to isomorphism, and we classify isomorphisms between them. Using the classification of isomorphisms, we compute the automorphism group of an arithmetic noncommutative projective line.

show abstract

Witt’s Theorem for Noncommutative Conic Curves

Nyman

2015

Appl Categor Struct

View full text Add to dashboard Cite

show abstract

Serre duality for non-commutative ${\mathbb {P}}^{1}$-bundles

Cited by 21 publications

References 16 publications

The birational geometry of noncommutative surfaces

The birational geometry of noncommutative surfaces

The geometry of arithmetic noncommutative projective lines

Witt’s Theorem for Noncommutative Conic Curves

Contact Info

Product

Resources

About