A noncommutative version of Beilinson's theorem

Minamoto, Hiroyuki

doi:10.1016/j.jalgebra.2008.03.014

Cited by 13 publications

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“…In particular, we show S nc (M ) is coherent, and its proj category P nc (M ) is derived equivalent to the corresponding bimodule species. This generalizes the main theorem of [8], which in turn is a generalization of Beilinson's derived equivalence. As corollaries, we show that P nc (M ) is hereditary and there is a structure theorem for sheaves on P nc (M ) analogous to that for P 1 .…”

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confidence: 71%

A Representation Theoretic Study of Non-Commutative Symmetric Algebras

Chan¹,

Nyman²

2019

Proceedings of the Edinburgh Mathematical Society

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We study Van den Bergh's noncommutative symmetric algebra S nc (M ) (over division rings) via Minamoto's theory of Fano algebras. In particular, we show S nc (M ) is coherent, and its proj category P nc (M ) is derived equivalent to the corresponding bimodule species. This generalizes the main theorem of [8], which in turn is a generalization of Beilinson's derived equivalence. As corollaries, we show that P nc (M ) is hereditary and there is a structure theorem for sheaves on P nc (M ) analogous to that for P 1 . 2010 Mathematics Subject Classification. Primary 14A22, 16S38; Secondary 16E35. Proof. To prove the first result (which appeared in [5]), one shows that there are D 1 − D 0 -bimodule isomorphisms Hom D1 (M D1 , D 1 ) → Hom k (M, k) and Hom D0 ( D0 M, D 0 ) → Hom k (M, k).The first one takes ψ : M D1 → D 1 to tr D1/k •ψ, and the second is similar.The proofs of the second and third results follow the proof of [4, Lemma 3.2].

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confidence: 71%

A Representation Theoretic Study of Non-Commutative Symmetric Algebras

Chan¹,

Nyman²

2019

Proceedings of the Edinburgh Mathematical Society

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“…Since X is the full subgraph of X with vertex set X, v( X) = |X|. It now follows from Remark 3.2.1 that the right-hand side of (3)(4)(5)(6)(7)(8) is p X (0). Hence by Lemma 3.5,…”

Section: If G = 4 Thenmentioning

confidence: 94%

“…If p ∈ Z[t ±1 ] and p(θ) > 0, then p = q M for some M ∈ gr(A) by Lemma 2.6 so p is in the right-hand side of (2)(3)(4)(5)(6)(7)(8). It is clear that 0 is in the right-hand side of (2-8).…”

Section: The Converse Of Proposition 23 Is Not Truementioning

confidence: 97%

“…Explicitly, if F, G ∈ qgr(A), then Ext 2 qgr(A) (F, G) = 0, dim k Ext * qgr(A) (F, G) < ∞, and Hom qgr(A) (F, G) ∼ = Ext 1 qgr(A) (G, F(−n − 1)) * . In [8], Minamoto gives additional evidence that qgr(A) is like coh(P 1 ) by proving an analogue of the well-known equivalence, D b (Qcoh(P 1 k )) ≡ D b (kQ 2 ), of bounded derived categories where kQ 2 is the path algebra of the quiver with two vertices and 2 arrows from the first vertex to the second. Minamoto shows that D b (Qcoh(A)) ≡ D b (kQ n+1 ) where Q n+1 is the path algebra of the quiver with two vertices and n + 1 arrows from the first vertex to the second.…”

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confidence: 99%

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The Grothendieck group of non-commutative non-noetherian analogues of P1 and regular algebras of global dimension two

Sisodia¹,

Smith²

2015

Journal of Algebra

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Let V be a finite-dimensional positively-graded vector space. Let b ∈ V ⊗ V be a homogeneous element whose rank is dim(V ). Let A = T V/(b), the quotient of the tensor algebra T V modulo the 2-sided ideal generated by b. Let gr(A) be the category of finitely presented graded left A-modules and fdim(A) its full subcategory of finite dimensional modules. Let qgr(A) be the quotient category gr(A)/fdim(A). We compute the Grothendieck group K 0 (qgr(A)). In particular, if the reciprocal of the Hilbert series of A, which is a polynomial, is irreducible, then K 0 (qgr(A)) ∼ = Z[θ] ⊂ R as ordered abelian groups where θ is the smallest positive real root of that polynomial. When dim k (V ) = 2, qgr(A) is equivalent to the category of coherent sheaves on the projective line, P 1 , or a stacky P 1 if V is not concentrated in degree 1. If dim k (V ) ≥ 3, results of Piontkovski and Minamoto suggest that qgr(A) behaves as if it is the category of "coherent sheaves" on a non-commutative, non-noetherian analogue of P 1 .

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“…The ε-norm of a triangulated category. Recall that for σ = (Z, P) ∈ Stab(T ) we denote (see [23,Section 3]): (43) P T σ = {exp(iπφ σ (X)) : X ∈ σ ss } = {exp(iπt) : t ∈ R and P(t) = {0}}, Here we will use also the notation:…”

Section: 2mentioning

confidence: 99%

Some new categorical invariants

Dimitrov

Katzarkov

2019

Sel. Math. New Ser.

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In this paper we introduce several new categorical notions and give many examples. We prove that the moduli space of stability conditions on the derived category of representations of K(l), the l-Kronecker quiver, is biholomorphic to C × H for l ≥ 3. This produces an example of semi-orthogonal decomposition, SOD, T = T1, T2 , where Stab(T ) is not biholomorphic to Stab(T1) × Stab(T2) (whereas Stab(T1 ⊕ T2) ∼ = Stab(T1) × Stab(T2) when rank(K0(Ti)) < +∞).These calculations suggest a new notion of a norm, which strictly increases on {D b (K(l))} l≥2 . To a triangulated category T which has property of a phase gap we attach a non-negative number T ε . Natural assumptions on a SOD T = T1, T2 imply T1, T2 ε ≥ max{ T1 ε , T2 ε }.Using the norm we define a non-trivial topology on the set of equivalence classes of triangulated categories with a phase gap, in which the set of discrete derived categories is a discrete subset, whereas the rationality of a smooth surface S ensures thatViewing D b (K(l)) as a non-commutative curve, we observe that it is reasonable to count noncommutative curves in any category which lies in a small neighborhood (w.r. to our topology) of a given non-commutative curve. Examples show that this idea (non-commutative curve-counting) opens directions to new categorical structures and connections to number theory and classical geometry. We give a general definition, which specializes to the non-commutative curve-counting invariants. In an example arising on the A side we specialize our definition to non-commutative Calabi-Yau curve-counting, where the entities we count are a Calabi-Yau modification of D b (K(l)).In the last section we speculate that one might consider a holomorphic family of categories, introduced by Kontsevich, as a non-commutative extension with the norm, introduced here, playing a role similar to the classical notion of degree of an extension in Galois theory.Aut(D b (P 2 )) l

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A noncommutative version of Beilinson's theorem

Abstract: We prove a noncommutative version of the John-Nirenberg theorem for nontracial filtrations of von Neumann algebras. As an application, we obtain an analogue of the classical large deviation inequality for elements of the associated BM O space.

Cited by 13 publications

References 14 publications

A Representation Theoretic Study of Non-Commutative Symmetric Algebras

A Representation Theoretic Study of Non-Commutative Symmetric Algebras

The Grothendieck group of non-commutative non-noetherian analogues of P1 and regular algebras of global dimension two

Some new categorical invariants

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