Regular modules over 2-dimensional quantum Beilinson algebras of Type $$S$$ S

Mori, Izuru

doi:10.1007/s00209-015-1407-x

Cited by 5 publications

(26 citation statements)

References 30 publications

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“…In fact, the exact sequence 0 → L → M → N → 0 induces an exact sequence Proof. The assumption on A implies that the isomorphism classes of simple objects in tails 0 A := {πM ∈ tails A | GKdim M ≤ 1} are given by {πM p } p∈E by [12,Lemma 3.5,Proposition 4.4]. We prove it by induction on r.…”

Section: An Application To Skew Exterior Algebrasmentioning

confidence: 96%

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Noncommutative matrix factorizations with an application to skew exterior algebras

Mori¹,

Ueyama²

2018

Preprint

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Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factorizations and totally reflexive modules (this result is analogous to the famous result by Eisenbud for commutative hypersurfaces). As an application, we describe indecomposable noncommutative graded matrix factorizations over skew exterior algebras.

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Section: An Application To Skew Exterior Algebrasmentioning

confidence: 96%

“…Proof. Note that if (1) α ij α jk α ki = 1 for all 1 ≤ i, j, k ≤ n, then E = P n−1 by [16, Theorem 4.1], and if (2) n = 3 and α 12 α 23 α 31 is not a root of unity, then ||τ || = ∞ by [12,Lemma 4.13]. In either case, A !…”

mentioning

confidence: 99%

Noncommutative matrix factorizations with an application to skew exterior algebras

Mori¹,

Ueyama²

2018

Preprint

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“…It is known that the isomorphism classes of simple regular modules over are parameterized by (cf. [13, Theorem 3.19]). For a three-dimensional quantum polynomial algebra A , we expect that the following are equivalent: is finite over its center. is -representation tame in the sense of [6].…”

Section: An Application To Beilinson Algebrasmentioning

confidence: 99%

“…These results are important to study representation theory of the Beilinson algebra , which is a typical example of a -representation infinite algebra defined in [6]. This was the original motivation of the paper [13].…”

Section: Introductionmentioning

confidence: 98%

Quantum projective planes finite over their centers

Itaba

Mori²

2022

Can. Math. Bull.

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For a 3-dimensional quantum polynomial algebra 𝐴 = A (𝐸 , 𝜎), Artin-Tate-Van den Bergh showed that 𝐴 is finite over its center if and only if | 𝜎 | < ∞. Moreover, Artin showed that if 𝐴 is finite over its center and 𝐸 ≠ P 2 , then 𝐴 has a fat point module, which plays an important role in noncommutative algebraic geometry, however the converse is not true in general. In this paper, we will show that, if 𝐸 ≠ P 2 , then 𝐴 has a fat point module if and only if the quantum projective plane Proj nc 𝐴 is finite over its center in the sense of this paper if and only if |𝜈 * 𝜎 3 | < ∞ where 𝜈 is the Nakayama automorphism of 𝐴. In particular, we will show that if the second Hessian of 𝐸 is zero, then 𝐴 has no fat point module.

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“…This result provides a strong connection between noncommutative projective geometry and representation theory of finite dimensional algebras. See [6], [9], [13] for details.…”

Section: ))mentioning

confidence: 99%

Cluster tilting modules and noncommutative projective schemes

Ueyama

2017

Pacific J. Math.

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In this paper, we study the relationship between equivalences of noncommutative projective schemes and cluster tilting modules. In particular, we prove the following result. Let $A$ be an AS-Gorenstein algebra of dimension $d\geq 2$ and ${\mathsf{tails}\,} A$ the noncommutative projective scheme associated to $A$. If $\operatorname{gldim}({\mathsf{tails}\,} A)< \infty$ and $A$ has a $(d-1)$-cluster tilting module $X$ satisfying that its graded endomorphism algebra is $\mathbb N$-graded, then the graded endomorphism algebra $B$ of a basic $(d-1)$-cluster tilting submodule of $X$ is a two-sided noetherian $\mathbb N$-graded AS-regular algebra over $B_0$ of global dimension $d$ such that ${\mathsf{tails}\,} B$ is equivalent to ${\mathsf{tails}\,} A$.Comment: 16 page

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Regular modules over 2-dimensional quantum Beilinson algebras of Type $$S$$ S

Cited by 5 publications

References 30 publications

Noncommutative matrix factorizations with an application to skew exterior algebras

Noncommutative matrix factorizations with an application to skew exterior algebras

Quantum projective planes finite over their centers

Cluster tilting modules and noncommutative projective schemes

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