Koszulity of splitting algebras associated with cell complexes☆☆Some of our results were later generalized by T. Cassidy, C. Phan, and B. Shelton in their paper “Noncommutative Koszul algebras from combinatorial topology” in arXiv:0811.3450.

Retakh, Vladimir; Serconek, Shirlei; Wilson, Robert Lee

doi:10.1016/j.jalgebra.2009.11.039

Journal of Algebra

2010

DOI: 10.1016/j.jalgebra.2009.11.039

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Koszulity of splitting algebras associated with cell complexes☆☆Some of our results were later generalized by T. Cassidy, C. Phan, and B. Shelton in their paper “Noncommutative Koszul algebras from combinatorial topology” in arXiv:0811.3450.

Vladimir Retakh¹,

Shirlei Serconek²,

Robert Lee Wilson³

Abstract: We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M. For a twodimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two.

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Cited by 4 publications

References 13 publications

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Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

Kloefkorn

Shelton

2015

Journal of Algebra

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Abstract. We study a finite dimensional quadratic graded algebra RΓ defined from a finite ranked poset Γ. This algebra has been central to the study of the splitting algebras AΓ introduced by Gelfand, Retakh, Serconek and Wilson, [4]. Those algebras are known to be quadratic when Γ satisfies a combinatorial condition known as uniform. A central question in this theory has been: when are the algebras Koszul? We prove that RΓ is Koszul and Γ is uniform if and only if the poset Γ is Cohen-Macaulay. We also show that the cohomology of the order complex of Γ can be identified with certain cohomology groups defined internally to the ring RΓ, HR Γ (n, 0) (introduced in [2]) whenever Γ is Cohen-Macualay. Finally, we settle in the negative the long-standing question: Does numerically Koszul imply Koszul for algebras of the form RΓ.

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Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

Kloefkorn

Shelton

2015

Journal of Algebra

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Hilbert series of algebras associated to directed graphs and order homology

Retakh

Serconek²,

Wilson

2012

Journal of Pure and Applied Algebra

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The Koszul property as a topological invariant and measure of singularities

Sadofsky¹,

Shelton²

2011

Pacific J. Math.

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have associated to any regular cell complex X a quadratic K -algebra R(X). They gave a combinatorial solution to the question of when this algebra is Koszul. The algebra R(X) is a combinatorial invariant but not a topological invariant. We show that, nevertheless, the property that R(X) be Koszul is a topological invariant.In the process, we establish some conditions on the types of local singularities that can occur in cell complexes X such that R(X) is Koszul, and more generally in cell complexes that are pure and connected by codimension-one faces.

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Koszulity of splitting algebras associated with cell complexes☆☆Some of our results were later generalized by T. Cassidy, C. Phan, and B. Shelton in their paper “Noncommutative Koszul algebras from combinatorial topology” in arXiv:0811.3450.

Abstract: We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M. For a twodimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two.

Cited by 4 publications

References 13 publications

Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

Hilbert series of algebras associated to directed graphs and order homology

The Koszul property as a topological invariant and measure of singularities

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