The Koszul property as a topological invariant and measure of singularities

Sadofsky, Hal; Shelton, Brad

doi:10.2140/pjm.2011.252.473

Cited by 3 publications

(3 citation statements)

References 6 publications

(18 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, the results here shed light on many of the previous known results about splitting algebras. In addition to the overlap with results from [9], [8] and [5] already mentioned, many of the results of [2], [11] and [12] follow immediately from the results here. The common maximal chain length in (2) above is the rank of x in Γ and is denoted rk Γ (x), or if there is no possiblity of confusion simply rk(x).…”

Section: Introductionsupporting

confidence: 85%

Splitting Algebras II: The Cohomology Algebra

Shelton¹

2012

Preprint

Self Cite

View full text Add to dashboard Cite

Gelfand, Retakh, Serconek and Wilson,in [3], defined a graded algebra AΓ attached to any finite ranked poset Γ -a generalization of the universal algebra of pseudo-roots of noncommutative polynomials. This algebra has since come to be known as the splitting algebra of Γ. The splitting algebra has a secondary filtration related to the rank function on the poset and the associated graded algebra is denoted here by A Γ . We calculate the cohomology algebra (and coalgebra) of A Γ explicitly. As a corollary to this calculation we have a proof that A Γ is Koszul (respectively quadratic) if and only if Γ is Cohen-Macaulay (respectively uniform). We show by example that the cohomology algebra (resp. coalgebra) of AΓ may be strictly smaller that the cohomology algebra (resp. coalgebra) of A Γ .

show abstract

Section: Introductionsupporting

confidence: 85%

Splitting Algebras II: The Cohomology Algebra

Shelton¹

2012

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…However, the conditions of that theorem can, with some small effort, be translated directly into the statement that Γ is Cohen-Macaulay, thereby relating that theorem directly to 6.2. Since Cohen-Macaulay is known to be a topological invariant, Theorem 6.2 also recaptures the results of [11].…”

Section: A Few Examplessupporting

confidence: 53%

“…The second main result of [2], Theorem 5.3, is an exact description, in combinatorial-topological terms, of when the algebras RΓ are Koszul (these conditions include uniform, expressed as a topological condition). In the paper [11] it was further shown that the conditions under whichΓ is Koszul are topological invariants rather than just combinatorial invariants. However, the conditions of that theorem can, with some small effort, be translated directly into the statement thatΓ is Cohen-Macaulay, thereby relating that theorem directly to 6.2.…”

Section: A Few Examplesmentioning

confidence: 99%

Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

Kloefkorn

Shelton

2015

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Weakly Cohen–Macaulay posets and a class of finite-dimensional graded quadratic algebras

Kloefkorn

2017

Journal of Algebra

View full text Add to dashboard Cite

Abstract. To a finite ranked poset Γ we associate a finite-dimensional graded quadratic algebra RΓ. Assuming Γ satisfies a combinatorial condition known as uniform, RΓ is related to a well-known algebra, the splitting algebra AΓ. First introduced by Gelfand, Retakh, Serconek, and Wilson, splitting algebras originated from the problem of factoring non-commuting polynomials. Given a finite ranked poset Γ, we ask: Is RΓ Koszul? The Koszulity of RΓ is related to a combinatorial topology property of Γ called Cohen-Macaulay. Kloefkorn and Shelton proved that if Γ is a finite ranked cyclic poset, then Γ is Cohen-Macaulay if and only if Γ is uniform and RΓ is Koszul. We define a new generalization of Cohen-Macaulay, weakly Cohen-Macaulay, and we note that this new class includes posets with disconnected open subintervals. We prove: if Γ is a finite ranked cyclic poset, then Γ is weakly Cohen-Macaulay if and only if RΓ is Koszul.

show abstract

The Koszul property as a topological invariant and measure of singularities

Cited by 3 publications

References 6 publications

Splitting Algebras II: The Cohomology Algebra

Splitting Algebras II: The Cohomology Algebra

Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

Weakly Cohen–Macaulay posets and a class of finite-dimensional graded quadratic algebras

Contact Info

Product

Resources

About