Algebraic Shifting and Exterior and Symmetric Algebra Methods

Nagel, Uwe; Römer, Tim; Vinai, Natale Paolo

doi:10.1080/00927870701665321

Cited by 5 publications

(8 citation statements)

References 34 publications

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“…which has been also obtained in another way in [16,Theorem 2.4(i)]. Furthermore, this bound is tight if and only if J is componentwise linear.…”

Section: Introductionsupporting

confidence: 51%

“…We formulate and prove the following result using the generic annihilator numbers. Plugging in the description of the generic annihilator numbers in terms of the minimal generators of J e and taking into account that we use the reversed order on n , our result is the same as [16,Theorem 2.4(i)], which is a direct consequence of the construction of the Cartan homology for stable ideals in [3, Proposition 3.1].…”

Section: Proposition 33 ([3 Theorem 22]) Let M ∈mentioning

confidence: 53%

“…Finally, we show in Theorem 3.15 that the so-called Cartan-Betti numbers (see [16,Definition 2.2]) can be bounded from above by a positive linear combination of the exterior generic annihilator numbers. In particular, this gives rise to an upper bound for the exterior Betti numbers,…”

Section: Introductionmentioning

confidence: 98%

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Exterior Depth and Exterior Generic Annihilator Numbers

Kämpf

Kubitzke

2012

Communications in Algebra

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We study the exterior depth of an E-module and its exterior generic annihilator numbers. For the exterior depth of a squarefree E-module, we show how it relates to the symmetric depth of the corresponding S-module and classify those simplicial complexes having a particular exterior depth in terms of their exterior algebraic shifting. We define exterior annihilator numbers analogously to the annihilator numbers over the polynomial ring introduced by Trung [19] and Conca et al. [9]. In addition to a combinatorial interpretation of the annihilator numbers, we show how they are related to the symmetric Betti numbers and the Cartan-Betti numbers, respectively. We finally conclude with an example which shows that neither the symmetric nor the exterior generic annihilator numbers are minimal among the annihilator numbers with respect to a sequence.

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“…which has been also obtained in another way in [16,Theorem 2.4(i)]. Furthermore, this bound is tight if and only if J is componentwise linear.…”

Section: Introductionsupporting

confidence: 51%

Section: Proposition 33 ([3 Theorem 22]) Let M ∈mentioning

confidence: 53%

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Exterior Depth and Exterior Generic Annihilator Numbers

Kämpf

Kubitzke

2012

Communications in Algebra

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“…The existence of the generic initial ideal gin(J) of a graded ideal J in the exterior algebra over an infinite field is proved by Aramova, Herzog and Hibi in [3, Theorem 1.6], analogously to the case of ideals in the polynomial ring. (See, e.g., also [11,Chapter 5] or [14] for related results. )…”

Section: Initial and Generic Initial Idealsmentioning

confidence: 99%

Homological properties of Orlik–Solomon algebras

Kämpf

Römer

2009

manuscripta math.

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ABSTRACT. The Orlik-Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first we study homological properties of E-modules as e.g. complexity, depth and regularity. In particular, we consider modules with linear injective resolutions. We apply our results to Orlik-Solomon algebras of matroids and give formulas for the complexity, depth and regularity of such rings in terms of invariants of the matroid. Moreover, we characterize those matroids whose Orlik-Solomon ideal has a linear projective resolution and compute in these cases the Betti numbers of the ideal.

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“…Theorem 4.4 and Corollary 4.5 provide the following new characterization of componentwise linear ideals in the exterior algebra. (See also [19] for other characterizations of componentwise linear ideals.) Theorem 4.7.…”

Section: It Is Easily Verified Thatmentioning

confidence: 99%

Rigidity of Linear Strands and Generic Initial Ideals

Murai

Singla²

2008

Nagoya Mathematical Journal

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Abstract. Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in S and E when passing to their generic initial ideals. First, we prove that if the graded Betti numbers β S ii+k (S/I) = β S ii+k`S / Gin(I)´for some i > 1 and k ≥ 0, then β S qq+k (S/I) = β S qq+k`S / Gin(I)´for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show that if β E ii+k (E/I) = β E ii+k`E / Gin(I)´for some i > 1 and k ≥ 0, then β E qq+k (E/I) = β E qq+k`E / Gin(I)´for all q ≥ 1, where I ⊂ E is a graded ideal. In addition, it will be shown that the graded Betti numbers β R ii+k (R/I) = β R ii+k`R / Gin(I)´for all i ≥ 1 if and only if I k and I k+1 have a linear resolution. Here I d is the ideal generated by all homogeneous elements in I of degree d, and R can be either the polynomial ring or the exterior algebra.

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Algebraic Shifting and Exterior and Symmetric Algebra Methods

Cited by 5 publications

References 34 publications

Exterior Depth and Exterior Generic Annihilator Numbers

Exterior Depth and Exterior Generic Annihilator Numbers

Homological properties of Orlik–Solomon algebras

Rigidity of Linear Strands and Generic Initial Ideals

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