Binomial ideals

Eisenbud, David; Sturmfels, Bernd

doi:10.1215/s0012-7094-96-08401-x

Cited by 303 publications

(475 citation statements)

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“…One case where an affirmative answer is known is when the CI ideal is a binomial ideal (that is, generated by binomials p u − αp v ). Here rationality holds because of the following result of commutative algebra [44].…”

Section: Conditional Independence Modelsmentioning

confidence: 99%

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Lectures on Algebraic Statistics

Drton

Sturmfels

Sullivant

2009

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332

423

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Section: Conditional Independence Modelsmentioning

confidence: 99%

“…Given any matrix Z ∈ R 4×4 , the matrix τ ∈ T Θ (Σ 0 ) minimizing the above norm must be equal to Z in the entries τ 11 , τ 22 , τ 33 , τ 34 , τ 44 . Also, τ 12 = 0.…”

mentioning

confidence: 99%

Lectures on Algebraic Statistics

Drton

Sturmfels

Sullivant

2009

Self Cite

332

423

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“…From an algebro-geometric point of view, the varieties associated to binomial ideals are unions of (translates of) toric varieties [4], and thus of much interest. Combinatorially, binomial ideals already contain the class of monomial ideals, a cornerstone of combinatorial commutative algebra [13,18].…”

Section: Introductionmentioning

confidence: 99%

“…The systematic study of the primary decomposition of binomial ideals was initiated by Eisenbud and Sturmfels in [4]. Over an algebraically closed field, they proved that the associated primes and primary components of binomial ideals can be chosen binomial.…”

Section: Introductionmentioning

confidence: 99%

Decompositions of cellular binomial ideals

Eser

Matusevich

2016

Journal of the London Mathematical Society

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“…Hence, by Corollary A.2, these are Koszul filtrations in all characteristics. We are left with two cases: the algebras (10) and (11). If char k = 2, one checks that the proposed filtrations for (10) and (11) work.…”

Section: Proof See Appendix Bmentioning

confidence: 99%

The Koszul property for spaces of quadrics of codimension three

D’Alì¹

2017

Journal of Algebra

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A b s t r ac t . In this paper we prove that, if k is an algebraically closed field of characteristic different from 2, almost all quadratic standard graded k-algebras R such that dim k R 2 = 3 are Koszul. More precisely, up to graded k-algebra homomorphisms and trivial fiber extensions, we find out that only two (or three, when the characteristic of k is 3) algebras of this kind are non-Koszul.Moreover, we show that there exist nontrivial quadratic standard graded kalgebras with dim k R 1 = 4, dim k R 2 = 3 that are Koszul but do not admit a Gröbner basis of quadrics even after a change of coordinates, thus settling in the negative a question asked by Conca. I n t ro d u c t i o nA (commutative) standard graded algebra R over a field k is a quotient of a polynomial ring over k in a finite number of variables by a homogeneous ideal I not containing any linear form. We say that R is Koszul when the minimal graded free resolution of k as an R-module is linear: for a survey about Koszulness in the commutative setting, see [7]. It is well-known that any Koszul algebra is quadratic, i.e. its defining ideal I is generated by quadrics. Note that, if R is a trivial fiber extension, i.e. it contains a linear form ∈ R 1 such that R 1 = 0, then R is Koszul if and only if R/ is Koszul.For the rest of this introduction, let R be a quadratic standard graded k-algebra, where k is an algebraically closed field of characteristic different from two. Backelin proved in [1] that, if dim k R 2 = 2, then R is Koszul (this actually holds for any field k). Conca then proved in [5] that, if dim k R 2 = 3 and R is Artinian, then R is Koszul. The main problem we are addressing in this paper is to find out what happens when, in the latter case, we drop the Artinian assumption. We will prove in Theorem 3.1 that, up to trivial fiber extension, the only non-Koszul algebras live in embedding dimension three. More precisely, these non-Koszul algebras in three variables are the two (or three, when char k = 3) objects exhibited by Backelin and Fröberg in [2]. As a byproduct, we shall also obtain a list of all possible Hilbert series when dim k R 2 = 3, see Theorem 3.3.Our results are in agreement with the numerical data presented by Roos in the characteristic 0 four-variable case, see [15].

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Binomial ideals

Cited by 303 publications

References 25 publications

Lectures on Algebraic Statistics

Lectures on Algebraic Statistics

Decompositions of cellular binomial ideals

The Koszul property for spaces of quadrics of codimension three

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