Generic Initial Ideals and Squeezed Spheres
نویسنده
چکیده
In 1988 Kalai construct a large class of simplicial spheres, called squeezed spheres, and in 1991 presented a conjecture about generic initial ideals of Stanley–Reisner ideals of squeezed spheres. In the present paper this conjecture will be proved. In order to prove Kalai’s conjecture, based on the fact that every squeezed (d−1)-sphere is the boundary of a certain d-ball, called a squeezed d-ball, generic initial ideals of Stanley–Reisner ideals of squeezed balls will be determined. In addition, generic initial ideals of exterior face ideals of squeezed balls are determined. On the other hand, we study the squeezing operation, which assigns to each Gorenstein* complex Γ having the weak Lefschetz property a squeezed sphere Sq(Γ), and show that this operation increases graded Betti numbers.
منابع مشابه
Reverse Lexicographic Initial Ideals of Generic Ideals Are Nitely Generated
This article generalizes the well-known notion of generic forms to the algebra R 0 , introduced in 27]. For the total degree, then reverse lexicographic order, we prove that the initial ideal of an ideal generated by nitely many generic forms (in countably innnitely many variables) is nitely generated. This contrasts to the lexicographic order, for which initial ideals of generic ideals in gene...
متن کاملNice Initial Complexes of Some Classical Ideals
This is a survey article on Gorenstein initial complexes of extensively studied ideals in commutative algebra and algebraic geometry. These include defining ideals of Segre and Veronese varieties, toric deformations of flag varieties known as Hibi ideals, determinantal ideals of generic matrices of indeterminates, and ideals generated by Pfaffians of generic skew symmetric matrices. We give a s...
متن کاملOn Some Partial Orders Associated to Generic Initial Ideals
We study two partial orders on [x1, . . . , xn], the free abelian monoid on {x1, . . . , xn}. These partial orders, which we call the “strongly stable” and the “stable” partial order, are defined by the property that their filters are precisely the strongly stable and the stable monoid ideals. These ideals arise in the study of generic initial ideals.
متن کاملGröbner Basis Techniques in Algebraic Combinatorics
Gröbner basis techniques in the algebraic study of triangulations of convex polytopes as well as of the number of faces of simplicial complexes will be discussed. Of these two traditional topics in combinatorics, the first will be studied by using initial ideals of toric ideals and the second will be studied by using generic initial ideals of monomial ideals.
متن کاملMonomial Ideals Arising from Flag Complexes Whose Generic Initial Ideals Do Not Depend on Term Orders
We will study monomial ideals I in the exterior algebra as well as in the polynomial ring whose generic initial ideal is constant for all term orders up to permutations of variables. First, in the exterior algebra, we determine all graphs and all flag complexes whose exterior face ideal satisfies the above condition. Second, in the polynomial ring, it will be shown that the generic initial idea...
متن کامل