Combinatorial Shifting and Graded Betti Numbers
نویسندگان
چکیده
Let ∆ be a simplicial complex and I∆ its Stanley–Reisner ideal. It has been conjectured that, for each i and j, the graded Betti number βii+j(I∆) of I∆ is smaller than or equal to that of I∆c , where ∆ c is a combinatorial shifted complex of ∆. In the present paper the conjecture will be proved affirmatively. In particular the inequalities βii+j(I∆) ≤ βii+j(I∆lex) hold for all i and j, where ∆ is the unique lexsegment simplicial complex with the same f -vector as ∆ and where the base field is of arbitrary characteristic.
منابع مشابه
The Behavior of Graded Betti Numbers via Algebraic Shifting and Combinatorial Shifting
Let ∆ be a simplicial complex and I∆ its Stanley–Reisner ideal. We write ∆ for the exterior algebraic shifted complex of ∆ and ∆ for a combinatorial shifted complex of ∆. It will be proved that for all i and j one has the inequalities βii+j(I∆e) ≤ βii+j(I∆c) on the graded Betti numbers of I∆e and I∆c . In addition, the bad behavior of graded Betti numbers of I∆c will be studied.
متن کاملShifting Operations and Graded Betti Numbers
The behaviour of graded Betti numbers under exterior and symmetric algebraic shifting is studied. It is shown that the extremal Betti numbers are stable under these operations. Moreover, the possible sequences of super extremal Betti numbers for a graded ideal with given Hilbert function are characterized. Finally it is shown that over a field of characteristic 0, the graded Betti numbers of a ...
متن کاملTight Combinatorial Manifolds and Graded Betti Numbers
In this paper, we study the conjecture of Kühnel and Lutz, who state that a combinatorial triangulation of the product of two spheres S×S with j ≥ i is tight if and only if it has exactly i+2j+4 vertices. To approach this conjecture, we use graded Betti numbers of Stanley–Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when ...
متن کاملOn a special class of Stanley-Reisner ideals
For an $n$-gon with vertices at points $1,2,cdots,n$, the Betti numbers of its suspension, the simplicial complex that involves two more vertices $n+1$ and $n+2$, is known. In this paper, with a constructive and simple proof, wegeneralize this result to find the minimal free resolution and Betti numbers of the $S$-module $S/I$ where $S=K[x_{1},cdots, x_{n}]$ and $I$ is the associated ideal to ...
متن کاملBetti Numbers of Graded Modules and the Multiplicity Conjecture in the Non-cohen-macaulay Case
Abstract. We use the results by Eisenbud and Schreyer [3] to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the Multiplicity Conjecture of Herzog, Huneke and Srinivasan [5] for modules that are not necessarily Cohen-Macaulay. We give a combinatorial proof of th...
متن کامل