Abstract
An ideal I subset mathbb {k}[x_1, ldots , x_n] is said to have linear powers if I^k has a linear minimal free resolution, for all integers k>0. In this paper, we study the Betti numbers of I^k, for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for d=2, 3 or n-1, and the product of all ideals generated by s variables, for s=n-1 or n-2. We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.
Highlights
Let S = k[x1, . . . , xn] be a polynomial ring over a field k
This follows from the result in [10], which states that monomial ideals generated in degree two, with linear resolution, have linear powers
Let S = k[x1, . . . , xn], and let I ⊂ S be a homogeneous ideal with linear powers, generated in degree d
Summary
Application of this result, together with a method involving Rees algebras, gives explicit expressions for βi (S/I k), as polynomials in k, for the ideals in A with d = 2, 3, and n − 1, and the ideals in B with s = n − 1 and n − 2 Both A and B are subclasses of a larger class of ideals with linear powers, namely the polymatroidal ideals. Result in [10], which states that monomial ideals generated in degree two, with linear resolution, have linear powers. 3 to the study of ideals with linear powers, not necessarily polymatroidal, with the property that their Rees ideals are generated in the degrees (0, 2) and (1, 1)
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
