Syzygies of codimension 2 lattice ideals
Let \(S= k[x_1,x_2,\dots, x_n]\) be the ring of polynomials of variables \(x_1,x_2,\dots, x_n\) over a field \(k\), \(L\) be a sublattice of the set of monomials from \(k[x_1^{\pm 1}, x_2^{\pm 1},\dots, x_n^{\pm 1}]\), \(I_L\) be the associated lattice ideal [see \textit{I. Peeva} and \textit{B. Sturmfels}, J. Am. Math. Soc. 11, No. 2, 363-373 (1998; Zbl 0905.13005)] of codimension 2. The minimal free resolution \(F\) of \(S/I_L\) over \(S\) is constructed. In so doing, computation of multigraded Betti numbers of resolutions by simplicial complexes, Gale diagrams, Scarf's theory [see the paper cited above as well as \textit{D. Bayer}, \textit{I. Peeva} and \textit{B. Sturmfels}, Math. Res. Lett. 5, No. 1-2, 31-46 (1998; Zbl 0909.13010) and \textit{D. Bayer} and \textit{B. Sturmfels}, J. Reine Angew. Math. 502, 123-140 (1998; Zbl 0909.13011)] are used. In addition, results concerning the Poincaré series of (infinite) minimal free resolutions of modules over \(S/I_L\), the rate of \(S/I_L\) [\textit{J. Backelin}, Reports., Univ. Stockholm 25 (1988)] are obtained. The Eisenbud-Goto regularity conjecture [\textit{D. Eisenbud} and \textit{S. Goto}, J. Algebra 88, 89-133 (1984; Zbl 0531.13015)] is proved in the case, where \(I_L\) is a toric ideal. A computational algorithm is presented.
scientific article; zbMATH DE number 1229586