Minimal Free Resolution of Distributive Lattice Ideals

Das, Priya (2018) Minimal Free Resolution of Distributive Lattice Ideals. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

Construction of free resolution of a module is an important problem in Commutative Algebra, as free resolution gives us a host of useful information on the algebraic and geometric properties of modules. Calculation of the first syzygy of a module is the first step towards finding its free resolution. However, it is, in general, difficult to compute the first syzygy module. This thesis calculates the first syzygy of Hibi ring and applies the result to answer other algebraic and geometric questions. Let L be a finite distributive lattice, let S = K[x1; x2; : : : ; xn] be a polynomial ring of |L|-many variables and let I = (xαxβ-xαvβxαvβ:α,β ∈ L,α ⍭ β . The ring R[L] = S/I is called Hibi ring. We describe all the generators of the first syzygy of Hibi ring in terms of sublattices of the distributive lattice. We have used the aforementioned knowledge to calculate the first Betti number of Hibi ring for a planar distributive lattice in terms of join-meet irreducible elements of the lattice. Finally, we describe the Gröbner basis for the first syzygy of Hibi ring for a planar distributive lattice.

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