May 25 2021 2:00 pm
Syzygies and Equification of Monomial Ideals
Dr. Shaheen Nazir
Zoom Meetings (Online)
MS Thesis defense
Linear resolution is an essential technique in commutative algebra to study invariants such as betti numbers, projective dimension, and regularity. These invariants play a considerable role in learning the topologies of different algebraic structures. Here, we study a construction known as linearization. Linearization is a way of constructing an ideal from a monomial ideal I in a single degree but in a larger polynomial ring. The new ideal Lin(I) has linear resolution and linear quotients. Furthermore, we discuss another technique, equification, to generalize the linearization to all the monomial ideals. Equification converts a non-equigenerated monomial ideal I to an equigenerated monomial ideal Ieq in a polynomial ring with one more variable. Moreover, we see the connection between the syzygies of I and I eq and investigate their betti numbers. Our aim is to introduce a class of ideals, for which the syzygy of Ieq is derived from the syzygy of I.