Abstract: Borel ideals are a well-studied class of ideals in Commutative Algebra. It is known that Borel ideals are minimally resolved by the Eliahou-Kervaire resolution and formulas are known for invariants such as regularity and graded Betti numbers. Borel ideals are particularly useful because in characteristic zero, generic initial ideals are Borel and can be used to bound the invariants of ideals that may not have such a nice structure. In 2013, Francisco, Mermin, and Schweig introduced $Q$-Borel ideals, a generalization of Borel ideals. $Q$-Borel ideals are not as well-understood as Borel ideals but some progress has been made for certain classes of these ideals. The goal of this talk is to give an overview of Borel ideals and give a brief introduction to $Q$-Borel ideals.
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