Initial Ideals of Borel Type

This paper provides a survey of results on regularity, Betti numbers and reduction numbers, showing their stability in passing from an ideal to its initial ideal if the latter is of Borel type. Similar results happen when one passes from an ideal to its generic initial ideal. We also present analogous results on extremal Betti numbers. © 2015, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore.