Commutative algebra 26 (Examples of Artinian rings)

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. In this lecture we give some examples or Artin rings and write them as products of local rings. The examples include some Artin rings of small length, and some tensor products of fields. We also show that there are more Artin rings than one might expect (in technical terms, the dimension of the Hilbert scheme of length n Artin rings with m generators can be more than the obvious guess mn if m is greater than 2). Reading: Section 2.4 Exercises: 2.24 Try to classify Artinian rings of length 3 or 4 over a field. (You do not need to succeed: the point is that you see how complicated this is.)