Failure of Integral Closure

The Failure of Integral Closure

What Singularities look like to Algebra

Recall that a ring  R is integrally closed if it is integral over its field of fractions e.g. any element of Frac(R) which is a root of a monic polynomial with coefficients in  R is contained in R. For example,  \mathbb{Z} is integrally closed.

Consider  \mathbb{C}[t^2,t^3] . The polynomial  t = t^3/t^2 \in \mathbb{C}(t) is a root of the (monic) polynomial  x^2 - t^2 and so is integral over  \mathbb{C}[t^2,t^3]. However,  t \notin \mathbb{C}[t^2,t^3]. And for those who like pictures: the geometric incarnation of this fact is  that the plane curve  Y^2 - X^3 = 0 has a singularity.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s