### Tight Closure

This is a small introduction on tight closure. This is an active field of research in commutative algebra, and this is essentially a survey article. This article will closely follow the paper “An introduction to tight closure” by Karen Smith.

Definition: Let $R$ be a Noetherian domain of prime characteristic $p$ (not that in general, $p$ need not be prime). Let $I\subset R$ be an ideal with generators $(y_1,y_2,\dots,y_r)$ Then an element $z$ is defined to be in the tight closure $I^*$ if $\exists c\in R$ such that $cz^{p^e}\in (y_1^{p^e},y_2^{p^e},\dots,y_r^{p^e})$.

What does this condition even mean? Let the ring under consideration be $\Bbb{Z_3}[x,y,z]$, and let the ideal $I$ be $(x,y)$. Does the tight closure $I^*$ contain $I$? For example, $x+y\in (x,y)$. Then is it true that $(x+y)^9\in (x^9, y^9)$? Yes! Because remember that the ring has characteristic $3$. Hence all the other terms in the binomial expansion are $0$. In general, $I\subset I^*$. It is easy to see why. What is an example of an element outside of $I$ that belongs to $I^*$? Clearly, $z\notin I^*$? Why? Why can we not have a value for $c$ such that $cz^{p^e}\in (x^{p^e}. y^{p^e})$? For example, $c=x$. However, the value for $c$ should remain the same for all prime powers $p^e$. Clearly, there is no such $c$.

Is $I^*$ an ideal? Yes. This is part is quite clear.

Properties of tight closure:

1. If $R$ is regular, then all ideals of $R$ are tightly closed. In fact, one of the most important uses of tight closure is to compensate for the fact that the ring under consideration may not be regular.

2. If $R\hookrightarrow S$ is an integral extension, then $IS\cap R\subset I^*$ for all ideals $I\subset R$. What does this condition mean? You’re multiplying $I$ with an ideal outside of $R$. It might create elements in $R$ that are outside of $I$, and even outside of $I^*$. The former is possible, but the latter is not.

3. If $R$ is local, with system of parameters $x_1,x_2,\dots,x_d$, then $(x_1,x_2,\dots,x_i):x_{i+1}\subset (x_1,x_2,\dots,x_i)^*$. This means that we start building an ideal with the element $x_1$, and then every subsequent element that we add is present in the closure of the pre-existing ideal. Hence, it is like we’re building an ideal up from $(x_1)$ to $(x_1)^*$.

4. If $\mu$ denotes the minimal number of generators of $I$, then $\overline{I^\mu}\subset I^*\subset\overline{I}$. Here $\overline{I}$ denotes the integral closure of $I$. Note that the number of generators of an ideal is generally not well defined. For instance the ideal $(x)\subset \Bbb{Q}[x]$ can also be written as $(x^2+2x,x^2)$. However, the minimal number of generators is well-defined, as we’re talking about a Noetherian ring. Hence, every ideal has a finite number of generators. Note that $I^*\subset \overline{I}$ is easy to see. For instance, let $I=(x,y)$ in $\Bbb{R}[x,y]$. Then $a\in I^*$ implies that $ca^{p^e}\in (x^{p^e},y^{p^e})$. Hence, $ca^{p^e}-(\text{polynomial in }x^{p^e}\text{ and }y^{p^e})=0$. This implies that $a$ is integral over $I$, and hence $I^*\subset \overline{I}$. What about $\overline{I^\mu}\subset I^*$?

5. If $\phi:R\to S$ is any ring map, $I^*S\subset (IS)^*$. Here $IS$ is actually $\phi(I)S$. This property is labelled as “persistence” in the paper. I suppose what this means is that it is good to persist (find the closure *after* you find the image) rather than throw up your hands at the beginning (finding the closure right at the beginning).

But I’m probably just putting words into Karen’s mouth. What do I know.

It seems to me that a tight closure is a “tighter” form of closure; tighter than integral closure for instance. And for a lot of analytic requirements, it is just the right size; integral closure would be too big.