Tight closure/Regular ring/Solid closure/Motivation/Section

An important property of tight closure is that it is trivial for regular rings, i.e.

${}I^{*}=I\,$

for every ideal ${}I$. This rests upon Kunz's theorem saying that the Frobenius homomorphism for regular rings is flat. This property implies the following cohomological property of torsors.

Corollary

Let ${}(R,{\mathfrak {m}})$ denote a regular local ring of dimension ${}d$ and of positive characteristic, let ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ be an ${}{\mathfrak {m}}$-primary ideal and ${}f\in R$ be an element with ${}f\notin I$. Let ${}B=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}$ be the corresponding forcing algebra. Then the extended ideal ${}{\mathfrak {m}}B$ satisfies

${}H_{{\mathfrak {m}}B}^{d}(B)=H^{d-1}(D({\mathfrak {m}}B),{\mathcal {O}}_{B})=0\,.$

Proof

This follows from fact and ${}f\notin I^{*}$.

$\Box$

In dimension two this is true in every (even mixed) characteristic.

Theorem

Let ${}(R,{\mathfrak {m}})$ denote a two-dimensional regular local ring, let ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ be an ${}{\mathfrak {m}}$-primary ideal and ${}f\in R$ an element with ${}f\notin I$. Let

${}B=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}\,$

be the corresponding forcing algebra. Then for the extended ideal ${}{\mathfrak {m}}B$ we have

${}H_{{\mathfrak {m}}B}^{2}(B)=H^{1}(D({\mathfrak {m}}B),{\mathcal {O}}_{B})=0\,.$

In particular, the open subset ${}D({\mathfrak {m}}B)$ is an affine scheme if and only if ${}f\notin I$.

The main point for the proof of this result is that for ${}f\notin I$, the natural mapping

$H^{1}(U,{\mathcal {O}}_{X})\longrightarrow H^{1}(T,{\mathcal {O}}_{T})$

is not injective by a Matlis duality argument. Since the local cohomology of a regular ring is explicitly known, this map annihilates some cohomology class of the form ${}{\frac {1}{fg}}$ where ${}f,g$ are parameters. But then it annihilates the complete local cohomology module and then ${}T$ is an affine scheme.

For non-regular two-dimensional rings it is a difficult question in general to decide whether a torsor is affine or not. A satisfactory answer is only known in the normal two-dimensional graded case over a field, which we will deal with in the final lectures.

In higher dimension in characteristic zero it is not true that a regular ring is solidly closed (meaning that every ideal equals its solid closure), as was shown by the following example of Paul Roberts.

Example

Let ${}K$ be a field of characteristic ${}0$ and let

${}B=K[X,Y,Z][U,V,W]/{\left(X^{3}U+Y^{3}V+Z^{3}W-X^{2}Y^{2}Z^{2}\right)}\,.$

Then the ideal ${}{\mathfrak {a}}=(X,Y,Z)B$ has the property that ${}H_{\mathfrak {a}}^{3}(B)\neq 0$. This means that in ${}R=K[X,Y,Z]$, the element ${}X^{2}Y^{2}Z^{2}$ belongs to the solid closure of the ideal ${}{\left(X^{3},Y^{3},Z^{3}\right)}$, and hence the three-dimensional polynomial ring is not solidly closed.

This example was the motivation for the introduction of parasolid closure, which has all the good properties of solid closure but which is also trivial for regular rings.