Forcing algebras/Relation to tight closure/Local cohomology/short description

Definition

Let ${}R$ be a commutative ring and let ${}f_{1},\ldots ,f_{n}$ and ${}f$ be elements in ${}R$. Then the ${}R$-algebra

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

is called the forcing algebra of these elements (or these data).

If we fix a point ${}P\in \operatorname {Spec} \,R$, then the fiber over ${}P$ of the spectrum of the forcing algebra, ${}\operatorname {Spec} \,A$, is just

$\operatorname {Spec} \,\kappa (P)[T_{1},\ldots ,T_{n}]/(f_{1}(P)T_{1}+\ldots +f_{n}(P)T_{n}+f(P)).$

This is the solution set to an inhomogeneous linear equation over ${}\kappa (P)$, so it is an affine space (in the sense of linear algebra) (may be empty). If the ideal ${}(f_{1},\ldots ,f_{n})$ is primary to ${}{\mathfrak {m}}$, and ${}f\in {\mathfrak {m}}$, then the fiber over ${}{\mathfrak {m}}$ has dimension ${}n$, whereas the dimensions of the other fibers are ${}n-1$.

The relation between tight closure and forcing algebras is given in the following theorem.

Theorem

Let ${}R$ be a normal excellent local domain with maximal ideal ${}{\mathfrak {m}}$ over a field of positive characteristic. Let ${}f_{1},\ldots ,f_{n}$ generate an ${}{\mathfrak {m}}$-primary ideal ${}I$ and let ${}f$ be another element in ${}R$. Then

${}f\in I^{*}$ if and only if

${}H_{\mathfrak {m}}^{\dim(R)}(A)\neq 0\,,$

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

denotes the forcing algebra of these elements.

If the dimension ${}d$ is at least two, then

$H_{\mathfrak {m}}^{d}(R)\longrightarrow H_{\mathfrak {m}}^{d}(A)\cong H_{{\mathfrak {m}}A}^{d}(A)\cong H^{d-1}(D({{\mathfrak {m}}A}),{\mathcal {O}}_{A}).$

This means that we have to look at the cohomological properties of the complement of the exceptional fiber over the closed point. If the dimension is two, then we have to look whether the first cohomology of the structure sheaf vanishes. This is true if and only if the open subset ${}D({\mathfrak {m}}A)$ is an affine scheme (the spectrum of a ring).