Tight closure/Solid closure/Introduction/Section

Let ${}R$ be a noetherian domain of positive characteristic, let

$F\colon R\longrightarrow R,f\longmapsto f^{p},$

be the Frobenius homomorphism and

$F^{e}\colon R\longrightarrow R,f\longmapsto f^{q},$

(mit ${}q=p^{e}$) its ${}e$th iteration. Let ${}I$ be an ideal and set

${}I^{[q]}={\text{ extended ideal of }}I{\text{ under }}F^{e}\,.$

Then define the tight closure of ${}I$ to be the ideal

${}I^{*}={\left\{f\in R\mid {\text{ there exists }}z\neq 0{\text{ such that }}zf^{q}\in I^{[q]}{\text{ for all }}q=p^{e}\right\}}\,.$

The element ${}f$ defines the cohomology class ${}c\in H^{1}(D(I),\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})$. Suppose that ${}R$ is normal and that ${}I$ has height at least ${}2$ (think of a local normal domain of dimension at least ${}2$ and an ${}{\mathfrak {m}}$-primary ideal ${}I$). Then the ${}e$th Frobenius pull-back of the cohomology class is

${}F^{e*}(c)\in H^{1}(D(I),F^{e*}{\left(\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}\right)}\cong H^{1}(D(I),\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)})\,$

(${}q=p^{e}$) and this is the cohomology class corresponding to ${}f^{q}$. By the height assumption, ${}zF^{e*}(c)=0$ if and only if ${}zf^{q}\in {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}$, and if this holds for all ${}e$ then ${}f\in I^{*}$ by definition. This shows already that tight closure under the given conditions does only depend on the cohomology class.

This is also a consequence of the following theorem of Hochster which gives a characterization of tight closure in terms of forcing algebra and local cohomology.

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

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

This means that we have to look at the cohomological properties of the complement of the exceptional fiber over the closed point, i.e. the torsor given by these data. If ${}H^{d-1}(D({\mathfrak {m}}B),{\mathcal {O}}_{B})=0$ then this is true for all quasicoherent sheaves instead of the structure sheaf. This property can be expressed by saying that the cohomological dimension of ${}D({\mathfrak {m}}B)$ is ${}\leq d-2$ and thus smaller than the cohomological dimension of the punctured spectrum ${}D({\mathfrak {m}})$, which is exactly ${}d-1$. So belonging to tight closure can be rephrased by saying that the formation of the corresponding torsor does not change the cohomological dimension.

If the dimension is two, then we have to look whether the first cohomology of the structure sheaf vanishes. This is true (by Serre's cohomological criterion for affineness) if and only if the open subset ${}D({\mathfrak {m}}B)$ is an affine scheme (the spectrum of a ring).

The right hand side of the equivalence in fact (the non-vanishing of the top-dimensional local cohomology) is independent of any characteristic assumption, and can be taken as the basis for the definition of another closure operation, called solid closure. So the theorem above says that in positive characteristic tight closure and solid closure coincide. There is also a definition of tight closure for algebras over a field of characteristic ${}0$ by reduction to positive characteristic.