It is shown that tight closure commutes with localization in any two-dimensional ringR of prime characteristic if eitherR is a Nagata ring orR possesses a weak test element. Moreover, it is proved that tight closure commutes with localization at height one prime ideals in any ring of prime characteristic.

Let 𝑅 be a (not necessarily local) Noetherian ring and 𝑀 a finitely generated 𝑅-module of finite dimension 𝑑. Let $\mathfrak{a}$ be an ideal of 𝑅 and $\mathfrak{M}$ denote the intersection of all prime ideals $\mathfrak{p}\in\mathrm{Supp}_R H^d_a(M)$. It is shown that

$$H^d_a(M)\simeq H^d_{\mathfrak{M}}(M)/\sum\limits_{n\in\mathbb{N}}\langle \mathfrak{M}\rangle(0:_{H^d_{\mathfrak{M}}(M)}a^n),$$

where for an Artinian 𝑅-module 𝐴 we put $\langle\mathfrak{M}\rangle A=\cap_{n\in\mathbb{N}}\mathfrak{M}^n A$. As a consequence, it is proved that for all ideals $\mathfrak{a}$ of 𝑅, there are only finitely many non-isomorphic top local cohomology modules $H^d_a(M)$ having the same support. In addition, we establish an analogue of the Lichtenbaum–Hartshorne vanishing theorem over rings that need not be local.