arXiv2020
An $R$-algebra $S$ is $R$-solid if there exists a nonzero $R$-linear map $S \rightarrow R$. In characteristic $p$, the study of $F$-singularities such as Frobenius splittings implicitly rely on the $R$-solidity of $R^{1/p}$. Following recent results of the first two authors on the Frobenius non-splitting of certain excellent $F$-pure rings, in this paper we use the notion of solidity to systematically study the notion of excellence, with an emphasis on $F$-singularities. We show that for rings $R$ essentially of finite type over complete local rings of characteristic $p$, reducedness implies the $R$-solidity of $R^{1/p}$, $F$-purity implies Frobenius splitting, and $F$-pure regularity implies split $F$-regularity. We demonstrate that Henselizations and completions are not solid, providing obstructions for the $R$-solidity of $R^{1/p}$ for arbitrary excellent rings. This also has negative consequences for the solidity of big Cohen-Macaulay algebras, an important example of which are absolute integral closures of excellent local rings in prime characteristic. We establish a close relationship between the solidity of absolute integral closures and the notion of Japanese rings. Analyzing the Japanese property reveals that Dedekind domains $R$ for which $R^{1/p}$ is $R$-solid are excellent, despite our recent examples of excellent Euclidean domains with no nonzero $p^{-1}$-linear maps. Additionally, we show that while perfect closures are often solid in algebro-geometric situations, there exist locally excellent domains with solid perfect closures whose absolute integral closures are not solid. In an appendix, Karen E. Smith uses the solidity of absolute integral closures to characterize the test ideal for a large class of Gorenstein domains of prime characteristic.