论文标题
泰特代数和弗罗贝尼乌斯不拆分出色的常规环
Tate algebras and Frobenius non-splitting of excellent regular rings
论文作者
论文摘要
在许多通常发生的情况下，在积极特征的交换代数和代数几何形状中，Frobenius映射为纯度纯度也是纯净的主要特征环。但是，使用刚性几何形状的基本结构，我们表明，即使对于欧几里得领域，优质的$ f $ pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure-pure the的主要特征也不是Frobenius分开的。我们的构造使用了特征性$ p $的完整的非Archimedean Field $ K $，而没有非零连续$ k $ -linear maps $ k^{1/p} \ to k $。根据Gabber的观念给出了这样一个领域的明确例子，并且可能具有独立的兴趣。我们的例子解决了一个长期以来的公开问题，这是$ f $ singularities的理论，可以追溯到Hochster和Roberts提出$ f $ purity的概念时。我们构建的出色欧几里得域也承认No nonzero $ r $ -linear Maps $ r^{1/p} \ rightarrow r $。这些是第一个说明$ f $ purity和frobenius拆分的例子，定义了出色域的不同类别的奇异性，也是第一个出色域的示例，没有非零$ p^{ - 1} $ - 线性地图。从测试理想理论的角度来看，后者特别有趣。
An excellent ring of prime characteristic for which the Frobenius map is pure is also Frobenius split in many commonly occurring situations in positive characteristic commutative algebra and algebraic geometry. However, using a fundamental construction from rigid geometry, we show that excellent $F$-pure rings of prime characteristic are not Frobenius split in general, even for Euclidean domains. Our construction uses the existence of a complete non-Archimedean field $k$ of characteristic $p$ with no nonzero continuous $k$-linear maps $k^{1/p} \to k$. An explicit example of such a field is given based on ideas of Gabber, and may be of independent interest. Our examples settle a long-standing open question in the theory of $F$-singularities whose origin can be traced back to when Hochster and Roberts introduced the notion of $F$-purity. The excellent Euclidean domains we construct also admit no nonzero $R$-linear maps $R^{1/p} \rightarrow R$. These are the first examples that illustrate that $F$-purity and Frobenius splitting define different classes of singularities for excellent domains, and are also the first examples of excellent domains with no nonzero $p^{-1}$-linear maps. The latter is particularly interesting from the perspective of the theory of test ideals.
