For a commutative local ring $R$, consider (noncommutative)$R$-algebras $Lambda$ of the form $Lambda = operatorname{End}_R(M)$where $M$ is a reflexive $R$-module with nonzero free direct summand.Such algebras $Lambda$ of finite global dimension can be viewed aspotential substitutes for, or analogues of, a resolution ofsingularities of $operatorname{Spec} R$. For example, Van den Berghhas shown that a three-dimensional Gorenstein normal$mathbb{C}$-algebra with isolated terminal singularities has acrepant resolution of singularities if and only if it has such analgebra $Lambda$ with finite global dimension and which is maximalCohen--Macaulay over $R$ (a ``noncommutative crepant resolution ofsingularities''). We produce algebras$Lambda=operatorname{End}_R(M)$ having finite global dimension intwo contexts: when $R$ is a reduced one-dimensional complete localring, or when $R$ is a Cohen--Macaulay local ring of finiteCohen--Macaulay type. If in the latter case $R$ is Gorenstein, thenthe construction gives a noncommutative crepant resolution ofsingularities in the sense of Van den Bergh.
展开▼
机译：对于可交换的局部环$ R $，考虑（非可交换的）$ R $-代数$ Lambda $的形式为$ Lambda =运算符{End} _R（M）$，其中$ M $是一个自反$ R $模块，且非零可以将这种有限全局维的代数$ Lambda $视为$ operatorname {Spec} R $奇异性解析的潜在替代物或类似物。例如，范登·伯格哈斯（Van den Berghhas）表明，具有且仅当末端具有奇异点的三维Gorenstein正态$ mathbb {C} $-代数具有奇异点的渐增解析度时-Macaulay超过$ R $（``非可交换的新奇解析度奇点''）。我们在两个上下文中生成具有有限全局维的代数$ Lambda = operatorname {End} _R（M）$：当$ R $是简化的一维完整本地化，或者$ R $是Cohen-Macaulay有限Cohen的局部环时--Macaulay类型。如果在后一种情况下$ R $是Gorenstein，则该构造在范登伯格的意义上给出了奇点的不可交换的新分辨率。
展开▼
