2 results
Annihilators and decompositions of singularity categories
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 3 / August 2024
- Published online by Cambridge University Press:
- 23 April 2024, pp. 699-713
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Given any commutative Noetherian ring R and an element x in R, we consider the full subcategory
$\mathsf{C}(x)$ of its singularity category consisting of objects for which the morphism that is given by the multiplication by x is zero. Our main observation is that we can establish a relation between 
$\mathsf{C}(x), \mathsf{C}(y)$ and 
$\mathsf{C}(xy)$ for any two ring elements x and y. Utilizing this observation, we obtain a decomposition of the singularity category and consequently an upper bound on the dimension of the singularity category.
Finite Cohen–Macaulay Type and Smooth Non-Commutative Schemes
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- Journal:
- Canadian Journal of Mathematics / Volume 60 / Issue 2 / 01 April 2008
- Published online by Cambridge University Press:
- 20 November 2018, pp. 379-390
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- 01 April 2008
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A commutative local Cohen–Macaulay ring
$R$ of finite Cohen–Macaulay type is known to be an isolated singularity; that is, 
$\text{Spec}(R)\backslash \{m\}$ is smooth. This paper proves a non-commutative analogue. Namely, if 
$A$ is a (non-commutative) graded Artin–Schelter Cohen–Macaulay algebra which is fully bounded Noetherian and has finite Cohen–Macaulay type, then the non-commutative projective scheme determined by $A$ is smooth.
