In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local … See more For a commutative Noetherian local ring R, a finite (i.e. finitely generated) R-module $${\displaystyle M\neq 0}$$ is a Cohen-Macaulay module if $${\displaystyle \mathrm {depth} (M)=\mathrm {dim} (M)}$$ (in general we have: See more There is a remarkable characterization of Cohen–Macaulay rings, sometimes called miracle flatness or Hironaka's criterion. Let R be a local ring which is finitely generated as a module over some regular local ring A contained in R. Such a subring exists for any localization R at a See more An ideal I of a Noetherian ring A is called unmixed in height if the height of I is equal to the height of every associated prime P of A/I. (This is stronger than saying that A/I is equidimensional; see below.) The unmixedness theorem is said to hold for the ring A if … See more Noetherian rings of the following types are Cohen–Macaulay. • Any regular local ring. This leads to various examples of Cohen–Macaulay rings, such as the … See more We say that a locally Noetherian scheme $${\displaystyle X}$$ is Cohen–Macaulay if at each point $${\displaystyle x\in X}$$ the local ring $${\displaystyle {\mathcal {O}}_{X,x}}$$ is … See more • A Noetherian local ring is Cohen–Macaulay if and only if its completion is Cohen–Macaulay. • If R is a Cohen–Macaulay ring, then the polynomial ring R[x] and the … See more 1. If K is a field, then the ring R = K[x,y]/(x ,xy) (the coordinate ring of a line with an embedded point) is not Cohen–Macaulay. This follows, for example, by Miracle Flatness: R is finite over the polynomial ring A = K[y], with degree 1 over points of the affine line Spec … See more WebOct 4, 2024 · 1 On a regular scheme, every nonzero locally free sheaf of finite rank is CM. So in particular, all line bundles on a regular scheme are Cohen-Macaulay. To find a counterexample to your claim (and your hoped-for improvement), it suffices to find a regular scheme with infinite Picard group.
18.726 Algebraic Geometry - MIT OpenCourseWare
WebMay 23, 2024 · A ring is called Cohen-Macaulay if its depth is equal to its dimension. More generally, a commutative ring is called Cohen-Macaulay if is Noetherian and all of its localizations at prime ideals are Cohen-Macaulay. In geometric terms, a scheme is called Cohen-Macaulay if it is locally Noetherian and its local ring at every point is … Webthe single question; if klt singularities are Cohen–Macaulay or not over ... Assume that X is an excellent scheme whose closed points have perfect residue field of characteristic p> 5. Let x∈ (X,∆) be a log–canonical threefold singularity which is not a log–canonical center. Suppose x∈ C, where Cis a curve and a mini- tickets reinhard mey 2022
algebraic geometry - Dualizing complex of Cohen-Macaulay variety ...
WebLocal Cohen-Macaulay rings are equidimensional. Proof. From the proof of the previous proposition we in fact get that any two maximal chains of prime ideals have the same … WebAug 31, 2013 · ag.algebraic geometry - cohen-macaulayness of reduced and non-reduced schemes - MathOverflow cohen-macaulayness of reduced and non-reduced schemes Asked 9 years, 6 months ago Modified 9 years, 6 months ago Viewed 656 times 9 Let $X$ be a Cohen-Macaulay scheme (let's say of finite type over a field). WebFeb 11, 2024 · A primitive multiple scheme is a Cohen–Macaulay scheme Y such that the associated reduced scheme \(X=Y_{\mathrm{red}}\) is smooth, irreducible, and Y can be locally embedded in a smooth variety of dimension \(\dim (X)+1\).If n is the multiplicity of Y, there is a canonical filtration \(X=X_1\subset X_2\subset \cdots \subset X_n=Y\), such … tickets remedy kion group