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Algebras and Representation Theory pp 1—30 Cite as. Article First Online: 14 March This is a preview of subscription content, log in to check access.

## Computational geometry pdf

Acknowledgments The author would like to thank Hailong Dao and Jeanne Duflot for their useful comments in the preparation of this manuscript. Bass, H. Benjamin, Inc.

Dao, H. Algebras Represent. Doherty, B. Dugger, D. Hiramatsu, Naoya.

### Architects Sketches: Dialogue and Design

Degenerations of graded Cohen-Macaulay modules. Algebra 7 , no. Read more about accessing full-text Buy article. Abstract Article info and citation First page References Abstract We introduce a notion of degenerations of graded modules. Article information Source J. Export citation. Export Cancel. References T. There is a remarkable characterization of Cohen—Macaulay rings, sometimes called miracle flatness or Hironaka's criterion.

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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 prime ideal of a finitely generated algebra over a field, by the Noether normalization lemma ; it also exists when R is complete and contains a field, or when R is a complete domain. A geometric reformulation is as follows.

Let X be a connected affine scheme of finite type over a field K for example, an affine variety. Let n be the dimension of X. By Noether normalization, there is a finite morphism f from X to affine space A n over K. Then X is Cohen—Macaulay if and only all fibers of f have the same degree. Finally, there is a version of Miracle Flatness for graded rings. Let R be a finitely generated commutative graded algebra over a field K ,. Again, it follows that this freeness is independent of the choice of the polynomial subring A.

## Representation-theoretic properties of balanced big Cohen–Macaulay modules

The unmixedness theorem is said to hold for the ring A if every ideal I generated by a number of elements equal to its height is unmixed. A Noetherian ring is Cohen—Macaulay if and only if the unmixedness theorem holds for it. The unmixed theorem applies in particular to the zero ideal an ideal generated by zero elements and thus it says a Cohen-Macaulay ring is an equidimensional ring ; in fact, in the strong sense: there is no embedded component and each component has the same codimension.