Dimension | Commutative algebra

Dimension theory (algebra)

In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of the dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety). A large part of dimension theory consists in studying the conditions under which several dimensions are equal, and many important classes of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological dimension is equal to the Krull dimension. The theory is simpler for commutative rings that are finitely generated algebras over a field, which are also quotient rings of polynomial rings in a finite number of indeterminates over a field. In this case, which is the algebraic counterpart of the case of affine algebraic sets, most of the definitions of the dimension are equivalent. For general commutative rings, the lack of geometric interpretation is an obstacle to the development of the theory; in particular, very little is known for non-noetherian rings. (Kaplansky's Commutative rings gives a good account of the non-noetherian case.) Throughout the article, denotes Krull dimension of a ring and the height of a prime ideal (i.e., the Krull dimension of the localization at that prime ideal.) Rings are assumed to be commutative except in the last section on dimensions of non-commutative rings. (Wikipedia).

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This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the dimension of a topological space, algebraic set, or ring.

From playlist Algebraic geometry I: Varieties

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This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We give an introductory survey of many different ways of defining dimension. Reading: Section Exercises:

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From playlist Linear Algebra (Entire Course)

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The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!

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The concept of “dimension” in measured signals

This is part of an online course on covariance-based dimension-reduction and source-separation methods for multivariate data. The course is appropriate as an intermediate applied linear algebra course, or as a practical tutorial on multivariate neuroscience data analysis. More info here:

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Michael Atiyah: Poincaré conjecture, Hodge conjecture, Yang-Mills, Navier-Stokes [2000]

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A mini-course on vertex operator algebras of N= 2 Superconformal Field (Lecture 2) by Madalena Lemos

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A mini-course on vertex operator algebras of N= 2 Superconformal... (Lecture 3) by Madalena Lemos

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Quotient ring | Global dimension | Dimension of an algebraic variety | Bass number | Minimal prime ideal | Valuation ring | Krull dimension | Regular local ring | Associated prime | Zariski tangent space | Hilbert–Poincaré series | Auslander–Buchsbaum formula | Injective hull | Commutative algebra | Artinian ring | Complete intersection ring | Perfect complex | Local criterion for flatness | Jacobson radical | Cohen–Macaulay ring | Chain complex | Depth (ring theory) | Regular sequence | Polynomial ring | Flat module | Mathematics | Gelfand–Kirillov dimension | Noetherian ring | Primary ideal | Associated graded ring | Regular ring | Exterior algebra | Krull's principal ideal theorem | Local cohomology | Scheme (mathematics) | Q.E.D. | Length of a module | Commutative ring