Editing Dimension (section)

===Varieties===
{{Main|Dimension of an algebraic variety}}
The dimension of an [[algebraic variety]] may be defined in various equivalent ways. The most intuitive way is probably the dimension of the [[tangent space]] at any [[Regular point of an algebraic variety]]. Another intuitive way is to define the dimension as the number of [[hyperplane]]s that are needed in order to have an intersection with the variety that is reduced to a finite number of points (dimension zero). This definition is based on the fact that the intersection of a variety with a hyperplane reduces the dimension by one unless if the hyperplane contains the variety.

An [[algebraic set]] being a finite union of algebraic varieties, its dimension is the maximum of the dimensions of its components. It is equal to the maximal length of the chains <math>V_0\subsetneq V_1\subsetneq \cdots \subsetneq V_d</math> of sub-varieties of the given algebraic set (the length of such a chain is the number of "<math>\subsetneq</math>").

Each variety can be considered as an [[stack (mathematics)|algebraic stack]], and its dimension as variety agrees with its dimension as stack. There are however many stacks which do not correspond to varieties, and some of these have negative dimension. Specifically, if ''V'' is a variety of dimension ''m'' and ''G'' is an [[algebraic group]] of dimension ''n'' [[Group action (mathematics)|acting on ''V'']], then the [[quotient stack]] [''V''/''G''] has dimension ''m''&nbsp;−&nbsp;''n''.<ref>{{citation |last=Fantechi|first=Barbara|chapter=Stacks for everybody|title=European Congress of Mathematics Volume I|volume=201|pages=349–359|series=Progr. Math.|publisher=Birkhäuser|year=2001|chapter-url=http://www.mathematik.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf|url-status=live|archive-url=https://web.archive.org/web/20060117052957/http://www.mathematik.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf|archive-date=2006-01-17}}
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