(Dimension)
The dimension of a topological space is the supremum of the lengths \(n\) of chains \[ Z_0 \subsetneq Z_1 \subsetneq \cdots \subsetneq Z_n \] of irreducible closed subsets.
If \(Y \subseteq \AA^n\) is an affine algebraic set, then \(\dim Y = \krulldim A(Y)\). [@Har10a, Proposition I.1.7]
Let \(k\) be any field and \(B\) a finitely generated \(k\)-algebra that is a domain. Then \(\krulldim B = \trdeg_k \Frac B\). [@Har10a, Theorem I.1.8A]
If \(Y\) is a quasi-affine variety, then \(\dim Y = \dim \closure{Y}\), where \(\closure{Y}\) is the closure of \(Y\) in \(\AA^n\). [@Har10a, Proposition I.1.10]
Hence for an affine variety \(X\) over \(k\), \[ \dim X = \krulldim A(X) = \trdeg_k k(X) , \] and in particular \(\dim \AA^n = n\) [@Har10a, Proposition I.1.9]. For nonconstant \(f\in k[x_1,\ldots,x_n]\), every irreducible component of \(V(f)\subseteq\AA^n\) has dimension \(n-1\) [@Har10a, Proposition I.1.13]. In terms of transcendence degree: if \(g\) is an irreducible factor of \(f\) of positive degree in \(x_n\), then the images of \(x_1,\ldots,x_{n-1}\) in \(k[x_1,\ldots,x_n]/(g)\) are algebraically independent, because \(g\) divides no nonzero polynomial in \(x_1,\ldots,x_{n-1}\), and the image of \(x_n\) is algebraic over them.
Dependencies
D-9DIKBIrreducible and Noetherian spacesuses
Backlinks
P-AGHHYPSECTIONA hypersurface section drops the dimension by exactly oneusesP-AGHDIMTOPOLOGYDimension under subspaces, open covers, and closed subsetsusesPR-VARHTKrull's principal ideal theorem, and codimension oneusesD-0SYCYSmooth and singular points; Jacobian and regularity criteriausesD-L6ERWThe Hilbert polynomial and its coefficientsusesP-AGHTWCUBICThe twisted cubic is an affine variety of dimension oneuses