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An affine algebraic curve is an affine variety of Krull dimension one. An irreducible affine curve has a finitely generated one-dimensional domain as its coordinate ring. Quotienting this ring by a nonzero element gives a zero-dimensional Noetherian ring, hence an Artinian ring. Consequently a nonzero regular function has finitely many zeros. No nonsingularity assumption is part of the definition.

An Artinian ring satisfies the descending chain condition on ideals. A commutative ring is Artinian exactly when it is Noetherian and has Krull dimension zero.

For a local ring $(R,\mathfrak{m})$ with residue field $\kappa =R/\mathfrak{m}$, its cotangent space is the $\kappa$-vector space $\mathfrak{m}/{\mathfrak{m}}^2$. In a Noetherian local ring, its dimension is the minimal number of generators of $\mathfrak{m}$, by the Nakayama lemma. Equality of this dimension with the Krull dimension defines a regular local ring.

The depth of a Noetherian local ring is the maximal length of an $R$-regular sequence in its maximal ideal. It is at most its Krull dimension. Quotienting by a nonzero divisor lowers depth by one. This invariant measures how many independent successive nonzero-divisor equations the local algebra supports.

If $f\in k[Y][X]$ is monic of positive degree in $X$, monic division makes $k[Y][X]/(f)$ free of rank ${\mathrm{deg}}_{X}f$ over $k[Y]$. The inclusion of $k[Y]$ is injective and the quotient is an integral extension, so its Krull dimension is one. This argument applies even when the polynomial factors or the field has positive characteristic.

The supremum of the lengths of strict chains of nonempty irreducible closed subsets. It is the maximum of the dimensions of the irreducible components. For an affine algebraic set it equals the Krull dimension of its coordinate ring; for a quasi-projective algebraic set it is the supremum of these dimensions over affine open subsets.

The dimension of a scheme is the supremum of the lengths of strict chains of irreducible closed subsets of its underlying topological space. On an affine scheme it is the Krull dimension of its coordinate ring. Taking the reduced subscheme leaves the dimension unchanged.

The embedding dimension of a Noetherian local ring $(R,\mathfrak{m})$ is ${\dim }_{R/\mathfrak{m}}(\mathfrak{m}/{\mathfrak{m}}^2)$. It counts the smallest number of local generators needed for the maximal ideal. It is at least the Krull dimension; equality characterizes a regular local ring.

Let $(A,\mathfrak{m})$ be a regular local ring of Krull dimension $d$. Its formal power series ring $B=A[[x]]$ is Noetherian and local, with maximal ideal $I=\mathfrak{m}B+(x)$ and embedding dimension $d+1$. Since $I$ is generated by $d+1$ elements, the Krull height theorem gives $\dim B\le d+1$. For the reverse inequality choose a strict prime chain ${\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_d=\mathfrak{m}$ in $A$. Finite generation of each ${\mathfrak{p}}_i$ gives $B/{\mathfrak{p}}_{i}B\cong (A/{\mathfrak{p}}_i)[[x]]$, an integral domain, so each extended ideal is prime. The inclusions remain strict by contraction to $A$, and $\mathfrak{m}B⊊I$ adds one more step since $x\notin \mathfrak{m}B$. This proves $\dim B=d+1$, equal to its embedding dimension, and hence regularity.

Let a commutative $k$-algebra $R$ be generated by $x_1,\dots ,x_n$, and let $R_j$ be the span of words of total degree at most $j$ in those generators. The associated graded ring of this filtration is a standard graded algebra, andThe Hilbert-Serre theorem therefore makes ${\dim }_k{R}_j$ eventually a polynomial in $j$. Its degree is $\dim R$; this follows by applying the Rees-ring deformation, whose special fibers are $R$ and the associated graded ring, so they have the same Krull dimension.

Height measures prime-chain length within Krull dimension theory.

The height of an ideal $I$ is the infimum of the heights of prime ideals containing it:The height of a prime is the supremum of the lengths of strict prime chains ending at that prime.

The height of a prime ideal is the supremum of lengths of strict prime chains ending at it, equivalently the Krull dimension of its localization at a prime ideal. The Krull height theorem bounds the height of a minimal prime over an $n$-generated ideal by $n$.

If $M$ has Krull dimension $d$, the coefficient of $n^d$ in its Hilbert-Samuel polynomial is $e_I(M)/d!$. The positive integer $e_I(M)$ is the Hilbert–Samuel multiplicity of $M$ with respect to $I$.

An ideal $I$ of a Noetherian local ring $(A,\mathfrak{m})$ is an ideal of definition when $A/I$ has Krull dimension zero. Equivalently, $I$ is $\mathfrak{m}$-primary, so $\sqrt{I}=\mathfrak{m}$ and ${\mathfrak{m}}^r\subseteq I$ for some positive integer $r$.

If a Noetherian ring has finite Krull dimension $d$, then it has infinitely many prime ideals of each height $n$ with $0<n<d$. In a prime chain of maximal length, apply prime ideals between a three-prime chain to the terms of heights $n-1$, $n$, and $n+1$.

The Krull dimension of ${\mathcal{O}}_{X,x}$ at a point $x$. For a closed point of an irreducible affine variety it equals $\dim X$, by the closed-point dimension lemma for affine domains. At a generic point of a codimension-one subvariety the local dimension is one.

The dimension of the fibre of a morphism locally at its source point $x$: the maximum of the dimensions of the fibre's irreducible components through $x$. This differs from the Krull dimension of the local ring at a nonclosed point.

The dimension of an algebraic set is the supremum of lengths of strict chains of nonempty irreducible closed subsets. It is the maximum of the dimensions of the irreducible components. On an affine variety $X=\mathrm{Spec}A$, it equals the Krull dimension of $A$; if $X$ is irreducible, Noether normalization identifies this with the transcendence degree of its function field over $k$. Nonempty Zariski-open subsets of an irreducible algebraic variety have the same dimension, so the definition is compatible with affine charts and with birational maps. In particular, $\dim {\mathbb{A}}^n=\dim {\mathbb{P}}^n=n$ and $\dim (X\times Y)=\dim X+\dim Y$ for irreducible algebraic varieties.

The fiber dimension theorem says that a dominant morphism of algebraic varieties $f:X\to Y$ between irreducible algebraic varieties has generic fibre dimensionEvery irreducible component of every nonempty fibre over a closed point has dimension at least $r$. There is a nonempty Zariski-open subset of $Y$ on which the fibres are nonempty and are equidimensional algebraic varieties of dimension $r$. Dominance is required; for a general morphism of algebraic varieties, replace $Y$ by the closure of its image. This gives the dimension of image of a morphism formula.

For the generic fibre, the formula is the additivity of transcendence degree in the tower $k\subset k(Y)\subset k(X)$. The lower bound on closed fibres comes from cutting by at most $\dim Y$ local parameters after a finite Noether normalization of an affine target chart: the principal hypersurface dimension lemma drops dimension by at most one per equation, and the finite normalization fibre separates the finitely many target points. For the generic upper bound, take finitely many affine source charts and apply Noether normalization to each coordinate ring over $k(Y)$. Clearing denominators in $k[Y]$ makes each chart finite over ${\mathbb{A}}^r$ relative to a suitable open target chart. Its fibres have dimension at most $r$. Intersecting these finitely many target opens and using the lower bound gives generic equidimensionality of fibres.

The upper semicontinuity of local fibre dimension is a statement on the source: for a finite-type morphism of schemes, the points $x$ satisfying ${\dim }_x{X}_{f(x)}\le d$ form an open set. For a proper morphism, the maximum fibre dimension is upper semicontinuous on the target as well. The properness qualification matters; one should not assert the latter for every finite-type morphism of schemes.

A first application is dimension counting for plane sections and intersections. On an irreducible affine variety, imposing $q$ polynomial equations gives each nonempty component algebraic codimension at most $q$, by repeated principal ideal theorem. On a smooth variety this underlies the intersection dimension bound on a smooth variety. A second application is to images: if a morphism of algebraic varieties has finite nonempty fibres, its image closure has the same dimension as the source. If it is also proper, that image is closed. Thus an isolated fibre point forces dominance in equal dimensions when the irreducible source and target have equal dimension.

For the cubic-surface application, projective lines in ${\mathbb{P}}^3$ form the Grassmannian $G=\mathrm{Gr}(2,4)$ of dimension four. Cubic forms form ${\mathbb{P}}^{19}$, since there are $\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$ degree-three monomials in four variables. The cubic surface line incidence varietyhas fibre ${\mathbb{P}}^{15}$ over every $L$: restricting a cubic to a projective line gives four independently prescribable coefficients. It is therefore an irreducible projective-space bundle of dimension $4+15=19$. The other projection $\pi :I\to {\mathbb{P}}^{19}$ is a proper morphism, so its image $Z$ is closed and irreducible.

To show $Z={\mathbb{P}}^{19}$, dimension counting alone is not enough; exhibit a fibre with an isolated point. TakeA nearby projective line is the graph $x_2=a{x}_0+b{x}_1$, $x_3=c{x}_0+d{x}_1$. Restriction givesIt vanishes only when $a=b=c=d=0$. Thus $L_0$ is an isolated reduced point of the fibre. Applying the lower bound in the fiber dimension theorem to $I\to Z$ gives $0\ge 19-\dim Z$. Since $Z\subseteq {\mathbb{P}}^{19}$, its dimension is exactly 19, and closedness forces $Z={\mathbb{P}}^{19}$. This is the incidence proof that a cubic surface contains a line. The auxiliary cubic $F_0$ need not be smooth: the argument proves existence for every cubic form, and hence in particular for every smooth cubic surface.

Choose a basis $x_1,x_2,x_3$ of the Lie algebra and filter its universal enveloping algebra by word length: $F_{0}R=\mathbb{C}$ and $F_{d}R$ is spanned by products of at most $d$ basis elements. The relations $x_j{x}_i-x_i{x}_j=[x_j,x_i]$ replace a degree-two commutator by degree one, so the degree-one symbols commute.

The Poincaré-Birkhoff-Witt theorem says the ordered monomials $x_1^{a_1}{x}_2^{a_2}{x}_3^{a_3}$ are a basis. Its relevant proof can be seen by reordering an inverted neighboring pair using the displayed relation. Reordering terminates because same-length terms have fewer inversions and bracket terms have smaller word length. Disjoint swaps commute; the only overlapping three-letter reorderings differ by $[x_i,[x_j,x_k]]+[x_j,[x_k,x_i]]+[x_k,[x_i,x_j]]=0$. The Jacobi identity therefore makes the reduced ordered expression independent of the reordering path. Ordered monomials have no remaining relations and form the basis. Their degree-$d$ symbols giveFor generators $m_1,\dots ,m_q$ of the finitely generated module $M$, put $M_d={\sum }_j{F}_{d}R{m}_j$. This is a good filtration, and $\mathrm{gr}M$ is finite over the three-variable polynomial ring. Define the dimension of a filtered module as the Krull dimension of its associated graded module, equivalently the degree of the eventual cumulative polynomial ${\dim }_{\mathbb{C}}{M}_d$, or the Gelfand–Kirillov dimension of a module. Two good filtrations have bounded shifts, so the growth degree is independent of the generators. Sinceits polynomial degree is at most three. Therefore every nonzero finitely generated $R$-module has dimension at most $3$. The zero module may be assigned the usual separate empty-support convention. This bound also follows immediately from its support of a module inside affine three-space.

A formal power series $f={\sum }_{n\ge 0}{a}_n{x}^n$ is a unit exactly when its constant coefficient is a unit in $A$. If $fg=1$, constant coefficients give $a_0{b}_0=1$. Conversely, for invertible $a_0$, setThen every positive-degree coefficient of $fg$ is zero and its constant coefficient is one, so $g=\sum b_n{x}^n$ is an inverse. This is a formal coefficient calculation and requires no convergence.

Since $A$ is a local ring, its nonunits are exactly $\mathfrak{m}$. The nonunits of $B$ are consequently the series with $a_0\in \mathfrak{m}$, which are exactly $I=\mathfrak{m}B+xB$: such a series is $a_0+xg$, and conversely an element of $I$ has constant coefficient in $\mathfrak{m}$. The map $B\to k$ taking the constant coefficient modulo $\mathfrak{m}$ is surjective with kernel $I$, so $I$ is a maximal ideal. Every proper ideal consists of nonunits and hence lies in $I$, proving that $I$ is the unique maximal ideal. ThusThis proves the description of formal power series over a local ring.

To compute the cotangent space of a formal power series ring, expandDefine the $k$-linear mapChanging a lift $a$ by ${\mathfrak{m}}^2$ or a lift $b$ by $\mathfrak{m}$ changes the image by $I^2$, so the map is well-defined. Every series in $I$ is congruent modulo $x^{2}B\subseteq I^2$ to its first two coefficients, proving surjectivity. If $a+bx\in I^2$, its constant coefficient lies in ${\mathfrak{m}}^2$ and its linear coefficient lies in $\mathfrak{m}$, so both input classes are zero. This proves injectivity. ThereforeFinally, part (a) gives generators $a_1,\dots ,a_d$ of $\mathfrak{m}$, so $I=(a_1,\dots ,a_d,x)$. Use the Krull height theorem in its lecture form: in a Noetherian ring, a prime minimal over an ideal generated by $r$ elements has height at most $r$. The problem allows Noetherianity of $B$ to be assumed. Apply the theorem to $I$ itself, whose height is the Krull dimension of the local ring $B$. Hence $\boxed{\dim B=\mathrm{ht}(I)\le d+1}$.