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.Theorem — If A → B {displaystyle A o B} is a morphism of noetherian local rings, thenProposition — If R is a noetherian ring, thenTheorem — Let R ⊂ R ′ {displaystyle Rsubset R'} be integral domains, p ′ ⊂ R ′ {displaystyle {mathfrak {p}}'subset R'} be a prime ideal and p = R ∩ p ′ {displaystyle {mathfrak {p}}=Rcap {mathfrak {p}}'} . If R is a Noetherian ring, thenLemma —  pd R ⁡ k = g l . d i m ⁡ R {displaystyle operatorname {pd} _{R}k=operatorname {gl.dim} R} (possibly infinite).Lemma — Let R 1 = R / f R {displaystyle R_{1}=R/fR} , f a non-zerodivisor of R. If f is a non-zerodivisor on M, thenTheorem of Serre — R regular ⇔ g l . d i m ⁡ R < ∞ ⇔ g l . d i m ⁡ R = dim ⁡ R . {displaystyle Leftrightarrow operatorname {gl.dim} R<infty Leftrightarrow operatorname {gl.dim} R=dim R.} Corollary — A regular local ring is a unique factorization domain.Theorem — Let R be a ring. Then g l . d i m ⁡ R [ x 1 , … , x n ] = g l . d i m ⁡ R + n {displaystyle operatorname {gl.dim} R=operatorname {gl.dim} R+n} .Proposition (Rees) — Let M be a finite R-module. Then depth ⁡ M = sup { n | Ext R i ⁡ ( k , M ) = 0 , i < n } {displaystyle operatorname {depth} operatorname {M} =sup{n|operatorname {Ext} _{R}^{i}(k,M)=0,i<n}} .Theorem — Let M be a finite module over a noetherian local ring R. If pd R ⁡ M < ∞ {displaystyle operatorname {pd} _{R}M<infty } , thenTheorem (Grothendieck) — Let M be a finite R-module. ThenTheorem — Suppose R is Noetherian, M is a finite module over R and x i {displaystyle x_{i}} are in the Jacobson radical of R. Then the following are equivalentCorollary — The sequence x i {displaystyle x_{i}} is M-regular if and only if any of its permutations is so.Corollary — If x 1 , … , x n {displaystyle x_{1},dots ,x_{n}} is an M-regular sequence, then x 1 j , … , x n j {displaystyle x_{1}^{j},dots ,x_{n}^{j}} is also an M-regular sequence for each positive integer j.Theorem — Assume R is local. Then letTheorem — R is a complete intersection ring if and only if its Koszul algebra is an exterior algebra.Theorem — For any ring R,Proposition — A ring has weak global dimension zero if and only if it is von Neumann regular.Bernstein's inequality — See 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, dim {displaystyle operatorname {dim} } denotes Krull dimension of a ring and ht {displaystyle operatorname {ht} } 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. Let R be a noetherian ring or valuation ring. Then If R is noetherian, this follows from the fundamental theorem below (in particular, Krull's principal ideal theorem), but it is also a consequence of a more precise result. For any prime ideal p {displaystyle {mathfrak {p}}} in R, This can be shown within basic ring theory (cf. Kaplansky, commutative rings). In addition, in each fiber of Spec ⁡ R [ x ] → Spec ⁡ R {displaystyle operatorname {Spec} R o operatorname {Spec} R} , one cannot have a chain of primes ideals of length ≥ 2 {displaystyle geq 2} . Since an artinian ring (e.g., a field) has dimension zero, by induction one gets a formula: for an artinian ring R, Let ( R , m ) {displaystyle (R,{mathfrak {m}})} be a noetherian local ring and I a m {displaystyle {mathfrak {m}}} -primary ideal (i.e., it sits between some power of m {displaystyle {mathfrak {m}}} and m {displaystyle {mathfrak {m}}} ). Let F ( t ) {displaystyle F(t)} be the Poincaré series of the associated graded ring gr I ⁡ R = ⊕ 0 ∞ I n / I n + 1 {displaystyle operatorname {gr} _{I}R=oplus _{0}^{infty }I^{n}/I^{n+1}} . That is, where ℓ {displaystyle ell } refers to the length of a module (over an artinian ring ( gr I ⁡ R ) 0 = R / I {displaystyle (operatorname {gr} _{I}R)_{0}=R/I} ). If x 1 , … , x s {displaystyle x_{1},dots ,x_{s}} generate I, then their image in I / I 2 {displaystyle I/I^{2}} have degree 1 and generate gr I ⁡ R {displaystyle operatorname {gr} _{I}R} as R / I {displaystyle R/I} -algebra. By the Hilbert–Serre theorem, F is a rational function with exactly one pole at t = 1 {displaystyle t=1} of order d ≤ s {displaystyle dleq s} . Since