En matemáticas , la teoría de la dimensión es el estudio en términos de álgebra conmutativa de la noción dimensión de una variedad algebraica (y por extensión la de un esquema ). La necesidad de una teoría para una noción aparentemente tan simple resulta de la existencia de muchas definiciones de la dimensión que son equivalentes sólo en los casos más regulares (ver Dimensión de una variedad algebraica ). Una gran parte de la teoría de la dimensión consiste en estudiar las condiciones bajo las cuales varias dimensiones son iguales, y muchas clases importantes de anillos conmutativos pueden definirse como los anillos en los que dos dimensiones son iguales; por ejemplo, unEl anillo regular es un anillo conmutativo tal que la dimensión homológica es igual a la dimensión de Krull .
La teoría es más simple para anillos conmutativos que se generan finitamente álgebra sobre un campo, que son también anillos de cociente de anillos de polinomios en un número finito de indeterminados más de un campo. En este caso, que es la contraparte algebraica del caso de conjuntos algebraicos afines , la mayoría de las definiciones de la dimensión son equivalentes. Para los anillos conmutativos generales, la falta de interpretación geométrica es un obstáculo para el desarrollo de la teoría; en particular, se sabe muy poco de los anillos no eterianos. ( Los anillos conmutativos de Kaplansky dan una buena descripción del caso no-noetheriano).
A lo largo del artículo, denota la dimensión Krull de un anillo yla altura de un ideal primo (es decir, la dimensión de Krull de la localización en ese ideal primo). Se supone que los anillos son conmutativos excepto en la última sección sobre dimensiones de anillos no conmutativos.
Resultados basicos
Sea R un anillo noetheriano o un anillo de valoración . Luego
Si R es noetheriano, esto se sigue del teorema fundamental siguiente (en particular, el teorema ideal principal de Krull ), pero también es una consecuencia de un resultado más preciso. Para cualquier ideal primordialen R ,
- .
- para cualquier ideal principal en que se contrae a .
Esto se puede demostrar dentro de la teoría básica del anillo (cf. Kaplansky, anillos conmutativos). Además, en cada fibra de, one cannot have a chain of primes ideals of length .
Since an artinian ring (e.g., a field) has dimension zero, by induction one gets a formula: for an artinian ring R,
Anillos locales
Fundamental theorem
Let be a noetherian local ring and I a -primary ideal (i.e., it sits between some power of and ). Let be the Poincaré series of the associated graded ring . That is,
where refers to the length of a module (over an artinian ring ). If generate I, then their image in have degree 1 and generate as -algebra. By the Hilbert–Serre theorem, F is a rational function with exactly one pole at of order . Since
- ,
we find that the coefficient of in is of the form
That is to say, is a polynomial in n of degree . P is called the Hilbert polynomial of .
We set . We also set to be the minimum number of elements of R that can generate an -primary ideal of R. Our ambition is to prove the fundamental theorem:
- .
Since we can take s to be , we already have from the above. Next we prove by induction on . Let be a chain of prime ideals in R. Let and x a nonzero nonunit element in D. Since x is not a zero-divisor, we have the exact sequence
- .
The degree bound of the Hilbert-Samuel polynomial now implies that . (This essentially follows from the Artin-Rees lemma; see Hilbert-Samuel function for the statement and the proof.) In , the chain becomes a chain of length and so, by inductive hypothesis and again by the degree estimate,
- .
The claim follows. It now remains to show More precisely, we shall show:
- Lemma: The maximal ideal contains elements , d = Krull dimension of R, such that, for any i, any prime ideal containing has height .
(Notice: is then -primary.) The proof is omitted. It appears, for example, in Atiyah–MacDonald. But it can also be supplied privately; the idea is to use prime avoidance.
Consequences of the fundamental theorem
Let be a noetherian local ring and put . Then
- , since a basis of lifts to a generating set of by Nakayama. If the equality holds, then R is called a regular local ring.
- , since .
- (Krull's principal ideal theorem) The height of the ideal generated by elements in a noetherian ring is at most s. Conversely, a prime ideal of height s is minimal over an ideal generated by s elements. (Proof: Let be a prime ideal minimal over such an ideal. Then . The converse was shown in the course of the proof of the fundamental theorem.)
Theorem — If is a morphism of noetherian local rings, then
The equality holds if is flat or more generally if it has the going-down property.
Proof: Let generate a -primary ideal and be such that their images generate a -primary ideal. Then for some s. Raising both sides to higher powers, we see some power of is contained in ; i.e., the latter ideal is -primary; thus, . The equality is a straightforward application of the going-down property.
Proposition — If R is a noetherian ring, then
- .
Proof: If are a chain of prime ideals in R, then are a chain of prime ideals in while is not a maximal ideal. Thus, . For the reverse inequality, let be a maximal ideal of and . Clearly, . Since is then a localization of a principal ideal domain and has dimension at most one, we get by the previous inequality. Since is arbitrary, it follows .
Nagata's altitude formula
Theorem — Let be integral domains, be a prime ideal and . If R is a Noetherian ring, then
where the equality holds if either (a) R is universally catenary and R' is finitely generated R-algebra or (b) R' is a polynomial ring over R.
Proof:[2] First suppose is a polynomial ring. By induction on the number of variables, it is enough to consider the case . Since R' is flat over R,
- .
By Noether's normalization lemma, the second term on the right side is:
Next, suppose is generated by a single element; thus, . If I = 0, then we are already done. Suppose not. Then is algebraic over R and so . Since R is a subring of R', and so since is algebraic over . Let denote the pre-image in of . Then, as , by the polynomial case,
Here, note that the inequality is the equality if R' is catenary. Finally, working with a chain of prime ideals, it is straightforward to reduce the general case to the above case.
See also: Quasi-unmixed ring.
Métodos homologicos
Regular rings
Let R be a noetherian ring. The projective dimension of a finite R-module M is the shortest length of any projective resolution of M (possibly infinite) and is denoted by . We set ; it is called the global dimension of R.
Assume R is local with residue field k.
Lemma — (possibly infinite).
Proof: We claim: for any finite R-module M,
- .
By dimension shifting (cf. the proof of Theorem of Serre below), it is enough to prove this for . But then, by the local criterion for flatness, Now,
completing the proof.
Remark: The proof also shows that if M is not free and is the kernel of some surjection from a free module to M.
Lemma — Let , f a non-zerodivisor of R. If f is a non-zerodivisor on M, then
- .
Proof: If , then M is R-free and thus is -free. Next suppose . Then we have: as in the remark above. Thus, by induction, it is enough to consider the case . Then there is a projective resolution: , which gives:
- .
But Hence, is at most 1.
Theorem of Serre — R regular
Proof:[3] If R is regular, we can write , a regular system of parameters. An exact sequence , some f in the maximal ideal, of finite modules, , gives us:
But f here is zero since it kills k. Thus, and consequently . Using this, we get:
The proof of the converse is by induction on . We begin with the inductive step. Set , among a system of parameters. To show R is regular, it is enough to show is regular. But, since , by inductive hypothesis and the preceding lemma with ,
The basic step remains. Suppose . We claim if it is finite. (This would imply that R is a semisimple local ring; i.e., a field.) If that is not the case, then there is some finite module with and thus in fact we can find M with . By Nakayama's lemma, there is a surjection from a free module F to M whose kernel K is contained in . Since , the maximal ideal is an associated prime of R; i.e., for some nonzero s in R. Since , . Since K is not zero and is free, this implies , which is absurd.
Corollary — A regular local ring is a unique factorization domain.
Proof: Let R be a regular local ring. Then , which is an integrally closed domain. It is a standard algebra exercise to show this implies that R is an integrally closed domain. Now, we need to show every divisorial ideal is principal; i.e., the divisor class group of R vanishes. But, according to Bourbaki, Algèbre commutative, chapitre 7, §. 4. Corollary 2 to Proposition 16, a divisorial ideal is principal if it admits a finite free resolution, which is indeed the case by the theorem.
Theorem — Let R be a ring. Then .
Depth
Let R be a ring and M a module over it. A sequence of elements in is called an M-regular sequence if is not a zero-divisor on and is not a zero divisor on for each . A priori, it is not obvious whether any permutation of a regular sequence is still regular (see the section below for some positive answer.)
Let R be a local Noetherian ring with maximal ideal and put . Then, by definition, the depth of a finite R-module M is the supremum of the lengths of all M-regular sequences in . For example, we have consists of zerodivisors on M is associated with M. By induction, we find
for any associated primes of M. In particular, . If the equality holds for M = R, R is called a Cohen–Macaulay ring.
Example: A regular Noetherian local ring is Cohen–Macaulay (since a regular system of parameters is an R-regular sequence.)
In general, a Noetherian ring is called a Cohen–Macaulay ring if the localizations at all maximal ideals are Cohen–Macaulay. We note that a Cohen–Macaulay ring is universally catenary. This implies for example that a polynomial ring is universally catenary since it is regular and thus Cohen–Macaulay.
Proposition (Rees) — Let M be a finite R-module. Then .
More generally, for any finite R-module N whose support is exactly ,
- .
Proof: We first prove by induction on n the following statement: for every R-module M and every M-regular sequence in ,
- (*)
The basic step n = 0 is trivial. Next, by inductive hypothesis, . But the latter is zero since the annihilator of N contains some power of . Thus, from the exact sequence and the fact that kills N, using the inductive hypothesis again, we get
- ,
proving (*). Now, if , then we can find an M-regular sequence of length more than n and so by (*) we see . It remains to show if . By (*) we can assume n = 0. Then is associated with M; thus is in the support of M. On the other hand, It follows by linear algebra that there is a nonzero homomorphism from N to M modulo ; hence, one from N to M by Nakayama's lemma.
The Auslander–Buchsbaum formula relates depth and projective dimension.
Theorem — Let M be a finite module over a noetherian local ring R. If , then
Proof: We argue by induction on , the basic case (i.e., M free) being trivial. By Nakayama's lemma, we have the exact sequence where F is free and the image of f is contained in . Since what we need to show is . Since f kills k, the exact sequence yields: for any i,
Note the left-most term is zero if . If , then since by inductive hypothesis, we see If , then and it must be
As a matter of notation, for any R-module M, we let
One sees without difficulty that is a left-exact functor and then let be its j-th right derived functor, called the local cohomology of R. Since , via abstract nonsense,
- .
This observation proves the first part of the theorem below.
Theorem (Grothendieck) — Let M be a finite R-module. Then
- .
- and if
- If R is complete and d its Krull dimension and if E is the injective hull of k, then is representable (the representing object is sometimes called the canonical module especially if R is Cohen–Macaulay.)
Proof: 1. is already noted (except to show the nonvanishing at the degree equal to the depth of M; use induction to see this) and 3. is a general fact by abstract nonsense. 2. is a consequence of an explicit computation of a local cohomology by means of Koszul complexes (see below).
Koszul complex
Let R be a ring and x an element in it. We form the chain complex K(x) given by for i = 0, 1 and for any other i with the differential
For any R-module M, we then get the complex with the differential and let be its homology. Note:
More generally, given a finite sequence of elements in a ring R, we form the tensor product of complexes:
and let its homology. As before,
- ,
- .
We now have the homological characterization of a regular sequence.
Theorem — Suppose R is Noetherian, M is a finite module over R and are in the Jacobson radical of R. Then the following are equivalent
- is an M-regular sequence.
- .
- .
Corollary — The sequence is M-regular if and only if any of its permutations is so.
Corollary — If is an M-regular sequence, then is also an M-regular sequence for each positive integer j.
A Koszul complex is a powerful computational tool. For instance, it follows from the theorem and the corollary
(Here, one uses the self-duality of a Koszul complex; see Proposition 17.15. of Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry.)
Another instance would be
Theorem — Assume R is local. Then let
- ,
the dimension of the Zariski tangent space (often called the embedding dimension of R). Then
- .
Remark: The theorem can be used to give a second quick proof of Serre's theorem, that R is regular if and only if it has finite global dimension. Indeed, by the above theorem, and thus . On the other hand, as , the Auslander–Buchsbaum formula gives . Hence, .
We next use a Koszul homology to define and study complete intersection rings. Let R be a Noetherian local ring. By definition, the first deviation of R is the vector space dimension
where is a system of parameters. By definition, R is a complete intersection ring if is the dimension of the tangent space. (See Hartshorne for a geometric meaning.)
Theorem — R is a complete intersection ring if and only if its Koszul algebra is an exterior algebra.
Injective dimension and Tor dimensions
Let R be a ring. The injective dimension of an R-module M denoted by is defined just like a projective dimension: it is the minimal length of an injective resolution of M. Let be the category of R-modules.
Theorem — For any ring R,
Proof: Suppose . Let M be an R-module and consider a resolution
where are injective modules. For any ideal I,
which is zero since is computed via a projective resolution of . Thus, by Baer's criterion, N is injective. We conclude that . Essentially by reversing the arrows, one can also prove the implication in the other way.
The theorem suggests that we consider a sort of a dual of a global dimension:
It was originally called the weak global dimension of R but today it is more commonly called the Tor dimension of R.
Remark: for any ring R, .
Proposition — A ring has weak global dimension zero if and only if it is von Neumann regular.
Teoría de la multiplicidad
Dimensiones de anillos no conmutativos
Let A be a graded algebra over a field k. If V is a finite-dimensional generating subspace of A, then we let and then put
- .
It is called the Gelfand–Kirillov dimension of A. It is easy to show is independent of a choice of V.
Example: If A is finite-dimensional, then gk(A) = 0. If A is an affine ring, then gk(A) = Krull dimension of A.
Bernstein's inequality — See [1]
See also: Goldie dimension, Krull–Gabriel dimension.
Ver también
- Bass number
- Perfect complex
- amplitude
Notas
- ^ Eisenbud, Theorem 10.10
- ^ Matsumura, Theorem 15.5.
- ^ Weibel 1994, Theorem 4.4.16
Referencias
- Bruns, Winfried; Herzog, Jürgen (1993), Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, ISBN 978-0-521-41068-7, MR 1251956
- Part II of Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, New York: Springer-Verlag, ISBN 0-387-94268-8, MR 1322960.
- Chapter 10 of Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8.
- Kaplansky, Irving, Commutative rings, Allyn and Bacon, 1970.
- H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.
- Serre, Jean-Pierre (1975), Algèbre locale. Multiplicités, Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel. Troisième édition, 1975. Lecture Notes in Mathematics (in French), 11, Berlin, New York: Springer-Verlag
- Weibel, Charles A. (1995). An Introduction to Homological Algebra. Cambridge University Press.