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Generalizations of Pauli matrices

2015-01-07T16:31:57Z

163.1.246.64: /* Generalized Gell-Mann matrices (Hermitian) */

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Pound–Drever–Hall technique

2014-01-24T18:15:43Z

163.1.246.64: /* PDH readout function */

In [[mathematics]], '''dimension theory''' is a branch of [[commutative algebra]] studying the notion of the [[Krull dimension|dimension]] of a [[commutative ring]], and by extension that of a [[Scheme (mathematics)|scheme]].

The theory is much simpler for an [[affine ring]]; i.e., an integral domain that is a finitely generated algebra over a field. By [[Noether's normalization lemma]], the Krull dimension of such a ring is the [[transcendence degree]] over the base field and the theory runs in parallel with the counterpart in algebraic geometry; cf. [[Dimension of an algebraic variety]]. The general theory tends to be less geometrical; in particular, very little works/is known for non-noetherian rings. (Kaplansky's commutative rings gives a good account of the non-noetherian case.) Today, a standard approach is essentially that of Bourbaki and EGA, which makes essential use of [[graded module]]s and, among other things, emphasizes the role of [[multiplicity of an ideal|multiplicities]], the generalization of the degree of a projective variety. In this approach, [[Krull's principal ideal theorem]] appears as a corollary.

Throughout the article, <math>\operatorname{dim}</math> denotes [[Krull dimension]] of a ring and <math>\operatorname{ht}</math> the [[height (ring theory)|height]] of a prime ideal (i.e., the Krull dimension of the localization at that prime ideal.)

== Basic results ==

Let ''R'' be a noetherian ring or [[valuation ring]]. Then
:<math>\operatorname{dim} R[x] = \operatorname{dim} R + 1.</math>
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 the more precise result. For any prime ideal <math>\mathfrak{p}</math> in ''R'',
:<math>\operatorname{ht}(\mathfrak{p} R[x]) = \operatorname{ht}(\mathfrak{p})</math>.
:<math>\operatorname{ht}(\mathfrak{q}) = \operatorname{ht}(\mathfrak{p}) + 1</math> for any prime ideal <math>\mathfrak{q} \supsetneq \mathfrak{p} R[x]</math> in <math>R[x]</math> that contracts to <math>\mathfrak{p}</math>.
This can be shown within basic ring theory (cf. Kaplansky, commutative rings). By the way, it says in particular that in each fiber of <math>\operatorname{Spec} R[x] \to \operatorname{Spec} R</math>, one cannot have a chain of primes ideals of length <math>\ge 2</math>.

Since an artinian ring (e.g., a field) has dimension zero, by induction, one gets the formula: for an artinian ring ''R'',
:<math>\operatorname{dim} R[x_1, \dots, x_n] = n.</math>

== Fundamental theorem ==
Let <math>(R, \mathfrak{m})</math> be a noetherian local ring and ''I'' a <math>\mathfrak{m}</math>-[[primary ideal]] (i.e., it sits between some power of <math>\mathfrak{m}</math> and <math>\mathfrak{m}</math>). Let <math>F(t)</math> be the [[Hilbert–Poincaré series|Poincaré series]] of the [[associated graded ring]] <math>\operatorname{gr}_I R = \oplus_0^\infty I^n / I^{n+1}</math>. That is,
:<math>F(t) = \sum_0^\infty \ell(I^n / I^{n+1}) t^n</math>
where <math>\ell</math> refers to the [[length of a module]] (over an artinian ring <math>(\operatorname{gr}_I R)_0 = R/I</math>). If <math>x_1, \dots, x_s</math> generate ''I'', then their image in <math>I/I^2</math> have degree 1 and generate <math>\operatorname{gr}_I R</math> as <math>R/I</math>-algebra. By the [[Hilbert–Serre theorem]], ''F'' is a rational function with exactly one pole at <math>t=1</math> of order, say, ''d''. It also says (contained in the proof) that <math>d \le s</math>. Since
:<math>(1-t)^{-d} = \sum_0^\infty \binom{d-1+j}{d-1} t^j</math>,
we find that, for ''n'' large, the coefficient of <math>t^n</math> in <math>F(t) = (1-t)^d F(t) (1 - t)^{-d}</math> is of the form
:<math>\sum_0^N a_k \binom{d-1+n - k}{d-1} = \left(\sum a_k \right) {n^{d-1} \over {d-1}!} + O(n^{d-2}).</math>
That is to say, <math>\ell(I^n / I^{n+1})</math> is a polynomial <math>P</math> in ''n'' of degree <math>d - 1</math> when ''n'' is large. ''P'' is called the [[Hilbert polynomial]] of <math>\operatorname{gr}_I R</math>.

We set <math>d(R) = d</math>. We also set <math>\delta(R)</math> to be the minimum number of elements of ''R'' that can generate a <math>\mathfrak{m}</math>-primary ideal of ''R''. Our ambition is to prove the '''fundamental theorem''':
:<math>\delta(R) = d(R) = \dim R</math>.
Since we can take ''s'' to be <math>\delta(R)</math>, we already have <math>\delta(R) \ge d(R)</math> from the above. Next we prove <math>d(R) \ge \operatorname{dim}R</math> by induction on <math>d(R)</math>. Let <math>\mathfrak{p}_0 \subsetneq \cdots \subsetneq \mathfrak{p}_m</math> be a chain of prime ideals in ''R''. Let <math>D = R/\mathfrak{p}_0</math> and ''x'' a nonzero nonunit element in ''D''. Since ''x'' is not a zero-divisor, we have the exact sequence
:<math>0 \to D \overset{x}\to D \to D/xD \to 0</math>.
The degree bound of the Hilbert-Samuel polynomial now implies that <math>d(D) > d(D/xD) \ge d(R/\mathfrak{p}_1)</math>. (This essentially follows from the [[Artin-Rees lemma]]; see [[Hilbert-Samuel function]] for the statement and the proof.) In <math>R/\mathfrak{p}_1</math>, the chain <math>\mathfrak{p}_i</math> becomes a chain of length <math>m-1</math> and so, by inductive hypothesis and again by the degree estimate,
:<math>m-1 \le \operatorname{dim}(R/\mathfrak{p}_1) \le d(R/\mathfrak{p}_1) \le d(D) - 1 \le d(R) - 1</math>.
The claim follows. It now remains to show <math>\operatorname{dim}R \ge \delta(R).</math> More precisely, we shall show:
:'''Lemma''': ''R'' contains elements <math>x_1, \dots, x_s</math> such that, for any ''i'', any prime ideal containing <math>(x_1, \dots, x_i)</math> has height <math>\ge i</math>.
(Notice: <math>(x_1, \dots, x_s)</math> is then <math>\mathfrak{m}</math>-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 <math>(R, \mathfrak{m})</math> be a noetherian local ring and put <math>k = R/\mathfrak{m}</math>. Then
*<math>\operatorname{dim}R \le \operatorname{dim}_k \mathfrak{m}/\mathfrak{m}^2</math>, since a basis of <math>\mathfrak{m}/\mathfrak{m}^2</math> lifts to a generating set of <math>\mathfrak{m}</math> by Nakayama. If the equality holds, then ''R'' is called a [[regular local ring]].
*<math>\operatorname{dim} \widehat{R} = \operatorname{dim} R</math>, since <math>\operatorname{gr}R = \operatorname{gr}\widehat{R}</math>.

([[Krull's principal ideal theorem]]) The height of the ideal generated by elements <math>x_1, \dots, x_s</math> in a noetherian ring ''R'' is at most ''s''. Conversely, a prime ideal of height ''s'' can be generated by ''s'' elements.

Proof: Let <math>\mathfrak{p}</math> be a prime ideal minimal over such an ideal. Then <math>s \ge \operatorname{dim} R_\mathfrak{p} = \operatorname{ht} \mathfrak{p}</math>. The converse was shown in the course of the proof of the fundamental theorem.

If <math>A \to B</math> is a morphism of noetherian local rings, then
:<math>\operatorname{dim}B/\mathfrak{m}_A B \ge \operatorname{dim}B - \operatorname{dim} A.</math><ref>{{harvnb|Eisenbud|loc=Theorem 10.10}}</ref>
The equality holds if <math>A \to B</math> is [[flat module|flat]] or more generally if it has the [[going-down property]]. (Here, <math>B/\mathfrak{m}_A B</math> is thought of as a [[special fiber]].)

Proof: Let <math>x_1, \dots, x_n</math> generate a <math>\mathfrak{m}_A</math>-primary ideal and <math>y_1, \dots, y_m</math> be such that their images generate a <math>\mathfrak{m}_B/\mathfrak{m}_A B</math>-primary ideal. Then <math>{\mathfrak{m}_B}^s \subset (y_1, \dots, y_m) + \mathfrak{m}_A B</math> for some ''s''. Raising both sides to higher powers, we see some power of <math>\mathfrak{m}_B</math> is contained in <math>(y_1, \dots, y_m, x_1, \dots, x_n)</math>; i.e., the latter ideal is <math>\mathfrak{m}_B</math>-primary; thus, <math>m + n \ge \dim B</math>. The equality is a straightforward application of the going-down property.

If ''R'' is a noetherian local ring, then
:<math>\dim R[x] = \dim R + 1</math>.
Proof: If <math>\mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n</math> are a chain of prime ideals in ''R'', then <math>\mathfrak{p}_iR[x]</math> are a chain of prime ideals in <math>R[x]</math> while <math>\mathfrak{p}_nR[x]</math> is not a maximal ideal. Thus, <math>\dim R + 1 \le \dim R[x]</math>. For the reverse inequality, let <math>\mathfrak{q}</math> be a maximal ideal of <math>R[x]</math> and <math>\mathfrak{p} = R \cap \mathfrak{q}</math>. Since <math>R[x] / \mathfrak{p} R[x] = (R/\mathfrak{p}) [x]</math> is a principal ideal domain, we get <math>1 + \operatorname{dim} R \ge 1 + \operatorname{dim} R_\mathfrak{p} \ge \operatorname{dim} R[x]_\mathfrak{q}</math> by the previous inequality. Since <math>\mathfrak{q}</math> is arbitrary, this implies <math>1 + \operatorname{dim} R \ge \operatorname{dim} R[x]</math>.

== 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 ''R'' (possibly infinite) and is denoted by <math>\operatorname{pd}_R M</math>. We set <math>\operatorname{gl.dim} R = \sup \{ \operatorname{pd}_R M | \text{M is a finite module} \}</math>; it is called the [[global dimension]] of ''R''.

Assume ''R'' is local with residue field ''k''.

{{math_theorem|name=Lemma|<math>\operatorname{pd}_R k = \operatorname{gl.dim} R</math> (possibly infinite).}}

Proof: We claim: for any finite ''R''-module ''M'',
:<math>\operatorname{pd}_R M \le n \Leftrightarrow \operatorname{Tor}^R_{n+1}(M, k) = 0</math>.
By dimension shifting (cf. the proof of Theorem of Serre below), it is enough to prove this for <math>n = 0</math>. But then, by the [[local criterion for flatness]], <math>\operatorname{Tor}^R_1(M, k) = 0 \Rightarrow M\text{ flat } \Rightarrow M\text{ free } \Rightarrow \operatorname{pd}_R(M) \le 0.</math>
Now,
:<math>\operatorname{gl.dim} R \le n \Rightarrow \operatorname{pd}_R k \le n \Rightarrow \operatorname{Tor}^R_{n+1}(-, k) = 0 \Rightarrow \operatorname{pd}_R - \le n \Rightarrow \operatorname{gl.dim} R \le n,</math>
completing the proof.

{{math_theorem|name=Lemma|Let <math>R_1 = R/fR</math>, ''f'' a non-zerodivisor of ''R''. If ''f'' is a non-zerodivisor on a finite module ''M'', then <math>\operatorname{pd}_R M \ge \operatorname{pd}_{R_1} (M \otimes R_1)</math>.}}

Proof: If <math>\operatorname{pd}_R M = 0</math>, then ''M'' is ''R''-free and thus <math>M \otimes R_1</math> is <math>R_1</math>-free. Next suppose <math>\operatorname{pd}_R M > 0</math>. Then we have: <math>\operatorname{pd}_R K = \operatorname{pd}_R M - 1</math> when <math>K</math> is the kernel of some surjection from a free module to ''M''. Thus, by induction, it is enough to consider the case <math>\operatorname{pd}_R M = 1</math>. Then there is a projective resolution:
:<math>0 \to P_1 \to P_0 \to M \to 0</math>,
which gives:
:<math>\operatorname{Tor}^R_1(M, R_1) \to P_1 \otimes R_1 \to P_0 \otimes R_1 \to M \otimes R_1 \to 0</math>.
But tensoring <math>0 \to R \overset{f}\to R \to R_1 \to 0</math> with ''M'' we see the first term vanishes. Hence, <math>\operatorname{pd}_R (M \otimes R_1)</math> is at most 1.

{{math_theorem|name=Theorem of Serre|''R'' regular <math>\Leftrightarrow \operatorname{gl.dim}R < \infty \Leftrightarrow \operatorname{gl.dim}R = \dim R.</math>}}

Proof:<ref>{{harvnb|Weibel|1994|loc=Theorem 4.4.16}}</ref> If ''R'' is regular, we can write <math>k = R/(f_1, \dots, f_n)</math>, <math>f_i</math> a regular system of parameters. An exact sequence <math>0 \to M \overset{f}\to M \to M_1 \to 0</math>, some ''f'' in the maximal ideal, of finite modules, <math>\operatorname{pd}_R M < \infty</math>, gives us:
:<math>0 = \operatorname{Tor}^R_{i+1}(M, k) \to \operatorname{Tor}^R_{i+1}(M_1, k) \to \operatorname{Tor}^R_i(M, k) \overset{f}\to \operatorname{Tor}^R_i(M, k), \quad i \ge \operatorname{pd}_R M.</math>
But ''f'' here is zero since it kills ''k''. Thus, <math>\operatorname{Tor}^R_{i+1}(M_1, k) \simeq \operatorname{Tor}^R_i(M, k)</math> and consequently <math>\operatorname{pd}_R M_1 = 1 + \operatorname{pd}_R M</math>. Using this, we get:
:<math>\operatorname{pd}_R k = 1 + \operatorname{pd}_R (R/(f_1, \dots, f_{n-1})) = \cdots = n.</math>
The proof of the converse is by induction on <math>\operatorname{dim}R</math>. We begin with the inductive step. Set <math>R_1 = R/f_1 R</math>, <math>f_1</math> among a system of parameters. To show ''R'' is regular, it is enough to show <math>R_1</math> is regular. But, since <math>\dim R_1 < \dim R</math>, by inductive hypothesis and the preceding lemma with <math>M = k</math>,
:<math>\operatorname{pd}_R k = \operatorname{gl.dim} R < \infty \Rightarrow \operatorname{pd}_{R_1} k = \operatorname{gl.dim} R_1 < \infty \Rightarrow R_1 \text{ regular}.</math>

The basic step remains. Suppose <math>\operatorname{dim}R = 0</math>. We claim <math>\operatorname{gl.dim}R = 0</math> if it is finite. (This would imply that ''R'' is a [[semisimple ring]]; i.e., a field.) If that is not the case, then there is some finite module <math>M</math> with <math>0 < \operatorname{pd}_R M < \infty</math> and thus in fact we can find ''M'' with <math>\operatorname{pd}_R M = 1</math>. By Nakayama's lemma, there is a surjection <math>u: F \to M</math> such that <math>u \otimes 1: F \otimes k \to M \otimes k</math> is an isomorphism. Denoting by ''K'' the kernel we have:
:<math>0 \to K \to F \overset{u}\to M \to 0</math>.
Since <math>\operatorname{pd}_R K = \operatorname{pd}_R M - 1 = 0</math>, ''K'' is free. Since <math>\operatorname{dim}R = 0</math>, the maximal ideal <math>\mathfrak{m}</math> is an [[associated prime]] of ''R''; i.e., <math>\mathfrak{m} = \operatorname{ann}(s)</math> for some ''s'' in ''R''. Since <math>K \subset \mathfrak{m} M</math>, <math>s K = 0</math>. Since ''K'' is not zero, this implies <math>s = 0</math>, which is absurd. The proof is complete.

== Depths ==

Let ''R'' be a ring and ''M'' a module over it. A sequence of elements <math>x_1, \dots, x_n</math> in <math>R</math> is called a [[regular sequence]] if <math>x_1</math> is not a zero-divisor on <math>M</math> and <math>x_i</math> is not a zero divisor on <math>M/(x_1, \dots, x_{i-1})M</math> for each <math>i = 2, \dots, n</math>.

Assume ''R'' is local with maximal ideal ''m''. Then the [[depth (ring theory)|depth]] of ''M'' is the supremum of any maximal regular sequence <math>x_i</math> in ''m''. It is easy to show (by induction, for example) that <math>\operatorname{depth} M \le \operatorname{dim} R</math>. If the equality holds, ''R'' is called the [[Cohen–Macaulay ring]].

{{math_theorem|name=Proposition|<math>\operatorname{depth} \operatorname{M} = \sup \{ n | \operatorname{Ext}_R^i(k, M) = 0, i < n. \}</math>}}

The [[Auslander–Buchsbaum formula]] relates depth and projective dimension.

{{math_theorem|Let ''M'' be a finite module over a noetherian local ring ''R''. If <math>\operatorname{pd}_R M < \infty</math>, then
:<math>\operatorname{pd}_R M + \operatorname{depth} M = \operatorname{depth} R.</math>}}

==References==
{{reflist}}
* Part II of {{Citation | last=Eisenbud | first=David | author-link=David Eisenbud | year=1995 | title=Commutative algebra. With a view toward algebraic geometry | volume=150 | series=Graduate Texts in Mathematics | place=New York | publisher=Springer-Verlag | mr=1322960 | isbn=0-387-94268-8}}.
* Chapter 10 of {{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Macdonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969}}.
* [[Irving Kaplansky|Kaplansky, Irving]], ''Commutative rings'', Allyn and Bacon, 1970.
* {{cite book |last=Weibel |first=Charles A. |authorlink=Charles Weibel |title=An Introduction to Homological Algebra |url= |accessdate= |year=1995 |publisher=Cambridge University Press |location= |isbn= |page=}}

[[Category:Dimension]]
[[Category:Commutative algebra]]

Rabi resonance method

2012-05-29T10:33:59Z

163.1.246.69: /* Theory */

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