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| 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]].
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| 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.
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| 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.)
| | They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. Medical word press themes give you the latest medical designs. These templates are professionally designed and are also Adsense ready. Word - Press also provides protection against spamming, as security is a measure issue. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>Most Word - Press web developers can provide quality CMS website solutions and they price their services at reasonable rates. WPTouch is among the more well known Word - Press smartphone plugins which is currently in use by thousands of users. This plugin allows a blogger get more Facebook fans on the related fan page. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. By using Word - Press, you can develop very rich, user-friendly and full-functional website. <br><br>The least difficult and very best way to do this is by acquiring a Word - Press site. s cutthroat competition prevailing in the online space won. all the necessary planning and steps of conversion is carried out in this phase, such as splitting, slicing, CSS code, adding images, header footer etc. The animation can be quite subtle these as snow falling gently or some twinkling start in the track record which are essentially not distracting but as an alternative gives some viewing enjoyment for the visitor of the internet site. If you have any issues about where by and how to use [http://by.ix-cafe.com/wordpressdropboxbackup86833 wordpress backup], you can get in touch with us at our own webpage. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. The company gains commission from the customers' payment. This includes enriching the content with proper key words, tactfully defining the tags and URL. <br><br>Under Settings —> Reading, determine if posts or a static page will be your home page, and if your home page is a static page, what page will contain blog posts. It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. Word - Press can also be quickly extended however improvement API is not as potent as Joomla's. Web developers and newbies alike will have the ability to extend your web site and fit other incredible functions with out having to spend more. Verify whether your company has a team of developers or programmers having hands-on experience and knowledge about all Word - Press concepts. |
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| == Basic results ==
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| Let ''R'' be a noetherian ring or [[valuation ring]]. Then
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| :<math>\operatorname{dim} R[x] = \operatorname{dim} R + 1.</math>
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| 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'',
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| :<math>\operatorname{ht}(\mathfrak{p} R[x]) = \operatorname{ht}(\mathfrak{p})</math>.
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| :<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>.
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| 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>.
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| Since an artinian ring (e.g., a field) has dimension zero, by induction, one gets the formula: for an artinian ring ''R'',
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| :<math>\operatorname{dim} R[x_1, \dots, x_n] = n.</math>
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| == Fundamental theorem ==
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| 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,
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| :<math>F(t) = \sum_0^\infty \ell(I^n / I^{n+1}) t^n</math>
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| 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
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| :<math>(1-t)^{-d} = \sum_0^\infty \binom{d-1+j}{d-1} t^j</math>,
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| 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
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| :<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>
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| 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>.
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| 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''':
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| :<math>\delta(R) = d(R) = \dim R</math>.
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| 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
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| :<math>0 \to D \overset{x}\to D \to D/xD \to 0</math>.
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| 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,
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| :<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>.
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| The claim follows. It now remains to show <math>\operatorname{dim}R \ge \delta(R).</math> More precisely, we shall show:
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| :'''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>.
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| (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]].
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| == Consequences of the fundamental theorem ==
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| Let <math>(R, \mathfrak{m})</math> be a noetherian local ring and put <math>k = R/\mathfrak{m}</math>. Then
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| *<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]].
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| *<math>\operatorname{dim} \widehat{R} = \operatorname{dim} R</math>, since <math>\operatorname{gr}R = \operatorname{gr}\widehat{R}</math>.
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| ([[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.
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| 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.
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| If <math>A \to B</math> is a morphism of noetherian local rings, then
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| :<math>\operatorname{dim}B/\mathfrak{m}_A B \ge \operatorname{dim}B - \operatorname{dim} A.</math><ref>{{harvnb|Eisenbud|loc=Theorem 10.10}}</ref>
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| 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]].)
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| 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.
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| If ''R'' is a noetherian local ring, then | |
| :<math>\dim R[x] = \dim R + 1</math>.
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| 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>.
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| == Regular rings ==
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| 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''.
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| Assume ''R'' is local with residue field ''k''.
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| {{math_theorem|name=Lemma|<math>\operatorname{pd}_R k = \operatorname{gl.dim} R</math> (possibly infinite).}}
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| Proof: We claim: for any finite ''R''-module ''M'',
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| :<math>\operatorname{pd}_R M \le n \Leftrightarrow \operatorname{Tor}^R_{n+1}(M, k) = 0</math>.
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| 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>
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| Now,
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| :<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>
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| completing the proof.
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| {{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>.}}
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| 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:
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| :<math>0 \to P_1 \to P_0 \to M \to 0</math>,
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| which gives:
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| :<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>.
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| 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.
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| {{math_theorem|name=Theorem of Serre|''R'' regular <math>\Leftrightarrow \operatorname{gl.dim}R < \infty \Leftrightarrow \operatorname{gl.dim}R = \dim R.</math>}}
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| 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:
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| :<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>
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| 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:
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| :<math>\operatorname{pd}_R k = 1 + \operatorname{pd}_R (R/(f_1, \dots, f_{n-1})) = \cdots = n.</math>
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| 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>,
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| :<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>
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| 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:
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| :<math>0 \to K \to F \overset{u}\to M \to 0</math>.
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| 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.
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| == Depths ==
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| 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>.
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| 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]].
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| {{math_theorem|name=Proposition|<math>\operatorname{depth} \operatorname{M} = \sup \{ n | \operatorname{Ext}_R^i(k, M) = 0, i < n. \}</math>}}
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| The [[Auslander–Buchsbaum formula]] relates depth and projective dimension.
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| {{math_theorem|Let ''M'' be a finite module over a noetherian local ring ''R''. If <math>\operatorname{pd}_R M < \infty</math>, then
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| :<math>\operatorname{pd}_R M + \operatorname{depth} M = \operatorname{depth} R.</math>}}
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| ==References==
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| {{reflist}}
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| * 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}}.
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| * 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}}.
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| * [[Irving Kaplansky|Kaplansky, Irving]], ''Commutative rings'', Allyn and Bacon, 1970.
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| * {{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=}}
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| [[Category:Dimension]]
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| [[Category:Commutative algebra]]
| |
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