Global dimension

<h2 id="examples">Examples</h2>
<ul><li>Let A = K[x1,...,xn] be the <a href="/facts/Ring_of_polynomials/ua3vk8Ih">ring of polynomials</a> in n variables over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K. Then the global dimension of A is equal to n. This statement goes back to <a href="/facts/David_Hilbert/1vhlHn4b">David Hilbert</a>'s foundational work on homological properties of polynomial rings; see <a href="/facts/Hilbert%2527s_syzygy_theorem/HtYbyvz2">Hilbert's syzygy theorem</a>. More generally, if R is a Noetherian ring of finite global dimension k and A = R[x] is a ring of polynomials in one variable over R then the global dimension of A is equal to k + 1.</li></ul>
<ul><li>A ring has global dimension zero if and only if it is <a href="/facts/Semisimple_ring/9KASYk35">semisimple</a>.</li></ul>
<ul><li>The global dimension of a ring A is less than or equal to one if and only if A is <a href="/facts/Hereditary_ring/w2yUaqFe">hereditary</a>. In particular, a commutative <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domain</a> which is not a field has global dimension one. For example 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 has global dimension one.</li></ul>
<ul><li>The first <a href="/facts/Weyl_algebra/PADSD0k9">Weyl algebra</a> A1 is a noncommutative Noetherian <a href="/facts/Domain_(ring_theory)/p4eKRC8W">domain</a> of global dimension one.</li></ul>
<ul><li>If a ring is right Noetherian, then the right global dimension is the same as the weak global dimension, and is at most the left global dimension. In particular if a ring is right and left Noetherian then the left and right global dimensions and the weak global dimension are all the same.</li>
<li>The <a href="/facts/Triangular_matrix_ring/L1NeGpTz">triangular matrix ring</a> 
 
 
 
 
 
 [
 
 
 
 
 Z
 
 
 
 
 Q
 
 
 
 
 
 0
 
 
 
 Q
 
 
 
 
 ]
 
 
 
 
 {\displaystyle {\begin{bmatrix}\mathbb {Z} &\mathbb {Q} \\0&\mathbb {Q} \end{bmatrix}}}
 
 has right global dimension 1, weak global dimension 1, but left global dimension 2. It is right Noetherian but not left Noetherian.</li></ul>
<h2 id="alternative-characterizations">Alternative characterizations</h2>
The right global dimension of a ring A can be alternatively defined as:

<ul><li>the supremum of the set of projective dimensions of all <a href="/facts/Cyclic_module/w9tMplLC">cyclic</a> right A-modules;</li>
<li>the supremum of the set of projective dimensions of all <a href="/facts/Finitely-generated_module/6Njcpgzr">finite</a> right A-modules;</li>
<li>the supremum of the <a href="/facts/Injective_dimension/1sAYFnyJ">injective dimensions</a> of all right A-modules;</li>
<li>when A is a commutative Noetherian <a href="/facts/Local_ring/gRTdg5Qx">local ring</a> with <a href="/facts/Maximal_ideal/nf4DnKVS">maximal ideal</a> m, the <a href="/facts/Projective_dimension/lHymqhNC">projective dimension</a> of the <a href="/facts/Residue_field/e6nflEmB">residue field</a> A/m.</li></ul>
The left global dimension of A has analogous characterizations obtained by replacing "right" with "left" in the above list.
Serre <a href="/facts/Mathematical_proof/7ECDrU80">proved</a> that a commutative Noetherian local ring A is <a href="/facts/Regular_local_ring/P9s5Fvrl">regular</a> if and only if it has finite global dimension, in which case the global dimension coincides with the <a href="/facts/Krull_dimension/KQb2uLtz">Krull dimension</a> of A. This theorem opened the door to application of homological methods to commutative algebra.

<ul><li><a href="/facts/David_Eisenbud/hw2yEnnx">Eisenbud, David</a> (1999), Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150 (3rd ed.), Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-94268-8.</li>
<li><a href="/facts/Irving_Kaplansky/nRLyp7K9">Kaplansky, Irving</a> (1972), Fields and Rings, Chicago Lectures in Mathematics (2nd ed.), University Of Chicago Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-226-42451-0, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1001.16500">1001.16500</a></li>
<li>Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-36764-6.</li>
<li>McConnell, J. C.; Robson, J. C.; Small, Lance W. (2001), Revised (ed.), Noncommutative Noetherian Rings, <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>, vol. 30, American Mathematical Society, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-2169-5.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Auslander, Maurice (1955). "On the dimension of modules and algebras. III. Global dimension". Nagoya Math J. 9: 67–77. doi:10.1017/S0027763000023291. <a href="http://projecteuclid.org/euclid.nmj/1118799684" target="_blank">http://projecteuclid.org/euclid.nmj/1118799684</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Global dimension open-in-new

Global dimension