View source for Depth of a module

One of the cohomological characteristics of a module over a commutative ring. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311801.png" /> be a Noetherian ring, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311802.png" /> be an ideal in it and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311803.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311804.png" />-module of finite type. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311806.png" />-depth of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311807.png" /> is the least integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311808.png" /> for which

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The depth of a module is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118010.png" /> or by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118011.png" />. A different definition can be given in terms of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118013.png" />-regular sequence, i.e. a sequence of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118016.png" /> is not a [[Zero divisor|zero divisor]] in the module

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The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118018.png" />-depth of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118019.png" /> is equal to the length of the largest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118020.png" />-regular sequence consisting of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118021.png" />. The maximal ideal is usually taken for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118022.png" /> in the case of a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118023.png" />. The following formula is valid:

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118025.png" /> denotes a prime ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118026.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118027.png" /> is considered as a module over the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118028.png" />.

The concept of the depth of a module was introduced in [[#References|[1]]] under the name of homological codimension. If the projective dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118029.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118030.png" /> over a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118031.png" /> is finite, then

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In general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118033.png" /> is not larger than the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118034.png" />.

The depth of a module is one of the basic tools in the study of modules. Thus, Cohen–Macaulay modules and rings (cf. [[Cohen–Macaulay ring|Cohen–Macaulay ring]]) have been defined in terms of the depth of modules. The Serre criterion (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118036.png" />) for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118037.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118038.png" />:

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for all prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118041.png" />, often proves to be useful. Finally, the depth of modules is closely connected with local cohomology modules: The statement

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is equivalent to saying that the local cohomology modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118043.png" /> vanish if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118044.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Auslander,   D.A. Buchsbaum,   "Homological dimension in Noetherian rings"  ''Proc. Nat. Acad. Sci. USA'' , '''42'''  (1956)  pp. 36–38</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,   "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Grothendieck,   "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux" , ''SGA 2'' , IHES  (1965)</TD></TR></table>