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(Redirected from Depth of a module)
In commutative algebra, if ''R'' is a commutative ring and ''M'' an ''R''-module, an element ''r'' in ''R'' is called 'M-regular' if ''r'' is a not a zerodivisor on ''M'', and ''M/rM'' is nonzero. An 'R-regular sequence' on ''M'' is a ''d''-tuple
:''r1, ..., rd'' in ''R''
such that for each ''i ≤ d'', ''ri'' is ''M''-regular on the quotient ''R''-module
:''M/(r1, ..., ri-1)M''.
Such a sequence is also called an ''M''-sequence.
It may be that ''r1, ..., rd'' may is an ''M''-sequence, and yet some permutation of the sequence is not. It is, however, a theorem that if ''R'' is a local ring, then an sequence is an ''R''-sequence only if every permutation of it is an ''R''-sequence.
The 'depth' of ''R'' is defined as the maximum length of a regular ''R''-sequence on ''R''. More generally, the depth of an ''R''-module ''M'' is the maximum length of an ''R''-regular sequence on ''M''. The concept is inherently module-theoretic and so there is no harm in approaching it from this point of view.
The depth of a module is always at least ''0'' and no greater than the dimension of the module.
# If ''k'' is a field, it possesses no non-zero non-unit elements so its depth as a ''k''-module is ''0''.
# If ''k'' is a field and ''X'' is an indeterminate, then ''X'' is a nonzerodivisor on the formal power series ring ''R = k''
# If ''k'' is a field and ''X1, X2, ..., Xd'' are indeterminates, then ''X1, X2, ..., Xd'' form a regular sequence of length ''d'' on the polynomial ring ''k''[''X1, X2, ..., Xd''] and there are no longer ''R''-sequences, so ''R'' has depth ''d'', as does the formal power series ring in ''d'' indeterminates over any field.
An important case is when the depth of a ring equals its Krull dimension: the ring is then said to be a Cohen-Macaulay ring. The three examples shown are all Cohen-Macaulay rings. Similarly in the case of modules, the module ''M'' is said to be Cohen-Macaulay if its depth equals its dimension.
★ David Eisenbud, ''Commutative Algebra with a View Toward Algebraic Geometry''. Springer Graduate Texts in Mathematics, no. 150. ISBN 0-387-94268-8
★ Winfried Bruns; Jürgen Herzog, ''Cohen-Macaulay rings''. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN 0-521-41068-1
In commutative algebra, if ''R'' is a commutative ring and ''M'' an ''R''-module, an element ''r'' in ''R'' is called 'M-regular' if ''r'' is a not a zerodivisor on ''M'', and ''M/rM'' is nonzero. An 'R-regular sequence' on ''M'' is a ''d''-tuple
:''r1, ..., rd'' in ''R''
such that for each ''i ≤ d'', ''ri'' is ''M''-regular on the quotient ''R''-module
:''M/(r1, ..., ri-1)M''.
Such a sequence is also called an ''M''-sequence.
It may be that ''r1, ..., rd'' may is an ''M''-sequence, and yet some permutation of the sequence is not. It is, however, a theorem that if ''R'' is a local ring, then an sequence is an ''R''-sequence only if every permutation of it is an ''R''-sequence.
The 'depth' of ''R'' is defined as the maximum length of a regular ''R''-sequence on ''R''. More generally, the depth of an ''R''-module ''M'' is the maximum length of an ''R''-regular sequence on ''M''. The concept is inherently module-theoretic and so there is no harm in approaching it from this point of view.
The depth of a module is always at least ''0'' and no greater than the dimension of the module.
| Contents |
| Examples |
| References |
Examples
# If ''k'' is a field, it possesses no non-zero non-unit elements so its depth as a ''k''-module is ''0''.
# If ''k'' is a field and ''X'' is an indeterminate, then ''X'' is a nonzerodivisor on the formal power series ring ''R = k''
# If ''k'' is a field and ''X1, X2, ..., Xd'' are indeterminates, then ''X1, X2, ..., Xd'' form a regular sequence of length ''d'' on the polynomial ring ''k''[''X1, X2, ..., Xd''] and there are no longer ''R''-sequences, so ''R'' has depth ''d'', as does the formal power series ring in ''d'' indeterminates over any field.
An important case is when the depth of a ring equals its Krull dimension: the ring is then said to be a Cohen-Macaulay ring. The three examples shown are all Cohen-Macaulay rings. Similarly in the case of modules, the module ''M'' is said to be Cohen-Macaulay if its depth equals its dimension.
References
★ David Eisenbud, ''Commutative Algebra with a View Toward Algebraic Geometry''. Springer Graduate Texts in Mathematics, no. 150. ISBN 0-387-94268-8
★ Winfried Bruns; Jürgen Herzog, ''Cohen-Macaulay rings''. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN 0-521-41068-1
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