Depth (commutative algebra)

The depth of a module , especially an ideal , is studied in commutative algebra . It is an important invariant that plays a role in various definitions and sentences.

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .

definition

If a module over a ring is, so that is depth of the thickness of a maximum - regular sequence of elements . ${\ displaystyle M}$ ${\ displaystyle R}$ ${\ displaystyle \ mathrm {tf} (M)}$ ${\ displaystyle M}$ ${\ displaystyle M}$ ${\ displaystyle R}$

The notation for the depth of a module in the literature are not uniform: in addition to , and is also , and to find. ${\ displaystyle M}$ ${\ displaystyle \ mathrm {tf} (M)}$ ${\ displaystyle \ mathrm {t} (M)}$ ${\ displaystyle \ mathrm {depth} (M)}$ ${\ displaystyle \ mathrm {prof} (M)}$

properties

If there is a local Noetherian ring with maximal ideal and a finite module (which is not trivial, i.e. not equal to 0) is over , then: ${\ displaystyle R}$ ${\ displaystyle M}$ ${\ displaystyle R}$

If the remainder class field is then: ${\ displaystyle k = R / m}$

{\ displaystyle \ mathrm {tf} (M) <\ infty}

{\ displaystyle \ mathrm {tf} (M) = \ mathrm {min} (\ {n \ in \ mathbb {N} | \ mathrm {Ext} ^ {n} (k, M) \ neq 0 \}}

The following applies:

{\ displaystyle \ mathrm {tf} (M) \ leq \ mathrm {dim} M}

Modules (or rings) for which equality applies are called Cohen-Macaulay modules (or Cohen-Macaulay rings).

${\ displaystyle \ mathrm {tf} (M) = 0 \ Leftrightarrow m \ in Ass (M)}$

( is the set of prime ideals to be associated with .) ${\ displaystyle \ mathrm {Ace} (M)}$ ${\ displaystyle M}$ ${\ displaystyle R}$

Has a finite projective dimension , then: ${\ displaystyle M}$

{\ displaystyle \ mathrm {tf} (M) + \ mathrm {proj.dim} (M) = \ mathrm {tf} (A)}

In particular is

{\ displaystyle \ mathrm {tf} (M) \ leq \ mathrm {tf} (A)}

Examples

If a vector space is over a body of vector space dimension , then its depth as module is equal . ${\ displaystyle V}$ ${\ displaystyle K}$ ${\ displaystyle n}$ ${\ displaystyle K}$ ${\ displaystyle n}$

The depth of a regular local ring is its Krull dimension.

Web links

Depth of a module in the Encyclopaedia of Mathematics (English)

literature

Michael F. Atiyah , Ian G. MacDonald : Introduction to Commutative Algebra. Addison-Wesley, Reading MA 1969, ISBN 0-201-00361-9 .
Rainer Brüske, Friedrich Ischebeck, Ferdinand Vogel: Commutative Algebra. BI-Wissenschaftsverlag, Mannheim et al. 1989, ISBN 3-411-14041-0 .
Robin Hartshorne : Algebraic Geometry (= Graduate Texts in Mathematics. 52). Springer, New York et al. 1977, ISBN 3-540-90244-9 .
Ernst Kunz : Introduction to commutative algebra and algebraic geometry (= Vieweg course. 46 advanced course in mathematics. ). Vieweg, Braunschweig et al. 1980, ISBN 3-528-07246-6 .