Tridiagonal matrix: Difference between revisions

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In [[mathematics]], the '''Koszul complex''' was first introduced to define a [[cohomology theory]] for [[Lie algebra]]s, by [[Jean-Louis Koszul]] (see [[Lie algebra cohomology]]). It turned out to be a useful general construction in [[homological algebra]].


==Introduction==
In [[commutative algebra]], if ''x'' is an element of the ring ''R'', multiplication by ''x'' is ''R''-linear and so represents an ''R''-[[module (mathematics)|module]] [[homomorphism]] ''x'':''R'' →''R'' from ''R'' to itself. It is useful to throw in zeroes on each end and make this a (free) ''R''-complex:


:<math>
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0\to R\xrightarrow{\ x\ }R\to0.
</math>
 
Call this [[chain complex]] ''K''<sub>•</sub>(''x'').
 
Counting the right-hand copy of ''R'' as the zeroth degree and the left-hand copy as the first degree, this chain complex neatly captures the most important facts about multiplication by ''x'' because its zeroth homology is exactly the homomorphic image of ''R'' modulo the multiples of ''x'', H<sub>0</sub>(''K''<sub>•</sub>(''x'')) = ''R''/''xR'', and its first homology is exactly the [[Annihilator (ring theory)|annihilator]] of ''x'', H<sub>1</sub>(''K''<sub>•</sub>(''x'')) = Ann<sub>''R''</sub>(''x'').
 
This chain complex ''K''<sub>•</sub>(''x'') is called the '''Koszul complex''' of ''R'' with respect to ''x''.
 
Now, if ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub> are elements of ''R'', the '''Koszul complex''' of ''R'' with respect to ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>, usually denoted ''K''<sub>•</sub>(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>), is the [[tensor product]] in the [[Category (mathematics)|category]] of ''R''-complexes of the Koszul complexes defined above individually for each ''i''.
 
The Koszul complex is a [[free module|free]] chain complex. There are exactly (''n'' choose ''j'') copies of the ring ''R'' in the ''j''th degree in the complex (0 ≤ ''j'' ≤ ''n''). The matrices involved in the maps can be written down precisely. Letting <math>e_{i_1...i_p}</math> denote a free-basis generator in
''K''<sub>''p''</sub>, ''d'': ''K''<sub>''p''</sub> {{mapsto}} ''K''<sub>''p'' &minus; 1</sub> is defined by:
 
:<math>
d(e_{i_1...i_p}) := \sum _{j=1}^{p}(-1)^{j-1}x_{i_j}e_{i_1...\widehat{i_j}...i_p}.
</math>
 
For the case of two elements ''x'' and ''y'', the Koszul complex can then be written down quite succinctly as
:<math>
0 \to R \xrightarrow{\ d_2\ }  R^2 \xrightarrow{\ d_1\ } R\to 0,
</math>
with the matrices <math>d_1</math> and <math>d_2</math> given by
 
:<math>
d_1 = \begin{bmatrix}
x & y\\
\end{bmatrix}
</math> and
:<math>
d_2 = \begin{bmatrix}
-y\\
x\\
\end{bmatrix}.
</math>
Note that ''d<sub>i</sub>'' is applied on the left. The [[cycle (homology theory)|cycle]]s in degree 1 are then exactly the linear relations on the elements ''x'' and ''y'', while the boundaries are the trivial relations. The first Koszul homology H<sub>1</sub>(''K''<sub>•</sub>(''x'', ''y'')) therefore measures exactly the relations mod the trivial relations. With more elements the higher-dimensional Koszul homologies measure the higher-level versions of this.
 
In the case that the elements ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub> form a [[Regular sequence (algebra)|regular sequence]], the higher homology modules of the Koszul complex are all zero.
 
==Example==
If ''k'' is a field and ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''d''</sub> are indeterminates and ''R'' is the polynomial ring ''k''[''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''d''</sub>], the Koszul complex ''K''<sub>•</sub>(''X''<sub>''i''</sub>) on the ''X''<sub>''i''</sub>'s forms a concrete free ''R''-resolution of ''k''.
 
==Theorem==
Let (''R'', ''m'') be a [[Noetherian]] [[local ring]] with maximal ideal ''m'', and let ''M'' be a finitely-generated ''R''-module. If ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub> are elements of the maximal ideal ''m'', then the following are equivalent:
 
# The (''x''<sub>''i''</sub>) form a [[Regular sequence (algebra)|regular sequence]] on ''M'',
# H<sub>1</sub>(''K''<sub>•</sub>(''x''<sub>''i''</sub>)) = 0,
# H<sub>''j''</sub>(''K''<sub>•</sub>(''x''<sub>''i''</sub>)) = 0 for all ''j'' ≥ 1.
 
==Applications==
 
The Koszul complex is essential in defining the [[joint spectrum of a tuple of bounded operators|joint spectrum of a tuple]] of [[bounded linear operator]]s in a [[Banach space]].
 
==See also==
 
*[[Koszul–Tate complex]]
 
==References==
* [[David Eisenbud]], ''Commutative Algebra. With a view toward algebraic geometry'', Graduate Texts in Mathematics, vol 150, [[Springer-Verlag]], New York, 1995. ISBN 0-387-94268-8
 
[[Category:Homological algebra]]

Revision as of 17:00, 26 February 2014


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