Koszul complex

In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its homology can be used to tell when a set of elements of a (local) ring is an M-regular sequence, and hence it can be used to prove basic facts about the depth of a module or ideal which is an algebraic notion of dimension that is related to but different from the geometric notion of Krull dimension. Moreover, in certain circumstances, the complex is the complex of syzygies, that is, it tells you the relations between generators of a module, the relations between these relations, and so forth.
Definition.
Let "R" be a commutative ring and "E" a free module of finite rank "r" over "R". We write formula_1 for the "i"-th exterior power of "E". Then, given an "R"-linear map formula_2,
the Koszul complex associated to "s" is the chain complex of "R"-modules:
where the differential formula_4 is given by: for any formula_5 in "E",
The superscript formula_7 means the term is omitted. (Showing formula_8 is straightforward; alternatively, this identity also follows using the #Self-duality of a Koszul complex.)
Note that formula_9 and formula_10. Note also that formula_11; this isomorphism is not canonical (for example, a choice of a volume form in differential geometry provides an example of such an isomorphism.)
If formula_12 (i.e., an ordered basis is chosen), then, giving an "R"-linear map formula_13 amounts to giving a finite sequence formula_14 of elements in "R" (namely, a row vector) and then one sets formula_15
If "M" is a finitely generated "R"-module, then one sets:
which is again a chain complex with the induced differential formula_17. 
The "i"-th homology of the Koszul complex
is called the "i"-th Koszul homology. For example, if formula_12 and formula_20 is a row vector with entries in "R", then formula_21 is
and so
Similarly,
Koszul complexes in low dimensions.
Given a commutative ring "R", an element "x" in "R", and an "R"-module "M", the multiplication by "x" yields a homomorphism of "R"-modules,
Considering this as a chain complex (by putting them in degree 1 and 0, and adding zeros elsewhere), it is denoted by formula_26. By construction, the homologies are
the annihilator of "x" in "M".
Thus, the Koszul complex and its homology encode fundamental properties of the multiplication by "x".
This chain complex formula_28 is called the Koszul complex of "R" with respect to "x", as in #Definition. 
The Koszul complex for a pair formula_29 is 
with the matrices formula_31 and formula_32 given by
Note that formula_35 is applied on the left. The cycles 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 H1("K"•("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 formula_36 form a regular sequence, the higher homology modules of the Koszul complex are all zero.
Example.
If "k" is a field and formula_37 are indeterminates and "R" is the polynomial ring formula_38, the Koszul complex formula_39 on the formula_40's forms a concrete free "R"-resolution of "k".
Properties of a Koszul homology.
Let "E" be a finite-rank free module over "R", let formula_41 be an "R"-linear map, and let "t" be an element of "R". Let formula_42 be the Koszul complex of formula_43.
Using formula_44,
there is the exact sequence of complexes:
where [-1] signifies the degree shift by -1 and formula_46. One notes: for formula_47 in formula_48,
In the language of homological algebra, the above means that formula_42 is the mapping cone of formula_51.
Taking the long exact sequence of homologies, we obtain:
Here, the connecting homomorphism
is computed as follows. By definition, formula_54 where "y" is an element of formula_42 that maps to "x". Since formula_42 is a direct sum, we can simply take "y" to be (0, "x"). Then the early formula for formula_57 gives formula_58.
The above exact sequence can be used to prove the following.
Proof by induction on "r". If "formula_59", then formula_60. Next, assume the assertion is true for "r" - 1. Then, using the above exact sequence, one sees formula_61 for any formula_62. The vanishing is also valid for formula_63, since formula_64is a nonzerodivisor on formula_65 formula_66
Proof: By the theorem applied to "S" and "S" as an "S"-module, we see "K"("y"1, ..., "y""n") is an "S"-free resolution of "S"/("y"1, ..., "y""n"). So, by definition, the "i"-th homology of formula_67 is the right-hand side of the above. On the other hand, formula_68 by the definition of the "S"-module structure on "M". formula_66
Proof: Let "S" = "R"["y"1, ..., "y""n"]. Turn "M" into an "S"-module through the ring homomorphism "S" → "R", "y""i" → "x""i" and "R" an "S"-module through . By the preceding corollary, formula_70 and then
For a local ring, the converse of the theorem holds. More generally,
Proof: We only need to show 2. implies 1., the rest being clear. We argue by induction on "r". The case "r" = 1 is already known. Let "x" denote "x"1, ..., "x""r"-1. Consider
Since the first formula_64 is surjective, formula_75 with formula_76. By Nakayama's lemma, formula_77 and so "x" is a regular sequence by the inductive hypothesis. Since the second formula_64 is injective (i.e., is a nonzerodivisor), formula_79 is a regular sequence. (Note: by Nakayama's lemma, the requirement formula_80 is automatic.) formula_66
Tensor products of Koszul complexes.
In general, if "C", "D" are chain complexes, then their tensor product formula_82 is the chain complex given by
with the differential: for any homogeneous elements "x", "y",
where |"x"| is the degree of "x".
This construction applies in particular to Koszul complexes. Let "E", "F" be finite-rank free modules, and let formula_85 and formula_86 be two "R"-linear maps. Let formula_42 be the Koszul complex of the linear map formula_88. Then, as complexes,
To see this, it is more convenient to work with an exterior algebra (as opposed to exterior powers). Define the graded derivation of degree formula_90
by requiring: for any homogeneous elements "x", "y" in Λ"E",
One easily sees that formula_95 (induction on degree) and that the action of formula_96 on homogeneous elements agrees with the differentials in #Definition.
Now, we have formula_97 as graded "R"-modules. Also, by the definition of a tensor product mentioned in the beginning,
Since formula_99 and formula_57 are derivations of the same type, this implies formula_101
Note, in particular,
The next proposition shows how the Koszul complex of elements encodes some information about sequences in the ideal generated by them.
Proof: (Easy but omitted for now)
As an application, we can show the depth-sensitivity of a Koszul homology. Given a finitely generated module "M" over a ring "R", by (one) definition, the depth of "M" with respect to an ideal "I" is the supremum of the lengths of all regular sequences of elements of "I" on "M". It is denoted by formula_103. Recall that an "M"-regular sequence "x"1, ..., "x""n" in an ideal "I" is maximal if "I" contains no nonzerodivisor on formula_104.
The Koszul homology gives a very useful characterization of a depth.
Proof: To lighten the notations, we write H(-) for H("K"(-)). Let "y"1, ..., "y""s" be a maximal "M"-regular sequence in the ideal "I"; we denote this sequence by formula_105. First we show, by induction on formula_106, the claim that formula_107 is formula_108 if formula_109 and is zero if formula_110. The basic case formula_111 is clear from #Properties of a Koszul homology. From the long exact sequence of Koszul homologies and the inductive hypothesis,
which is
formula_113 Also, by the same argument, the vanishing holds for formula_110. This completes the proof of the claim.
Now, it follows from the claim and the early proposition that formula_115 for all "i" > "n" - "s". To conclude "n" - "s" = "m", it remains to show that it is nonzero if "i" = "n" - "s". Since formula_105 is a maximal "M"-regular sequence in "I", the ideal "I" is contained in the set of all zerodivisors on formula_117, the finite union of the associated primes of the module. Thus, by prime avoidance, there is some nonzero "v" in formula_117 such that formula_119, which is to say,
Self-duality.
There is an approach to a Koszul complex that uses a cochain complex instead of a chain complex. As it turns out, this results essentially in the same complex (the fact known as the self-duality of a Koszul complex).
Let "E" be a free module of finite rank "r" over a ring "R". Then each element "e" of "E" gives rise to the exterior left-multiplication by "e":
Since formula_123, we have: formula_124; that is,
is a cochain complex of free modules. This complex, also called a Koszul complex, is a complex used in . Taking the dual, there is the complex:
Using an isomorphism formula_127, the complex formula_128 coincides with the Koszul complex in #Definition.
Use.
The Koszul complex is essential in defining the joint spectrum of a tuple of commuting bounded linear operators in a Banach space.