Editing Koszul complex

{{Short description|Construction in homological algebra}}
{{context|date=November 2016}}
{{technical|date=November 2016}}
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]]. As a tool, its homology can be used to tell when a set of elements of a (local) ring is an [[Regular sequence#Definitions|M-regular sequence]], and hence it can be used to prove basic facts about the [[Depth (ring theory)|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 [[Hilbert's syzygy theorem#Syzygies (relations)|syzygies]], that is, it tells you the relations between generators of a module, the relations between these relations, and so forth.

== Definition == 
Let ''A'' be a commutative ring and ''s: A<sup>r</sup> → A'' an ''A''-linear map. Its '''Koszul complex''' ''K<sub>s</sub>'' is
:<math> \bigwedge^r A^r\ \to\ \bigwedge^{r-1}A^r\ \to\ \cdots\ \to \ \bigwedge^1 A^r\ \to\ \bigwedge^0 A^r\simeq A</math>
where the maps send
:<math> \alpha_1\wedge\cdots\wedge\alpha_k\ \mapsto\ \sum_{i=1}^{k}(-1)^{i+1}s(\alpha_i)\ \alpha_1\wedge\cdots\wedge\hat{\alpha}_i\wedge\cdots\wedge\alpha_k </math>
where <math>\hat{\ }</math> means the term is omitted and <math>\wedge</math> means the [[wedge product]]. One may replace <math>A^r</math> with any ''A''-module.

== Motivating example ==
Let ''M'' be a manifold, variety, scheme, ..., and ''A'' be the ring of functions on it, denoted <math>\mathcal{O}(M)</math>.

The map <math>s\colon A^r \to A</math> corresponds to picking ''r'' functions <math>f_1,...,f_r</math>. When ''r = 1'', the Koszul complex is
:<math>\mathcal{O}(M)\ \stackrel{\cdot f}{\to}\ \mathcal{O}(M)</math>
whose [[cokernel]] is the ring of functions on the zero locus ''f = 0''. In general, the Koszul complex is
:<math>\mathcal{O}(M)\ \stackrel{\cdot (f_1,...,f_r)}{\to}\ \mathcal{O}(M)^r\ \to\ \cdots\ \to\ \mathcal{O}(M)^r\ \stackrel{\cdot (f_1,\dots,f_r)}{\to}\ \mathcal{O}(M).</math>
The cokernel of the last map is again functions on the zero locus <math>f_1 = \cdots = f_r = 0</math>. It is the tensor product of the ''r'' many Koszul complexes for <math>f_i = 0</math>, so its dimensions are given by binomial coefficients. 

In pictures: given functions <math>s_i</math>, how do we define the locus where they all vanish?
[[File:Koszul1.png|center|thumb|upright=1.5]]
In algebraic geometry, the ring of functions of the zero locus is <math>A/(s_1,\dots , s_r)</math>. In [[Derived algebraic geometry|''derived'' algebraic geometry]], the ''dg'' ring of functions is the Koszul complex. If the loci <math>s_i=0</math> [[regular sequence|intersect transversely]], these are equivalent. 
[[File:Koszul2.png|center|thumb|upright=1.5]]
Thus: Koszul complexes are ''derived intersections'' of zero loci.

== Properties ==

=== Algebra structure ===
First, the Koszul complex ''K<sub>s</sub>'' of ''(A,s)'' is a [[chain complex]]: the composition of any two maps is zero. Second, the map 
:<math> K_s\otimes K_s\ \to\ K_s \ \ \ \, \ \ \ \, \ \ \ \,  \ \ \ (\alpha_1\wedge\cdots \wedge\alpha_k)\otimes (\beta_1\wedge\cdots\wedge\beta_\ell)\ \mapsto\ \alpha_1\wedge\cdots\wedge \alpha_k\wedge\beta_1\wedge\cdots\wedge\beta_\ell </math>
makes it into a [[Differential graded algebra|dg algebra]].<ref>[http://stacks.math.columbia.edu The Stacks Project], section [https://stacks.math.columbia.edu/tag/0621 0601]</ref>

=== As a tensor product ===
The Koszul complex is a tensor product: if <math>s=(s_1,\dots , s_r)</math>, then
:<math>K_{s}\ \simeq\ K_{s_1}\otimes\cdots\otimes K_{s_r}</math>
where <math>\otimes</math> denotes the [[derived tensor product]] of chain complexes of ''A''-modules.<ref>[http://stacks.math.columbia.edu The Stacks Project], section [https://stacks.math.columbia.edu/tag/0621 0601], Lemma 15.28.12</ref> 

=== Vanishing in regular case ===
When <math>s_1, \dots, s_r</math> form a [[regular sequence]], the map <math>K_s \to A/(s_1, \dots, s_r)</math> is a quasi-isomorphism, i.e. 
:<math> \operatorname{H}^i(K_s)\ =\ 0,\qquad i\ne 0,</math>
and as for any ''s'', <math>H^0(K_s) = A/(s_1,\dots, s_r)</math>.

== History ==
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]]. As a tool, its homology can be used to tell when a set of elements of a (local) ring is an [[Regular sequence#Definitions|M-regular sequence]], and hence it can be used to prove basic facts about the [[Depth (ring theory)|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 [[Hilbert's syzygy theorem#Syzygies (relations)|syzygies]], that is, it tells you the relations between generators of a module, the relations between these relations, and so forth.

== Detailed Definition ==
Let ''R'' be a commutative ring and ''E'' a free module of finite rank ''r'' over ''R''. We write <math>\bigwedge^i E</math> for the ''i''-th [[exterior power]] of ''E''. Then, given an [[module homomorphism|''R''-linear map]] <math>s\colon E \to R</math>,
the '''Koszul complex associated to ''s''''' is the [[chain complex]] of ''R''-modules:
:<math>K_{\bullet}(s)\colon 0 \to \bigwedge^r E \overset{d_r} \to \bigwedge^{r-1} E \to \cdots \to \bigwedge^1 E \overset{d_1}\to R \to 0</math>,
where the differential <math>d_k</math> is given by: for any <math>e_i</math> in ''E'',
:<math>d_k (e_1 \wedge \dots \wedge e_k) = \sum_{i=1}^k (-1)^{i+1} s(e_i) e_1 \wedge \cdots \wedge \widehat{e_i} \wedge \cdots \wedge e_k</math>.
The superscript <math>\widehat{\cdot}</math> means the term is omitted. To show that <math>d_k \circ d_{k+1} = 0</math>, use the [[#Self-duality|self-duality]] of a Koszul complex.

Note that <math>\bigwedge^1 E = E</math> and <math>d_1 = s</math>. Note also that <math>\bigwedge^r E \simeq R</math>; this isomorphism is not canonical (for example, a choice of a [[volume form]] in differential geometry provides an example of such an isomorphism).

If <math>E = R^r</math> (i.e., an ordered basis is chosen), then, giving an ''R''-linear map <math>s\colon R^r\to R</math> amounts to giving a finite sequence <math>s_1, \dots, s_r</math> of elements in ''R'' (namely, a row vector) and then one sets <math>K_{\bullet}(s_1, \dots, s_r) = K_{\bullet}(s).</math>

If ''M'' is a finitely generated ''R''-module, then one sets:
:<math>K_{\bullet}(s, M) = K_{\bullet}(s) \otimes_R M</math>,
which is again a chain complex with the induced differential <math>(d \otimes 1_M)(v \otimes m) = d(v) \otimes m</math>. 

The ''i''-th homology of the Koszul complex
:<math>\operatorname{H}_i(K_{\bullet}(s, M)) = \operatorname{ker}(d_i \otimes 1_M)/\operatorname{im}(d_{i+1}  \otimes 1_M)</math>
is called the '''''i''-th Koszul homology'''. For example, if <math>E = R^r</math> and <math>s = [s_1 \cdots s_r]</math> is a row vector with entries in ''R'', then <math>d_1 \otimes 1_M</math> is
:<math>s \colon M^r \to M, \, (m_1, \dots, m_r) \mapsto s_1 m_1 + \dots + s_r m_r</math>
and so
:<math>\operatorname{H}_0(K_{\bullet}(s, M)) = M/(s_1, \dots, s_r)M = R/(s_1, \dots, s_r) \otimes_R M.</math>
Similarly,
:<math>\operatorname{H}_r(K_{\bullet}(s, M)) = \{ m \in M : s_1 m = s_2 m = \dots = s_r m = 0 \} = \operatorname{Hom}_R(R/(s_1, \dots, s_r), M).</math>

===Koszul complexes in low dimensions===
Given a commutative ring ''R'', an element ''x'' in ''R'', and an ''R''-[[module (mathematics)|module]] ''M'', the multiplication by ''x'' yields a [[homomorphism]] of ''R''-modules,
:<math>M \to M.</math>
Considering this as a [[chain complex]] (by putting them in degree 1 and 0, and adding zeros elsewhere), it is denoted by <math>K(x, M)</math>. By construction, the homologies are
:<math>H_0(K(x, M)) = M/xM, H_1(K(x,M)) = \operatorname{Ann}_M(x) = \{m \in M , xm = 0 \},</math>
the [[Annihilator (ring theory)|annihilator]] of ''x'' in ''M''.
Thus, the Koszul complex and its homology encode fundamental properties of the multiplication by ''x''. This chain complex <math>K_{\bullet}(x)</math> is called the '''Koszul complex''' of ''R'' with respect to ''x'', as in [[#Definition]].

<!--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 <math>x_1, x_2, \ldots, x_n</math>, usually denoted <math>K_{\bullet}(x_1, x_2, \ldots, x_n)</math>, is the [[tensor product]] (in the [[Category (mathematics)|category]] of ''R''-complexes) <math> K_\bullet(x_1) \otimes K_\bullet(x_2) \otimes \cdots \otimes K_\bullet(x_n) </math> of the Koszul complexes defined above individually for each ''i''.

The Koszul complex is a [[free module|free]] chain complex. For every ''p'', its ''p''th degree entry <math>K_p</math> is a free ''R''-module of rank <math>\dbinom{n}{p}</math> (thus, it is zero unless <math>0 \le p \le  n)</math>; this module has a basis <math>\left(e_{i_1,\ldots ,i_p}\right)_{1 \leq i_1 < i_2 < \cdots < i_p \leq n}</math>. The element <math>e_{i_1,\ldots ,i_p}</math> is defined as the pure tensor <math>f_1 \otimes f_2 \otimes \cdots \otimes f_n \in K_\bullet(x_1) \otimes K_\bullet(x_2) \otimes \cdots \otimes K_\bullet(x_n)</math>, where for every <math>1 \le j\le n</math>, we let <math>f_j</math> be the generator 1 of <math>K_1(x_j)</math> if <math>j \in \{i_1, \ldots , i_p\}</math> and the generator 1 of <math>K_0(x_j)</math> otherwise.

The boundary map of the Koszul complex can be written explicitly with respect to this basis. Namely, the ''R''-linear map <math>d \colon K_p \to K_{p-1} </math> is defined by:

:<math>
d(e_{i_1,\dots,i_p}) := \sum _{j=1}^{p}(-1)^{j-1}x_{i_j}e_{i_1,\dots,\widehat{i_j},...,i_p},
</math>

where <math>i_1,\dots,\widehat{i_j},\dots,i_p</math> means <math>i_1,\dots,i_{j-1},i_{j+1},\dots,i_p</math> (that is, the ''j''-th term is being omitted). -->
The Koszul complex for a pair <math>(x, y) \in R^2</math> is 
:<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 <math>d_i</math> is applied on the right. 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 <math>H_1(K_{\bullet}(x, y)) </math> 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 <math>x_1, x_2, \dots, x_n</math> 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 <math>X_1, X_2,\dots, X_d</math> are indeterminates and ''R'' is the polynomial ring <math>k[X_1, X_2,\dots, X_d]</math>, the Koszul complex <math>K_{\bullet}(X_i)</math> on the <math>X_i</math>'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 <math>s\colon E\to  R</math> be an ''R''-linear map, and let ''t'' be an element of ''R''. Let <math>K(s, t)</math> be the Koszul complex of <math>(s, t)\colon E \oplus R \to R</math>.

Using <math>\bigwedge^k(E \oplus R) = \oplus_{i=0}^k \bigwedge^{k-i} E \otimes \bigwedge^i R = \bigwedge^k E \oplus \bigwedge^{k-1} E</math>,
there is the exact sequence of complexes:
:<math>0 \to K(s) \to K(s, t) \to K(s)[-1] \to 0</math>,
where <math>[-1]</math> signifies the degree shift by <math>-1</math> and <math>d_{K(s)[-1]} = -d_{K(s)}</math>. One notes:<ref>Indeed, by linearity, we can assume <math>(x, y) = (e_1 + \epsilon) \wedge e_2 \wedge \cdots \wedge e_k \in \wedge^k (E \oplus R)</math> where <math>R \simeq R \epsilon \subset E \oplus R</math>. Then
:<math>d_{K(s, t)}((x, y)) = (s(e_1) + t) e_2 \wedge \cdots \wedge e_{k} + \sum_{i = 2}^k (-1)^{i + 1} s(e_i) (e_1 + \epsilon) \wedge e_2 \wedge \cdots \widehat{e_i} \cdots \wedge e_k</math>,
which is <math>(d_{K(s)}x + ty, -d_{K(s)} y)</math>.</ref> for <math>(x, y)</math> in <math>\bigwedge^k E \oplus \bigwedge^{k-1} E</math>,
:<math>d_{K(s, t)}((x, y)) = (d_{K(s)} x + ty, d_{K(s)[-1]} y).</math>
In the language of [[homological algebra]], the above means that <math>K(s, t)</math> is the [[mapping cone (homological algebra)|mapping cone]] of <math>t\colon K(s) \to K(s)</math>.

Taking the long exact sequence of homologies, we obtain:
:<math>\cdots \to \operatorname{H}_i(K(s)) \overset{t} \to \operatorname{H}_i(K(s)) \to \operatorname{H}_i(K(s, t)) \to \operatorname{H}_{i-1}(K(s)) \overset{t}\to \cdots.</math>
Here, the connecting homomorphism
:<math>\delta: \operatorname{H}_{i+1}(K(s)[-1]) = \operatorname{H}_{i}(K(s)) \to \operatorname{H}_{i}(K(s))</math>
is computed as follows. By definition, <math>\delta([x]) = [d_{K(s,t)}(y)]</math> where ''y'' is an element of <math>K(s, t)</math> that maps to ''x''. Since <math>K(s, t)</math> is a direct sum, we can simply take ''y'' to be (0, ''x''). Then the early formula for <math>d_{K(s, t)}</math> gives <math>\delta([x]) = t[x]</math>.

The above exact sequence can be used to prove the following.

{{math_theorem
| math_statement =<ref>{{harvnb|Matsumura|1989|loc=Theorem 16.5. (i)}}</ref> Let ''R'' be a ring and ''M'' a module over it. If a sequence <math> x_1, x_2, \cdots, x_r </math> of elements of ''R''  is a [[Regular sequence (algebra)|regular sequence]] on ''M'', then
:<math>\operatorname{H}_i(K(x_1, \dots, x_r) \otimes M) = 0</math>
for all <math>i \geq 1</math>. In particular, when ''M'' = ''R'', this is to say
:<math>0 \to \bigwedge^r R^r \overset{d_r} \to \bigwedge^{r-1} R^r \to \cdots \to \bigwedge^2 R^r \overset{d_2} \to R^r \overset{[x_1 \cdots x_r]}\to R \to R/(x_1, \cdots, x_r) \to 0</math>
is exact; i.e., <math>K(x_1, \dots, x_r)</math> is an ''R''-[[free resolution]] of <math>R/(x_1, \dots, x_r)</math>.
}}

Proof by induction on ''r''. If ''<math>r=1</math>'', then <math>\operatorname{H}_1(K(x_1;M)) = \operatorname{Ann}_M(x_1) = 0</math>. Next, assume the assertion is true for ''r'' - 1. Then, using the above exact sequence, one sees <math>\operatorname{H}_i(K(x_1, \dots, x_r; M)) = 0</math> for any <math>i \geq 2</math>. The vanishing is also valid for <math>i=1</math>, since <math>x_r</math> is a nonzerodivisor on <math>\operatorname{H}_0(K(x_1, \dots, x_{r-1}; M)) = M/(x_1, \dots, x_{r-1})M.</math> <math>\square</math>

{{math_theorem
| name = Corollary
| math_statement =<ref>{{harvnb|Eisenbud|1995|loc=Exercise 17.10.}}</ref> Let ''R'', ''M'' be as above and <math> x_1, x_2, \cdots, x_n </math> a sequence of elements of ''R''. Suppose there are a ring ''S'', an ''S''-regular sequence <math> y_1, y_2, \cdots, y_n </math> in ''S'' and a ring homomorphism ''S'' → ''R'' that maps <math> y_i </math> to <math> x_i </math>. (For example, one can take <math> S=\Z[y_1,\cdots,y_n] </math>.) Then
:<math>\operatorname{H}_i(K(x_1, \dots, x_n) \otimes_R M) = \operatorname{Tor}^S_i(S/(y_1, \dots, y_n),  M).</math>
where Tor denotes the [[Tor functor]] and ''M'' is an ''S''-module through <math>S\to R</math>.
}}
Proof: By the theorem applied to ''S'' and ''S'' as an ''S''-module, we see that <math>K(y_1, \dots, y_n)</math> is an ''S''-free resolution of <math>S/(y_1, \dots, y_n)</math>. So, by definition, the ''i''-th homology of <math>K(y_1, \dots, y_n) \otimes_S M</math> is the right-hand side of the above. On the other hand, <math>K(y_1, \dots, y_n) \otimes_S M = K(x_1, \dots, x_n) \otimes_R M</math> by the definition of the ''S''-module structure on ''M''. <math>\square</math>

{{math_theorem
| name = Corollary
| math_statement =<ref>{{harvnb|Serre|1975|loc=Ch IV, A § 2, Proposition 4.}}</ref> Let ''R'', ''M'' be as above and <math> x_1, x_2, \cdots, x_n </math> a sequence of elements of ''R''. Then both the ideal <math> I=(x_1, x_2, \cdots, x_n) </math> and the annihilator of ''M'' annihilate
:<math>\operatorname{H}_i(K(x_1, \dots, x_n) \otimes M)</math>
for all ''i''.
}}
Proof: Let ''S'' = ''R''[''y''<sub>1</sub>, ..., ''y''<sub>''n''</sub>]. Turn ''M'' into an ''S''-module through the ring homomorphism ''S'' → ''R'', ''y''<sub>''i''</sub> → ''x''<sub>''i''</sub> and ''R'' an ''S''-module through {{nowrap|''y''<sub>''i''</sub> → 0}}. By the preceding corollary, <math>\operatorname{H}_i(K(x_1, \dots, x_n) \otimes M) = \operatorname{Tor}_i^S(R, M)</math> and then
:<math>\operatorname{Ann}_S\left(\operatorname{Tor}_i^S(R, M)\right) \supset \operatorname{Ann}_S(R) + \operatorname{Ann}_S(M) \supset (y_1, \dots, y_n) + \operatorname{Ann}_R(M) + (y_1 - x_1, ..., y_n - x_n).</math> <math>\square</math>

For a [[local ring]], the converse of the theorem holds. More generally,

{{math_theorem|math_statement=<ref>{{harvnb|Matsumura|1989|loc=Theorem 16.5. (ii)}}</ref> Let ''R'' be a ring and ''M'' a nonzero finitely generated module over ''R'' . If <math>x_1, x_2, \dots, x_r</math> are elements of the [[Jacobson radical]] of ''R'', then the following are equivalent:
# The sequence <math>x_1, \dots, x_r</math> is a [[Regular sequence (algebra)|regular sequence]] on ''M'',
# <math>\operatorname{H}_1(K(x_1, \dots, x_r) \otimes M) = 0</math>,
# <math>\operatorname{H}_i(K(x_1, \dots, x_r) \otimes M) = 0</math> for all ''i'' ≥ 1.}}

Proof: We only need to show 2. implies 1., the rest of the [[Ringschluss|cycle of implications]] <math>1.\Rightarrow 3.\Rightarrow 2.\Rightarrow 1.</math> being clear. We argue by induction on ''r''. The case ''r'' = 1 is already known. Let ''x{{'}}'' denote ''x''<sub>1</sub>, ..., ''x''<sub>''r''-1</sub>. Consider
:<math>\cdots \to \operatorname{H}_1(K(x'; M)) \overset{x_r} \to \operatorname{H}_1(K(x'; M)) \to \operatorname{H}_1(K(x_1, \dots, x_r; M)) = 0 \to M/x'M \overset{x_r}\to \cdots.</math>
Since the first <math>x_r</math> is surjective, <math>N = x_r N</math> with <math>N = \operatorname{H}_1(K(x'; M))</math>. By [[Nakayama's lemma]], <math>N = 0</math> and so ''x{{'}}'' is a regular sequence by the inductive hypothesis. Since the second <math>x_r</math> is injective (i.e., is a nonzerodivisor), <math>x_1, \dots, x_r</math> is a regular sequence. (Note: by Nakayama's lemma, the requirement <math>M/(x_1, \dots, x_r)M \ne 0</math> is automatic.) <math>\square</math>

== Tensor products of Koszul complexes ==
In general, if ''C'', ''D'' are chain complexes, then their tensor product <math>C \otimes D</math> is the chain complex given by

:<math>(C \otimes D)_n = \sum_{i + j = n} C_i \otimes D_j</math>
with the differential: for any homogeneous elements ''x'', ''y'',
:<math>d_{C \otimes D} (x \otimes y) = d_C(x) \otimes y + (-1)^{|x|} x \otimes d_D(y)</math>
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 <math>s\colon E\to R</math> and <math>t\colon F\to R</math> be two ''R''-linear maps. Let <math>K(s, t)</math> be the Koszul complex of the linear map <math>(s, t)\colon E \oplus F \to R</math>. Then, as complexes,
:<math>K(s, t) \simeq K(s) \otimes K(t).</math>
To see this, it is more convenient to work with an [[exterior algebra]] (as opposed to exterior powers). Define the graded derivation of degree <math>-1</math>
:<math>d_s: \wedge E \to \wedge E</math>
by requiring: for any homogeneous elements ''x'', ''y'' in Λ''E'',
*<math>d_s(x) = s(x)</math> when <math>|x| = 1</math>
*<math>d_s(x \wedge y) = d_s(x) \wedge y + (-1)^{|x|}x \wedge d_s(y)</math>
One easily sees that <math>d_s \circ d_s = 0</math> (induction on degree) and that the action of <math>d_s</math> on homogeneous elements agrees with the differentials in [[#Definition]].

Now, we have <math>\wedge(E \oplus F) = \wedge E \otimes \wedge F</math> as graded ''R''-modules. Also, by the definition of a tensor product mentioned in the beginning,
:<math>d_{K(s) \otimes K(t)}(e \otimes 1 + 1 \otimes f) = d_{K(s)}(e) \otimes 1 + 1 \otimes d_{K(t)}(f) = s(e) + t(f) = d_{K(s, t)}(e + f).</math>
Since <math>d_{K(s) \otimes K(t)}</math> and <math>d_{K(s, t)}</math> are derivations of the same type, this implies <math>d_{K(s) \otimes K(t)} = d_{K(s, t)}.</math>

Note, in particular,
:<math>K(x_1, x_2, \dots, x_r) \simeq K(x_1) \otimes K(x_2) \otimes \cdots \otimes K(x_r)</math>.

The next proposition shows how the Koszul complex of elements encodes some information about sequences in the ideal generated by them.

{{math_theorem|name=Proposition|math_statement=Let ''R'' be a ring and ''I'' = (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) an ideal generated by some ''n''-elements. Then, for any ''R''-module ''M'' and any elements ''y''<sub>1</sub>, ..., ''y''<sub>''r''</sub> in ''I'',
:<math>\operatorname{H}_i(K(x_1, \dots, x_n, y_1, \dots, y_r; M)) \simeq \bigoplus_{i = j + k} \operatorname{H}_j(K(x_1, \dots, x_n; M)) \otimes \wedge^k R^r.</math>
where <math>\wedge^k R^r</math> is viewed as a complex with zero differential. (In fact, the decomposition holds on the chain-level).}}
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 (ring theory)|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 <math>\operatorname{depth}(I, M)</math>. Recall that an ''M''-regular sequence ''x''<sub>1</sub>, ...,  ''x''<sub>''n''</sub> in an ideal ''I'' is maximal if ''I'' contains no nonzerodivisor on <math>M/(x_1, \dots, x_n) M</math>.

The Koszul homology gives a very useful<!-- this is factually correct, I think --> characterization of a depth.

{{math_theorem|note=depth-sensitivity|math_statement=Let ''R'' be a Noetherian ring, ''x''<sub>1</sub>, ...,  ''x''<sub>''n''</sub> elements of ''R'' and ''I'' = (''x''<sub>1</sub>, ...,  ''x''<sub>''n''</sub>) the ideal generated by them. For a finitely generated module ''M'' over ''R'', if, for some integer ''m'',
:<math>\operatorname{H}_i(K(x_1, \dots, x_n) \otimes M) = 0</math> for all ''i'' > ''m'',
while
:<math>\operatorname{H}_m(K(x_1, \dots, x_n) \otimes M) \ne 0,</math>
then every maximal ''M''-regular sequence in ''I'' has length ''n'' - ''m'' (in particular, they all have the same length). As a consequence,
:<math>\operatorname{depth}(I, M) = n - m</math>.}}
Proof: To lighten the notations, we write H(-) for H(''K''(-)). Let ''y''<sub>1</sub>, ..., ''y''<sub>''s''</sub> be a maximal ''M''-regular sequence in the ideal ''I''; we denote this sequence by <math>\underline{y}</math>. First we show, by induction on <math>l</math>, the claim that <math>\operatorname{H}_i(\underline{y}, x_1, \dots, x_l; M)</math> is <math>\operatorname{Ann}_{M/\underline{y} M}(x_1, \dots, x_l)</math> if <math>i = l</math> and is zero if <math>i > l</math>. The basic case <math>l = 0</math> is clear from [[#Properties of a Koszul homology]]. From the long exact sequence of Koszul homologies and the inductive hypothesis,
:<math>\operatorname{H}_l\left(\underline{y}, x_1, \dots, x_l; M \right) = \operatorname{ker}\left(x_l: \operatorname{Ann}_{M/\underline{y}M}(x_1, \dots, x_{l-1}) \to \operatorname{Ann}_{M/\underline{y}M}(x_1, \dots, x_{l-1}) \right)</math>,
which is
<math>\operatorname{Ann}_{M/\underline{y}M}(x_1, \dots, x_l).</math> Also, by the same argument, the vanishing holds for <math>i > l</math>. This completes the proof of the claim.

Now, it follows from the claim and the early proposition that <math>\operatorname{H}_i(x_1, \dots, x_n; M) = 0</math> for all ''i'' > ''n'' - ''s''. To conclude ''n'' - ''s'' = ''m'', it remains to show that it is nonzero if ''i'' = ''n'' - ''s''. <!-- this part is not really needed? We first note that <math>M/I M \ne 0</math> as follows. If <math>M = IM</math>, then, by the variant of the Cayley-Hamilton theorem used in the proof of [[Nakayama's lemma]], we can find an element ''z'' in ''I'' such that 1 - ''z'' is in the annihilator of ''M''. But this contradicts the fact that ''I'' and the annihilator of ''M'' annihilate <math>\operatorname{H}_m(x_1, \dots, x_n; M) \ne 0</math> (see a corollary in [[#Properties of a Koszul homology]]). Hence, <math>M/IM \ne 0</math>.--> Since <math>\underline{y}</math> is a maximal ''M''-regular sequence in ''I'', the ideal ''I'' is contained in the set of all zerodivisors on <math>M/\underline{y}M</math>, the finite union of the associated primes of the module. Thus, by prime avoidance, there is some nonzero ''v'' in <math>M/\underline{y}M</math> such that <math>I \subset \mathfrak{p} = \operatorname{Ann}_R(v)</math>, which is to say,
:<math>0 \ne v \in \operatorname{Ann}_{M/\underline{y}M}(I) \simeq \operatorname{H}_n\left(x_1, \dots, x_n, \underline{y}; M\right) = \operatorname{H}_{n-s}(x_1, \dots, x_n; M) \otimes \wedge^s R^s.</math> <math>\square</math>

== 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'':
:<math>l_e: \wedge^k E \to \wedge^{k+1} E, \, x \mapsto e \wedge x.</math>
Since <math>e \wedge e = 0</math>, we have: <math>l_e \circ l_e = 0</math>; that is,
:<math>0 \to R \overset{1 \mapsto e}\to \wedge^1 E \overset{l_e}\to \wedge^2 E \to \cdots \to \wedge^r E \to 0</math>
is a cochain complex of free modules. This complex, also called a Koszul complex, is a complex used in {{harv|Eisenbud|1995}}. Taking the dual, there is the complex:
:<math>0 \to (\wedge^r E)^* \to (\wedge^{r-1} E)^* \to \cdots \to (\wedge^2 E)^* \to (\wedge^1E)^* \to R \to 0</math>.
Using an isomorphism <math>\wedge^k E \simeq (\wedge^{r-k} E)^* \simeq \wedge^{r-k} (E^*)</math>, the complex <math>(\wedge E, l_e)</math> coincides with the Koszul complex in the [[#Definition|definition]].

==Use==

The Koszul complex is essential in defining the joint spectrum of a tuple of commuting [[bounded linear operator]]s in a [[Banach space]].{{fact|date=November 2016}}

==See also==
*[[Koszul–Tate complex]]
*[[Syzygy (mathematics)]]

== Notes ==
{{reflist}}

==References==
* {{cite book |last1=Eisenbud |first1=David |author-link1=David Eisenbud |title=Commutative algebra: with a view toward algebraic geometry |series=Graduate Texts in Mathematics |volume=150 |date=1995 |publisher=Springer |location=New York |isbn=0-387-94268-8}}
* {{Citation | title=Intersection theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]]. 3. Folge. | isbn=978-3-540-62046-4 | mr=1644323 | year=1998 | volume=2 | edition=2nd | author=William Fulton |author1-link = William Fulton (mathematician)}} 
* {{Citation | last1=Matsumura | first1=Hideyuki | title=Commutative Ring Theory | publisher=[[Cambridge University Press]] | edition=2nd | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-36764-6 | year=1989}}
* {{Citation | last1=Serre | first1=Jean-Pierre | author1-link = Jean-Pierre Serre | title=Algèbre locale, Multiplicités | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel. Troisième édition, 1975. Lecture Notes in Mathematics | year=1975 | volume=11 |language=French}}
* [http://stacks.math.columbia.edu The Stacks Project], section [https://stacks.math.columbia.edu/tag/0621 0601]

==External links==
* [[Melvin Hochster]], [http://www.math.lsa.umich.edu/~hochster/711F07/L10.03.pdf Math 711: Lecture of October 3, 2007] (especially the very last part).

[[Category:Homological algebra]]