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In mathematics, the Koszul complex was first introduced to define a cohomology theory for
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s, by
Jean-Louis Koszul Jean-Louis Koszul (; January 3, 1921 – January 12, 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex. He was a second generation member of Bourbaki. Biography Koszul was educated at the in ...
(see
Lie algebra cohomology In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to prope ...
). It turned out to be a useful general construction in
homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...
. 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 In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
. 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 \bigwedge^i E for the ''i''-th
exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
of ''E''. Then, given an ''R''-linear map s\colon E \to R, the Koszul complex associated to ''s'' is the chain complex of ''R''-modules: :K_(s)\colon 0 \to \bigwedge^r E \overset \to \bigwedge^ E \to \cdots \to \bigwedge^1 E \overset\to R \to 0, where the differential d_k is given by: for any e_i in ''E'', :d_k (e_1 \wedge \dots \wedge e_k) = \sum_^k (-1)^ s(e_i) e_1 \wedge \cdots \wedge \widehat \wedge \cdots \wedge e_k. The superscript \widehat means the term is omitted. To show that d_k \circ d_ = 0, use the self-duality of a Koszul complex. Note that \bigwedge^1 E = E and d_1 = s. Note also that \bigwedge^r E \simeq R; this isomorphism is not canonical (for example, a choice of a
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of t ...
in differential geometry provides an example of such an isomorphism.) If E = R^r (i.e., an ordered basis is chosen), then, giving an ''R''-linear map s\colon R^r\to R amounts to giving a finite sequence s_1, \dots, s_r of elements in ''R'' (namely, a row vector) and then one sets K_(s_1, \dots, s_r) = K_(s). If ''M'' is a finitely generated ''R''-module, then one sets: :K_(s, M) = K_(s) \otimes_R M, which is again a chain complex with the induced differential (d \otimes 1_M)(v \otimes m) = d(v) \otimes m. The ''i''-th homology of the Koszul complex :\operatorname_i(K_(s, M)) = \operatorname(d_i \otimes 1_M)/\operatorname(d_ \otimes 1_M) is called the ''i''-th Koszul homology. For example, if E = R^r and s = _1 \cdots s_r/math> is a row vector with entries in ''R'', then d_1 \otimes 1_M is :s \colon M^r \to M, \, (m_1, \dots, m_r) \mapsto s_1 m_1 + \dots + s_r m_r and so :\operatorname_0(K_(s, M)) = M/(s_1, \dots, s_r)M = R/(s_1, \dots, s_r) \otimes_R M. Similarly, :\operatorname_r(K_(s, M)) = \ = \operatorname_R(R/(s_1, \dots, s_r), M).


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 In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same ...
of ''R''-modules, :M \to M. Considering this as a chain complex (by putting them in degree 1 and 0, and adding zeros elsewhere), it is denoted by K(x, M). By construction, the homologies are :H_0(K(x, M)) = M/xM, H_1(K(x,M)) = \operatorname_M(x) = \, the annihilator of ''x'' in ''M''. Thus, the Koszul complex and its homology encode fundamental properties of the multiplication by ''x''. This chain complex K_(x) is called the Koszul complex of ''R'' with respect to ''x'', as in #Definition. The Koszul complex for a pair (x, y) \in R^2 is : 0 \to R \xrightarrow R^2 \xrightarrow R\to 0, with the matrices d_1 and d_2 given by : d_1 = \begin x & y\\ \end and : d_2 = \begin -y\\ x\\ \end. Note that d_i 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 x_1, x_2, \dots, x_n form a
regular sequence In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection. Definitions F ...
, the higher homology modules of the Koszul complex are all zero.


Example

If ''k'' is a field and X_1, X_2,\dots, X_d are indeterminates and ''R'' is the polynomial ring k _1, X_2,\dots, X_d/math>, the Koszul complex K_(X_i) on the X_i'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 s\colon E\to R be an ''R''-linear map, and let ''t'' be an element of ''R''. Let K(s, t) be the Koszul complex of (s, t)\colon E \oplus R \to R. Using \wedge^k(E \oplus R) = \oplus_^k \bigwedge^ E \otimes \bigwedge^i R = \bigwedge^k E \oplus \bigwedge^ E, there is the exact sequence of complexes: :0 \to K(s) \to K(s, t) \to K(s) 1\to 0 where 1signifies the degree shift by -1 and d_ = -d_. One notes:Indeed, by linearity, we can assume (x, y) = (e_1 + \epsilon) \wedge e_2 \wedge \cdots \wedge e_k \in \wedge^k (E \oplus R) where R \simeq R \epsilon \subset E \oplus R. Then :d_((x, y)) = (s(e_1) + t) e_2 \wedge \cdots \wedge e_ + \sum_^k (-1)^ s(e_i) (e_1 + \epsilon) \wedge e_2 \wedge \cdots \widehat \cdots \wedge e_k, which is (d_x + ty, -d_ y). for (x, y) in \wedge^k E \oplus \bigwedge^ E, :d_((x, y)) = (d_ x + ty, d_ y). In the language of
homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...
, the above means that K(s, t) is the mapping cone of t\colon K(s) \to K(s). Taking the long exact sequence of homologies, we obtain: :\cdots \to \operatorname_i(K(s)) \overset \to \operatorname_i(K(s)) \to \operatorname_i(K(s, t)) \to \operatorname_(K(s)) \overset\to \cdots. Here, the connecting homomorphism :\delta: \operatorname_(K(s) 1 = \operatorname_(K(s)) \to \operatorname_(K(s)) is computed as follows. By definition, \delta( = _(y)/math> where ''y'' is an element of K(s, t) that maps to ''x''. Since K(s, t) is a direct sum, we can simply take ''y'' to be (0, ''x''). Then the early formula for d_ gives \delta( = t /math>. The above exact sequence can be used to prove the following. Proof by induction on ''r''. If ''r=1'', then \operatorname_1(K(x_1;M)) = \operatorname_M(x_1) = 0. Next, assume the assertion is true for ''r'' - 1. Then, using the above exact sequence, one sees \operatorname_i(K(x_1, \dots, x_r; M)) = 0 for any i \geq 2. The vanishing is also valid for i=1, since x_ris a nonzerodivisor on \operatorname_0(K(x_1, \dots, x_; M)) = M/(x_1, \dots, x_)M. \square 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 K(y_1, \dots, y_n) \otimes_S M is the right-hand side of the above. On the other hand, K(y_1, \dots, y_n) \otimes_S M = K(x_1, \dots, x_n) \otimes_R M by the definition of the ''S''-module structure on ''M''. \square 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, \operatorname_i(K(x_1, \dots, x_n) \otimes M) = \operatorname_i^S(R, M) and then :\operatorname_S\left(\operatorname_i^S(R, M)\right) \supset \operatorname_S(R) + \operatorname_S(M) \supset (y_1, \dots, y_n) + \operatorname_R(M) + (y_1 - x_1, ..., y_n - x_n). \square For a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic nu ...
, 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 :\cdots \to \operatorname_1(K(x'; M)) \overset \to \operatorname_1(K(x'; M)) \to \operatorname_1(K(x_1, \dots, x_r; M)) = 0 \to M/x'M \overset\to \cdots. Since the first x_r is surjective, N = x_r N with N = \operatorname_1(K(x'; M)). By
Nakayama's lemma In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and ...
, N = 0 and so ''x'' is a regular sequence by the inductive hypothesis. Since the second x_r is injective (i.e., is a nonzerodivisor), x_1, \dots, x_r is a regular sequence. (Note: by Nakayama's lemma, the requirement M/(x_1, \dots, x_r)M \ne 0 is automatic.) \square


Tensor products of Koszul complexes

In general, if ''C'', ''D'' are chain complexes, then their tensor product C \otimes D is the chain complex given by :(C \otimes D)_n = \sum_ C_i \otimes D_j with the differential: for any homogeneous elements ''x'', ''y'', :d_ (x \otimes y) = d_C(x) \otimes y + (-1)^ x \otimes d_D(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 s\colon E\to R and t\colon F\to R be two ''R''-linear maps. Let K(s, t) be the Koszul complex of the linear map (s, t)\colon E \oplus F \to R. Then, as complexes, :K(s, t) \simeq K(s) \otimes K(t). To see this, it is more convenient to work with an
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
(as opposed to exterior powers). Define the graded derivation of degree -1 :d_s: \wedge E \to \wedge E by requiring: for any homogeneous elements ''x'', ''y'' in Λ''E'', *d_s(x) = s(x) when , x, = 1 *d_s(x \wedge y) = d_s(x) \wedge y + (-1)^x \wedge d_s(y) One easily sees that d_s \circ d_s = 0 (induction on degree) and that the action of d_s on homogeneous elements agrees with the differentials in #Definition. Now, we have \wedge(E \oplus F) = \wedge E \otimes \wedge F as graded ''R''-modules. Also, by the definition of a tensor product mentioned in the beginning, :d_(e \otimes 1 + 1 \otimes f) = d_(e) \otimes 1 + 1 \otimes d_(f) = s(e) + t(f) = d_(e + f). Since d_ and d_ are derivations of the same type, this implies d_ = d_. Note, in particular, :K(x_1, x_2, \dots, x_r) \simeq K(x_1) \otimes K(x_2) \otimes \cdots \otimes K(x_r). 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 \operatorname(I, M). Recall that an ''M''-regular sequence ''x''1, ..., ''x''''n'' in an ideal ''I'' is maximal if ''I'' contains no nonzerodivisor on M/(x_1, \dots, x_n) M. 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 \underline. First we show, by induction on l, the claim that \operatorname_i(\underline, x_1, \dots, x_l; M) is \operatorname_(x_1, \dots, x_l) if i = l and is zero if i > l. The basic case l = 0 is clear from #Properties of a Koszul homology. From the long exact sequence of Koszul homologies and the inductive hypothesis, :\operatorname_l\left(\underline, x_1, \dots, x_l; M \right) = \operatorname\left(x_l: \operatorname_(x_1, \dots, x_) \to \operatorname_(x_1, \dots, x_) \right), which is \operatorname_(x_1, \dots, x_l). Also, by the same argument, the vanishing holds for i > l. This completes the proof of the claim. Now, it follows from the claim and the early proposition that \operatorname_i(x_1, \dots, x_n; M) = 0 for all ''i'' > ''n'' - ''s''. To conclude ''n'' - ''s'' = ''m'', it remains to show that it is nonzero if ''i'' = ''n'' - ''s''. Since \underline is a maximal ''M''-regular sequence in ''I'', the ideal ''I'' is contained in the set of all zerodivisors on M/\underlineM, the finite union of the associated primes of the module. Thus, by prime avoidance, there is some nonzero ''v'' in M/\underlineM such that I \subset \mathfrak = \operatorname_R(v), which is to say, :0 \ne v \in \operatorname_(I) \simeq \operatorname_n\left(x_1, \dots, x_n, \underline; M\right) = \operatorname_(x_1, \dots, x_n; M) \otimes \wedge^s R^s. \square


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'': :l_e: \wedge^k E \to \wedge^ E, \, x \mapsto e \wedge x. Since e \wedge e = 0, we have: l_e \circ l_e = 0; that is, :0 \to R \overset\to \wedge^1 E \overset\to \wedge^2 E \to \cdots \to \wedge^r E \to 0 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: :0 \to (\wedge^r E)^* \to (\wedge^ E)^* \to \cdots \to (\wedge^2 E)^* \to (\wedge^1E)^* \to R \to 0. Using an isomorphism \wedge^k E \simeq (\wedge^ E)^* \simeq \wedge^ (E^*), the complex (\wedge E, l_e) coincides with the Koszul complex in the
definition A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definiti ...
.


Use

The Koszul complex is essential in defining the joint spectrum of a tuple of commuting bounded linear operators in a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
.


See also

* Koszul–Tate complex * Syzygy (mathematics)


Notes


References

*
David Eisenbud David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and Director of the Mathematical Sciences Research Institute (MSRI); he previously s ...
, ''Commutative Algebra. With a view toward algebraic geometry'',
Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) ( ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standa ...
, vol 150,
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, New York, 1995. * * * {{Citation , last1=Serre , first1=Jean-Pierre , author1-link = Jean-Pierre Serre , title=Algèbre locale, Multiplicités , publisher=
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, 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


External links

* Melvin Hochster
Math 711: Lecture of October 3, 2007
(especially the very last part). Homological algebra