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 A be a commutative ring and s: Ar → A an A-linear map. Its Koszul complex Ks is
⋀ r A r → ⋀ r − 1 A r → ⋯ → ⋀ 1 A r → ⋀ 0 A r ≃ A {\displaystyle \bigwedge ^{r}A^{r}\ \to \ \bigwedge ^{r-1}A^{r}\ \to \ \cdots \ \to \ \bigwedge ^{1}A^{r}\ \to \ \bigwedge ^{0}A^{r}\simeq A}
where the maps send
α 1 ∧ ⋯ ∧ α k ↦ ∑ i = 1 k ( − 1 ) i + 1 s ( α i ) α 1 ∧ ⋯ ∧ α ^ i ∧ ⋯ ∧ α k {\displaystyle \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}}
where ^ {\displaystyle {\hat {\ }}} means the term is omitted and ∧ {\displaystyle \wedge } means the wedge product. One may replace A r {\displaystyle A^{r}} with any A-module.
Motivating example Let M be a manifold, variety, scheme, ..., and A be the ring of functions on it, denoted O ( M ) {\displaystyle {\mathcal {O}}(M)} . The map s : A r → A {\displaystyle s\colon A^{r}\to A} corresponds to picking r functions f 1 , . . . , f r {\displaystyle f_{1},...,f_{r}} , where the correspondence is given by f 1 = s ( 1 , 0 , . . . , 0 ) , . . . , f r = s ( 0 , . . . , 0 , 1 ) . {\displaystyle f_{1}=s(1,0,...,0),...,f_{r}=s(0,...,0,1).} When r = 1, the Koszul complex is
O ( M ) → ⋅ f O ( M ) {\displaystyle {\mathcal {O}}(M)\ {\stackrel {\cdot f}{\to }}\ {\mathcal {O}}(M)}
whose cokernel is the ring of functions on the zero locus f = 0. In general, the Koszul complex is
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