Koshulya complex

The Koszul complex was first introduced in mathematics by to define the cohomology theory of Lie algebras . Subsequently, he proved to be a useful general construction of homological algebra . Its homology can be used to determine whether a sequence of elements of a ring is , and, as a consequence, it can be used to prove the basic properties of the module or ideal .

Content

Definition

Let R be a commutative ring and E be a free R -module of finite rank r . We denote by $\wedge ^{i}E$ ${\ displaystyle \ wedge ^ {i} E}$ ith external degree E. Then for an R- linear map $s:E\to R$ ${\ displaystyle s: E \ to R}$ the Koszul complex associated with s is a chain complex of R -modules

K_{\bullet }(s):0\to \wedge ^{r}E{\overset {d_{r}}{\to }}\wedge ^{r-1}E\to \cdots \to \wedge ^{1}E{\overset {d_{1}}{\to }}R\to 0,

{\ displaystyle K _ {\ bullet} (s): 0 \ to \ wedge ^ {r} E {\ overset {d_ {r}} {\ to}} \ wedge ^ {r-1} E \ to \ cdots \ to \ wedge ^ {1} E {\ overset {d_ {1}} {\ to}} R \ to 0,}

in which the differential d _k is given by the rule: for any e _i from E

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}

{\ displaystyle 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}}

Superscript ${\widehat {\cdot }}$ ${\ displaystyle {\ widehat {\ cdot}}}$ means that the multiplier is skipped.

notice, that $\wedge ^{1}E=E$ ${\ displaystyle \ wedge ^ {1} E = E}$ and $d_{1}=s$ ${\ displaystyle d_ {1} = s}$ . We also note that $\wedge ^{r}E\simeq R$ ${\ displaystyle \ wedge ^ {r} E \ simeq R}$ ; this isomorphism is not canonical (for example, the choice of the volume form in differential geometry is an example of such an isomorphism).

If E = R ^r (i.e., a basis is chosen), then defining an R- linear mapping s : R ^r → R is equivalent to defining a finite sequence s ₁ , ..., s _{r of} elements R (row vectors) and in this case denote $K_{\bullet }(s_{1},\dots ,s_{r})=K_{\bullet }(s).$ ${\ displaystyle K _ {\ bullet} (s_ {1}, \ dots, s_ {r}) = K _ {\ bullet} (s).}$

If M is a finitely generated R -module, then

K_{\bullet }(s,M)=K_{\bullet }(s)\otimes _{R}M

{\ displaystyle K _ {\ bullet} (s, M) = K _ {\ bullet} (s) \ otimes _ {R} M}

.

ith homology of the Koszul complex

\operatorname {H} _{i}(K_{\bullet }(s,M))=\operatorname {ker} (d_{i}\otimes 1_{M})/\operatorname {im} (d_{i+1}\otimes 1_{M})

{\ displaystyle \ operatorname {H} _ {i} (K _ {\ bullet} (s, M)) = \ operatorname {ker} (d_ {i} \ otimes 1_ {M}) / \ operatorname {im} (d_ {i + 1} \ otimes 1_ {M})}

are called the i-th homology of Koszul . For example, if E = R ^r and $s=[s_{1}\cdots s_{r}]$ ${\ displaystyle s = [s_ {1} \ cdots s_ {r}]}$ Is a row vector of elements of R , then the differential of the Koszul complex $d_{1}\otimes 1_{M}$ ${\ displaystyle d_ {1} \ otimes 1_ {M}}$ there is

s:M^{r}\to M,\,(m_{1},\dots ,m_{r})\mapsto s_{1}m_{1}+\dots +s_{r}m_{r}

{\ displaystyle s: M ^ {r} \ to M, \, (m_ {1}, \ dots, m_ {r}) \ mapsto s_ {1} m_ {1} + \ dots + s_ {r} m_ { r}}

and

\operatorname {H} _{0}(K_{\bullet }(s,M))=M/(s_{1},\dots ,s_{r})M=R/(s_{1},\dots ,s_{r})\otimes _{R}M.

{\ displaystyle \ operatorname {H} _ {0} (K _ {\ bullet} (s, M)) = M / (s_ {1}, \ dots, s_ {r}) M = R / (s_ {1} , \ dots, s_ {r}) \ otimes _ {R} M.}

Also

\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).

{\ displaystyle \ 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).}

Small Koszul Complexes

Given an element x of the ring R and an R -module M , multiplication by x gives a homomorphism of R -modules

M\to M.

{\ displaystyle M \ to M.}

If we consider it as a chain complex (concentrated in degrees 1 and 0), it is denoted $K(x,M)$ ${\ displaystyle K (x, M)}$ . His homologies are equal

H_{0}(K(x,M))=M/xM,H_{1}(K(x,M))=Ann_{M}(x)=\{m\in M,xm=0\},

{\ displaystyle H_ {0} (K (x, M)) = M / xM, H_ {1} (K (x, M)) = Ann_ {M} (x) = \ {m \ in M, xm = 0 \},}

Thus, the Koszul complex and its homology store the basic information about the properties of multiplication by x .

The chain complex K _• ( x ) is called the Koszul complex of the element x of the ring R. If x ₁ , x ₂ , ..., x _n are elements of R , the Koszul complex of the sequence x ₁ , x ₂ , ..., x _n , usually denoted by K _• ( x ₁ , x ₂ , ..., x _n ), is the tensor product $K_{\bullet }(x_{1})\otimes K_{\bullet }(x_{2})\otimes \cdots \otimes K_{\bullet }(x_{n})$ ${\ displaystyle K _ {\ bullet} (x_ {1}) \ otimes K _ {\ bullet} (x_ {2}) \ otimes \ cdots \ otimes K _ {\ bullet} (x_ {n})}$ Koszul complexes for each i .

Koszul complex for couples $(x,y)\in R^{2}$ ${\ displaystyle (x, y) \ in R ^ {2}}$ has the form

0\to R{\xrightarrow {\ d_{2}\ }}R^{2}{\xrightarrow {\ d_{1}\ }}R\to 0,

{\ displaystyle 0 \ to R {\ xrightarrow {\ d_ {2} \}} R ^ {2} {\ xrightarrow {\ d_ {1} \}} R \ to 0,}

where are the matrices $d_{1}$ ${\ displaystyle d_ {1}}$ and $d_{2}$ ${\ displaystyle d_ {2}}$ are set as

d_{1}={\begin{bmatrix}x&y\\\end{bmatrix}}

{\ displaystyle d_ {1} = {\ begin {bmatrix} x & y \\\ end {bmatrix}}}

and

d_{2}={\begin{bmatrix}-y\\x\\\end{bmatrix}}.

{\ displaystyle d_ {2} = {\ begin {bmatrix} -y \\ x \\\ end {bmatrix}}.}

Then cycles of degree 1 are exactly linear relations between the elements x and y , while boundaries are trivial relations. The first Koszul homology H ₁ ( K _• ( x , y )), thus, describe relations modulo trivial relations.

In the case when the elements x ₁ , x ₂ , ..., x _n form a regular sequence, all higher Koszul homologies are nullified.

Example

If k is a field, X ₁ , X ₂ , ..., X _d are unknown and R is a ring of polynomials k [ X ₁ , X ₂ , ..., X _d ], the Koszul complex K _• ( X _i ) of the sequence X _i is a concrete example of the free resolvent of an R -module k .

Literature

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