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Let $R$ be a commutative ring and $x$ and $y$ two elements in $R$. I want to construct the Koszul complex on $x$ and $y$. We start by the following two chain complexes

$$C_2=0\to C_1=R\xrightarrow{\ x\ } C_0= R\to C_{-1}=0$$ $$D_2=0\to D_1=R\xrightarrow{\ y\ } D_0= R\to D_{-1}=0$$ Now we construct the tensor product chain complex which we denote $CD:=C\otimes D$: $$CD_2=C_1\otimes D_1=R\otimes R$$ $$CD_1=C_1\otimes D_0 \oplus C_0\otimes D_1 =R\otimes R \oplus R\otimes R $$ $$CD_0=C_0\otimes D_0=R\otimes R$$ and we get the chain complex $$CD_3=0 \to CD_2=R \otimes R\xrightarrow{\ \partial_2\ } CD_1= R \otimes R \oplus R \otimes R \xrightarrow{\ \partial_1\ } CD_0= R \otimes R \to CD_{-1} =0$$ We now compute $\partial_1$ and $\partial_2$:

$$\partial_2 (c_1\otimes d_1)=(xc_1)\otimes d_1-c_1\otimes (yd_1)$$ and $$\partial_1 (c_1\otimes d_0+c_0\otimes d_1)=(xc_1)\otimes d_0+c_0\otimes (yd_1).$$

Now I want to move from here to express $\partial_1$ and $\partial_2$ in the way expressed in the wikipedia page (section Introduction). I don't understand the notation $R^2$ and the matrix expression of the differentials and where did the tensor product disappear from the final result.

Thank you for your help!!

  • 2
    $R^2 = R \oplus R$, which explains the matrices. The tensor product disappeared because $R\otimes_R R\cong R$.2012-12-14
  • 0
    I still don't see how to get matrix expression for $\partial_1$ and $\partial_2$, can we say for example that $$\partial_2(c_1\otimes d_1)=\mathbf (x, -y) \begin{bmatrix} c_1\otimes d_1 & 0\\0 & c_1\otimes d_1 \end{bmatrix}$$2012-12-14
  • 0
    and $$\partial_1 (c_1\otimes d_0+c_0\otimes d_1)=\mathbf (x, y) \begin{bmatrix} c_1\otimes d_0\\ c_0\otimes d_1 \end{bmatrix}$$2012-12-14
  • 0
    Also why is that $R\otimes_R R \cong R$?2012-12-14
  • 0
    @palio: as for your last question: convince yourself that the map $R \rightarrow R \otimes_R R$, $r \mapsto 1 \otimes r$ is an isomorphism.2012-12-14
  • 0
    is this an isomorphism of $R-$modules? and is this an isomorphism because of $\otimes_R$ taken over $R$? what about $R\otimes_{\mathbb Z} R$ for example?2012-12-14

2 Answers 2