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Example 49.11.2. Let $d \geq 1$ be an integer. Consider variables $a_{ij}^ l$ for $1 \leq i, j, l \leq d$ and denote

\[ A_ d = \mathbf{Z}[a_{ij}^ k]/J \]

where $J$ is the ideal generated by the elements

\[ \left\{ \begin{matrix} \sum _ l a_{ij}^ la_{lk}^ m - \sum _ l a_{il}^ ma_{jk}^ l & \forall i, j, k, m \\ a_{ij}^ k - a_{ji}^ k & \forall i, j, k \\ a_{i1}^ j - \delta _{ij} & \forall i, j \end{matrix} \right. \]

where $\delta _{ij}$ indices the Kronecker delta function. We define an $A_ d$-algebra $B_ d$ as follows: as an $A_ d$-module we set

\[ B_ d = A_ d e_1 \oplus \ldots \oplus A_ d e_ d \]

The algebra structure is given by $A_ d \to B_ d$ mapping $1$ to $e_1$. The multiplication on $B_ d$ is the $A_ d$-bilinar map

\[ m : B_ d \times B_ d \longrightarrow B_ d, \quad m(e_ i, e_ j) = \sum a_{ij}^ k e_ k \]

It is straightforward to check that the relations given above exactly force this to be an $A_ d$-algebra structure. The morphism

\[ \pi _ d : Y_ d = \mathop{\mathrm{Spec}}(B_ d) \longrightarrow \mathop{\mathrm{Spec}}(A_ d) = X_ d \]

is the “universal” finite free morphism of rank $d$.

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