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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