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

where $J$ is the ideal generated by the elements

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

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

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

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

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