
Remark 10.106.12. Let $R \to S$ be a ring map. Sometimes the set of elements $g \in S$ such that $g \otimes 1 = 1 \otimes g$ is called the epicenter of $S$. It is an $R$-algebra. By the construction of Lemma 10.106.11 we get for each $g$ in the epicenter a matrix factorization

$(g) = Y X Z$

with $X \in \text{Mat}(n \times n, R)$, $Y \in \text{Mat}(1 \times n, S)$, and $Z \in \text{Mat}(n \times 1, S)$. Namely, let $x_{i, j}, y_ i, z_ j$ be as in part (2) of the lemma. Set $X = (x_{i, j})$, let $y$ be the row vector whose entries are the $y_ i$ and let $z$ be the column vector whose entries are the $z_ j$. With this notation conditions (b) and (c) of Lemma 10.106.11 mean exactly that $Y X \in \text{Mat}(1 \times n, R)$, $X Z \in \text{Mat}(n \times 1, R)$. It turns out to be very convenient to consider the triple of matrices $(X, YX, XZ)$. Given $n \in \mathbf{N}$ and a triple $(P, U, V)$ we say that $(P, U, V)$ is a $n$-triple associated to $g$ if there exists a matrix factorization as above such that $P = X$, $U = YX$ and $V = XZ$.

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