Lemma 15.81.3. Let $R \to A$ be a ring map of finite type. Let $K^\bullet$ be a complex of $A$-modules. Let $m \in \mathbf{Z}$. The following are equivalent

1. for some presentation $\alpha : R[x_1, \ldots , x_ n] \to A$ the complex $K^\bullet$ is an $m$-pseudo-coherent complex of $R[x_1, \ldots , x_ n]$-modules,

2. for all presentations $\alpha : R[x_1, \ldots , x_ n] \to A$ the complex $K^\bullet$ is an $m$-pseudo-coherent complex of $R[x_1, \ldots , x_ n]$-modules.

In particular the same equivalence holds for pseudo-coherence.

Proof. If $\alpha : R[x_1, \ldots , x_ n] \to A$ and $\beta : R[y_1, \ldots , y_ m] \to A$ are presentations. Choose $f_ j \in R[x_1, \ldots , x_ n]$ with $\alpha (f_ j) = \beta (y_ j)$ and $g_ i \in R[y_1, \ldots , y_ m]$ with $\beta (g_ i) = \alpha (x_ i)$. Then we get a commutative diagram

$\xymatrix{ R[x_1, \ldots , x_ n, y_1, \ldots , y_ m] \ar[d]^{x_ i \mapsto g_ i} \ar[rr]_-{y_ j \mapsto f_ j} & & R[x_1, \ldots , x_ n] \ar[d] \\ R[y_1, \ldots , y_ m] \ar[rr] & & A }$

After a change of coordinates the ring homomorphism $R[x_1, \ldots , x_ n, y_1, \ldots , y_ m] \to R[x_1, \ldots , x_ n]$ is isomorphic to the ring homomorphism which maps each $y_ i$ to zero. Similarly for the left vertical map in the diagram. Hence, by induction on the number of variables this lemma follows from Lemma 15.81.2. The pseudo-coherent case follows from this and Lemma 15.64.5. $\square$

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