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
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,
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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