## 15.81 Relatively pseudo-coherent modules

This section is the analogue of Section 15.80 for pseudo-coherence.

Lemma 15.81.1. Let $R$ be a ring. Let $K^\bullet$ be a complex of $R$-modules. Consider the $R$-algebra map $R[x] \to R$ which maps $x$ to zero. Then

$K^\bullet \otimes _{R[x]}^{\mathbf{L}} R \cong K^\bullet \oplus K^\bullet $

in $D(R)$.

Proof. Choose a K-flat resolution $P^\bullet \to K^\bullet$ over $R$ such that $P^ n$ is a flat $R$-module for all $n$, see Lemma 15.59.10. Then $P^\bullet \otimes _ R R[x]$ is a K-flat complex of $R[x]$-modules whose terms are flat $R[x]$-modules, see Lemma 15.59.3 and Algebra, Lemma 10.39.7. In particular $x : P^ n \otimes _ R R[x] \to P^ n \otimes _ R R[x]$ is injective with cokernel isomorphic to $P^ n$. Thus

$P^\bullet \otimes _ R R[x] \xrightarrow {x} P^\bullet \otimes _ R R[x]$

is a double complex of $R[x]$-modules whose associated total complex is quasi-isomorphic to $P^\bullet$ and hence $K^\bullet$. Moreover, this associated total complex is a K-flat complex of $R[x]$-modules for example by Lemma 15.59.4 or by Lemma 15.59.5. Hence

\begin{align*} K^\bullet \otimes _{R[x]}^{\mathbf{L}} R & \cong \text{Tot}(P^\bullet \otimes _ R R[x] \xrightarrow {x} P^\bullet \otimes _ R R[x]) \otimes _{R[x]} R = \text{Tot}(P^\bullet \xrightarrow {0} P^\bullet ) \\ & = P^\bullet \oplus P^\bullet  \cong K^\bullet \oplus K^\bullet  \end{align*}

as desired. $\square$

Lemma 15.81.2. Let $R$ be a ring and $K^\bullet$ a complex of $R$-modules. Let $m \in \mathbf{Z}$. Consider the $R$-algebra map $R[x] \to R$ which maps $x$ to zero. Then $K^\bullet$ is $m$-pseudo-coherent as a complex of $R$-modules if and only if $K^\bullet$ is $m$-pseudo-coherent as a complex of $R[x]$-modules.

Proof. This is a special case of Lemma 15.64.11. We also prove it in another way as follows.

Note that $0 \to R[x] \to R[x] \to R \to 0$ is exact. Hence $R$ is pseudo-coherent as an $R[x]$-module. Thus one implication of the lemma follows from Lemma 15.64.11. To prove the other implication, assume that $K^\bullet$ is $m$-pseudo-coherent as a complex of $R[x]$-modules. By Lemma 15.64.12 we see that $K^\bullet \otimes ^{\mathbf{L}}_{R[x]} R$ is $m$-pseudo-coherent as a complex of $R$-modules. By Lemma 15.81.1 we see that $K^\bullet \oplus K^\bullet $ is $m$-pseudo-coherent as a complex of $R$-modules. Finally, we conclude that $K^\bullet$ is $m$-pseudo-coherent as a complex of $R$-modules from Lemma 15.64.8. $\square$

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. $\square$

Definition 15.81.4. Let $R \to A$ be a finite type ring map. Let $K^\bullet$ be a complex of $A$-modules. Let $M$ be an $A$-module. Let $m \in \mathbf{Z}$.

1. We say $K^\bullet$ is $m$-pseudo-coherent relative to $R$ if the equivalent conditions of Lemma 15.81.3 hold.

2. We say $K^\bullet$ is pseudo-coherent relative to $R$ if $K^\bullet$ is $m$-pseudo-coherent relative to $R$ for all $m \in \mathbf{Z}$.

3. We say $M$ is $m$-pseudo-coherent relative to $R$ if $M$ is $m$-pseudo-coherent relative to $R$.

4. We say $M$ is pseudo-coherent relative to $R$ if $M$ is pseudo-coherent relative to $R$.

Part (2) means that $K^\bullet$ is pseudo-coherent as a complex of $R[x_1, \ldots , x_ n]$-modules for any surjection $R[y_1, \ldots , y_ m] \to A$, see Lemma 15.64.5. This definition has the following pleasing property.

Lemma 15.81.5. Let $R$ be a ring. Let $A \to B$ be a finite map of finite type $R$-algebras. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $B$-modules. Then $K^\bullet$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$ if and only if $K^\bullet$ seen as a complex of $A$-modules is $m$-pseudo-coherent (pseudo-coherent) relative to $R$.

Proof. Choose a surjection $R[x_1, \ldots , x_ n] \to A$. Choose $y_1, \ldots , y_ m \in B$ which generate $B$ over $A$. As $A \to B$ is finite each $y_ i$ satisfies a monic equation with coefficients in $A$. Hence we can find monic polynomials $P_ j(T) \in R[x_1, \ldots , x_ n][T]$ such that $P_ j(y_ j) = 0$ in $B$. Then we get a commutative diagram

$\xymatrix{ & R[x_1, \ldots , x_ n, y_1, \ldots , y_ m] \ar[d] \\ R[x_1, \ldots , x_ n] \ar[d] \ar[r] & R[x_1, \ldots , x_ n, y_1, \ldots , y_ m]/(P_ j(y_ j)) \ar[d] \\ A \ar[r] & B }$

The top horizontal arrow and the top right vertical arrow satisfy the assumptions of Lemma 15.64.11. Hence $K^\bullet$ is $m$-pseudo-coherent (resp. pseudo-coherent) as a complex of $R[x_1, \ldots , x_ n]$-modules if and only if $K^\bullet$ is $m$-pseudo-coherent (resp. pseudo-coherent) as a complex of $R[x_1, \ldots , x_ n, y_1, \ldots , y_ m]$-modules. $\square$

Lemma 15.81.6. Let $R$ be a ring. Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $(K^\bullet , L^\bullet , M^\bullet , f, g, h)$ be a distinguished triangle in $D(A)$.

1. If $K^\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$ and $L^\bullet$ is $m$-pseudo-coherent relative to $R$ then $M^\bullet$ is $m$-pseudo-coherent relative to $R$.

2. If $K^\bullet , M^\bullet$ are $m$-pseudo-coherent relative to $R$, then $L^\bullet$ is $m$-pseudo-coherent relative to $R$.

3. If $L^\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$ and $M^\bullet$ is $m$-pseudo-coherent relative to $R$, then $K^\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$.

Moreover, if two out of three of $K^\bullet , L^\bullet , M^\bullet$ are pseudo-coherent relative to $R$, the so is the third.

Proof. Follows immediately from Lemma 15.64.2 and the definitions. $\square$

Lemma 15.81.7. Let $R \to A$ be a finite type ring map. Let $M$ be an $A$-module. Then

1. $M$ is $0$-pseudo-coherent relative to $R$ if and only if $M$ is a finite type $A$-module,

2. $M$ is $(-1)$-pseudo-coherent relative to $R$ if and only if $M$ is a finitely presented relative to $R$,

3. $M$ is $(-d)$-pseudo-coherent relative to $R$ if and only if for every surjection $R[x_1, \ldots , x_ n] \to A$ there exists a resolution

$R[x_1, \ldots , x_ n]^{\oplus a_ d} \to R[x_1, \ldots , x_ n]^{\oplus a_{d - 1}} \to \ldots \to R[x_1, \ldots , x_ n]^{\oplus a_0} \to M \to 0$

of length $d$, and

4. $M$ is pseudo-coherent relative to $R$ if and only if for every presentation $R[x_1, \ldots , x_ n] \to A$ there exists an infinite resolution

$\ldots \to R[x_1, \ldots , x_ n]^{\oplus a_1} \to R[x_1, \ldots , x_ n]^{\oplus a_0} \to M \to 0$

by finite free $R[x_1, \ldots , x_ n]$-modules.

Proof. Follows immediately from Lemma 15.64.4 and the definitions. $\square$

Lemma 15.81.8. Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $K^\bullet , L^\bullet \in D(A)$. If $K^\bullet \oplus L^\bullet$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$ so are $K^\bullet$ and $L^\bullet$.

Proof. Immediate from Lemma 15.64.8 and the definitions. $\square$

Lemma 15.81.9. Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a bounded above complex of $A$-modules such that $K^ i$ is $(m - i)$-pseudo-coherent relative to $R$ for all $i$. Then $K^\bullet$ is $m$-pseudo-coherent relative to $R$. In particular, if $K^\bullet$ is a bounded above complex of $A$-modules pseudo-coherent relative to $R$, then $K^\bullet$ is pseudo-coherent relative to $R$.

Proof. Immediate from Lemma 15.64.9 and the definitions. $\square$

Lemma 15.81.10. Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $K^\bullet \in D^{-}(A)$ such that $H^ i(K^\bullet )$ is $(m - i)$-pseudo-coherent (resp. pseudo-coherent) relative to $R$ for all $i$. Then $K^\bullet$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$.

Proof. Immediate from Lemma 15.64.10 and the definitions. $\square$

Lemma 15.81.11. Let $R$ be a ring, $f \in R$ an element, $R_ f \to A$ is a finite type ring map, $g \in A$, and $K^\bullet$ a complex of $A$-modules. If $K^\bullet$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R_ f$, then $K^\bullet \otimes _ A A_ g$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$.

Proof. First we show that $K^\bullet$ is $m$-pseudo-coherent relative to $R$. Namely, suppose $R_ f[x_1, \ldots , x_ n] \to A$ is surjective. Write $R_ f = R[x_0]/(fx_0 - 1)$. Then $R[x_0, x_1, \ldots , x_ n] \to A$ is surjective, and $R_ f[x_1, \ldots , x_ n]$ is pseudo-coherent as an $R[x_0, \ldots , x_ n]$-module. Hence by Lemma 15.64.11 we see that $K^\bullet$ is $m$-pseudo-coherent as a complex of $R[x_0, x_1, \ldots , x_ n]$-modules.

Choose an element $g' \in R[x_0, x_1, \ldots , x_ n]$ which maps to $g \in A$. By Lemma 15.64.12 we see that

\begin{align*} K^\bullet \otimes _{R[x_0, x_1, \ldots , x_ n]}^{\mathbf{L}} R[x_0, x_1, \ldots , x_ n, \frac{1}{g'}] & = K^\bullet \otimes _{R[x_0, x_1, \ldots , x_ n]} R[x_0, x_1, \ldots , x_ n, \frac{1}{g'}] \\ & = K^\bullet \otimes _ A A_ f \end{align*}

is $m$-pseudo-coherent as a complex of $R[x_0, x_1, \ldots , x_ n, \frac{1}{g'}]$-modules. write

$R[x_0, x_1, \ldots , x_ n, \frac{1}{g'}] = R[x_0, \ldots , x_ n, x_{n + 1}]/(x_{n + 1}g' - 1).$

As $R[x_0, x_1, \ldots , x_ n, \frac{1}{g'}]$ is pseudo-coherent as a $R[x_0, \ldots , x_ n, x_{n + 1}]$-module we conclude (see Lemma 15.64.11) that $K^\bullet \otimes _ A A_ g$ is $m$-pseudo-coherent as a complex of $R[x_0, \ldots , x_ n, x_{n + 1}]$-modules as desired. $\square$

Lemma 15.81.12. Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $A$-modules which is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$. Let $R \to R'$ be a ring map such that $A$ and $R'$ are Tor independent over $R$. Set $A' = A \otimes _ R R'$. Then $K^\bullet \otimes _ A^{\mathbf{L}} A'$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R'$.

Proof. Choose a surjection $R[x_1, \ldots , x_ n] \to A$. Note that

$K^\bullet \otimes _ A^{\mathbf{L}} A' = K^\bullet \otimes _ R^{\mathbf{L}} R' = K^\bullet \otimes _{R[x_1, \ldots , x_ n]}^{\mathbf{L}} R'[x_1, \ldots , x_ n]$

by Lemma 15.61.2 applied twice. Hence we win by Lemma 15.64.12. $\square$

Lemma 15.81.13. Let $R \to A \to B$ be finite type ring maps. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $A$-modules. Assume $B$ as a $B$-module is pseudo-coherent relative to $A$. If $K^\bullet$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$, then $K^\bullet \otimes _ A^{\mathbf{L}} B$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$.

Proof. Choose a surjection $A[y_1, \ldots , y_ m] \to B$. Choose a surjection $R[x_1, \ldots , x_ n] \to A$. Combined we get a surjection $R[x_1, \ldots , x_ n, y_1, \ldots y_ m] \to B$. Choose a resolution $E^\bullet \to B$ of $B$ by a complex of finite free $A[y_1, \ldots , y_ n]$-modules (which is possible by our assumption on the ring map $A \to B$). We may assume that $K^\bullet$ is a bounded above complex of flat $A$-modules. Then

\begin{align*} K^\bullet \otimes _ A^{\mathbf{L}} B & = \text{Tot}(K^\bullet \otimes _ A B) \\ & = \text{Tot}(K^\bullet \otimes _ A A[y_1, \ldots , y_ m] \otimes _{A[y_1, \ldots , y_ m]} B) \\ & \cong \text{Tot}\left( (K^\bullet \otimes _ A A[y_1, \ldots , y_ m]) \otimes _{A[y_1, \ldots , y_ m]} E^\bullet \right) \\ & = \text{Tot}(K^\bullet \otimes _ A E^\bullet ) \end{align*}

in $D(A[y_1, \ldots , y_ m])$. The quasi-isomorphism $\cong$ comes from an application of Lemma 15.59.7. Thus we have to show that $\text{Tot}(K^\bullet \otimes _ A E^\bullet )$ is $m$-pseudo-coherent as a complex of $R[x_1, \ldots , x_ n, y_1, \ldots y_ m]$-modules. Note that $\text{Tot}(K^\bullet \otimes _ A E^\bullet )$ has a filtration by subcomplexes with successive quotients the complexes $K^\bullet \otimes _ A E^ i[-i]$. Note that for $i \ll 0$ the complexes $K^\bullet \otimes _ A E^ i[-i]$ have zero cohomology in degrees $\leq m$ and hence are $m$-pseudo-coherent (over any ring). Hence, applying Lemma 15.81.6 and induction, it suffices to show that $K^\bullet \otimes _ A E^ i[-i]$ is pseudo-coherent relative to $R$ for all $i$. Note that $E^ i = 0$ for $i > 0$. Since also $E^ i$ is finite free this reduces to proving that $K^\bullet \otimes _ A A[y_1, \ldots , y_ m]$ is $m$-pseudo-coherent relative to $R$ which follows from Lemma 15.81.12 for instance. $\square$

Lemma 15.81.14. Let $R \to A \to B$ be finite type ring maps. Let $m \in \mathbf{Z}$. Let $M$ be an $A$-module. Assume $B$ is flat over $A$ and $B$ as a $B$-module is pseudo-coherent relative to $A$. If $M$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$, then $M \otimes _ A B$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$.

Proof. Immediate from Lemma 15.81.13. $\square$

Lemma 15.81.15. Let $R$ be a ring. Let $A \to B$ be a map of finite type $R$-algebras. Let $m \in \mathbf{Z}$. Let $K^\bullet$ be a complex of $B$-modules. Assume $A$ is pseudo-coherent relative to $R$. Then the following are equivalent

1. $K^\bullet$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $A$, and

2. $K^\bullet$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$.

Proof. Choose a surjection $R[x_1, \ldots , x_ n] \to A$. Choose a surjection $A[y_1, \ldots , y_ m] \to B$. Then we get a surjection

$R[x_1, \ldots , x_ n, y_1, \ldots , y_ m] \to A[y_1, \ldots , y_ m]$

which is a flat base change of $R[x_1, \ldots , x_ n] \to A$. By assumption $A$ is a pseudo-coherent module over $R[x_1, \ldots , x_ n]$ hence by Lemma 15.64.13 we see that $A[y_1, \ldots , y_ m]$ is pseudo-coherent over $R[x_1, \ldots , x_ n, y_1, \ldots , y_ m]$. Thus the lemma follows from Lemma 15.64.11 and the definitions. $\square$

Lemma 15.81.16. Let $R \to A$ be a finite type ring map. Let $K^\bullet$ be a complex of $A$-modules. Let $m \in \mathbf{Z}$. Let $f_1, \ldots , f_ r \in A$ generate the unit ideal. The following are equivalent

1. each $K^\bullet \otimes _ A A_{f_ i}$ is $m$-pseudo-coherent relative to $R$, and

2. $K^\bullet$ is $m$-pseudo-coherent relative to $R$.

The same equivalence holds for pseudo-coherence relative to $R$.

Proof. The implication (2) $\Rightarrow$ (1) is in Lemma 15.81.11. Assume (1). Write $1 = \sum f_ ig_ i$ in $A$. Choose a surjection $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r] \to A$. such that $y_ i$ maps to $f_ i$ and $z_ i$ maps to $g_ i$. Then we see that there exists a surjection

$P = R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r]/(\sum y_ iz_ i - 1) \longrightarrow A.$

Note that $P$ is pseudo-coherent as an $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r]$-module and that $P[1/y_ i]$ is pseudo-coherent as an $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r, 1/y_ i]$-module. Hence by Lemma 15.64.11 we see that $K^\bullet \otimes _ A A_{f_ i}$ is an $m$-pseudo-coherent complex of $P[1/y_ i]$-modules for each $i$. Thus by Lemma 15.64.14 we see that $K^\bullet$ is pseudo-coherent as a complex of $P$-modules, and Lemma 15.64.11 shows that $K^\bullet$ is pseudo-coherent as a complex of $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r]$-modules. $\square$

Lemma 15.81.17. Let $R$ be a Noetherian ring. Let $R \to A$ be a finite type ring map. Then

1. A complex of $A$-modules $K^\bullet$ is $m$-pseudo-coherent relative to $R$ if and only if $K^\bullet \in D^{-}(A)$ and $H^ i(K^\bullet )$ is a finite $A$-module for $i \geq m$.

2. A complex of $A$-modules $K^\bullet$ is pseudo-coherent relative to $R$ if and only if $K^\bullet \in D^{-}(A)$ and $H^ i(K^\bullet )$ is a finite $A$-module for all $i$.

3. An $A$-module is pseudo-coherent relative to $R$ if and only if it is finite.

Proof. Immediate consequence of Lemma 15.64.17 and the definitions. $\square$

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