The Stacks project

15.75 Pseudo-coherent and perfect ring maps

We can define these types of ring maps as follows.

Definition 15.75.1. Let $A \to B$ be a ring map.

  1. We say $A \to B$ is a pseudo-coherent ring map if it is of finite type and $B$, as a $B$-module, is pseudo-coherent relative to $A$.

  2. We say $A \to B$ is a perfect ring map if it is a pseudo-coherent ring map such that $B$ as an $A$-module has finite tor dimension.

This terminology may be nonstandard. Using Lemma 15.74.7 we see that $A \to B$ is pseudo-coherent if and only if $B = A[x_1, \ldots , x_ n]/I$ and $B$ as an $A[x_1, \ldots , x_ n]$-module has a resolution by finite free $A[x_1, \ldots , x_ n]$-modules. The motivation for the definition of a perfect ring map is Lemma 15.69.2. The following lemmas gives a more useful and intuitive characterization of a perfect ring map.

Lemma 15.75.2. A ring map $A \to B$ is perfect if and only if $B = A[x_1, \ldots , x_ n]/I$ and $B$ as an $A[x_1, \ldots , x_ n]$-module has a finite resolution by finite projective $A[x_1, \ldots , x_ n]$-modules.

Proof. If $A \to B$ is perfect, then $B = A[x_1, \ldots , x_ n]/I$ and $B$ is pseudo-coherent as an $A[x_1, \ldots , x_ n]$-module and has finite tor dimension as an $A$-module. Hence Lemma 15.71.8 implies that $B$ is perfect as a $A[x_1, \ldots , x_ n]$-module, i.e., it has a finite resolution by finite projective $A[x_1, \ldots , x_ n]$-modules (Lemma 15.69.3). Conversely, if $B = A[x_1, \ldots , x_ n]/I$ and $B$ as an $A[x_1, \ldots , x_ n]$-module has a finite resolution by finite projective $A[x_1, \ldots , x_ n]$-modules then $B$ is pseudo-coherent as an $A[x_1, \ldots , x_ n]$-module, hence $A \to B$ is pseudo-coherent. Moreover, the given resolution over $A[x_1, \ldots , x_ n]$ is a finite resolution by flat $A$-modules and hence $B$ has finite tor dimension as an $A$-module. $\square$

Lots of the results of the preceding sections can be reformulated in terms of this terminology. We also refer to More on Morphisms, Sections 36.52 and 36.53 for the corresponding discussion concerning morphisms of schemes.

Lemma 15.75.3. A finite type ring map of Noetherian rings is pseudo-coherent.

Lemma 15.75.4. A ring map which is flat and of finite presentation is perfect.

Proof. Let $A \to B$ be a ring map which is flat and of finite presentation. It is clear that $B$ has finite tor dimension. By Algebra, Lemma 10.162.1 there exists a finite type $\mathbf{Z}$-algebra $A_0 \subset A$ and a flat finite type ring map $A_0 \to B_0$ such that $B = B_0 \otimes _{A_0} A$. By Lemma 15.74.17 we see that $A_0 \to B_0$ is pseudo-coherent. As $A_0 \to B_0$ is flat we see that $B_0$ and $A$ are tor independent over $A_0$, hence we may use Lemma 15.74.12 to conclude that $A \to B$ is pseudo-coherent. $\square$

Lemma 15.75.5. Let $A \to B$ be a finite type ring map with $A$ a regular ring of finite dimension. Then $A \to B$ is perfect.

Proof. By Algebra, Lemma 10.109.8 the assumption on $A$ means that $A$ has finite global dimension. Hence every module has finite tor dimension, see Lemma 15.63.19, in particular $B$ does. By Lemma 15.75.3 the map is pseudo-coherent. $\square$

Lemma 15.75.6. A local complete intersection homomorphism is perfect.

Proof. Let $A \to B$ be a local complete intersection homomorphism. By Definition 15.32.2 this means that $B = A[x_1, \ldots , x_ n]/I$ where $I$ is a Koszul ideal in $A[x_1, \ldots , x_ n]$. By Lemmas 15.75.2 and 15.69.3 it suffices to show that $I$ is a perfect module over $A[x_1, \ldots , x_ n]$. By Lemma 15.69.11 this is a local question. Hence we may assume that $I$ is generated by a Koszul-regular sequence (by Definition 15.31.1). Of course this means that $I$ has a finite free resolution and we win. $\square$

Lemma 15.75.7. Let $R \to A$ be a pseudo-coherent ring map. Let $K \in D(A)$. The following are equivalent

  1. $K$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$, and

  2. $K$ is $m$-pseudo-coherent (resp. pseudo-coherent) in $D(A)$.

Proof. Reformulation of a special case of Lemma 15.74.15. $\square$

Lemma 15.75.8. Let $R \to B \to A$ be ring maps with $\varphi : B \to A$ surjective and $R \to B$ and $R \to A$ flat and of finite presentation. For $K \in D(A)$ denote $\varphi _*K \in D(B)$ the restriction. The following are equivalent

  1. $K$ is pseudo-coherent,

  2. $K$ is pseudo-coherent relative to $R$,

  3. $K$ is pseudo-coherent relative to $A$,

  4. $\varphi _*K$ is pseudo-coherent,

  5. $\varphi _*K$ is pseudo-coherent relative to $R$.

Similar holds for $m$-pseudo-coherence.

Proof. Observe that $R \to A$ and $R \to B$ are perfect ring maps (Lemma 15.75.4) hence a fortiori pseudo-coherent ring maps. Thus (1) $\Leftrightarrow $ (2) and (4) $\Leftrightarrow $ (5) by Lemma 15.75.7.

Using that $A$ is pseudo-coherent relative to $R$ we use Lemma 15.74.15 to see that (2) $\Leftrightarrow $ (3). However, since $A \to B$ is surjective, we see directly from Definition 15.74.4 that (3) is equivalent with (4). $\square$


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