The Stacks project

86.11 Finite type morphisms

In Formal Spaces, Section 85.20 we have defined finite type morphisms of formal algebraic spaces. In this section we study the corresponding types of continuous ring maps of adic topological rings which have a finitely generated ideal of definition. We strongly suggest the reader skip this section.

Lemma 86.11.1. Let $A$ and $B$ be adic topological rings which have a finitely generated ideal of definition. Let $\varphi : A \to B$ be a continuous ring homomorphism. The following are equivalent:

  1. $\varphi $ is adic and $B$ is topologically of finite type over $A$,

  2. $\varphi $ is taut and $B$ is topologically of finite type over $A$,

  3. there exists an ideal of definition $I \subset A$ such that the topology on $B$ is the $I$-adic topology and there exist an ideal of definition $I' \subset A$ such that $A/I' \to B/I'B$ is of finite type,

  4. for all ideals of definition $I \subset A$ the topology on $B$ is the $I$-adic topology and $A/I \to B/IB$ is of finite type,

  5. there exists an ideal of definition $I \subset A$ such that the topology on $B$ is the $I$-adic topology and $B$ is in the category (86.2.0.2),

  6. for all ideals of definition $I \subset A$ the topology on $B$ is the $I$-adic topology and $B$ is in the category (86.2.0.2),

  7. $B$ as a topological $A$-algebra is the quotient of $A\{ x_1, \ldots , x_ r\} $ by a closed ideal,

  8. $B$ as a topological $A$-algebra is the quotient of $A[x_1, \ldots , x_ r]^\wedge $ by a closed ideal where $A[x_1, \ldots , x_ r]^\wedge $ is the completion of $A[x_1, \ldots , x_ r]$ with respect to some ideal of definition of $A$, and

  9. add more here.

Moreover, these equivalent conditions define a local property of morphisms of $\text{WAdm}^{adic*}$ as defined in Formal Spaces, Remark 85.17.4.

Proof. Taut ring homomorphisms are defined in Formal Spaces, Definition 85.4.11. Adic ring homomorphisms are defined in Formal Spaces, Definition 85.19.1. The lemma follows from a combination of Formal Spaces, Lemmas 85.25.6, 85.25.7, and 85.19.2. We omit the details. To be sure, there is no difference between the topological rings $A[x_1, \ldots , x_ n]^\wedge $ and $A\{ x_1, \ldots , x_ r\} $, see Formal Spaces, Remark 85.24.2. $\square$

Remark 86.11.2. Let $A \to B$ be an arrow of $\text{WAdm}^{adic*}$ which is adic and topologically of finite type (see Lemma 86.11.1). Write $B = A\{ x_1, \ldots , x_ r\} /J$. Then we can set1

\[ \mathop{N\! L}\nolimits _{B/A}^\wedge = \left(J/J^2 \longrightarrow \bigoplus B\text{d}x_ i\right) \]

Exactly as in the proof of Lemma 86.3.1 the reader can show that this complex of $B$-modules is well defined up to (unique isomorphism) in the homotopy category $K(B)$. Now, if $A$ is Noetherian and $I \subset A$ is an ideal of definition, then this construction reproduces the naive cotangent complex of $B$ over $(A, I)$ defined by Equation (86.3.0.1) in Section 86.3 simply because $A[x_1, \ldots , x_ n]^\wedge $ agrees with $A\{ x_1, \ldots , x_ r\} $ by Formal Spaces, Remark 85.24.2. In particular, we find that, still when $A$ is an adic Noetherian topological ring, the object $\mathop{N\! L}\nolimits _{B/A}^\wedge $ is independent of the choice of the ideal of definition $I \subset A$.

Lemma 86.11.3. Consider the property $P$ on arrows of $\textit{WAdm}^{adic*}$ defined in Lemma 86.11.1. Then $P$ is stable under base change as defined in Formal Spaces, Remark 85.17.8.

Proof. The statement makes sense by Lemma 86.11.1. To see that it is true assume we have morphisms $B \to A$ and $B \to C$ in $\textit{WAdm}^{adic*}$ and that as a topological $B$-algebra we have $A = B\{ x_1, \ldots , x_ r\} /J$ for some closed ideal $J$. Then $A \widehat{\otimes }_ B C$ is isomorphic to the quotient of $C\{ x_1, \ldots , x_ r\} /J'$ where $J'$ is the closure of $JC\{ x_1, \ldots , x_ r\} $. Some details omitted. $\square$

Lemma 86.11.4. Consider the property $P$ on arrows of $\textit{WAdm}^{adic*}$ defined in Lemma 86.11.1. Then $P$ is stable under composition as defined in Formal Spaces, Remark 85.17.13.

Proof. The statement makes sense by Lemma 86.11.1. The easiest way to prove it is true is to show that (a) compositions of adic ring maps between adic topological rings are adic and (b) that compositions of continuous ring maps preserves the property of being topologically of finite type. We omit the details. $\square$

The following lemma says that morphisms of adic* formal algebraic spaces are locally of finite type if and only if they are étale locally given by the types of maps of topological rings described in Lemma 86.11.1.

Lemma 86.11.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally adic* formal algebraic spaces over $S$. The following are equivalent

  1. for every commutative diagram

    \[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

    with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and étale, the morphism $U \to V$ corresponds to an arrow of $\textit{WAdm}^{adic*}$ which is adic and topologically of finite type,

  2. there exists a covering $\{ Y_ j \to Y\} $ as in Formal Spaces, Definition 85.7.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Formal Spaces, Definition 85.7.1 such that each $X_{ji} \to Y_ j$ corresponds to an arrow of $\textit{WAdm}^{adic*}$ which is adic and topologically of finite type,

  3. there exist a covering $\{ X_ i \to X\} $ as in Formal Spaces, Definition 85.7.1 and for each $i$ a factorization $X_ i \to Y_ i \to Y$ where $Y_ i$ is an affine formal algebraic space, $Y_ i \to Y$ is representable by algebraic spaces and étale, and $X_ i \to Y_ i$ corresponds to an arrow of $\textit{WAdm}^{adic*}$ which is adic and topologically of finite type, and

  4. $f$ is locally of finite type.

Proof. Immediate consequence of the equivalence of (1) and (2) in Lemma 86.11.1 and Formal Spaces, Lemma 85.25.9. $\square$

[1] In fact, this construction works for arrows of $\text{WAdm}^{count}$ satisfying the equivalent conditions of Formal Spaces, Lemma 85.25.6.

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