Definition 10.7.1. Let \varphi : R \to S be a ring map. We say \varphi : R \to S is finite if S is finite as an R-module.
10.7 Finite ring maps
Here is the definition.
Lemma 10.7.2. Let R \to S be a finite ring map. Let M be an S-module. Then M is finite as an R-module if and only if M is finite as an S-module.
Proof. One of the implications follows from Lemma 10.5.5. To see the other assume that M is finite as an S-module. Pick x_1, \ldots , x_ n \in S which generate S as an R-module. Pick y_1, \ldots , y_ m \in M which generate M as an S-module. Then x_ i y_ j generate M as an R-module. \square
Lemma 10.7.3. Suppose that R \to S and S \to T are finite ring maps. Then R \to T is finite.
Proof. If t_ i generate T as an S-module and s_ j generate S as an R-module, then t_ i s_ j generate T as an R-module. (Also follows from Lemma 10.7.2.) \square
Lemma 10.7.4. Let \varphi : R \to S be a ring map.
If \varphi is finite, then \varphi is of finite type.
If S is of finite presentation as an R-module, then \varphi is of finite presentation.
Proof. For (1) if x_1, \ldots , x_ n \in S generate S as an R-module, then x_1, \ldots , x_ n generate S as an R-algebra. For (2), suppose that \sum r_ j^ ix_ i = 0, j = 1, \ldots , m is a set of generators of the relations among the x_ i when viewed as R-module generators of S. Furthermore, write 1 = \sum r_ ix_ i for some r_ i \in R and x_ ix_ j = \sum r_{ij}^ k x_ k for some r_{ij}^ k \in R. Then
S = R[t_1, \ldots , t_ n]/ (\sum r_ j^ it_ i,\ 1 - \sum r_ it_ i,\ t_ it_ j - \sum r_{ij}^ k t_ k)
as an R-algebra which proves (2). \square
For more information on finite ring maps, please see Section 10.36.
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