arXiv:1503.02876v8 [] 19 Mar 2018

ON FINITE TYPE EPIMORPHISMS OF RINGS
ABOLFAZL TARIZADEH
Abstract. In this note, finite type epimorphisms of rings are
characterized.

1. Introduction
In this paper we investigate an important class of epimorphisms of
rings, namely finite type epimorphisms. Needless to say that finite type
ring maps have very closed connections with the geometric structures
and essentially originate from algebraic geometry. Meanwhile, finite
type epimorphisms of commutative rings have very interesting properties and they are important from various aspects. For example, “finite
type monomorphisms of schemes” can be considered as the geometric
interpretation of finite type epimorphisms of rings.
In the realm of epimorphisms of commutative rings there are some
highly non-trivial results in the literature. For instance, “every finite
type epimorphism of rings which is also injective and flat then it is of
finite presentation”, see [2, Theorem 1.1]. A special case of this result
was announced in [7, Corollary 3.4.7]. Another important result is due
to Lazard, see Theorem 2.2. In the present article, partially motivated
by the above results, we have obtained two new and non-trivial results.
In fact, Theorems 3.2 and 3.3 are the main results of this note. These
results can be considered as the analogous of Theorem 2.2 in the finite
type case whose hypotheses have been as much as possible weakened.
In this article, all of the rings are commutative.

2. preliminaries
Here we recall some material which are needed in the next section.
0

2010 Mathematics Subject Classification: 13B10, 13B30, 13A99, 13C11.
Key words and phrases: finite type epimorphism; module of differentials; universally
injective.
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ABOLFAZL TARIZADEH

By an epimorphism ϕ : R → S we mean it is an epimorphism in
the category of (commutative) rings. Surjective ring maps are special
cases of epimorphisms. As a specific example, the canonical ring map
Z → Q is an epimorphism of rings while it is not surjective.
Remark 2.1. It is easy to see that a ring map ϕ : R → S is an epimorphism of rings if and only if s ⊗ 1 = 1 ⊗ s for all s ∈ S. It is also
equivalent to the condition that S ⊗R Coker ϕ = 0. It follows that every
faithfully flat epimorphism of rings is an isomorphism. In particular,
every non-zero epimorphism of rings with source a field is an isomorphism. We refer to [8], specially [4], [5] and [6] for a comprehensive
discussion of epimorphisms of commutative rings. Also see [1].
Theorem 2.2. A ring map ϕ : R → S is an epimorphism of rings if
and only if the following conditions hold.
(i) The induced map ϕ∗ : Spec(S) → Spec(R) is injective.
(ii) For each prime ideal q of S the induced map κ(p) → κ(q) is an
isomorphism where p = ϕ∗ (q).
(iii) The kernel of the canonical ring map S ⊗R S → S is a finitely
generated ideal.
(iv) The module of Kähler differentials ΩS/R is zero.
Proof. See [4, Proposition 1.5]. 
Remark 2.3. It is well-known that if ϕ : R → A is a ring map and p
a prime ideal of R then p ∈ Im ϕ∗ if and only if A ⊗R κ(p) 6= 0 where
κ(p) is the residue field of R at p.
Definition 2.4. The map ϕ∗ : Spec(S) → Spec(R) induced by a ring
map ϕ : R → S is said to be universally injective if for any ring map
R → R′ then the induced map ψ ∗ : Spec(R′ ⊗R S) → Spec(R) is injective where ψ : R′ → R′ ⊗R S is the base change map.

3. Main results
Let ϕ : R → S be a ring map. Consider the canonical ring map
π : S ⊗R S → S which maps each pure tensor s ⊗ s′ of S ⊗R S into ss′ .
The kernel of π is generated by elements of the form s⊗1−1⊗s. Because

ON FINITE TYPE EPIMORPHISMS OF RINGS

if

P
i

si s′i = 0 then we may write

P
i

si ⊗ s′i =

P
i

3

(1 ⊗ s′i )(si ⊗ 1 − 1 ⊗ si).

To prove the main results of this note we need the following lemma.
Lemma 3.1. If a ring map ϕ : R → S is of finite type then Ker π is a
finitely generated ideal.
Proof. By the hypothesis there are elements s1 , ..., sn ∈ S such that
S = R[s1 , ..., sn ]. Consider the ideal J = (si ⊗ 1 − 1 ⊗ si : 1 ≤ i ≤ n) of
S ⊗R S. Clearly J ⊆ Ker π. To prove the reverse inclusion it suffices
to show that
sd11 ...sdnn ⊗ 1 − 1 ⊗ sd11 ...sdnn ∈ J.
We use an induction argument over n. If n = 1 then we have sd ⊗ 1 −
1 ⊗ sd = (s
⊗ 1)d − (1 ⊗ s)d = (s ⊗ 1)d−1 + (s ⊗ 1)d−2 (1 ⊗ s) + ... +

(1 ⊗ s)d−1 (s ⊗ 1 − 1 ⊗ s) which belongs to J. Let n > 1. Then we
dn−1
may write sd11 ...sdnn ⊗ 1 − 1 ⊗ sd11 ...sdnn = sdnn ⊗ 1 sd11 ...sn−1
⊗1−1⊗


dn−1
dn−1
d1
d1
dn
dn
s1 ...sn−1 + 1 ⊗ s1 ...sn−1 sn ⊗ 1 − 1 ⊗ sn which is, by the induction
hypothesis and the induction step, belonging to J. 
Theorem 3.2. A finite type ring map ϕ : R → S is an epimorphism of
rings if and only if the induced map ϕ∗ is injective and for each prime
ideal q of S the base change map κ(p) → Sq ⊗R κ(p) is an isomorphism
where p = ϕ∗ (q).
Proof. “⇒” By Theorem 2.2, ϕ∗ is injective. The composition
of ϕ with the canonical ring map S → Sq gives us the epimorphism
R → Sq . Thus the map κ(p) → Sq ⊗R κ(p) is an epimorphism since
every epimorphism of rings is stable under the base change. By Remark
2.3, Sq ⊗R κ(p) 6= 0 and so by Remark 2.1 the base change map is an
isomorphism. To prove the converse implication we shall use Theorem
2.2. We have Sq ⊗R κ(p) ≃ Sq /pSq . Thus Sq /pSq is a field and so
pSq = qSq . Hence the induced map κ(p) → κ(q) is an isomorphism.
Moreover, by [3, Tag 00RV], we have
ΩSq /R ⊗R κ(p) ≃ ΩSq ⊗R κ(p)/κ(p) = 0.
It follows that (qSq )ΩSq /R = ΩSq /R . By [3, Tag 00RZ], ΩS/R is finitely
generated S−module. Therefore (ΩS/R )q ≃ ΩSq /R is a finitely generated Sq −module. Then Nakayama implies that ΩSq /R = 0. It follows
that ΩS/R = 0 because (ΩS/R )q ≃ ΩSq /R = 0. Now using Lemma 3.1

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ABOLFAZL TARIZADEH

and Theorem 2.2 then we conclude that ϕ is an epimorphism. 

Theorem 3.3. Let ϕ : R → S be a finite type ring map. Then the
following conditions are equivalent.
(i) ϕ is an epimorphism of rings.
(ii) ϕ∗ is universally injective and ΩS/R = 0.
(iii) ϕ is formally unramified and for any field K and any ring maps
f, g : S → K if f ◦ ϕ = g ◦ ϕ then f = g.
(iv) ϕ∗ is injective, ΩS/R = 0 and for each prime ideal q of S the field
extension κ(p) ⊆ κ(q) is purely inseparable where p = ϕ∗ (q).

Proof. (i) ⇒ (ii) : Epimorphisms are stable under the base change.
Then apply Theorem 2.2.
(ii) ⇒ (iii) : The ring map ϕ is formally unramified if and only if
ΩS/R = 0, see [3, Tag 00UO]. There exist ring maps f ′ , g ′ : K ⊗R S → K
which map each pure tensor a ⊗ s of K ⊗R S into af (s) and ag(s), respectively. Let ψ : K → K ⊗R S be the base change of f ◦ ϕ : R → K.
Then, by the hypotheses, ψ ∗ is injective. Moreover K ⊗R S is a nontrivial ring. Therefore the prime spectrum of K ⊗R S is a single-point
set. This implies that
(f ′ )−1 (0) = (g ′)−1 (0). But for each s ∈ S,

′
f f (s) ⊗ 1 − 1 ⊗ s = 0. Thus g ′ f (s) ⊗ 1 − 1 ⊗ s = 0. Therefore
f (s) = g(s) for all s ∈ S.
(iii) ⇒ (i) : Let p be a prime ideal of S ⊗R S. The hypotheses imply
that η ◦ i = η ◦ j where η : S ⊗R S → κ(p) and i, j : S → S ⊗R S
are the canonical ring maps. Therefore for each s ∈ S, s ⊗ 1 − 1 ⊗ s
is a nilpotent element. Let J be the kernel of the canonical ring map
S ⊗R S → S. By Lemma 3.1, J is a finitely generated ideal. It follows
that J is a nilpotent ideal. But ΩS/R ≃ J/J 2 . Thus J = J 2 and so
J = 0. Therefore s ⊗ 1 = 1 ⊗ s for all s ∈ S. Hence ϕ is an epimorphism. (i) ⇒ (iv) : See Theorem 2.2.
(iv) ⇒ (iii) : We have f −1 (0) = g −1 (0) since f ◦ ϕ = g ◦ ϕ and ϕ∗ is
injective. Let q = f −1 (0). For each s ∈ S, by the hypotheses, there is
n
a natural number n ≥ 0 such that (s/1 + qSq )p is in the image of the
induced map κ(p) → κ(q) where p = ϕ∗ (q) and p is the characteristic
of κ(p). Therefore there are elements r ∈ R and t ∈ R \ p such that
n
n
n
ϕ(t)sp − ϕ(r) ∈ q. Thus f (s)p = (f ◦ ϕ)(r)(f ◦ ϕ)(t)−1 = g(s)p . This
n
p
= 0 since the characteristic of K is equal
implies that f (s) − g(s)
to p. Therefore f (s) = g(s) for all s ∈ S. 

ON FINITE TYPE EPIMORPHISMS OF RINGS

5

References
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Algebra, Volume 32, Issue 1, 1974, p. 44-50.
[3] Aise Johan de Jong et al. Stacks Project, see http://stacks.math.columbia.edu.
[4] Lazard, Daniel. Épimorphismes plats, Séminaire Samuel. Algèbre commutative, tomme 2 (1967-1968).
[5] Olivier, Jean-Pierre. Anneaux absolument plats universels et épimorphismes à
buts réduits, Séminaire Samuel. Algèbre commutative, tomme 2 (1967-1968).
[6] Roby, Norbert. Diverses caractérisations des épimorphismes, Séminaire
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[7] Raynaud, M. and Gruson, L. Critères de platitude et de projectivité, Inventiones Mathematicae, 13 (1971), 1-89.
[8] Séminaire Samuel. Les épimorphismes d’anneaux, 2, 1967-1968, Paris,
Secrétariat mathématique, 1968.
Department of Mathematics, Faculty of Basic Sciences, University
of Maragheh, P. O. Box 55136-553, Maragheh, Iran.
E-mail address: ebulfez1978@gmail.com

