DESINGULARIZATION OF REGULAR ALGEBRAS
MOHSEN ASGHARZADEH

arXiv:1310.1862v2 [] 14 Aug 2017

Abstract. We identify families of commutative rings that can be written as a direct limit of a directed
system of noetherian regular rings and investigate the homological properties of such rings.

1. Introduction
The goal of this work is to identify rings R that can be realized as a direct limit of a directed system
{Ri : i ∈ Γ} of noetherian regular rings (which we then call a densingularization of R), and to investigate
the homological properties of such an R. We emphasize that the poset Γ is filtered. A paradigm for this,
and one of the motivation for this work, is a result of Zariski [29] (and Popescu [24]):
Theorem 1.1. (Zariski-Popescu) Let (V, m) be a valuation domain containing a field k of zero characteristic. Then V has a densingularization.
It may be interesting to mention that the construction of densingularizations goes back to Akizuki [1]
and Nagata [20]. Recall from [6] that a ring is said to be regular, if each finitely generated ideal has finite
projective dimension. A ring is called coherent, if its finitely generated ideals are finitely presented. Our
first result in Section 2 is:
Proposition 1.2. Let R be a ring that has a desingularization and p a finitely generated prime ideal in
R. If Rp is coherent, then Rp is regular.
Also, Section 2 is devoted to computing the homological dimensions of an ideal I of a ring with a
desingularization {Ri : i ∈ Γ}. We do this by imposing some additional assumptions both on the ideal I,
the rings Ri and the poset Γ.
A quasilocal ring is a ring with a unique maximal ideal. A local ring is a noetherian quasilocal ring.
There are many definitions for the regularity condition in non-noetherian rings (see e.g. [15]). One of
these is the notion of super regularity. This notion was first introduced by Vasconcelos [28]. A coherent
quasilocal ring is called super regular if its global dimension is finite and equal to its weak dimension.
Section 3 deals with a desingularization of super regular rings. Our first result in this direction is
Proposition 3.3:
Proposition 1.3. Let {(Ri , mi )} be a directed system of local rings with the property that m2i = mi ∩m2i+1 .
If R := limRi is coherent and super regular, then each Ri is regular.
−→
We present a nice application of the notion of super regularity: we compute the global dimension of
certain perfect algebras. To this end, suppose R is a complete local domain which is not a field and
suppose that its perfect closure R∞ is coherent. In Proposition 3.4 we show that
gl. dim(R∞ ) = dim R + 1.
2010 Mathematics Subject Classification. Primary 13H05; Secondary 18A30.
Key words and phrases. Coherent rings; direct limits; homological dimensions; perfect rings; regular rings.

1

2

Let {Ri } be a pure directed system of local rings and suppose that the maximal ideal of R := lim Ri
−→
has a finite free resolution. In Proposition 4.1, we show R is noetherian and regular. Let F be a finite
Q
field. In Proposition 5.3 we present the desingularization of N F . This has some applications. For
Q
example, N F is stably coherent.
We cite [13] as a reference book on commutative coherent rings.
2. Homological properties of a desingularization
We start by introducing some notation. By p. dimR (−) (resp. fl. dimR (−)), we mean projective
dimension (resp. flat dimension) of an R-module. Denote the ith Koszul homology module of R with
respect to x := x1 , . . . , xn by Hi (x; R).
Remark 2.1. Let {Ri : i ∈ Γ} be a directed system of rings and let x := x1 , . . . , xn be in R := limRi . Let
−→
i0 be such that x ⊂ Ri0 . Then limi≥i H• (x, Ri ) ≃ H• (x, R).
−→ 0
Proof. This is straightforward and we leave it to the reader.



Definition 2.2. Let (R, m) be a quasilocal ring. Suppose m is generated by a finite sequence of elements
x := x1 , . . . , xn . Recall from Kabele [15] that R is H1 -regular, if H1 (x, R) = 0. Also, R is called Koszul
regular, if Hi (x, R) = 0 for all i > 0.
In general, H1 -regular rings are not Koszul regular, see [15].
Lemma 2.3. Let (R, m) be a coherent H1 -regular ring. Then R is Koszul regular.
Proof. Coherence regular local rings are integral domains. Recall from Definition 2.2 that m is finitely
generated. Let x := x1 , . . . , xn be a generating set for m. Since R is coherent and in view of [3, Lemma
3.7], the R-module Hi (y, R) is finitely generated, where y is a finite sequence of elements. By using basic
properties of Koszul homologies and by an easy induction one may show that Hi (x1 , . . . , xj ; R) = 0 for
all i > 0 and all j. We left the routine details to the reader, please see [19, Page 127-128 ].



Proposition 2.4. Let R be a ring with a desingularization and let p ∈ Spec(R) be finitely generated. If
Rp is coherent, then Rp is regular.
Proof. Let {Ri : i ∈ Γ} be a directed system of noetherian regular rings such that R := limRi . To
−→
simplify the notation, we replace Rp with (R, m) and (Ri )p∩Ri with (Ri , mi ). In view of the natural
isomorphism Rp ≃ lim(Ri )p∩Ri , we may do such a replacement. Let x := x1 , . . . , xn be a generating set
−→
for m. Without loss of the generality, we can assume that xi ∈ Rj for all i and all j. Set Ai := Ri /(x).
In the light of [17, Lemma 2.5.1],
0 → H2 (Ri , Ai , Ai ) → H1 (x, Ri ) → Ani → (x)/(x)2 → 0,
where H∗ (−, −, −) is the André-Quillen homology. By Remark 2.1, Koszul homology behaves well with
respect to direct limits. Recall from [17, Proposition 1.4.8] that André-Quillen homology behaves well
with respect to direct limits. These induce the following exact sequence
π

0 −→ H2 (R, k, k) −→ H1 (x, R) −→ Rn /mRn −→ m/m2 −→ 0,
where the map π induced from the natural surjective homomorphism that sends the canonical basis of
Ani to x. We view π as a surjective map of finite dimensional vector spaces with the same dimension. In
particular, π is an isomorphism.

3

Recall that k (resp. ki ) is the residue field of R (resp. Ri ). In the light of [17, Proposition 1.4.8,
Corollary 2.5.3], H2 (R, k, k) ≃ limH2 (Ri , ki , ki ) = 0. Thus H1 (x, R) = 0, because π is an isomorphism.
−→
Due to Lemma 2.3, fl. dimR (k) < ∞. Again, as R is coherent and in view of [13, Corollary 2.5.10], any
finitely generated ideal of R has finite projective dimension, i.e., R is regular.



By w. dim(R), we mean the weak dimension of R. By definition
w. dim(R) := sup{fl. dim(M ) : M is an R-module},
see [13, Page 20].
Lemma 2.5. Let {Ri : i ∈ Γ} be a directed system of rings such that their weak dimension is bounded
by an integer n. Set R := limRi . The following assertions hold:
−→
(i) The flat dimension of an R-module is bounded above by n.
(ii) If R is coherent, then projective dimension of any finitely presented R-module is bounded above
by n.
i
Proof. (i): Let M and N be two R-modules. By [12, VI, Exercise 17], TorR
TorR
j (M, N ) ≃ lim
j (M, N ),
−→
i
which is zero for all j > n and this is the thing that we search for.

(ii): This follows by [26, Corollary 11.5].



Let a be an ideal of a ring R and M an R-module. Let Σ be the family of all finitely generated
subideals b of a. The Koszul grade of a finitely generated ideal a := (x1 , . . . , xn ) on M is defined by
K. gradeR (a, M ) := inf{i ∈ N ∪ {0}|H i (HomR (K• (x), M )) 6= 0}.
Note that by [9, Corollary 1.6.22] and [9, Proposition 1.6.10 (d)], this does not depend on the choice of
generating sets of a. For an ideal a (not necessarily finitely generated), Koszul grade of a on M can be
defined by K. gradeR (a, M ) := sup{K. gradeR (b, M ) : b ∈ Σ}. By using [9, Proposition 9.1.2 (f)], this
definition coincides with the original definition for finitely generated ideals.
Corollary 2.6. Let {Ri : i ∈ Γ} be a directed system of coherent regular quasilocal rings such that their
Krull dimension is bounded by an integer. Suppose each Ri is noetherian and Γ is countable, or R is
coherent. Then R := limRi is regular.
−→
Proof. First, suppose that each Ri is noetherian and Γ is countable. Any ideal of R is countably generated.
It follows by the proof of [23, Corollary 2.47], that p. dim(−) ≤ fl. dim(−) + 1. It remains to recall
w. dim(Ri ) = dim(Ri ), because Ri is noetherian.
Now, suppose that R is coherent. Let I be a finitely generated ideal of R generated by x := x1 , . . . , xn .
There is j ∈ Γ such that x ⊆ Ri for all i ≥ j. Denote xRi by Ii and define mi := m ∩ Ri . In view of [3,
Lemma 3.2], K. gradeRi (mi , Ri ) ≤ dim Ri . Note that Ri /Ii has a finite free resolution. By [21, Chap. 6,
Theorem 2],
fl. dim(Ri /Ii )

≤ p. dim(Ri /Ii )
= K. grade(mi , Ri ) − K. grade(mi , Ri /Ii )
≤ K. grade(mi , Ri )
≤ dim Ri .

Thus,
{w. dim Ri : i ∈ Γ} ≤ sup{dim Ri : i ∈ Γ} < ∞.
So, Lemma 2.5 completes the proof.



4

Proposition 2.7. Let R be a quasilocal ring with a desingularization {Ri : i ∈ Γ}. The following holds:
(i) Any two-generated ideal of R has flat dimension bounded by 1.
(ii) If Γ is countable, then any two-generated ideal of R has projective dimension bounded by 2.
Proof. (i): Let I = (a, b) be a two-generated ideal of R. Without loss of generality, we may assume that
Ri is local. There is i0 ∈ Γ such that {a, b} ⊂ Ri for all i > i0 . We now apply an idea of BuchsbaumEisenbud. As, Ri is an unique factorization domain and in view of [10, Corollary 5.3], {a, b} has a
greatest common divisor c such that {a/c, b/c} is a regular sequence. Thus, p. dimRi ((a/c, b/c)Ri ) < 2.
Multiplication by c shows that (a/c, b/c) ≃ (a, b). Conclude that p. dimRi ((a, b)Ri ) < 2. Then by the
same reasoning as Lemma 2.5(i), fl. dimR (I) < 2.
(ii): In view of part (i) the claim follows by the argument of Corollary 2.6.



We will use the following result several times.
Lemma 2.8. (See [19, Theorem 23.1]) Let ϕ be a local map from a regular local ring (R, m) to a CohenMacaulay local ring (S, n). Suppose dim R + dim S/mS = dim S. Then ϕ is flat.
Example 2.9. The conclusion of Proposition 2.7 can not carry over three-generated ideals.
Proof. Let k be any field. For each n ≥ 3, set Rn := k[x1 , . . . , x2n−4 ]. Define
In := (x1 , x2 ) ∩ . . . ∩ (x2n−5 , x2n−4 ),
fn := x1 x3 . . . x2n−5 , and gn := x2 x4 . . . x2n−4 . Let hn be such that ((fn , gn ) : hn ) = In . It is proved in
[11] that
p. dimRn (Rn /(fn , gn , hn )) = n (∗)
The assignments x1 7→ x1 x2n−3 , x2 7→ x2 x2n−2 , and xi 7→ xi (for i 6= 1, 2) defines the ring homomorphism
ϕn,n+1 : Rn −→ Rn+1 . This has the following properties: ϕn,n+1 (fn ) = fn+1 , ϕn,n+1 (gn ) = gn+1 , and
ϕn,n+1 (In ) ⊆ In+1 . By using this we can choose hn+1 be such that ϕn,n+1 (hn ) = hn+1 . Look at the
directed system {Rn , ϕn,n+1 } and denote the natural map from Rn to R := limRn by ϕn . In view of the
−→
following commutative diagram,
ϕn,n+1

/ Rn+1
②
②
②
ϕn
②②ϕn+1
②
②
 ②
|②
R
Rn

the ideal I := (ϕn (fn ), ϕn (gn ), ϕn (hn ))R is independent of n.
Claim A. The extension Rn → Rn+1 is flat.
Indeed, denote the unique graded maximal ideal of Rn by mn . Set An := (Rn )mn . In view of [19,
Page 178], Rn → Rn+1 is flat provided the induced map ψn : An → An+1 is flat. In order
to prove ψn is flat, we note that dim(An ) = 2n − 4 and dim(An+1 ) = 2n − 2. Let m be the
n+1
m
unique graded maximal ideal of S := k[x1 , x2 , x2n−3 , x2n−2 ]. Then mA
.
≃ (x1 x2n−3S,x
n An+1
2 x2n−2 )
S
Since x2 x2n−2 ∈
/ p∈Ass(Sm /(x1 x2n−3 )) p, the sequence x1 x2n−3 , x2 x2n−2 is regular over Sm . In

n+1
n+1
particular, dim( mA
) = 2. Thus, dim(An+1 ) = dim(An ) + dim( mA
). In view of Lemma
n An+1
n An+1

2.8 we observe that An → An+1 is flat. This finishes the proof of the claim.

5

n
Set Tn := TorR
n (Rn /(fn , gn , hn ), k). Due to (∗), Tn 6= 0. Since Tn is graded (see [9, Page 33]), one

has (Tn )mn 6= 0 (see e.g., [9, Proposition 1.5.15(c)]). By [19, Exercise 7.7], Tor-modules compute with
localization. In the light of the rigidity property of Tor-modules over equal-characteristic regular local
n
rings (Auslander-Lichtenbaum) we see that TorR
n−i (Rn /(fn , gn , hn ), k)mn 6= 0 for all i ≥ 0. For more
n
details, please see [5]. In particular, TorR
n−i (Rn /(fn , gn , hn ), k) 6= 0 for all i ≥ 0. The map ϕn,n+1 induces

the following map:
R

n+1
ℓ
n
τn,n+1
: TorR
(Rn+1 /(fn+1 , gn+1 , hn+1 ), Rn+1 /mn Rn+1 ).
ℓ (Rn /(fn , gn , hn ), Rn /mn ) → Torℓ

ℓ
Claim B. The map τn,n+1
is one to one.
n
Indeed, in view of Claim A, the extension Rn → Rn+1 is flat. Set T := TorR
ℓ (Rn /(fn , gn , hn ), Rn /mn ).

This is a graded module over Rn . Recall from [19, Exercise 7.7] that
R

Torℓ n+1 (Rn+1 /(fn+1 , gn+1 , hn+1 ), Rn+1 /mn Rn+1 ) ≃ T ⊗Rn Rn+1 .
ℓ
is one to
Due to the proof of [19, Theorem 7.4(i)], T ֒→ T ⊗Rn Rn+1 is one to one. Thus, τn,n+1

one. This finishes the proof of the claim.
Let ℓ ≤ n. We combine Claim B along with [12, VI, Exercise 17] to construct the following injection
Rn /mℓ Rn

≃ (Rℓ /mℓ ) ⊗Rℓ Rn
n
֒→ TorR
ℓ (Rn /(fn , gn , hn ), Rn /mℓ Rn )
n
֒→ lim TorR
ℓ (Rn /(fn , gn , hn ), Rn /mℓ Rn )
−→

i

≃ TorR
ℓ (R/I, R/mℓ R).
Thus, 0 6= R/mℓR ֒→ TorR
ℓ (R/I, R/mℓ R). So p. dimR (R/I) = ∞, as claimed.



3. Desingularization via super regularity
We will use the following result several times.
Lemma 3.1. (See [28]) Let (R, m) be a super regular ring. Then m can be generated by a regular sequence.
In particular, m is finitely generated.
The notation SymR (−) stands for the symmetric algebra of an R-module. Also, we set GrR (I) :=
L∞ i i+1
, where I is an ideal of R.
i=0 I /I
Lemma 3.2. Let {(Ri , mi , ki ) : i ∈ Γ} be a directed system of local rings. Set R := lim Ri , m = lim mi
−→
−→
and k = lim ki . The following holds:
−→
i) GrR (m) ≃ limi GrRi (mi ).
−→
⊕µ(mi )
).
ii) Symk (k ⊕µ(m) ) ≃ limi Symki (ki
−→
Proof. i) Taking colimit of the following exact sequence of directed systems
0 −→ {mn+1
}i −→ {mni }i −→ {mn+1
/mni }i −→ 0,
i
i
and using 5-lemma, yields that limi mni /mn+1
≃ mn /mn+1 . In particular, GrR (m) ≃ limi GrRi (mi ).
i
−→
−→
ii) This is in [13, 8.3.3].



Proposition 3.3. Let {(Ri , mi ) : i ∈ Γ} be a directed system of local rings with the property that
m2i = mi ∩ m2i+1 . If R := limRi is coherent and super regular, then each Ri is regular.
−→

6

Proof. Denote the maximal ideal of R by m and denote the residue field of R (resp. Ri ) by k (resp. ki ).
In view of Lemma 3.1, m is generated by a regular sequence. Thus things equipped with the following
isomorphism
θ : Symk (k ⊕µ(m) ) −−−−→ GrR (m) :=

L∞

i=0

mi /mi+1 .

Look at the natural epimorphism
⊕µ(mi )

θi : Symki (ki

) ։ GrRi (mi ),

and the natural map ϕi : Vi := mi /m2i ֒→ Vi+1 := mi+1 /m2i+1 . We claim that:
Claim A. The map ϑi := Sym(ϕi ) is monomorphism.
Indeed, we look at the following diagram:
f

Symki (Vi ) ֒→

g

≃

Symki (Vi+1 ) ֒→ Symki (Vi+1 ) ⊗ki ki+1

❚❚❚❚
❚❚❚❚
❚❚❚❚
h
❚❚❚❚
ϑi
❚❚❚*

Symki+1 (Vi+1 )

/ Sym
ki+1 (Vi+1 ⊗ki ki+1 )
≃

i


/ Symk (L
dim
i+1
k

i

(ki+1 )

Vi+1 )

Remark that
1) Since Vi is a direct summand of Vi+1 as a ki -vector space, f is a monomorphism.
2) The map g is monomorphism, because ki is a field.
3) The horizontal isomorphism follows by [13, 8.3.2].
L
4) The vertical isomorphism follows by dimk (ki+1 ) Vi+1 ≃ Vi+1 ⊗ki ki+1 .
i
L
5) Since Vi+1 is a direct summand of
Vi+1 as a ki+1 -vector space, i is a monomorphism.
By these, we conclude that the map h is a monomorphism. So ϑi : Symki (Vi ) → Symki+1 (Vi+1 ) is
monomorphism. This completes the proof of Claim A.
Set Ki := ker θi . Also, remark that ϑi (Ki ) ⊆ Ki+1 . Again we denote the restriction map by ϑi : Ki ֒→
Ki+1 . Recall that limki ≃ k and lim(mni /mn+1
) ≃ mn /mn+1 . In view of Lemma 3.2, Sym(−) and Gr(−)
i
−→
−→
behave well with respect to direct limits. Hence, θ = limθi . Put all of these together to observe
−→
Ki ֒→ limKi ≃ ker θ = 0.
−→
So, θi is an isomorphism. This means that mi is generated by a regular sequence. The regularity of Ri
follows by this, because Ri is noetherian.



By gl. dim(R) we mean the global dimension of R. Let R be a noetherian local domain of prime
characteristic p. Recall that the perfect closure of R is defined by adjoining to R all higher p-power roots
of all elements of R and denote it by R∞ .
Proposition 3.4. Let R be a local domain of prime characteristic which is either excellent or homomorphic image of a Gorenstein local ring and suppose that its perfect closure is coherent (e.g., R is regular).
If R is not a field, then gl. dim(R∞ ) = dim R + 1.
Proof. Let x := x1 , . . . , xd be a system of parameters for R and set p := char R.
Claim A. One has x is a regular sequence on R∞ .

7

Indeed, this is in [25, Theorem 3.10] when R is homomorphic image of a Gorenstein ring. The argument
of [25, Theorem 3.10] is based on Almost Ring Theory. The claim in the excellent case is in [4,
Lemma 3.1]. This uses non-noetherian Tight Closure Theory.
In particular, fl. dim(R∞ /xR∞ ) = d. Combining this with [2, Theorem 1.2], w. dim(R∞ ) = dim R < ∞.
The same citation yields that gl. dim(R∞ ) ≤ dim R + 1. Suppose on the contrary that
gl. dim(R∞ ) 6= dim R + 1.
This says that gl. dim(R∞ ) = w. dim R. Note that R∞ is coherent and quasilocal. Denote its maximal
ideal by mR∞ . By definition, R∞ is super regular. In the light of Lemma 3.1, mR∞ is finitely generated.
We bring the following claim.
Claim B. One has mR∞ = mpR∞ .
Indeed, clearly, mpR∞ ⊂ mR∞ . Conversely, let r ∈ mR∞ . Since R is perfect, any polynomial such as
f (X) := X p − r has a root. Let r1/p ∈ R∞ be a root of f . We have (r1/p )p ∈ mR∞ . Since mR∞
is prime, r1/p ∈ mR∞ . We conclude from this that r = (r1/p )p ∈ mpR∞ , as claimed.
By Nakayama’s Lemma, mR∞ = 0, i.e., R∞ is a field. So, R is a field. This is a contradiction.



Question 3.5. Let R be a local domain of prime characteristic. What is gl. dim(R∞ )?
Proposition 3.6. Let R be a quasilocal containing a field of prime characteristic p which is integral and
purely inseparable extension of an F -finite regular local ring R0 . If R contains all roots of R0 , then R
has a desingularization with respect to a flat directed system of noetherian regular rings.
Proof. Note that R0 contains a field. Write R as a directed union of a filtered system {Ri } of its subrings
which are finitely generated algebras over R0 .
Claim A. The ring Ri is local.
Indeed, by assumption R0 is local. Denote its maximal ideal by m0 . Let mi and ni be two maximal ideals
of Ri . Both of them lying over m0 . Let r ∈ mi . Since the extension R0 → Ri is integral and
n

n

purely inseparable, we observe that rp ∈ R0 for some n ∈ N. Thus rp ∈ m0 = ni ∩ R0 . Since
ni is prime and r ∈ Ri , we deduce that r ∈ ni . Hence mi ⊂ ni . Therefore mi = ni , because mi is
maximal. So, Ri is local, as claimed.
We denote the unique maximal ideal of Ri by mi . Since R0 → R1 is integral, d := dim R0 = dim R1 .
n1

Remark that if y ∈ R1 , there is n1 ∈ N such that y p

∈ R0 . Since R0 → R1 is integral, R1 is finitely
n

generated as an R0 -module. From this we can pick a uniform n such that for any y ∈ R1 , y p ∈ R0 .
n

1/p

After adding R0

n

1/p

to R1 and denoting the new ring again by R1 , we may assume that R0
n

1/p

is a place that we use the assumptions R0∞ ⊆ R and that R0

⊂ R1 , here

is finite over R0 . Now, let x be a minimal

generating set for m0 . In particular, x is a regular system of parameters on R0 .
1/pn

Claim B. Let n be as the above paragraph. Then m1 = (x1
Indeed, let y ∈ m1 . Then y

pn

∈ R0 . In particular, y
n

pn

1/pn

, . . . , xd

∈ m0 . Then y

pn

)R1 .
P
=
ri xi where ri ∈ R0 . We are

in a situation to take p -th root in R1 , this is due to the choose of n. Taking pn -th roots, we have
P 1/pn 1/pn
1/pn
1/pn
1/pn
1/pn
∈ m1 . Thus y ∈ (x1 , . . . , xd )R1 . Therefore,
∈ R1 and xi
xi , where ri
y=
ri
1/pn

m1 =

1/pn

)R1 $ R1 . The
1/pn
1/pn
(x1 , . . . , xd )R1 as claimed.

m1 ⊂ (x1

, . . . , xd

reverse inclusion is trivial, because m1 is maximal. So,

8

In view of the claim, R1 is regular. Since m0 R1 is m1 -primary, the extension R0 → R1 is flat, please
see Lemma 2.8. Repeating this, one may observe that {Ri } is a desingularization for R and that Ri → Rj
is flat.



Lemma 3.7. One has limi∈Γ Ri [X] ≃ (limi∈Γ Ri )[X].
−→
−→
Proof. This is straightforward and we leave it to the reader.



Corollary 3.8. Adopt the notation of Proposition 3.6. Then R is stably coherent.
Proof. There is a flat directed system {Ri } of noetherian regular rings such that its direct limit is R. In
particular, Ri [X] → Rj [X] is flat. In view of [13, Theorem 2.3.3] and Lemma 3.7, R is stably coherent. 
The assumption R0∞ ⊆ R in Proposition 3.6 is really needed:
Example 3.9. Let F be a field of characteristic 2 with [F : F 2 ] = ∞. Let R̂0 = F [[x, y]] be the formal
power series on variables {x, y} and look at R0 := F 2 [[x, y]][F ]. Let {bi : i ∈ N} ⊂ F be an infinite set of
2-independent elements. Set
en :=
fn :=

P∞

i

bi

P∞

,

i

bi

.

i=n (xy)
yn

i=n (xy)
xn

Define R := R0 [ei , fi : i ∈ N]. This is quasilocal. Denote its unique maximal ideal by m. Recall from [20,
Page 206] that R0 → Rˆ0 is integral and purely inseparable. Since R0 ⊂ R ⊂ R̂0 , we get that R0 ⊂ R is
integral and purely inseparable. By [15, Example 1], p. dimR (m) = ∞ and that m is finitely generated.
In particular, R is not regular. We conclude from Proposition 2.3 that R is not coherent. So, R has no
desingularization with respect to its noetherian regular subrings.
4. Desingularization via purity
We begin by recalling the notion of the purity. Let M ⊂ N be modules over a ring R. Recall that M
is pure in N if M ⊗R L → N ⊗R L is monomorphism for every R-module L. We say a directed system
{Ri : i ∈ Γ} is pure if Ri −→ Rj is pure for all i, j ∈ Γ with i ≤ j.
Proposition 4.1. Let {(Ri , mi ) : i ∈ Γ} be a pure directed system of local rings and such that the maximal
ideal of (R, m) := limi∈I Ri has a finite free resolution. Then the following assertions are true:
−→
(i) There exists an i ∈ Γ such that Rj is regular for all i ≤ j.
(ii) There exists an i ∈ Γ such that Rj → Rk is flat for all i ≤ j ≤ k.
(iii) The ring R is noetherian and regular.
Proof. (i): Look at the following finite free resolution of m:
0

/ FN

/ ...

/ Fj+1

fj

/ Fj

/ ...

/ F0

/m

/ 0,

where Fj is finite free and fj is given by a matrix with finite rows and finite columns. By It (fj ), we
mean the ideal generated by t × t minors of fj . Let rj be the expected rank of fj , see [9, Section 9.1] for
its definition. By [9, Theorem 9.1.6], K. gradeR (Irj (fj ), R) ≥ j. There is an index i ∈ Γ such that all of
components of {fj } are in Ri . Let Fj (i) be the free Ri -module with the same rank as Fj . Consider fj
as a matrix over Ri , and denote it by fj (i). Recall that m is finitely generated. Choosing i sufficiently
large, we may assume that m = mi R. For the simplicity of the reader, we bring the following claim.

9

Claim A. Let A be a subring of a commutative ring B. Let X ∈ Matrs (A) and Y ∈ Matst (A) be matrices.
Look at X ∈ Matrs (B) and Y ∈ Matst (B) as matrices whose entries coming from B. If XY ∈
Matrt (B) is zero as a matrix over B, then XY ∈ Matrt (A) is zero as a matrix over A.
Indeed, since A is a subring of B, the claim is trivial.
Thus, fj (i)fj+1 (i) = 0. Look at the following complex of finite free modules:
0

/ FN (i)

/ ...

/ Fj+1 (i)

fj (i)

/ Fj (i)

/ ...

/ F0 (i)

/ mi

/ 0.

We are going to show that this is exact. Recall that It (fj (i)) is the ideal generated by t × t minors of
fj (i). Clearly, rj is the expected rank of fj (i). Let z := z1 , . . . , zs be a generating set for It (fj (i)). In
view of the purity, there are monomorphisms 0 −→ Hj (z, Ri ) −→ Hj (z, R) for all i and j, see [9, Exercise
10.3.31]. Then,
K. gradeR (Irj (fj ), R) ≤ K. gradeRi (Irj (fj (i)), Ri ).
Thus, K. gradeRi (Irj (fj ), Ri ) ≥ j. Again, due to [9, Theorem 9.1.6],
0 −→ FN (i) −→ · · · −→ F0 (i)
is acyclic. Thus, p. dim(Ri /mi ) < ∞. By Local-Global-Principle (please see [9, Theorem 2.2.7]), Ri is
regular.
(ii): By purity, dim Rm ≥ dim Rn for all n ≤ m, see [8, Remark 4 and Corollary 5]. Again, in the light
of purity,
mm = (mm R) ∩ Rm = (mn R) ∩ Rm = (mn Rm )R ∩ Rm = mn Rm ,
for all n ≤ m. Thus mn Rm = mm . Denote the minimal number of elements required to generate the
ideal mm by µ(mm ). Consequently, µ(mm ) ≤ µ(mn ). By part (i), (Ri , mi ) is regular. Hence
dim Rm = µ(mm ) ≤ µ(mn ) = dim Rn .
Therefore, dim Rm = dim Rn . In view of Lemma 2.8, Rn → Rm is flat for all n ≤ m, as claimed.
(iii): Recall from (ii) that mm = mn Rm for all n < m. In view of [22], R is noetherian. The ring R is
regular, because p. dimR (R/m) < ∞.



Example 4.2. Here, we present a desingularization with respect to a non-pure directed system. To this
Pℓ
end let R := {n + i=1 ni ti : n ∈ Z, ni ∈ Z[1/2]}. Then R has a desingularization {(Ri , φi,j )} where
φi,j : Ri → Rj is not pure.
Proof. For each i ∈ N, set Ri := Z[t/2i ]. Note that Ri is a noetherian regular ring. The system {Ri }i∈N
Pℓ
is directed with respect to the inclusion. Let f ∈ R. Then f = n + i=1 ni ti where ni ∈ Z[1/2]. There

is k ∈ Z such that ni = mi /2ik for some mi ∈ Z and for all i. Deduce by this that f ∈ Rk . Thus, {Ri }
gives a desingularization for R. Now, we look at the following equation 2X = t/2i . Clearly, t/2i+1 ∈ Ri+1
is a solution. The equation has no solution in Ri . In the light of [19, Theorem 7.13], the map Ri → Ri+1
is not pure, as claimed.



Also, we present the following example.
Example 4.3. Let D be a noetherian regular integral domain with a fraction field Q. Let R := {f ∈
Q[X1 , . . . , Xn ] : f (0, . . . , 0) ∈ D}. Then R has a desingularization.

10

Proof. Without loss of the generality we may assume that D =
6 Q. Recall that
O
R = {f ∈ Q[X1 , . . . , Xn ] : f (0, . . . , 0) ∈ D} ≃
{f ∈ Q[Xi ] : f (0) ∈ D}.
1≤i≤n

Let F := Q

⊕n

. This is flat as a D-module. Under the identification SymQ (Q) = Q[X], the image of

SymD (Q) in the natural map SymD (Q) → SymQ (Q) is D + XQ[X]. So
N
SymD (F ) ≃ n SymD (Q)
N
≃ n {f ∈ Q[Xi ] : f (0) ∈ D}.
Due to the Lazard’s theorem, there is a directed system of finitely generated free modules {Fi : i ∈ Γ}
with direct limit F . In view of [13, 8.3.3],
R ≃ SymD (F ) ≃ limi∈Γ SymD (Fi ).
−→
Since Ri := SymD (Fi ) is a noetherian regular ring, R can be realized as a direct limit of the directed
system {Ri : i ∈ Γ} of noetherian regular rings.



5. Desingularization of products
The following definition is taken from [17].
Definition 5.1. A ring is called DLFPF, if it is a direct limit of a finite product of fields.
Question 5.2. (See [17, Question 10]) Let E be a field and let R be a maximal DLFPF subring of

Q

N

E.

Does R contain a field isomorphic to E?
The following answers Question 5.2 in the finite-field case.
L
Q
Proposition 5.3. Let F be a finite field. Then N F ≃ lim ( f inite Fi ) where Fi is a field.
−→
Q
Proof. Let S ⊂ N F be the subring consisting of all elements that have only finitely many distinct
S
Q
coordinates. Since F is finite, S = N F . By [14, Proposition 5.2], S = Aj where Aj is an artinian
Q
regular subring of N F . It remains to note that any artinian regular ring is isomorphic to a finite direct
product of fields.
Corollary 5.4. Let F be a finite field. Then

Q

N


F is stably coherent.

Proof. We adopt the notation of Proposition 5.3 and we denote a finite family of variables by X. Let
L
Rγ := ( 1≤i≤nγ Fi ). Let γ ≤ δ. Since Rγ is of zero weak dimension, Rγ → Rδ is flat. It turns out that
Rγ [X] → Rδ [X] is flat. In view of Lemma 3.7 and by Proposition 5.3,
Y
( F )[X] ≃ (limγ∈Γ Rγ )[X] ≃ limγ∈Γ (Rγ [X]).
−→
−→
Q
is a flat direct limit of noetherian regular rings. By [13, Theorem 2.3.3], N F is stably coherent.



In the following item we collect some homological properties of the products of rings.
Fact 5.5. i) Let R be a noetherian local ring. Combining [13, Theorem 6.1.2] and [13, Theorem 6.1.20],
Q
yields that N R is coherent if and only if dim R < 3.
Q
ii) Let {RN : n ∈ N} be a family of rings such that R :=
Rn is coherent. Then w. dim R =
sup{w. dim R}, please see [13, Theorem 6.3.6].
iii) (See the proof of [23, Corollary 2.47]) Let a be an ideal of a ℵn -noetherian ring A.
p. dimA (A/a) ≤ fl. dimA (A/a) + n + 1.
iv) (See [17, Introduction]) The ring

Q

N

Then

Q is not written as a direct limit of a finite product of fields.

11

Example 5.6. The ring R :=

Q

N

Q is coherent and regular. But, R has no desingularization with respect

to its noetherian regular subrings.
Q
Proof. By Fact 5.5 i), N Q is coherent. In the light of Fact 5.5 ii), R is von Neumann regular, i.e., R is
of zero weak dimension. We deduce from Fact 5.5 iii) that gl. dim(R) < 3. Therefore, R is coherent and
regular. Suppose on the contrary that R can be written as a direct limit of its noetherian regular subrings
{Ri : i ∈ I}. Due to [9, Corollary 2.2.20], Ri ≃ Ri1 × . . .× Rini , where Rij is a noetherian regular domain.
By Qij we mean the fraction field of Rij . Recall that limi limj Qij is a direct limit of finite product of
−→ −→
its subfields. This is a consequence of the fact that any double direct limit is a direct limit, please see
[7, III) Proposition 9]. Recall from [17, Corollary 4] that a von Neumann regular subring of DLF P F is
Q
DLF P F . We conclude by this that N Q can be written as a direct limit of a finite product of fields.
This is a contradiction, please see Fact 5.5 iv).



Acknowledgement . I would like to thank M. Dorreh, S. M. Bhatwadekar and H. Brenner, because of a
talk on direct limits. I thank the anonymous referee for various suggestions.
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E-mail address: mohsenasgharzadeh@gmail.com

