#  On a question of Swan
with an Appendix by Kęstutis Česnavičius

Dorin Popescu  In the memory of Hideyuki Matsumura.  Dorin Popescu, Simion
Stoilow Institute of Mathematics of the Romanian Academy, Research unit 5,
University of Bucharest, P.O.Box 1-764, Bucharest 014700, Romania
<EMAIL_ADDRESS>
###### Abstract.

We show that a regular local ring is a filtered inductive limit of regular
local rings, essentially of finite type over $\bf Z$.

Key words  : Regular Rings, Smooth Morphisms, Regular Morphisms, Discrete
Valuation Rings, Projective Modules
2010 Mathematics Subject Classification: Primary 13B40, Secondary
14B25,13H05,13C10.

## Introduction

Let $R$ be a regular ring. The following is a well known question concerning
finitely generated projective modules over polynomial $R$-algebras.

######  Conjecture 1.

(Bass-Quillen Conjecture, [3, Problem IX], [27]) Every finitely generated
projective module $P$ over a polynomial $R$-algebra, $R[T]$,
$T=(T_{1},\ldots,T_{n})$ is extended from $R$, i.e. $P\cong
R[T]\otimes_{R}(P/(T)P)$.

The Bass-Quillen Conjecture (shortly the BQ Conjecture) has positive answers
in the following cases

i) if $\dim R\leq 1$ by Quillen and Suslin (see [27], [29]),

ii) if $R$ is essentially of finite type over a field by H. Lindel (see [13]),

iii) if $R$ is a local ring of unequal characteristic, essentially of finite
type over $\bf Z$, let us say $(R,{\mathfrak{m}},k)$ with $p=$char$\
k\not\in{\mathfrak{m}}^{2}$ by Swan [18].

Swan noticed that it will be useful for the general question to have a
positive answer to the following one.

######  Question 2.

(Swan [18]) A regular local ring is a filtered inductive limit of regular
local rings, essentially of finite type over $\bf Z$.

A partial positive answer is given below.

######  Theorem 3.

([25, Theorem 3.1]) The Swan question has a positive answer for a regular
local ring $(R,{\mathfrak{m}},k)$ in the following cases

  1. (1)

$p=\mbox{char}\ k\not\in{\mathfrak{m}}^{2}$,

  2. (2)

$R$ contains a field,

  3. (3)

$R$ is excellent Henselian.

Using the above theorem we showed using Lindel’s and Swan’s results that the
BQ Conjecture holds for regular local ring containing a field, or regular
local rings $(R,{\mathfrak{m}},k)$ with $p=$char$\ k\not\in{\mathfrak{m}}^{2}$
(see [25, Theorem 4.1]). After almost 30 years, we noticed that complete
positive answers of the above questions are missing and there are still people
interested to have them.

The purpose of the present paper is to give a complete positive answer to the
Swan Question (see Theorem 17 where important is the last sentence, which
follows from the first part of the statement necessary only for the proof).
When $\dim R=1$ our Theorem 8 uses a condition of separability which we remove
it in Theorem 15. Unfortunately, our result seems to be not very useful for
the Bass-Quillen Conjecture (see Remark 18). The appendix shows that our
Corollary 19 could be used to reduce a well known conjecture on purity to the
complete case (see Proposition 21).

##  1\. Discrete Valuation Rings of unequal characteristic.

Let $(A,{\mathfrak{m}})$ be a discrete valuation ring (a DVR for short) of
unequal characteristic. Then $A$ dominates ${\bf Z}_{(p)}$, where $p=$char
$k$, $k:=A/{\mathfrak{m}}$. We suppose that $p\in{\mathfrak{m}}^{2}$ and the
field extension $k\supset{\bf F}_{p}$ is separably generated.

######  Remark 4.

Suppose that the fraction field $K=\mbox{Fr}(A)$ of $A$ is a finite type
extension of $\bf Q$. Then it is possible that $k/{\bf F}_{p}$ is not a finite
type field extension (see [1, Theorem 6.1]). Thus $k/{\bf F}_{p}$ need not be
separably generated.

######  Lemma 5.

There exists a DVR subring $B$ of $A$ and a regular parameter $x$ of $A$ such
that

  1. (1)

$B\subset A$ is a ramified extension inducing an algebraic separable extension
on the residue fields,

  2. (2)

B contains a power $b=x^{e}$, $1<e\in{\bf N}$ of $x$, which is a regular
parameter in $B$,

  3. (3)

$C=B[x]_{(x)}$ is a DVR subring of $A$ such that
$C\cong(B[X]/(X^{e}-b))_{(X)}$,

  4. (4)

the extension $C\subset A$ is unramified.

###### Proof.

A lifting of a separable transcendence base ${\bar{y}}=({\bar{y}}_{i})_{i\in
I}$ of $k$ over ${\bf F}_{p}$ induces a system of algebraically independent
elements $y:=(y_{i})_{i\in I}$, $y_{i}\in A$ of $K$ over ${\bf Q}$. Then the
ring $C_{0}={\bf Z}[(Y_{i})_{i\in I}]_{p{\bf Z}[(Y_{i})_{i\in I}]}$ is a DVR
(see e.g [15, Theorem 83]). Consider the flat map $\psi_{0}:C_{0}\rightarrow
A$ given by $Y\rightarrow y$. Note that $\psi_{0}$ is ramified and induces an
algebraic separable extension on the residue fields. Suppose that $p=x^{s}t$
for some regular parameter $x$ of $A$ and an unit element $t\in A$,
$1<s\in{\bf N}$.

We claim that $t$ is not transcendental over ${\bf Q}(y)$. Indeed, choose $r$
such that $p^{r}>s$. Then $B=B^{\prime}\cap{\bf Q}(y,t^{p^{r}})$ is a DVR and
the residue field extension induced by $B\subset B^{\prime}$ is pure
inseparable because $u^{p^{r}}\in B$ modulo ${\mathfrak{m}}\cap B$ for every
$u\in B^{\prime}$. As it is also algebraic separable by hypothesis, we see
that this residue field extension is trivial and so $B^{\prime}$ must be a
ramified extension of $B$ of order $e_{B^{\prime}/B}=p^{r}>s$ which is false.

The DVR $B^{\prime}=A\cap{\bf Q}(y,t)$ contains $x^{s}=p/t$. If $B^{\prime}$
is an unramified extension of $C_{0}$ then the polynomial $X^{s}-p/t$ is
irreducible in $B^{\prime}[X]$,

$C=B^{\prime}[x]_{(x)}\cong(B^{\prime}[X]/(X^{s}-p/t))_{(X)}$

is a DVR subring of $A$, the extension $C\subset A$ is unramified and induces
an algebraic separable extension on the residue fields.

Otherwise (this is possible as shows Example 6), choose a regular parameter
$z^{\prime}$ of $B^{\prime}$. Then $z^{\prime}=x^{\prime
s^{\prime}}t^{\prime}$ for some regular parameter $x^{\prime}$ of $A$, $1\leq
s^{\prime}<s$ and an invertible element $t^{\prime}\in A$. Note that the field
extensions ${\bf Q}(y)\subset{\bf Q}(y,t^{\prime})\subset{\bf Q}(y,t)$ are
finite and the last one of degree $s^{\prime}$ because ${\bf
Q}(y,t^{\prime})={\bf Q}(y,s^{\prime})$, ${\bf Q}(y,t)={\bf Q}(y,s)$.

If $s^{\prime}=1$ then $B^{\prime}\subset A$ is unramified. Using again this
procedure we arrive in some steps to a DVR subring $B^{\prime\prime}$ of
$B^{\prime}$ such that the extension $B^{\prime\prime}\subset B^{\prime}$ is
ramified. This is because the degree of the corresponding fraction field
extensions over ${\bf Q}(y)$ decreases in each step, in the worst case
$B^{\prime\prime}=C_{0}$.

Thus we may assume that $s^{\prime}>1$. Repeating this procedure for
$A\cap{\bf Q}(y,t,t^{\prime})$ and so on, we arrive in some steps to a DVR
subring $(B,(z))$ of $A$ containing a power $b=x^{e}$, $e>1$ of a regular
parameter $x$ of $A$ with $z\not\in x^{e^{\prime}}A$ for $e^{\prime}<e$. Then
$C=B[x]_{(x)}\cong(B[X]/(X^{e}-b))_{(X)}$ is a DVR subring of $A$, the
extension $C\subset A$ is unramified and induces an algebraic separable
extension on the residue fields.

######  Example 6.

Note that $B=({\bf Z}[Y]/(Y^{2}-5))_{(Y)}$ is a DVR, ramified extension of
$C={\bf Z}_{(5)}$. The polynomial $f=X^{4}-5/(1+Y)\in B[X]$ is irreducible and
$D=(B[X]/(f))_{(X)}$ is a DVR, ramified extension of $B$. Note that $D\cap{\bf
Q}(Y)=B$ is a ramified extension of $C$. Thus in the above proof it is
possible that $B^{\prime}$ could be indeed a ramified extension of $C_{0}$.

Using by recurrence the above lemma we get the following proposition.

######  Proposition 7.

In the notations of the above lemma, there exists extensions of DVR subrings
of $A$,

$C_{0}\subset B_{1}\subset C_{1}\subset\ldots B_{r}\subset C_{r}\subset
B_{r+1}=A,$

$C_{0}$ being defined in Lemma 5, such that

  1. (1)

$C_{i}\subset B_{i+1}$ is an unramified extension for any $0\leq i\leq r$
inducing an algebrac separable extension on the residue fields,

  2. (2)

$C_{i}\cong(B_{i}[X]/(X^{e_{i}}-b_{i}))_{(X)}$ for some $b_{i}\in B_{i}$,
$1<e_{i}\in{\bf N}$, which is a regular parameter in $B_{i}$.

######  Theorem 8.

Suppose that $K=$Fr$A$ is a field extension of $\bf Q$ not necessarily of
finite type. Then $A$ is a filtered inductive union of regular local subrings
$(R_{i})_{i\in I}$ of $A$, essentially of finite type over $\bf Z$.

###### Proof.

It is enough to show that for a finite type $\bf Z$-algebra $E\subset A$ there
exists a regular local subring $R\subset A$ which contains $E$ and it is
essentially of finite type over $\bf Z$. By Proposition 7 there exists
extensions of DVR subrings of $A$,

$C_{0}\subset B_{1}\subset C_{1}\subset\ldots B_{r}\subset C_{r}\subset
B_{r+1}=A$

such that (1), (2) hold above. Apply the classical Néron Desingularization
(see [19], [10, Theorem 1]) for the case $C_{r}\subset B_{r+1}=A$. Then there
exists a regular local subring $R_{r}\subset A$, which contains $C_{r}[E]$ and
it is essentially of finite type over $C_{r}$. If $r=0$ then $A$ is an
unramified extension of $C_{0}$ inducing an algebraic separable extension on
residue fields and we are done.

Suppose that $r>0$. Using again the Néron Desingularization for the extension
$C_{r-1}\subset B_{r}$ we see that $B_{r}$ is a filtered inductive union of
regular local subrings essentially of finite type over $C_{r-1}$. Then there
exists a regular local subring $T_{r-1}$ of $B_{r}$ essentially of finite type
over $C_{r-1}$ such that

  1. (1)

$b_{r}$ belongs to a regular system of parameters $z_{r-1}$ of $T_{r-1}$ and
there exist a regular local subring $T^{\prime}_{r-1}$ of $B_{r}$ isomorphic
to $(T_{r-1}[X]/(X^{e_{r-1}}-b_{r-1}))_{(z_{r-1},X)}$,

  2. (2)

$R_{r}$ is defined over $T^{\prime}_{r-1}$, that is there exists a regular
local subring $R_{r-1}\supset T^{\prime}_{r-1}$ of $B_{r}$ such that
$R_{r}\cong R_{r-1}\otimes_{T^{\prime}_{r-1}}C_{r}$.

Applying by recurrence this procedure we find a regular local subring $R_{0}$
of $A$, which is essentially of finite type over $C_{0}$ and contains
$C_{0}[E]$. This is enough.

######  Remark 9.

If $p\not\in{\mathfrak{m}}^{2}$ then the problem is easier (see [25, Theorem
3.1]).

##  2\. Regular local rings of unequal characteristic

Let $(R,{\mathfrak{m}},k)$ be a regular local ring of dimension $n$ and
$p=$char $k$. We suppose that $0\not=p\in{\mathfrak{m}}^{2}$.

######  Lemma 10.

There exists a regular system of parameters $x=(x_{1},\ldots,x_{n})$ for $R$
such that $(p,x_{2},\ldots,x_{n})$ is a system of parameters for $R$. For any
such, the map ${\bf Z}[X_{2},\ldots,X_{n}]_{(p,X_{2},\ldots,X_{n})}\rightarrow
R$ given by $X_{i}\rightarrow x_{i}$, $1<i\leq n$ is flat and induces a
ramified extension of DVRs modulo $(x_{2},\ldots,x_{n})$.

###### Proof.

The second statement follows from the first and the flatness criterion (see
e.g. [15, 20.C]) Suppose that $n>1$ and let $z=(z_{1},\ldots,z_{n})$ be a
regular system of parameters of $R$. Using induction on $n$ it is enough to
choose $x_{n}$ from the infinite set $z_{n}+{\mathfrak{m}}^{2}$ which does not
divide $p$.

For the next results we need some preparations.

A ring morphism $u:A\rightarrow A^{\prime}$ of Noetherian rings has regular
fibers if for all prime ideals $p\in\operatorname{Spec}A$ the ring
$A^{\prime}/pA^{\prime}$ is a regular ring. It has geometrically regular
fibers if for all prime ideals $p\in\operatorname{Spec}A$ and all finite field
extensions $K$ of the fraction field of $A/p$ the ring
$K\otimes_{A/p}A^{\prime}/pA^{\prime}$ is regular. A flat morphism of
Noetherian rings $u$ is regular if its fibers are geometrically regular. If
$u$ is regular of finite type then $u$ is called smooth.

The following theorem extends Néron’s desingularization (see [19], [10]) and
it was useful to solve different problems concerning the projective modules
over regular rings, or from the Artin Approximation Theory (see [21], [23],
[25], [26], [30]).

######  Theorem 11.

(General Néron Desingularization, Popescu [20], [21], [22], [23], [24], André
[2], Swan [30]) Let $u:A\rightarrow A^{\prime}$ be a regular morphism of
Noetherian rings and $B$ an $A$-algebra of finite type. Then any $A$-morphism
$v:B\rightarrow A^{\prime}$ factors through a smooth $A$-algebra $C$, that is,
$v$ is a composite $A$-morphism $B\rightarrow C\rightarrow A^{\prime}$.

Let $A$ be a Noetherian ring, $E=A[Y]/I$, $Y=(Y_{1},\ldots,Y_{q})$. If
$f=(f_{1},\ldots,f_{r})$, $r\leq q$ is a system of polynomials from $I$ then
we can define the ideal $\Delta_{f}$ generated by all $r\times r$-minors of
the Jacobian matrix $(\partial f_{i}/\partial Y_{j})$. After Elkik [7] let
$H_{E/A}$ be the radical of the ideal $\sum_{f}((f):I)\Delta_{f}B$, where the
sum is taken over all systems of polynomials $f$ from $I$ with $r\leq q$.
$H_{E/A}$ defines the non smooth locus of $E$ over $A$.

######  Proposition 12.

In the notation and hypotheses of Lemma 10, let $E\subset R$ be a $C:=({\bf
Z}[x_{2},\ldots,x_{n}])_{(p,x_{2},\ldots,x_{n})}\cong({\bf
Z}[X_{2},\ldots,X_{n}])_{(p,X_{2},\ldots,X_{n})}$-subalgebra of finite type.
Suppose that $n>1$. Then the inclusion $v:E\rightarrow R$ factors through a
finite type $C$-algebra $F$, let us say $v$ is the composite map $E\rightarrow
F\xrightarrow{w}R$ such that $w(H_{F/C})R$ contains a power of $p$.

###### Proof.

Let $q$ be a minimal prime ideal of $h_{E}=\sqrt{v(H_{E/C})R}$ which does not
contain $p$. Then $C\rightarrow R_{q}$ is regular and using Theorem 11, or
[24, Lemma 8] we see that $v$ factors through a finite type $C$-algebra
$F_{1}$, let us say $v$ is the composite map $E\rightarrow
F\xrightarrow{w_{1}}R$, such that $h_{F_{1}}=\sqrt{w_{1}(H_{F_{1}/C})R}$
strictly contains $q$. Note that $w_{1}$ is not necessarily injective. Step by
step we arrive in this way to some new $F_{r}$ and $w_{r}:F_{r}\rightarrow R$
such that all minimal prime ideals of $h_{F_{r}}:=\sqrt{w_{r}(H_{F_{r}/C})R}$
contain $p$, that is, $p\in h_{F_{r}}$. We are done.

We will need the next result, which is in fact [24, Proposition 5] (see also
[11, Proposition 3]) written in our special case.

######  Proposition 13.

Let $A$ and $A^{\prime}$ be Noetherian rings and $u:A\rightarrow A^{\prime}$
be a ring morphism. Suppose that $A^{\prime}$ is local and let $E=A[Y]/I$,
$Y=(Y_{1},\ldots,Y_{s})$, $f=(f_{1},\ldots,f_{r})$, $r\leq s$ be a system of
polynomials from $I$, $(M_{j})_{j\in[l]}$ some $r\times r$-minors of the
Jacobian matrix $(\partial f_{i}/\partial Y_{j^{\prime}})$,
$(N_{j})_{j\in[l]}\in((f):I)$ and set $P:=\sum_{j=1}^{l}N_{j}M_{j}$. Let
$v:E\rightarrow A^{\prime}$ be an $A$-morphism. Suppose that

  1. (1)

there exists a non zero divisor $d\in A$ such that $d\equiv P$ modulo $I$,
which is also non zero divisor in $A^{\prime}$, and

  2. (2)

there exist an $A$-algebra $D$ of finite type and an $A$-morphism
$\omega:D\rightarrow A^{\prime}$ such that $d$ is a non zero divisor in $D$,
$\operatorname{Im}v\subset\operatorname{Im}\omega+d^{3}A^{\prime}$ and for
${\bar{A}}=A/(d^{3})$ the map
$\bar{v}={\bar{A}}{\otimes}_{A}v:\bar{E}=E/d^{3}E\rightarrow\bar{A}^{\prime}=A^{\prime}/d^{3}A^{\prime}$
factors through $\bar{D}=D/d^{3}D$.

Then there exists a smooth $D$-algebra $D^{\prime}$ such that $v$ factors
through $D^{\prime}$, let us say $v$ is the composite map $D\rightarrow
D^{\prime}\xrightarrow{w}A^{\prime}$ and
$h_{D}=\sqrt{\omega(H_{D/A})A^{\prime}}\subset
h_{D^{\prime}}=\sqrt{w(H_{D^{\prime}/A})A^{\prime}}$.

######  Remark 14.

Actually the proof from [24, Proposition 5] asks also for $D$ and $A^{\prime}$
to be flat over $A$. In our case, it is only necessary to assume that $d$ is a
non zero divisor in $D$ and $A^{\prime}$. We can even assume that $d$ is not
regular in $A^{\prime}$, but in this case we should change $d^{3}$ by $d^{3c}$
for some $c$ with $(0:_{A^{\prime}}d^{c})=(0:_{A^{\prime}}d^{c+1})$.

Next theorem extends Theorem 8 in the case when $k$ is not separable generated
over ${\bf F}_{p}$.

######  Theorem 15.

Let $(A,{\mathfrak{m}},k)$ be a Noetherian local ring,
$s=(s_{1},\ldots,s_{m})$ some positive integers and
$\gamma=(\gamma_{1},\ldots,\gamma_{m})$ a system of nilpotents of $A$. Suppose
that $0\not=p\in{\mathfrak{m}}^{2}$, $R=A/(\gamma)$ is a DVR and $A$ is a flat
${\mathfrak{N}}={\bf Z}_{(p)}[\Gamma]/(\Gamma^{s})$-algebra,
$\Gamma\rightarrow\gamma$ with $\Gamma=(\Gamma_{1},\ldots,\Gamma_{m})$ some
variables, and $(\Gamma^{s})$ denotes the ideal
$(\Gamma_{1}^{s_{1}},\ldots,\Gamma_{m}^{s_{m}})$. Then $A$ is a filtered
inductive limit of some Noetherian local ${\mathfrak{N}}$-algebras
$(A_{i})_{i}$ essentially of finite type with $A_{i}/\Gamma A_{i}$ regular
local rings. In particular, a DVR is a filtered inductive limit of some
regular local rings, essentially of finite type over $\bf Z$.

###### Proof.

Let $x\in A$ be an element inducing a local parameter of $R$, $E\subset A$ be
a finite type ${\mathfrak{N}}$-algebra and $v:E\rightarrow A$ be the
inclusion. Actually, we could simply take a morphism (not necessarily
injective) $v:E\rightarrow A$ for some finite type ${\mathfrak{N}}$-algebra
$E$. We may assume that $p\equiv x^{e}t_{0}$ modulo $(\gamma)$ for some
$e\in{\bf N}$ and $t_{0}\in A\setminus{\mathfrak{m}}$. Moreover, we may
suppose that $p=x^{e}t_{0}+\sum_{j=1}^{m}t_{j}\gamma_{j}$ for some $t_{j}\in
A$, $j\in[m]$. As in Proposition 12 we may assume that
$v(H_{E/{\mathfrak{N}}})A$ contains a power $b$ of $p$ because the map
${\mathfrak{N}}\rightarrow A_{q}$ is a regular map for all
$q\in\operatorname{Spec}A$ with $p\not\in q$.

Following the proof of [11, Theorem 2], we assume that
$b=\sum_{i=1}^{q}v(a_{i})z_{i}$, where $z_{i}\in A$, $a_{i}\in
H_{E/{\mathfrak{N}}}$. Set $E_{0}=E[Z]/(f)$, where
$f=-b+\sum_{i=1}^{q}a_{i}Z_{i}\in E[Z]$, $Z=(Z_{1},\ldots,Z_{q})$, and let
$v_{0}:E_{0}\rightarrow A$ be the map of $E$-algebras given by $Z\rightarrow
z$. Changing $E$ by $E_{0}$ we may assume that $b\in H_{E/{\mathfrak{N}}}$.

Let $E\cong{\mathfrak{N}}[Y]/I$, $Y=(Y_{1},\ldots Y_{m})$. Using [11, Lemma 4]
we may change again $E$ to assume that a power $d$ of $b$ is in
$((f):I)\Delta_{f}$ for some $f=(f_{1},\ldots,f_{r})$, $r\leq s$ from $I$,
where $\Delta_{f}$ denotes the ideal generated by the $r\times r$-minors of
$(\partial f/\partial Y)$. Then $d$ has the form $d\equiv
P=\sum_{i=1}^{q}M_{i}L_{i}\ \mbox{modulo}\ I$, for some $r\times r$ minors
$M_{i}$ of $(\partial f/\partial Y)$ and $L_{i}\in((f):I)$.

Let $(\Lambda,(p,\Gamma),k)$ be the (unique) Cohen ${\mathfrak{N}}$-algebra
with residue field $k$, that is $\Lambda$ is complete and the map
${\mathfrak{N}}\rightarrow\Lambda$ is flat (even regular) (see e.g. [15,
Theorem 83]). $\Lambda$ is a filtered inductive limit of some Noetherian local
${\mathfrak{N}}$-algebras $(C_{j})_{j}$ essentially smooth (see Theorem 11).
Then the completion of the DVR $A/(\gamma)$ is a finite free module over the
DVR $\Lambda/\Gamma\Lambda$ with the base $\\{1,x,\ldots,x^{e-1}\\}$ and so
$A/(\gamma,d^{3})$ is a finite free module over $\Lambda/(\Gamma,d^{3})$ with
the same base. Using [15, (20.G)] applied to
${\mathfrak{N}}\rightarrow\Lambda\rightarrow A$ we get $A$ flat over
$\Lambda$. It follows that $A/(d^{3})$ is a finite free module over
$\Lambda/(d^{3})$ with the same base and
$t_{j}\equiv\sum_{i=0}^{e-1}t_{ji}x^{i}$ modulo $d^{3}$ for some
$t_{ji}\in\Lambda$. Thus there exists a smooth ${{\mathfrak{N}}}$-algebra $C$
such that $(t_{ji})_{ji}$ are contained in the image of the canonical map
$\psi:C\rightarrow\Lambda$, let us say $t_{ji}$ are the images of some
$t^{\prime}_{ji}\in C$. Let $\varphi$ be the composite map

$C[X]/(d^{3})\rightarrow\Lambda[X]/(d^{3})\rightarrow{\tilde{A}}=A/(d^{3})$

induced by $\psi$ and $X\rightarrow x$. Actually, $\tilde{A}$ is a filtered
inductive limit of ${\mathfrak{N}}$-algebras of type
${\tilde{B}}=C[X]/(d^{3},p-t^{\prime}_{0}X^{e}-\sum_{j=1}^{m}t^{\prime}_{j}\Gamma_{j})$,
where $t^{\prime}_{j}=\sum_{i=0}^{e-1}t^{\prime}_{ji}X^{i}$ and $C$ is smooth
over ${\mathfrak{N}}$ and containing $(t^{\prime}_{ji})$.

We may assume that $C=({{\mathfrak{N}}}[U]/(g))_{ag^{\prime}}$,
$U=(U_{1},\ldots,U_{l})$ for a monic polynomial $g$ (in $U_{1}$),
$a\in{{\mathfrak{N}}}[U]$, where
${\bar{g}}^{\prime}=\partial{\bar{g}}/\partial U_{1}$ (see e.g [30, Theorem
2.5]). We may take $C$ such that the map
${\tilde{v}}:{\tilde{E}}:=E/d^{3}E\rightarrow{\tilde{A}}$ induced by $v$,
factors through $\tilde{B}$, let us say $\tilde{v}$ is the composite map
${\tilde{E}}\rightarrow{\tilde{B}}\xrightarrow{\tilde{w}}{\tilde{A}}$, the
last map being the limit map.

Choose a lifting $u=(u_{i})_{1\leq i\leq l}$ in $A$ of
$({\tilde{w}}(U_{i}))_{1\leq i\leq l}$. We have $g(u)=d^{3}z$ for some $z\in
A$. Set $D_{1}=C[X,U,Z]/(g-d^{3}Z)$ and let $w:D_{1}\rightarrow A$ be given by
$(X,U,Z)\rightarrow(x,u,z)$. Clearly, $D_{2}=(D_{1}/(\Gamma))_{w^{-1}((x))}$
is a regular local ring being essentially smooth over a polynomial ring over
$\bf Z$ and $p,X$ are contained in a regular system of parameters of $D_{2}$.
Then $D_{2}/(p-t_{0}^{\prime}X^{e})$ is a regular local ring and denote the
map

$D=D_{1}/(p-t_{0}^{\prime}X^{e}-\sum_{j=1}^{m}t^{\prime}_{j}\Gamma_{j})\rightarrow
A$

induced by $w$ also by $w$.

We have $\operatorname{Im}v\subset\operatorname{Im}w+d^{3}A$ and $v$ factors
modulo $d^{3}$ through $w$. Since $d$ is a non zero divisor in $D$, $A$ we may
apply Proposition 13 (see also Remark 14) for ${\mathfrak{N}}\rightarrow A$,
$E$, $D$ and $\omega=w$. We get an essentially smooth $D$-algebra $D^{\prime}$
such that $v$ factors through $D^{\prime}$. Since $D/\Gamma D$ is regular
local we get $D^{\prime}/\Gamma D^{\prime}$ regular too. This is enough using
e.g. [30, Lemma 1.5] (see below), $\mathcal{S}$ being the set of the regular
local ${\mathfrak{N}}$-algebras.

######  Lemma 16.

Let $\mathcal{S}$ be a class of finitely presented ${\mathfrak{N}}$-algebras.
Let $A$ be a ${\mathfrak{N}}$-algebra. Then the following statements are
equivalent:

(1) $A$ is a filtered inductive limit of algebras from $\mathcal{S}$,

(2) If $E$ is a finitely presented ${\mathfrak{N}}$-algebra and
$v:E\rightarrow A$ is a morphism then $v$ factors through an algebra from
$\mathcal{S}$.

The following theorem is a positive answer of a question of Swan [18].

######  Theorem 17.

Let $(A,{\mathfrak{m}},k)$ be a Noetherian local ring,
$s=(s_{1},\ldots,s_{m})$ some positive integers and
$\gamma=(\gamma_{1},\ldots,\gamma_{m})$ a system of nilpotents of $A$. Suppose
that $0\not=p\in{\mathfrak{m}}^{2}$, $R=A/(\gamma)$ is a regular local ring
and $A$ is a flat ${\mathfrak{N}}={\bf Z}_{(p)}[\Gamma]/(\Gamma^{s})$-algebra,
$\Gamma\rightarrow\gamma$ with $\Gamma=(\Gamma_{1},\ldots,\Gamma_{m})$ some
variables, and $(\Gamma^{s})$ denotes the ideal
$(\Gamma_{1}^{s_{1}}\cdots\Gamma_{m}^{s_{m}})$. Then $A$ is a filtered
inductive limit of some Noetherian local ${\mathfrak{N}}$-algebras
$(F_{i})_{i}$ essentially of finite type with $F_{i}/\Gamma F_{i}$ regular
local rings. In particular, a regular local ring $(R,{\mathfrak{m}},k)$ of
unequal characteristic is a filtered inductive limit of regular local rings
essentially of finite type over $\bf Z$.

###### Proof.

Let $x_{1},\ldots,x_{n}\in A$, $n=\dim A$ be a system of elements defining a
regular system of parameters in R, $E\subset A$ a finite type
${\mathfrak{N}}$-algebra and $v:E\rightarrow A$ be the inclusion. Suppose that
$n>1$ by Theorem 15 and let $p\equiv
t_{0}\Pi_{i=1}^{l_{1}}z_{1i}^{\alpha_{1i}}$ modulo $(\gamma)$ for some
$\alpha_{1i}\in{\bf N}$, $t_{0}\in A\setminus{\mathfrak{m}}$ and some elements
$z_{1i}$ of $A$ inducing irreducible elements in $R$, with
$(z_{1i},\gamma)\not=(z_{1i^{\prime}},\gamma)$ for $i\not=i^{\prime}$, $R$
being a unique factorization domain. Moreover, we may suppose that
$p=t_{0}\Pi_{i=1}^{l_{1}}z_{1i}^{\alpha_{1i}}+\sum_{j=1}^{m}t_{j}\gamma_{j}$
for some $t_{j}\in A$, $j\in[m]$.

As in Proposition 12 we may assume that $v(H_{E/{\mathfrak{N}}})A$ contains a
power of $p$ and as in Theorem 15 we may assume that
$E\cong{\mathfrak{N}}[Y]/I$, $Y=(Y_{1},\ldots Y_{m})$ and that a power $d$ of
$p$ is in $((f):I)\Delta_{f}$ for some $f=(f_{1},\ldots,f_{r})$, $r\leq s$
from $I$. Then $d$ has the form $d\equiv P=\sum_{i=1}^{q}M_{i}L_{i}\
\mbox{modulo}\ I$, for some $r\times r$ minors $M_{i}$ of $(\partial
f/\partial Y)$ and $L_{i}\in((f):I)$.

Consider as in Theorem 15 the Cohen ${\mathfrak{N}}$-algebra
$(\Lambda_{1i},(p,\Gamma),\mbox{Fr}(A/(z_{1i},\gamma)))$ with residue field
Fr$(A/(z_{1i},\gamma))$, that is $\Lambda_{1i}$ is complete and the map
${\mathfrak{N}}\rightarrow\Lambda_{1i}$ is flat (even regular). $\Lambda_{1i}$
is a filtered inductive limit of some Noetherian local
${\mathfrak{N}}$-algebras $(C_{ij})_{j}$ essentially smooth (see Theorem 11).
Then the completion of the DVR $R_{(z_{1i})}$ is a finite free module over the
DVR $\Lambda_{1i}/\Gamma\Lambda_{1i}$ with the base
$\\{1,z_{1i},\ldots,z_{1i}^{\alpha_{1i}-1}\\}$ and so
$(A/(\gamma,d^{3}))_{(z_{1i})}$ is a finite free module over
$\Lambda_{1i}/(\Gamma,d^{3})$ with the same base. It follows that
$(A/(d^{3}))_{(z_{1i},\gamma)}$ is a finite free module over
$\Lambda_{1i}/(d^{3})$ with the same base.

Set $S=A\setminus\cup_{i=1}^{l_{i}}(z_{1i},\gamma)A$ and
${\tilde{A}}:=A/(d^{3})$,
${\tilde{{\mathfrak{N}}}}:={{\mathfrak{N}}}/(d^{3})$. Then $S^{-1}{\tilde{A}}$
is a product of the local Artinian rings ${\tilde{A}}_{(z_{1i},\gamma)}$ which
are finite free modules over $({\tilde{\Lambda}}_{1i})=\Lambda_{1i}/(d^{3})$,
the base of $S^{-1}{\tilde{A}}$ over
${\tilde{\Lambda}}_{1}:=\Pi_{i=1}^{l_{1}}{\tilde{\Lambda}}_{1i}$ is given by

$\\{z_{11}^{\varepsilon_{11}}\cdots
z_{1l_{1}}^{\varepsilon_{1l_{1}}}:0\leq\varepsilon_{11}<\alpha_{11},\ldots,0\leq\varepsilon_{1l_{1}}<\alpha_{1l_{1}}\\}.$

Let $t_{j}\equiv\sum
t_{j\varepsilon_{11}\ldots\varepsilon_{1l_{1}}}z_{11}^{\varepsilon_{11}}\cdots
z_{1l_{1}}^{\varepsilon_{1l_{1}}}$ modulo $(d^{3})$ for some
$t_{j\varepsilon_{11}\ldots\varepsilon_{1l_{1}}}\in{\tilde{\Lambda}}_{1}$.

Then we may choose some smooth $\tilde{{\mathfrak{N}}}$-algebras of type
$({\tilde{C}}_{1i})_{i}$ such that the limit maps give a map
${\tilde{\psi}}:{\tilde{C}}_{1}:=\Pi_{i=1}^{l_{1}}{\tilde{C}}_{1i}\rightarrow{\tilde{\Lambda}}_{1}$
whose image contains $(t_{j\varepsilon_{11}\ldots\varepsilon_{1l_{1}}})$, let
us say
$t_{j\varepsilon_{11}\ldots\varepsilon_{1l_{1}}}={\tilde{\psi}}(t^{\prime}_{j\varepsilon_{11}\ldots\varepsilon_{1l_{1}}})$
for some
$t^{\prime}_{j\varepsilon_{11}\ldots\varepsilon_{1l_{1}}}\in{\tilde{C}}_{1}$.
Let
${\tilde{\varphi}}_{1}:{\tilde{C}}_{1}[Z_{11},\ldots,Z_{1l_{1}}]\rightarrow
S^{-1}{\tilde{A}}$ for some new variables $Z_{11},\ldots,Z_{1l_{1}}$ be the
map induced by $\tilde{\psi}$ and $Z_{1i}\rightarrow z_{1i}$. Note that

$t^{\prime}_{j}=\sum
t^{\prime}_{j\varepsilon_{11}\ldots\varepsilon_{1l_{1}}}Z_{11}^{\varepsilon_{11}}\cdots
Z_{1l_{1}}^{\varepsilon_{1l_{1}}}\in{\tilde{C}}_{1}[Z_{11},\ldots,Z_{1l_{1}}]$

is mapped to $t_{j}$ modulo $d^{3}$ and $S^{-1}({\tilde{A}})$ is a filtered
inductive limit of localizations of $\tilde{{\mathfrak{N}}}$-algebras of
finite type

${\tilde{B}}_{1}:={\tilde{C}}_{1}[Z_{11},\ldots,Z_{1l_{1}}]/(p-t_{0}^{\prime}Z_{11}^{\alpha_{11}}\cdots
Z_{1l_{1}}^{\alpha_{1l_{1}}}-\sum_{j\in[m]}t^{\prime}_{j}\Gamma_{j}).$

Thus the composite map $E\xrightarrow{v}A\rightarrow S^{-1}{\tilde{A}}$
factors through such an algebra ${\tilde{B}}_{1}$.

Clearly, ${\tilde{C}}_{1}$ is a localization of a finite type
$\tilde{{\mathfrak{N}}}$-algebra ${\tilde{C}}$ with
${\tilde{\varphi}}_{1}({\tilde{C}})\subset{\tilde{A}}$ and
${\tilde{\varphi}}_{1}(H_{{\tilde{C}}/{\tilde{{\mathfrak{N}}}}})\not\subset\cup_{i=1}^{l_{1}}(z_{1i},\gamma){\tilde{A}}$.
Unfortunately we cannot assume that
$t_{j}^{\prime}\in{\tilde{C}}[Z_{11},\ldots,Z_{1l_{1}}]$ but we may suppose
that there exist $c\in{\tilde{C}}$ and
$t_{j}^{\prime\prime}\in{\tilde{C}}[Z_{11},\ldots,Z_{1l_{1}}]$ such that
$ct_{j}^{\prime}=t_{j}^{\prime\prime}$ and
${\tilde{\varphi}}_{1}(c)\in{\tilde{A}}\setminus\cup_{i=1}^{l_{i}}(z_{1i},\gamma){\tilde{A}}$.
It follows that ${\tilde{B}}_{1}$ is a localization of

${\tilde{B}}={\tilde{C}}[T^{\prime},Z_{11},\ldots,Z_{1l_{1}}]/(cT_{0}^{\prime}-t_{0}^{\prime\prime},\ldots,cT_{m}^{\prime}-t_{m}^{\prime\prime},p-T_{0}^{\prime}Z_{11}^{\alpha_{11}}\cdots
Z_{1l_{1}}^{\alpha_{1l_{1}}}-\sum_{j\in[m]}T^{\prime}_{j}\Gamma_{j}),$

$T^{\prime}=(T^{\prime}_{0},\ldots,T^{\prime}_{m})$ being some new variables,
and we may consider the map
${\tilde{B}}\xrightarrow{{\tilde{\varphi}}}{\tilde{A}}$ given by
$T^{\prime}\rightarrow(t_{0},\ldots,t_{m})$ and $Z_{1j}\rightarrow z_{1j}$
modulo $d^{3}$.

Moreover, we may assume that the composite map
$E\xrightarrow{v}A\rightarrow{\tilde{A}}$ factors through $\tilde{\varphi}$.
The map
${\tilde{G}}_{1}:={\tilde{C}}[T^{\prime},Z_{11},\ldots,Z_{1l_{1}}]/(cT_{0}^{\prime}-t_{0}^{\prime\prime},\ldots,cT^{\prime}_{m}-t^{\prime\prime}_{m})\xrightarrow{{\tilde{w}}_{1}}{\tilde{A}}$
given by $T^{\prime}\rightarrow(t_{0},\ldots,t_{m})$, $Z_{1j}\rightarrow
z_{1j}$ modulo $d^{3}$ satisfies
${\tilde{w}}_{1}(H_{{\tilde{G}}_{1}/{\tilde{{\mathfrak{N}}}}})\not\subset\cup_{i=1}^{l_{i}}(z_{1i},\gamma){\tilde{A}}$.
Clearly,

${\tilde{B}}={\tilde{G}}_{1}/(p-T_{0}^{\prime}Z_{11}^{\alpha_{11}}\cdots
Z_{1l_{1}}^{\alpha_{1l_{1}}}-\sum_{j\in[m]}T^{\prime}_{j}\Gamma_{j})).$

We show by induction on $n$, that there exist an essentially smooth local
${\mathfrak{N}}$-algebra $(D,{\mathfrak{q}})$ with $p$ from a system of
regular parameters modulo $(\Gamma)$ of $D/\Gamma D$ and
$b\in{\mathfrak{q}}^{2}$ such that $v$ factors through a smooth
$D/(p-b)$-algebra $D^{\prime}$. This is enough because $D/(p-b,\Gamma)$ is
still regular and so $D^{\prime}/\Gamma D^{\prime}$ is too. The case $n=1$ is
done in Theorem 15.

Case $n=2$

We may suppose that $x_{2}$ is regular in $\tilde{A}$. If
${\tilde{w}}_{1}^{\prime}(H_{{\tilde{G}}_{1}/{\tilde{{\mathfrak{N}}}}}){\tilde{A}}={\tilde{A}}$
then we may change ${\tilde{G}}_{1}$ to be essentially smooth over
$\tilde{{\mathfrak{N}}}$ (see e.g. [20, Lemma 2.4]). Thus we may consider that
${\tilde{G}}_{1}/\Gamma{\tilde{G}}_{1}$ and hence also
${\tilde{B}}/\Gamma{\tilde{B}}$ are regular local rings which ends the proof
using Proposition 13 as in Theorem 15.

Otherwise, the ideal
${\tilde{w}}_{1}^{\prime}(H_{{\tilde{G}}_{1}/{\tilde{{\mathfrak{N}}}}}){\tilde{A}}$
is ${\mathfrak{m}}$-primary and a power $x_{2}^{e}$ of $x_{2}$ is in
${\tilde{w}}_{1}^{\prime}(H_{{\tilde{G}}_{1}/{\tilde{{\mathfrak{N}}}}}){\tilde{A}}$.
Moreover as above we may assume that
$x_{2}^{e}\equiv\sum_{i}M^{\prime}_{i}N^{\prime}_{i}$ modulo $I^{\prime}$ for
some $M^{\prime}_{i},N^{\prime}_{i},I^{\prime}$ associated to
${\tilde{G}}_{1}$, which has the form
${{\tilde{{\mathfrak{N}}}}}[Y^{\prime}]/{\tilde{I}}$,
$Y^{\prime}=(Y^{\prime}_{1},\ldots,Y^{\prime}_{m^{\prime}})$.

Note that $\bar{A}:=A/(x_{2}^{3e})$ is a flat
$\bar{{\mathfrak{N}}}={\mathfrak{N}}[X_{2}]/(X_{2}^{3e})$-algebra,
$X_{2}\rightarrow x_{2}$. By Theorem 15 applied to $(\Gamma,X_{2})$, $(s,3e)$
we see that $\bar{A}:=A/(x_{2}^{3e})$ is a filtered inductive limit of some
Noetherian local $\bar{{\mathfrak{N}}}$-algebras of type $G/(p-b)$, where
$(G,{\mathfrak{a}})$ is essentially smooth over $\bar{{\mathfrak{N}}}$, $p$ is
from a regular system of parameters modulo $(\Gamma,X_{2})$ of
$G/(\Gamma,X_{2})G$ and $b\in{\mathfrak{a}}^{2}$. Let
$\bar{\omega}:G/(p-b)\rightarrow\bar{A}$ be the limit map. We may suppose that
the composite map
${\tilde{G}}_{1}\xrightarrow{{\tilde{w}}_{1}}{\tilde{A}}\rightarrow{\tilde{A}}/(x_{2}^{3e})={\bar{A}}/(d^{3})$
factors through
${\tilde{\mathfrak{N}}}\otimes_{{\mathfrak{N}}}{\bar{\omega}}$.

We claim that there exists a Noetherian local ${\mathfrak{N}}$-algebra $D$
such that $D/(\Gamma)$ is regular and $\bar{\omega}$ factors through
$D/X_{2}^{3e}D$. Indeed, $G$ is a localization of a
$\bar{\mathfrak{N}}$-algebra of type $({\bar{\mathfrak{N}}}[W]/(g))_{\partial
g/\partial W_{1}}$ for some variables $W=(W_{1},\ldots,W_{\lambda})$ and $g$ a
monic polynomial in $W_{1}$ since $G$ is essentially smooth over
$\bar{\mathfrak{N}}$. Take $\varepsilon$ in $A$ lifting ${\bar{\omega}}(W)$
and $\delta,\delta^{\prime}\in A$ such that $g(\varepsilon)=x_{2}^{3e}\delta$,
$(p-b)(\varepsilon)=x_{2}^{3e}\delta^{\prime}$ and let $D$ be the
corresponding localization of

$({\mathfrak{N}}[X_{2},W,\Delta,\Delta^{\prime}]/(g-X_{2}^{3e}\Delta,p-b-
X_{2}^{3e}\Delta^{\prime}))_{\partial g/\partial W_{1}},$

$\Delta,\Delta^{\prime}$ being new variables. Let $\rho:D\rightarrow A$ be the
map given by $W\rightarrow g(W)$, $\Delta\rightarrow\delta$,
$\Delta^{\prime}\rightarrow\delta^{\prime}$. Then $\bar{\omega}$ factors
through $D/X_{2}^{3e}D\otimes\rho$. Since
$({\bar{\mathfrak{N}}}[W,\Delta]/(g-X_{2}^{3e}\Delta))_{\partial g/\partial
W_{1}}$ is smooth over $\bar{\mathfrak{N}}$, the corresponding localization
$C$ of $({\mathfrak{N}}[X_{2},W,\Delta]/(g-X_{2}^{3e}\Delta))_{\partial
g/\partial W_{1}}$ is such that $C/\Gamma C$ is regular and $p$ is from a
regular local system of parameters of $C/\Gamma C$. It follows that $D/\Gamma
D$ is regular local, which shows our claim.

Using Proposition 13 for
${\tilde{\mathfrak{N}}}[X_{2}]\rightarrow{\tilde{A}}$, $X_{2}\rightarrow
x_{2}$,
${\tilde{\mathfrak{N}}}[X_{2}]\otimes_{\tilde{\mathfrak{N}}}{\tilde{G}}_{1}$,
${\tilde{D}}=D/d^{3}D$ and
${\tilde{{\mathfrak{N}}}}\otimes_{{\mathfrak{N}}}\rho$ we see that
${\tilde{w}}_{1}$ factors through an essentially smooth $\tilde{D}$-algebra
${\tilde{D}}^{\prime}$. Then we may find an essentially smooth $D$-algebra
$D^{\prime}$ such that
${\tilde{{\mathfrak{N}}}}\otimes_{{\mathfrak{N}}}D^{\prime}\cong{\tilde{D}}^{\prime}$.
Clearly, $D^{\prime}/\Gamma D^{\prime}$ is regular because $D/\Gamma D$ is.
The map ${\tilde{D}}^{\prime}\rightarrow{\tilde{A}}$ lifts by the Implicit
Function Theorem to a $D$-morphism $\rho^{\prime}:D^{\prime}\rightarrow A$
such that $\tilde{{\mathfrak{N}}}\otimes_{{\mathfrak{N}}}v$ factors through
$\rho^{\prime}$ modulo $d^{3}$. Applying again Proposition 13 for
${{\mathfrak{N}}}\rightarrow A$, $E,D^{\prime},\rho^{\prime}$ we see that $v$
factors through an essentially smooth $D^{\prime}$-algebra $D^{\prime\prime}$,
such that $D^{\prime\prime}/\Gamma D^{\prime\prime}$ is regular since
$D^{\prime}/\Gamma D^{\prime}$ is.

Case $n>2$

As above we may suppose that
${\tilde{w}}_{1}^{\prime}(H_{{\tilde{G}}_{1}/{\tilde{{\mathfrak{N}}}}}){\tilde{A}}\subset{\mathfrak{m}}{\tilde{A}}$.
We will show that given a prime ideal $q\in\operatorname{Spec}A$ such that
$q{\tilde{A}}$ is a minimal prime ideal of
${\tilde{w}}_{1}^{\prime}(H_{{\tilde{G}}_{1}/{\tilde{{\mathfrak{N}}}}})$, the
map ${\tilde{w}}_{1}^{\prime}$ factors through a finite type
$\tilde{{\mathfrak{N}}}$-algebra of the form ${\tilde{P}}/(p-b)$,
$b\in{\tilde{P}}$, let us say ${\tilde{w}}_{1}^{\prime}$ is the composite map
${\tilde{G}}_{1}\rightarrow{\tilde{P}}/(p-b)\xrightarrow{{\tilde{\mu}}}{\tilde{A}}^{\prime}$,
where ${\tilde{A}}^{\prime}$ is a factor of $\tilde{A}$, ${\tilde{\mu}}$ is
induced by a map
${\tilde{\mu}}^{\prime}:{\tilde{P}}\rightarrow{\tilde{A}}^{\prime}$ with
${\tilde{w}}_{1}^{\prime}(H_{{\tilde{G}}_{1}/{\tilde{{\mathfrak{N}}}}}){\tilde{A}}^{\prime}\subset\sqrt{{\tilde{\mu}}(H_{{\tilde{P}}/{\tilde{{\mathfrak{N}}}}}){\tilde{A}}^{\prime}}\not\subset
q{\tilde{A}}^{\prime}$.

For the beginning we assume that height $q=2$. Then
$qA_{q}=(z_{1},z_{2},\gamma)A_{q}$ for some $z_{1},z_{2}\in q$ because $R_{q}$
is a regular local ring of dimension $2$. Assume that $(p,z_{2},\gamma)A_{q}$
is a $qA_{q}$-primary ideal. Let
${\mathfrak{N}}_{2}={\mathfrak{N}}[Z_{2}]_{(p,Z_{2},\Gamma)}$ and
$\Lambda_{2}$ be the (unique) Cohen ${\mathfrak{N}}_{2}$-algebra with the
residue field Fr$(A/q)$, that is $\Lambda_{2}$ is complete and the map
${\mathfrak{N}}_{2}\rightarrow\Lambda_{2}$ is flat (even regular). Then the
completion ${\hat{R}}_{q}$ of the regular local ring $R_{q}$ is a finite free
module over $\Lambda_{2}/(\Gamma)$. We have
$\lambda_{2}z_{2}^{e_{2}}\in{\tilde{w}}_{1}^{\prime}(H_{{\tilde{G}}_{1}/{\tilde{{\mathfrak{N}}}}}){\tilde{A}}$
for some $\lambda_{2}\in A\setminus q$ and $e_{2}\in{\bf N}$. Changing $z_{2}$
by $\lambda_{2}z_{2}$ we may suppose that $\lambda_{2}=1$. Set
${\tilde{\mathfrak{N}}}_{2}={\mathfrak{N}}_{2}/(d^{3})$. Changing
${\tilde{G}}_{1}$ by
${\tilde{\mathfrak{N}}}_{2}\otimes_{{\tilde{\mathfrak{N}}}}{\tilde{G}}_{1}$
and ${\tilde{w}}_{1}^{\prime}$ canonically we may suppose that
${\tilde{G}}_{1}$ is a ${\tilde{\mathfrak{N}}}_{2}$-algebra.

Similar to Case $n=2$ we may assume that
${\tilde{G}}_{1}\cong{\tilde{{\mathfrak{N}}}_{2}}[Y^{\prime}]/{\tilde{I}}$ and
that $z_{2}^{e_{2}}\equiv\sum_{j}M_{j}L_{j}$ modulo ${\tilde{I}}$ for a system
of polynomials $f^{(2)}$ from ${\tilde{I}}$, some polynomials
$L_{j}\in((f^{(2)}):{\tilde{I}})$ and some minors $M_{j}$ of $(\partial
f^{(2)}/\partial Y^{\prime})$. Set
${\tilde{\Lambda}_{2}}=\Lambda_{2}/(d^{3})$. Note that
${\tilde{A}}_{q}/(z_{2}^{3e_{2}},\gamma)\cong\hat{A_{q}}/(d^{3},z_{2}^{3e_{2}},\gamma)$
is finite free over $\tilde{\Lambda}_{2}/(Z_{2}^{3e_{2}},\Gamma)$ with the
basis $\\{z_{1}^{i}z_{2}^{j}:(i,j)\in N_{2}\\}$ for some
$N_{2}\subset[0,3e_{1}]\times[0,3e_{2}]$ and $e_{1}\in{\bf N}$ (we may suppose
$p\not\in(z_{1}^{e_{1}},z_{2}^{e_{2}})$ increasing $e_{1}$ and $e_{2}$
changing $L_{i}$ if necessary). It follows that
${\tilde{A}}_{q}/(z_{2}^{3e_{2}})$ is finite free over
$\tilde{\Lambda}_{2}/(Z_{2}^{3e_{2}})$ with the same basis. We have

$p\equiv\sum_{(i,j)\in
N_{2}^{\prime}}t_{20ij}z_{1}^{i}z_{2}^{j}+\sum_{\lambda\in[m],(i,j)\in
N_{2}^{\prime\prime}}t_{2\lambda ij}z_{1}^{i}z_{2}^{j}\gamma_{\lambda}$

modulo $(d^{3},z_{2}^{3e_{2}})$ for some $t_{20ij}\in A\setminus q$,
$t_{2\lambda ij}\in A$ and some subsets
$N_{2}^{\prime},N_{2}^{\prime\prime}\subset[0,3e_{1}]\times[0,3e_{2}]$.
$\Lambda_{2}$ is a filtered inductive limit of smooth
${{\mathfrak{N}}}_{2}$-algebras $C_{2}$ (see Theorem 11) and let
${\tilde{\psi}}_{2}:{\tilde{C}}_{2}/(Z_{2}^{3e_{2}})\rightarrow{\tilde{\Lambda}}_{2}/(Z_{2}^{3e_{2}})$,
${\tilde{C}}_{2}:=C_{2}/(d^{3})$ be the limit map. We may choose
${\tilde{C}}_{2}$ such that the image of the map

${\tilde{\psi}}_{2}^{\prime}:{\tilde{C}}_{2}[Z_{1}]/(Z_{1}^{3e_{1}},Z_{2}^{3e_{2}})\rightarrow{\tilde{A}}_{q}/(z_{1}^{3e_{1}},z_{2}^{3e_{2}}),$

$Z_{1}\rightarrow z_{1}$ contains $(t_{2\lambda ij})$, let us say $t_{2\lambda
ij}={\tilde{\psi}}_{2}^{\prime}(t_{2\lambda ij}^{\prime})$ for some
$t^{\prime}_{2\lambda ij}\in{\tilde{C}}_{2}[Z_{1}]$, $0\leq\lambda\leq m$.
Then

$t_{2}^{\prime}:=\sum_{(i,j)\in
N_{2}^{\prime}}t_{20ij}^{\prime}Z_{1}^{i}Z_{2}^{j}+\sum_{(i,j)\in
N_{2}^{\prime\prime},\lambda\in[m]}t_{2\lambda
ij}^{\prime}Z_{1}^{i}Z_{2}^{j}\Gamma_{\lambda}$

is mapped in $p$ modulo $(d^{3},z_{2}^{3se_{2}})$ by
${\tilde{\psi}_{2}^{\prime}}$.

Note that ${\tilde{C}}_{2}$ is a localization of a finite type
$\tilde{{\mathfrak{N}}}_{2}$-algebra ${\tilde{C}}_{2}^{\prime}$ with
${\tilde{\psi}}_{2}^{\prime}({\tilde{C}}_{2}^{\prime})\subset{\tilde{A}}/(z_{2}^{e_{2}})$
and
${\tilde{\psi}}_{2}^{\prime}(H_{{\tilde{C}}_{2}^{\prime}/{\tilde{{\mathfrak{N}}}_{2}}})\not\subset
q{\tilde{A}}/(z_{2}^{3e_{2}})$. Usually we have $t_{2\lambda
ij}^{\prime}\not\in{\tilde{C}}_{2}^{\prime}[Z_{1}]$ but there exist
$c,t^{\prime\prime}_{2\lambda ij}\in{\tilde{C}}_{2}^{\prime}[Z_{1}]$ such that
${\tilde{\psi}}_{2}^{\prime}(c)\in{\tilde{A}}/(z_{1}^{3e_{1}},z_{2}^{3e_{2}})\setminus
q{\tilde{A}}/(z_{1}^{3se_{1}},z_{2}^{3e_{2}})$ and $ct^{\prime}_{2\lambda
ij}=t^{\prime\prime}_{2\lambda ij}$. Since
${\tilde{A}}_{q}/(z_{1}^{3e_{1}},z_{2}^{3e_{2}})$ is a filtered inductive
limit of ${\mathfrak{N}}_{2}$-algebras of type
${\tilde{B}}_{2}={\tilde{C}}_{2}[Z_{1}]/(Z_{1}^{3e_{1}},Z_{2}^{3e_{2}},p-t^{\prime}_{2})$
we see that the composite map
${\tilde{G}}_{1}\xrightarrow{{\tilde{w}}^{\prime}_{1}}{\tilde{A}}\rightarrow{\tilde{A}}_{q}/(z_{1}^{3e_{1}},z_{2}^{3e_{2}})$
factors to a such ${\mathfrak{N}}_{2}$-algebra ${\tilde{B}}_{2}$.

Set
${\tilde{P}}_{2}:={\tilde{C}}_{2}^{\prime}[Z_{1},T^{\prime}]/(cT^{\prime}_{(2\lambda
ij}-t^{\prime\prime}_{20\lambda ij})_{\lambda
ij},Z_{1}^{3e_{1}},Z_{2}^{3e_{2}})$ for some new variables
$T^{\prime}=(T_{2\lambda ij}^{\prime})_{\lambda ij}$. Let
${\tilde{\mu}}_{2}^{\prime}:{\tilde{P}}_{2}\rightarrow{\tilde{A}}/(z_{1}^{3e_{1}},z_{2}^{3e_{2}})$
given by $T^{\prime}\rightarrow(t_{2\lambda ij})$. Note that
${\tilde{\mu}}_{2}^{\prime}(H_{{\tilde{P}}_{2}/{\tilde{\mathfrak{N}}}_{2}})\not\subset
q{\tilde{A}}/(z_{1}^{3e_{1}},z_{2}^{3e_{2}})$. We may assume that the
composite map
${\tilde{G}}_{1}\xrightarrow{{\tilde{w}}^{\prime}_{1}}{\tilde{A}}\rightarrow{\tilde{A}}/(z_{1}^{3e_{1}},z_{2}^{3e_{2}})$
factors to a ${\mathfrak{N}}_{2}$-algebra

${\tilde{B}}_{2}^{\prime}={\tilde{P}}_{2}/(p-\sum_{(i,j)\in
N_{2}^{\prime}}T_{20ij}^{\prime}Z_{1}^{i}Z_{2}^{j}+\sum_{(i,j)\in
N_{2}^{\prime\prime},1\leq\lambda\leq m}T_{2\lambda
ij}^{\prime}Z_{1}^{i}Z_{2}^{j}\Gamma_{\lambda}).$

Step by step, using this procedure we may find some elements $(z_{i})_{2\leq
i\leq\nu}$, some numbers $(e_{i})_{i}$, some finite type
$\tilde{{\mathfrak{N}}}_{i}={\tilde{\mathfrak{N}}}_{2}[Z_{3},\ldots,Z_{i}]/(Z_{2}^{3e_{2}},\ldots,Z_{i}^{3e_{i}})$-algebras
${\tilde{P}}_{i}$, some morphisms
${\tilde{w}}^{\prime}_{i}:{\tilde{P}}_{i}\rightarrow
A_{i}:={\tilde{A}}/(z_{3}^{3e_{2}},\ldots,z_{i}^{3e_{i}})$ and some elements
$b_{i}\in{\tilde{P}}_{i}$, $3\leq i\leq\nu$ such that for all $3\leq i\leq\nu$

  1. (1)

the composite map
${\tilde{P}}_{i-1}\xrightarrow{{\tilde{w}}^{\prime}_{i-1}}A_{i-1}\rightarrow
A_{i}$ factors by ${\tilde{P}}_{i}/(p-b_{i})$ namely
${\tilde{w}}^{\prime}_{i-1}$ is the composite map
${\tilde{P}}_{i-1}\rightarrow{\tilde{P}}_{i}/(p-b_{i})\xrightarrow{{\tilde{w}}_{i}}A_{i}$,
where ${\tilde{w}}_{i}$ is induced by ${\tilde{w}}^{\prime}_{i}$,

  2. (2)

there exists a strict inclusion
$\sqrt{{\tilde{w}}^{\prime}_{i-1}(H_{{\tilde{P}}_{i-1}/{\tilde{\mathfrak{N}}}_{i-1}})A_{i}}\subset\sqrt{{\tilde{w}}^{\prime}_{i}(H_{{\tilde{P}}_{i}/{\tilde{{\mathfrak{N}}}_{i}}}A_{i})}$,

  3. (3)

$(0:_{A_{i-1}}z_{i}^{e_{i}})=(0:_{A_{i-1}}z_{i}^{e_{i}+1})$ and
$z_{i}^{e_{i}}\in{\tilde{w}}^{\prime}_{i-1}(H_{{\tilde{P}}_{i-1}/{\tilde{{\mathfrak{N}}}}_{i-1}})A_{i}$.

By Noetherian induction we may assume that
${\tilde{w}}^{\prime}_{\nu}(H_{{\tilde{P}}_{\nu}/{\tilde{{\mathfrak{N}}}}_{\nu}}){\tilde{A}}_{\nu}$
is ${\mathfrak{m}}{\tilde{A}}$-primary. We may choose $x_{n}$ such that it is
regular in $(R_{i})_{i\in[\nu]}$ and a power $x_{n}^{e^{\prime}}$ of $x_{n}$
is in
${\tilde{w}}^{\prime}_{\nu}(H_{{\tilde{P}}_{\nu}/{\tilde{{\mathfrak{N}}}}_{\nu}}){\tilde{A}}_{\nu}$
(we can assume that $x_{n}^{e^{\prime}}\in\Delta_{f}((f):J)$ for some $f,J$
defining $P_{\nu}$ as usual).

Induct on $n>1$. Using the induction hypothesis, as in Case $n=2$,
$A/(x_{n}^{3e^{\prime}})$ is a filtered inductive limit of some Noetherian
local
${\tilde{\mathfrak{N}}}^{\prime}={\tilde{\mathfrak{N}}}[X_{n}]/(X_{n}^{3e^{\prime}})$-algebras
of type $G/(p-b)$, where $(G,{\mathfrak{a}})$ is essentially smooth over
${\tilde{\mathfrak{N}}}^{\prime}$, $p$ is from a system of regular parameters
of $G/(\Gamma,X_{n})$ and $b\in{\mathfrak{a}}^{2}$. Let
${\bar{\omega}}:G/(p-b)\rightarrow A/(x_{n}^{3e^{\prime}})$ be the limit map.
Then we find as in Case $n=2$ a Noetherian local ${\mathfrak{N}}$-algebra $D$
such that $D/(X_{n}^{3e^{\prime}})\cong/(p-b)$, $D/(\Gamma)$ is regular and
the composite map
${\tilde{P}}_{\nu}\xrightarrow{{\tilde{w}}^{\prime}_{\nu}}A_{\nu}\rightarrow
A_{\nu}/(x_{n}^{3e^{\prime}})$ factors through
${\tilde{\mathfrak{N}}}_{\nu}\otimes_{\tilde{\mathfrak{N}}}{\bar{\omega}}$. As
in Case $n=2$ using again Proposition 13 we find a Noetherian
${\mathfrak{N}}$-algebra $D^{\prime}$ and a map
$\rho^{\prime}:D^{\prime}\rightarrow A$ such that
$D^{\prime}/(\Gamma)D^{\prime}$ is regular and ${\tilde{w}}^{\prime}_{\nu}$
factors through
${\tilde{\mathfrak{N}}}_{\nu}\otimes_{{\mathfrak{N}}}D^{\prime}$.

In particular, the composite map
${\tilde{P}}_{\nu-1}\xrightarrow{{\tilde{w}}^{\prime}_{\nu-1}}A_{\nu-1}\rightarrow
A_{\nu}$ factors through
${\tilde{\mathfrak{N}}}_{\nu}\otimes_{\varphi}D^{\prime}$ and the procedure
continues. By recurrence using Proposition 13 and Remark 14 we find a
Noetherian local ${\mathfrak{N}}$-algebra $D^{\prime\prime}$ essentially
smooth over $D$ and a map $\rho^{\prime\prime}:D^{\prime\prime}\rightarrow A$
such that $D^{\prime\prime}/\Gamma D^{\prime\prime}$ is regular and
${\tilde{w}}_{1}^{\prime}$ factors through
${\tilde{\mathfrak{N}}}\otimes_{{\mathfrak{N}}}D^{\prime\prime}$. Finally the
same procedure shows that $v$ factors trough a ${\mathfrak{N}}$-algebra $F$
essentially smooth over $D$, with $F/\Gamma F$ regular.

######  Remark 18.

To show the BQ Conjecture for a regular local ring of unequal characteristic
applying Theorem 17 we need to prove the following facts:

i) if the BQ Conjecture holds for $\Lambda^{\prime}_{i-1}$ then it holds for
$\Lambda_{i}$, $1\leq i\leq r$,

ii) if the BQ Conjecture holds for $\Lambda_{i}$ then it holds for
$\Lambda^{\prime}_{i}$, $1\leq i<r$.

The first fact could be proved using the ideas of Lindel and Swan. For the
second fact we have to show the following result

iii) if the BQ Conjecture holds for a regular local ring $A$ and $b\in A$
belongs to a regular system of parameters of $A$ then it holds for a
localization of $A[W]/(W^{e}-b)$, $e\in{\bf N}$ (in fact for $A[W]/(W^{e}-b)$
using [28]).

This might be wrong as an example from [12] suggests. If $A={\bf
R}[X_{1}]_{(X_{1})}$, $B=(A[X_{2}]/(X_{2}^{3}-X_{1}^{2})))_{(X_{1},X_{2})}$
then there exist projective modules over $B[T]$ of rank one, which are not
free. On the other hand, Vorst shows in [31] that if the BQ Conjecture holds
for a regular local ring $A$ then it holds for a factor of a polynomial
$A$-algebra by a monomial ideal.

######  Corollary 19.

All regular local rings are filtered inductive limits of excellent regular
local rings.

##  3\. Appendix by Kęstutis Česnavičius111Laboratoire de Mathématiques
d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France;
email<EMAIL_ADDRESS>A purity conjecture for flat cohomology

The following purity conjecture seems to be a part of the folklore.

######  Conjecture 20.

For a regular local ring $(R,{\mathfrak{m}})$ of dimension $d$ and a
commutative, finite, flat $R$-group scheme $G$, the fppf cohomology with
supports vanishes in low degrees as follows

$H^{i}_{{\mathfrak{m}}}(R,G)=0\ \ \mbox{for}\ \ i<d;$

equivalently, setting
$U_{R}:=\operatorname{Spec}(R)\setminus\\{{\mathfrak{m}}\\}$, the pullback map

$H^{i}(R,G)\rightarrow H^{i}(U_{R},G)\ \ \mbox{is an isomorphism for}\ \
i<d-1\ \ \mbox{and injective for}\ \ i=d-1.$

As usual, we form the fppf cohomology of a scheme $S$ in the site whose
objects are all the $S$-schemes and coverings are jointly surjective morphisms
that are flat and locally of finite presentation.

Examples of known cases

  1. (1)

In its equivalent form that involves $U_{R}$, the conjecture is known for
cohomological degrees $i=0$ and $i=1$. More precisely, the map

$\ \ H^{0}(R,G)\rightarrow H^{0}(U_{R},G)\ \ \text{is bijective when}\ \ d>0$

by a well-known lemma on sections of finite schemes over normal bases (for the
lack of a better reference, see [5, Lemma 3.1.9]). This lemma and the
representability of $G$-torsors also show that the map

$\ \ H^{1}(R,G)\rightarrow H^{1}(U_{R},G)$

is injective when $d>0$; by a result of Moret-Bailly [17, Lemme 2] (whose
detailed proof may be found, for instance, in [14, §3.1]), this map is
bijective when $d>1$. Of course, there is no need to assume that $G$ is
commutative for these results in low degrees.

  2. (2)

The case when the order of $G$ is invertible in $R$ is also known; in fact, in
this case, by the absolute cohomological purity conjecture proved by Gabber,
$H^{i}_{{\mathfrak{m}}}(R,G)=0$ for $i<2d$, see [8].

  3. (3)

We have $H^{i}_{{\mathfrak{m}}}(R,\mathbb{G}_{a})=0$ for $i<d$ because the
depth of $R$ is $d$ (see [9, III.3.4]), so, in the case when $R$ is an
$\bf{F}_{p}$-algebra, the conjecture holds for $G=\alpha_{p}$ (where
$\alpha_{p}$ is the kernel of the absolute Frobenius morphism of
$\mathbb{G}_{a}$).

  4. (4)

By the purity for the Brauer group, whose proof was finished in [4], the
groups $H^{i}_{{\mathfrak{m}}}(R,\mathbb{G}_{m})$ vanish when $d\geq 2$ and
$i\leq 3$; thus, $H^{i}_{{\mathfrak{m}}}(R,\mu_{p})=0$ when $i\leq 3$ with
$i<d$ (where $\mu_{p}$ is the kernel of the absolute Frobenius morphism of
$\mathbb{G}_{m}$).

The goal of this appendix is to illustrate the main results of the paper with
the following reduction.

######  Proposition 21.

For every regular local ring $(R,{\mathfrak{m}})$ that is a filtered direct
limit of _excellent_ regular local rings, the conjecture above reduces to the
case when $R$ is complete.

###### Proof.

By descending the elements that comprise an $(R/{\mathfrak{m}})$-basis of
${\mathfrak{m}}/{\mathfrak{m}}^{2}$ to a finite level in the direct limit, we
may assume that $(R,{\mathfrak{m}})$ is a filtered direct limit of excellent
regular local rings each of whose dimension is at least that of $R$. Thus, the
limit formalism reduces us to the case when $R$ is excellent. Excision (see
[16, proof of III.1.27]) and [6, 18.7.6] then reduce us further to the case
when $R$ is both excellent and Henselian. We then use the Néron–Popescu
desingularization (see Theorem 11) to express the ${\mathfrak{m}}$-adic
completion $(\widehat{R},\widehat{{\mathfrak{m}}})$ of $(R,{\mathfrak{m}})$ as
a filtered direct limit of essentially smooth, local $R$-algebras
$(R_{j},{\mathfrak{m}}_{j})$. Since $R$ is Henselian local and shares the
residue field with $\widehat{R}$, Hensel’s lemma [6, 18.5.17] ensures that the
structure maps $R\rightarrow R_{j}$ have sections $R_{j}\rightarrow R$. In
particular, they induce the injections

$H^{i}_{{\mathfrak{m}}}(R,G)\hookrightarrow
H^{i}_{{\mathfrak{m}}_{j}}(R_{j},G),\ \ \mbox{which give the injection}\ \
H^{i}_{\mathfrak{m}}(R,G)\hookrightarrow
H^{i}_{\widehat{{\mathfrak{m}}}}(\widehat{R},G).$

This achieves the promised reduction to the case when $R$ is complete.

######  Remark 22.

By Corollary 19, we see that the above Conjecture is reduced to the complete
case.

Acknowledgements:

We owe thanks to Kęstutis Česnavičius for some helpful remarks on Lemmas 5, 9,
Proposition 11, Remark 15 and especially for the appendix and to M. Vâjâitu
and A. Zaharescu who hinted us Remark 4. Also we owe thanks to Moret-Bailly
for some useful remarks on the appendix and to the Referee for many
corrections. This work has been partially elaborated in the frame of the
International Research Network ECO-Math.

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