Unimodular rows over monoid rings

TL;DR

This paper proves that the elementary group acts transitively on unimodular rows over monoid rings for sufficiently large n, extending classical results from polynomial rings and advancing K-theory of monoid rings.

Contribution

It generalizes Suslin's polynomial ring results to monoid rings, completing a long-standing project and opening new avenues in K-theory research.

Findings

Elementary action is transitive on unimodular rows over monoid rings for n >= max(d+2,3)

Extends classical polynomial ring results to more general monoid rings

Provides new insights into the structure of K-theory of monoid rings

Abstract

For a commutative Noetherian ring R of dimension d and a commutative cancellative monoid M, the elementary action on unimodular n-rows over the monoid ring R[M] is transitive for n>=max(d+2,3). The starting point is the case of polynomial rings, considered by A. Suslin in the 1970s. The main result completes a project, initiated in the early 1990s, and suggests a new direction in the study of K-theory of monoid rings.

Equations

$$M(P)=\{m\in M\setminus \{0\}|\Phi (m)\in P\}\cup \{0\}\subset M;$$

$$M(P)=\{z\in {\mathbb{Z}}^r\setminus \{0\}|({\mathbb{R}}_{+}z)\cap \mathcal{H}\in P\}\cup \{0\}.$$

$$\displaystyle\deg\bigg{(}\sum_{j=1}^{r}d_{1j}c_{j}n_{1}\bigg{)}>\deg\bigg{(}\sum_{j=1}^{r}(a_{ij}n_{j}+b_{ij}c_{j}n_{1})\bigg{)},\quad i=2,\ldots,s,$$

$$\displaystyle\deg\bigg{(}\sum_{j=1}^{r}d_{1j}c_{j}n_{1}\bigg{)}>\deg\bigg{(}\sum_{j=1}^{r}d_{ij}c_{j}n_{1}\bigg{)},\quad i=2,\ldots,s,$$

$$\sum _{j=1}^r{d}_{1j}{c}_j>\sum _{j=1}^r{d}_{ij}{c}_j,i=2,\dots ,s,$$

$$\mathcal{H}=\mathrm{Aff}(m,n_1-(k+1)m,\dots ,n_{r-1}-(k+1)m)\subset {\mathbb{R}}^r,$$

$$\displaystyle\Delta_{1}=\big{(}{\mathbb{R}}_{+}m+{\mathbb{R}}_{+}(n_{1}-km)+\cdots+{\mathbb{R}}_{+}(n_{r-1}-km)\big{)}\ \cap\ \mathcal{H},$$

$${\Delta }_2=\mathrm{conv}(m,n_1-(k+1)m,\dots ,n_{r-1}-(k+1)m).$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\begin{aligned} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 45.09969pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\\&\crcr}}}\ignorespaces{\hbox{\kern-45.09969pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R\left[\left(L\setminus U(L)\right)\cup\{1\}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 76.9476pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 0.0pt\raise-60.16669pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 76.9476pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R[L]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 89.96324pt\raise-35.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 89.96324pt\raise-59.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-35.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-6.83507pt\raise-70.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 69.09969pt\raise-70.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 69.09969pt\raise-70.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R[U(L)]}$}}}}}}}\ignorespaces}}}}\ignorespaces,\end{aligned}\qquad\ker\pi=R(L\setminus U(L)).$$

$$(\mathbb{D}_{i})\quad\begin{aligned} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 23.17891pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\\\&&\crcr}}}\ignorespaces{\hbox{\kern-23.17891pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R[L(F_{i})_{*}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 77.17891pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 0.0pt\raise-69.16681pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 47.17891pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 77.17891pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R[L]\big{/}\sum_{j>i}R\operatorname{int}(L(F_{j}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 129.13908pt\raise-39.50006pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.84743pt\hbox{$\scriptstyle{\pi_{i}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 129.13908pt\raise-67.00012pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-39.50006pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-6.83507pt\raise-79.00012pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 77.17891pt\raise-79.00012pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 47.17891pt\raise-79.00012pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 77.17891pt\raise-79.00012pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R[L]\big{/}\sum_{j\geq i}R\operatorname{int}(L(F_{j}))}$}}}}}}}\ignorespaces}}}}\ignorespaces,\\ \\ \ker\pi_{i}=R(L\cap\operatorname{int}(F_{i})).\end{aligned}$$

$$b_1+b{y}^c\notin {\nu }_i,i=1,\dots ,p.$$

$$R[{\mathbb{Z}}_{+}^r]l_1+R[{\mathbb{Z}}_{+}^r]t_1=\cdots =R[{\mathbb{Z}}_{+}^r]l_{n-1}+R[{\mathbb{Z}}_{+}^r]t_1=R[{\mathbb{Z}}_{+}^r].$$

$$\{{\nu }_1,\dots ,{\nu }_p\}=\{\begin{array}{l}\text{the minimal primes over R[Z+r]h1′+⋯+R[Z+r]hk−1′, if k>1},\\ \text{the minimal primes in R[Z+r], if k=1}.\end{array}$$

$$\tilde{h}=f_1{h}_1^{\prime }+\cdots +$$

$$f_1,\dots ,f_{k-1},f_{k+1},\dots ,f_n\in \mathbb{R}[{\mathbb{Z}}_{+}^r],$$

$$h_k^{\prime }+\tilde{h}{t}_1^c\notin {\nu }_s,s=1,\dots ,p.$$

$$h_k^{\prime }+h^{\prime }{t}_1^c\notin {\nu }_s,s=1,\dots ,p,c\in \mathbb{N}.$$

$$h^{\prime }{t}_1^c\in A{h}_{k+1}^{\prime }+\cdots +A{h}_n^{\prime }.$$

$${\mathbf{h}}^{\prime }\underset{A}{\sim }(h_1^{\prime },\dots ,h_{k-1}^{\prime },h_k^{\prime }+h^{\prime }{t}_1^c,h_{k+1}^{\prime },\dots ,h_n^{\prime }).$$

$$R[{\mathbb{Z}}_{+}^r]h_1^{\prime }+\cdots +R[{\mathbb{Z}}_{+}^r]h_{k-1}^{\prime }+R[{\mathbb{Z}}_{+}^r]\left(h_k^{\prime }+h^{\prime }{t}_1^{c_j}\right)\subset R[{\mathbb{Z}}_{+}^r]$$

$$A_2\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Phi (m),\Phi (h_1(t_j)),\Phi (h_2(t_j))\in \mathcal{H},j=2,\dots ,r,$$

$$g_n\text{is quasi-monic, not in}R.$$

$${\Delta }_0=\mathrm{conv}\left(\Phi (t_1),\Phi (t_1^k{t}_2),\dots ,\Phi (t_1^k{t}_d)\right).$$

$$\tau =\tau (c_1,\dots ,c_r):R[{\mathbb{Z}}_{+}^r]⟶R[{\mathbb{Z}}_{+}^r],\tau (t_j)=t_j+t_1^{c_j},j=1,\dots ,r.$$

$$\alpha :={\epsilon }^{-1}\tau (\epsilon )\in {\mathrm{GL}}_n(R[{\mathbb{Z}}_{+}^d({\Delta }_0)]\subset {\mathrm{GL}}_n(A_2)\text{for}{c}_1,\dots ,c_r\gg 0.$$

$$\alpha ={\epsilon }_0\left(\begin{array}{cc}{\alpha }_0& 0\\ 0& 1\end{array}\right).$$

$$(h_1,\dots ,h_{n-1},\tau (g_n)):=\tau (\mathbf{g})\cdot \left(\begin{array}{cc}{\alpha }_0^{-1}& 0\\ 0& 1\end{array}\right)$$

$$=\mathbf{g}\cdot {\epsilon }_0\underset{A_2}{\sim }\mathbf{g}\in {\mathrm{Um}}_n(A_2).$$

$$\mathbf{f}\underset{R[M]}{\sim }(f_1^{\prime },\dots ,f_{n-1}^{\prime },{\pi }_2(\tau (g_n))\in {\mathrm{Um}}_n(R[M])$$

Full text

1. Introduction

An $n$-row $\mathbf{a}=(a_{1},\ldots,a_{n})$ with entries in a commutative ring $R$ is called unimodular if $Ra_{1}+\cdots+Ra_{n}=R$. If $R$ is Noetherian then, using prime avoidance, one can pass from $\mathbf{a}$ by elementary transformations to a unimodular row $\mathbf{b}=(b_{1},\ldots,b_{n})$, such that the height of the ideal $Rb_{1}+\cdots+Rb_{i}\subset R$ is at least $i$ for $i=1,\ldots,n$; see Section 2.1. Here an ‘elementary transformation’ means adding a multiple of a component to another component. In particular, if the (Krull) dimension of $R$ is $d$ and $n\geq d+2$ then every unimodular $n$-row over $R$ can be reduced, by elementary transformations, to $(1,0,\ldots,0)$. This is the basis of the classical Serre Splitting and Bass Cancellation Theorems [1, Ch. 4]. Our main result is

Theorem 1.1**.**

Let $R$ be a commutative Noetherian ring of dimension $d$ and $M$ be a commutative cancellative (not necessarily torsion free) monoid. Then the elementary action on unimodular $n$-rows over $R[M]$ is transitive for $n\geq\max(d+2,3)$.

Observe that the class of monoids in Theorem 1.1 is exactly the class of submonoids of arbitrary abelian groups.

The starting point is Suslin’s result in [20] that, for $(R,d)$ as above and arbitrary $r\in{\mathbb{N}}$, the elementary action on the set of unimodular $n$-rows over the polynomial ring $R[t_{1},\ldots,t_{r}]$ is transitive whenever $n\geq\max(d+2,3)$. The polynomial ring $R[t_{1},\ldots,t_{r}]$ is the monoid ring, corresponding to a free commutative monoid of rank $r$. Theorem 1.1 completes the project, initiated in [8, 9], where the transitivity was shown for a restricted class of monoids. Namely, in the torsion free case, Theorem 1.1 extends the class of monoids in [8, 9] in the same way as the general convex polytopes extend the class of stacked polytopes [3] (called the ‘polytopes of simplicial growth’ in [9]); see Section 2.5 for the relationship between monoids and polytopes. The class of stacked polytopes is negligibly small within general convex polytopes. But also, monoids with torsion have not been considered before.

We now describe consequences and research directions, suggested by Theorem 1.1.

Theorem 1.1 is new already when $R$ is a field, in which case it implies that $K_{1}(R[M])$ is described by Mennicke symbols [1, Ch. 6]. One would like to know what properties of $K_{1}(R[M])$ can be inferred from this fact – as the works [10, 15] show, $K_{1}(R[M])$ exhibits many interesting phenomena.

Using the Quillen induction for projective modules [16, Ch. V.3], one easily deduces from Theorem 1.1 that, for $R$ and $M$ as in the statement and $M$ torsion free without nontrivial units, every finitely generated projective $R[M]$-module of rank greater than $d$, which is stably extended from $R$, is in fact extended from $R$. For regular $R$ and seminormal $M$ this is shown in [22, Corolary 1.4], and for a Dedekind ring $R$ and a torsion free monoid $M$ this follows from [22, Theorem 1.5]. It is very likely that the techniques of [2, 19] can be combined with the proof of Theorem 1.1 to yield the full blown $K_{0}$-part of [11, Conjecture 2.4], claiming that finitely generated projective $R[M]$-modules of rank $\geq\max(d+1,2)$ are cancellative and split off free summands. This would extend [2, 19] in the same way as Theorem 1.1 extends Suslin’s mentioned result. It is also worth mentioning that a new proof of Anderson’s conjecture [6] can be derived from Theorem 1.1, in combination with [14], following the outline in [4, Exercises 8.7, 8.8, 9.4]. Ideologically, this is the same approach, though.

Theorem 1.1 is proved by developing a unimodular row version of the polytopal/algebraic framework, which we call the pyramidal descent. Three versions of this techniques were developed for projective modules [6], classical $K$-groups [7, 14], and higher stable $K$-groups [13]. However, putting the elementary rows in the same framework has resisted the attempts so far – only initial steps were accomplished in [8, 9]. The main obstruction for proving the general transitivity of the elementary action had been the non-existence of special non-monomial endomorphisms in monoid rings for producing monoid ring counterparts of multivariate monic polynomials. Here we find a new approach, allowing to circumvent this difficulty by suitably lifting critical steps to covering polynomial rings, where there is a ubiquity of endomorphisms, and even drop the torsion freeness assumption for monoids. The unimodular row version of the pyramidal descent is different from the previous versions in that no lifting to polynomial rings was used before. More importanty, it also suggests a possible approach to [11, Conjecture 2.4] on stabilizations of all higher $K$-groups of $R[M]$: one could try to show, based on [21, 24], that the mentioned stabilizations are no worse than those for $R[t_{1},\ldots,t_{r}]$, $r\in{\mathbb{N}}$. This would reduce the general case to the free commutative monoids. At present, the appropriate techniques for polynomial rings only exists for projective modules (unstable $K_{0}$) [2, 19], invertible matrices (unstable $K_{1}$) [20], and unstable Milnor groups [23]. Of course, Theorem 1.1 already implies the conjectured surjective $K_{1}$-stabilization.

Finally, by combining the techniques in Section 6.1 and [22, §15], one easily extends Theorem 1.1 to rings of the form $R[M]/I$, where $I$ is generated by a subset of $M$. Also, if $R$ is a field, the proof of Theorem 1.1 can be converted, along the lines in [17], into an algorithm that, for a given unimodular row over $R[M]$, finds elementary transformations, leading to $(1,0,\ldots,0)$.

A word on the organization of the paper: in Sections 2–6 we overview previous results and make series of reductions in the general case; Section 7 allows involvement of the general coefficient rings, without which Section 8 would lead to the special case of Theorem 1.1 when $R$ is field; in Section 9 we explain how to involve the monoids with torsion.

Notation: ${\mathbb{N}}=\{1,2,\ldots\}$ and $\gg$ is for ‘sufficiently larger than’.

Acknowledgment. I am very grateful to the referee for (i) detecting a mistake in the concluding inductive step at the end of Section 8.2 and suggesting a correction (see Remark 8.6), and (ii) suggesting the current form of Lemma 7.1.

2. Unimodular rows, monoids, polytopes

2.1. Unimodular rows

All information we need on unimodular rows (except the case $l>0$ of Theorem 2.2 below), including a detailed exposition of [20], is in [16].

All our rings are commutative and with unit. The elementary subgroup $\operatorname{E}_{n}(A)\subset\operatorname{GL}_{n}(A)$ of the general linear group over $A$ is generated by the elementary $n\times n$-matrices, i.e., the matrices which differ from the identity matrix in at most one nonzero off-diagonal entry. The following two statements are, respectively, [20, Corollary 1.4] and [20, Theorem 7.2]:

Lemma 2.1**.**

For any ring $A$ and $n\geq 3$, $\operatorname{E}_{n}(A)$ is normal in $\operatorname{GL}_{n}(A)$.

Theorem 2.2**.**

Assume $R$ is a Noetherian ring of dimension $d$ and $k,l,n\in{\mathbb{N}}$ with $n\geq\max(d+2,3)$. Then the elementary action on unimodular $n$-rows over the Laurant polynomial ring $R[t_{1},\ldots,t_{k},s_{1}^{\pm 1},\ldots,s_{l}^{\pm 1}]$ is transitive.

For a ring $A$, the set of unimodular rows of length $n$ will be denoted by $\operatorname{Um}_{n}(A)$. The group $\operatorname{E}_{n}(A)$ acts on $\operatorname{Um}_{n}(A)$ by multiplication on the right. More precisely, the right multiplication of an elementary matrix $I_{n}+aE_{ij}$ corresponds to adding the $a$-multiple of the $i$-th component to the $j$-th component.

For two elements $\mathbf{f},\mathbf{g}\in\operatorname{Um}_{n}(A)$ and a subring $S\subset A$, we write $\mathbf{f}\underset{S}{\sim}\mathbf{g}$ if $\mathbf{f}$ and $\mathbf{g}$ are in a same orbit of the $\operatorname{E}_{n}(S)$-action. The standard row $(1,0,\ldots,0)$ will be denoted by $\mathbf{e}$. In particular, $\mathbf{f}\underset{S}{\sim}\mathbf{e}$ means that there a matrix $\varepsilon\in\operatorname{E}_{n}(S)$, whose first row equals $\mathbf{f}$.

In order to avoid confusion between rows and ideals, the ideal in $A$, generated by elements $a_{1},\ldots,a_{n}$, will be denoted by $Aa_{1}+\cdots+Aa_{n}$.

For an ideal $I$ in a ring $A$, its height $\operatorname{ht}I=\operatorname{ht}_{A}I$ is the minimal height of a prime ideal over $I$. The height of the unit ideal is set to be $\infty$. For an ideal $J\subset A[t]$, the ideal of leading coefficients of elements in $J$ will be denoted by $L(I)$.

Lemma 2.3**.**

Let $A$ be a Noetherian ring.

1. (a)

For any element $\mathbf{a}\in\operatorname{Um}_{n}(A)$, there exists $\mathbf{b}=(b_{1},\ldots,b_{n})\in\operatorname{Um}_{n}(A)$, such that $\mathbf{a}\underset{A}{\sim}\mathbf{b}$ and \operatorname{ht}\big{(}Ab_{1}+\cdots+Ab_{i}\big{)}\geq i for all $i$. 2. (b)

For an ideal $J\subset A[t]$, we have $\operatorname{ht}_{A}L(J)\geq\operatorname{ht}_{A[t]}J$.

These are, respectively, Lemmas 3.4 and 3.2 in [16, Ch. III].

2.2. Polytopes and cones

For generalities on polytopes and cones, we refer the reader to [4, Ch. 1]. Here we only recall a few basic facts and conventions.

For a subset $X\subset{\mathbb{R}}^{r}$, its convex hull will be denoted by $\operatorname{conv}(X)$ and the affine hull will be denoted by $\operatorname{Aff}(X)$.

All our polytopes are assumed to be convex.

The empty set is the polytope of dimension $-1$.

A simplex is a polytope of the form $\Delta=\operatorname{conv}(x_{1},\ldots,x_{n})$ such that $\dim\Delta=n-1$.

A pyramid with apex $v$ and base $P$ means $\operatorname{conv}(v,P)$, where $v\notin\operatorname{Aff}(P)$.

The relative interior of a polytope $P$ will be denoted by $\operatorname{int}(P)$. By convention, $\operatorname{int}(P)=P$ when $P$ is a point.

Let ${\mathbb{R}}_{+}$ denote the non-negative reals. For two polytopes $P\subset Q$, sharing a vertex $v$, we say that $Q$ is tangent to P at $v$ if $\dim P=\dim Q$ and we have the equality of sets ${\mathbb{R}}_{+}(P-v)={\mathbb{R}}_{+}(Q-v)$.

A cone will always refer to a rational, finite, pointed cone in ${\mathbb{R}}^{r}$ for some $r$, i.e., a subset of the form $C={\mathbb{R}}_{+}z_{1}+\cdots+{\mathbb{R}}_{+}z_{n}\subset{\mathbb{R}}^{r}$ for some $z_{1},\ldots,z_{n}\in{\mathbb{Z}}^{r}$, which contains no nontrivial subspace.

For a nonzero cone $C\subset{\mathbb{R}}^{r}$, there is a rational affine hyperplane $\mathcal{H}\subset{\mathbb{R}}^{r}\setminus\{0\}$, such that $C={\mathbb{R}}_{+}(C\cap\mathcal{H})$ [4, Proposition 1.21]. In this case $C\cap\mathcal{H}$ is a rational polytope of dimension $\dim C-1$; i.e., the vertices of $C\cap\mathcal{H}$ belong to ${\mathbb{Q}}^{r}$.

2.3. Affine monoids

A detailed information on the monoids of interest can be found in [4, Ch. 2]. Below we give a quick review.

All our monoids are commutative, cancellative, and with unit.

The maximal subgroup of a monoid $M$ will be denoted by $U(M)$.

The group of differences of a monoid $M$, also known as the Grothendieck group of $M$, will be denoted by $\operatorname{gp}(M)$. Thus $M$ embeds into $\operatorname{gp}(M)$ and $\operatorname{gp}(-)$ is a left adjoint of the embedding of the category of monoids into that of abelian groups.

The additive monoid of nonnegative integers will be denoted by ${\mathbb{Z}}_{+}$.

Unless specified otherwise, (i) we use additive notation for the operation in a monoid $M$ but switch to the multiplicative notation when $M$ is considered inside the monoid ring $R[M]$, and (ii) we will make the natural identifications $R[{\mathbb{Z}}_{+}]=R[t]$ and $R[{\mathbb{Z}}_{+}^{r}]=R[t_{1},\ldots,t_{r}]$.

The submonoid of a monoid $M$, generated by elements $m_{1},\ldots,m_{n}$, will be denoted by ${\mathbb{Z}}_{+}m_{1}+\cdots+{\mathbb{Z}}_{+}m_{n}$.

A monoid $M$ is torsion free if $\operatorname{gp}(M)$ has no nonzero torsion. An affine monoid is a finitely generated torsion free monoid. Every affine monoid $M$ isomorphically embeds into ${\mathbb{Z}}^{r}$, where $r=\operatorname{rank}(\operatorname{gp}(M))$. We put $\operatorname{rank}(M)=\operatorname{rank}(\operatorname{gp}(M)$.

If $M$ is affine and $U(M)=0$, then $M$ is said to be positive.

Lemma 2.4**.**

Assume $M$ is a monoid, $\operatorname{gp}(M)={\mathbb{Z}}^{r}$, and $U(M)=0$.

1. (a)

(Gordan Lemma)* $M$ is affine if and only if ${\mathbb{R}}_{+}M$ is a cone in ${\mathbb{R}}^{r}$.* 2. (b)

If $M$ is affine then, for any ring $R$, the monoid ring $R[M]$ admits a grading $R[M]=R\oplus R_{1}\oplus\cdots$ where the elements of $M$ are homogeneous.

These are proved in Proposition 2.17(f) and Corollary 2.10(a) in [4].

An affine positive monoid has the smallest generating set – the set of indecomposable elements in $M$. It is called the Hilbert basis of $M$ and denoted by $\operatorname{Hilb}(M)$.

2.4. Normal and seminormal monoids

A torsion free monoid $M$ is normal if $x\in\operatorname{gp}(M)$ and $nx\in M$ for some $n\in{\mathbb{N}}$ imply $x\in M$. The normalization of a torsion free monoid $M$ is the smallest normal submonoid $\operatorname{n}(M)\subset\operatorname{gp}(M)$, containing $M$, i.e., $\operatorname{n}(M)=\{x\in\operatorname{gp}(M)\ |\ nx\in M\ \text{for some}\ n\in{\mathbb{N}}\}$. If $M$ is affine then $\operatorname{n}(M)$ is also affine.

Lemma 2.5**.**

Let $M$ be an affine positive monoid and $\operatorname{gp}(M)={\mathbb{Z}}^{r}$.

1. (a)

There exists $m\in M$ such that $m+\operatorname{n}(M)\subset M$. 2. (b)

$M$* is normal if and only if $M=({\mathbb{R}}_{+}M)\cap{\mathbb{Z}}^{r}$.* 3. (c)

There exists a basis $n_{1},\ldots,n_{r}\in{\mathbb{Z}}^{r}$, such that $M\subset{\mathbb{Z}}_{+}n_{1}+\cdots+{\mathbb{Z}}_{+}n_{r}$. 4. (d)

The relative interior $\operatorname{int}({\mathbb{R}}_{+}M)$ contains a basis $\{m_{1},\ldots,m_{r}\}\subset{\mathbb{Z}}^{r}$.

These are the statements 2.33, 2.24, 2.17(e), and 2.74 in [4], respectively.

A torsion free monoid $M$ is seminormal if $x\in\operatorname{gp}(M)$ and $2x,3x\in M$ imply $x\in M$ or, equivalently, $x\in\operatorname{gp}(M)$ and $nx\in M$ for all $n\gg 0$ imply $x\in M$. The seminormalization $\operatorname{sn}(M)$ is the smallest seminormal monoid in $\operatorname{gp}(M)$, containing $M$. Explicitly, $\operatorname{sn}(M)=\{x\in\operatorname{gp}(M)\ |\ nx\in M\ \text{for all}\ n\gg 0\}$. For $M$ affine, $\operatorname{sn}(M)$ is also affine.

2.5. $\Phi$-correspondence

To describe relations between submonoids of an affine positive monoid $M$ we will follow the following conventions:

1. $\centerdot$

The group $\operatorname{gp}(M)$ will be thought of as ${\mathbb{Z}}^{r}$, where $r=\operatorname{rank}(M)$; 2. $\centerdot$

In ${\mathbb{R}}^{r}$ there will be implicitly (sometimes explicitly) chosen a rational affine hyperplane $\mathcal{H}\subset{\mathbb{R}}^{r}\setminus\{0\}$, such that {\mathbb{R}}_{+}M={\mathbb{R}}_{+}\big{(}({\mathbb{R}}_{+}M)\cap\mathcal{H}\big{)}; 3. $\centerdot$

$\Phi(L):=({\mathbb{R}}_{+}L)\cap\mathcal{H}$ for a submonoid $L\subset M$ and $\Phi(m):=({\mathbb{R}}_{+}m)\cap\mathcal{H}$ for a nonzero element $m\in M$; 4. $\centerdot$

For a convex subset $P\subset\Phi(M)$ we introduce the submonoid

[TABLE] 5. $\centerdot$

If $M$ is normal and $P\subset\mathcal{H}$ is a convex subset then we introduce the submonoid

[TABLE]

(By Lemma 2.5(b), the last two notations are compatible when $P\subset\Phi(M)$.)

Gordan Lemma can be rephrased as follows: a nonzero submonoid $L$ of an affine positive monoid $M$ is affine if and only if $\Phi(L)$ is a rational polytope.

For an affine positive normal monoid $M$, an element $m\in M$ is an extremal generator if $\Phi(m)$ is a vertex of $\Phi(M)$ and $m$ is the generator of $({\mathbb{R}}_{+}m)\cap M\cong{\mathbb{Z}}_{+}$. Thus, $M$ has as many extremal generators as there are the vertices of $\Phi(M)$.

For an affine positive nonzero monoid $M$, we define the interior submonoid and interior ideal of $M$ by $M_{*}=M(\operatorname{int}(\Phi(M))$ and $\operatorname{int}(M)=M_{*}\setminus\{0\}$, respectively. Because of the convention $\operatorname{int}(P)=P$ for $P$ a point (Section 2.2), we have $M_{*}=M$ when $\operatorname{rank}(M)=1$.

Lemma 2.6**.**

1. Let $M$ be an affine positive monoid of rank $r$.
2. (a)

If $M$ is normal and $m\in M$ is an extremal generator then ${\mathbb{Z}}m+M={\mathbb{Z}}m+M_{0}\cong{\mathbb{Z}}m\times M_{0}$ for an affine positive normal submonoid $M_{0}\subset{\mathbb{Z}}^{r}$ of rank $r-1$. 3. (b)

$M$* is seminormal if and only if $M(F)_{*}=\operatorname{n}(M(F))_{*}$ for every face $F\subset\Phi(M)$ (including $\Phi(M)$ itself).*

These are, respectively, Propositions 2.32 and 2.40 in [4].

3. Pyramids and monoids

3.1. Pyramidal decomposition

Pyramidal descent is based on the following

Definition 3.1**.**

Let $M$ be an positive affine normal monoid and $m\in M$ be an extremal generator. A representation $M=M(\Delta)\cup M(\Gamma)$ is called a pyramidal decomposition with vertex $m$ if $\Delta,\Gamma\subset\Phi(M)$ are rational polytopes, such that:

1. (a)

$\Phi(M)=\Delta\cup\Gamma$, 2. (b)

$\Delta$ is a pyramid with base $\Delta\cap\Gamma$ and $\dim\Delta=\dim\Gamma$. 3. (c)

$\Phi(M)$ is tangent to $\Delta$ at $\Phi(m)$ (Section 2.2).

(Our pyramidal decompositions are called non-degenerate in [22], the degenerate ones corresponding to the case $\dim\Gamma<\dim\Delta$, i.e., when $\Gamma$ is a facet of $\Delta$.)

For a pyramidal decomposition $M=M(\Delta)\cup M(\Gamma)$, there exists a monoid homomorphism $\deg:M\to{\mathbb{Z}}$, such that $0\not=\deg(M(\Delta))\subset{\mathbb{Z}}_{+}$ and $\deg(M(\Gamma))\subset{\mathbb{Z}}_{-}$, where ${\mathbb{Z}}_{-}$ is the set of nonnegative integers. Such a map will be called a pyramidal degree, associated to the given decomposition. We always have if $0\not=\deg(M(\Gamma))$.

Lemma 3.2**.**

Let $M$ be an affine positive normal monoid, $M=M(\Delta)\cup M(\Gamma)$ be a pyramidal decomposition, and $\deg:M\to{\mathbb{Z}}$ be an associated pyramidal degree. Consider a system of linear combinations $l_{i}=\sum_{j=1}^{r}d_{ij}n_{j}$, $i=1,\ldots,s$, where

1. $\centerdot$

$n_{j}\in M\setminus\{0\}$* and $d_{ij}\in{\mathbb{Z}}_{+}$ for all $i,j$,* 2. $\centerdot$

$n_{1}$* is the vertex of the decomposition,* 3. $\centerdot$

$(d_{11},\ldots,d_{1r})>(d_{21},\ldots,d_{2r})>\ldots$* in the lexicographical order,*

Assume $d_{ij}=a_{ij}+b_{ij}$, where $a_{ij},b_{ij}\in{\mathbb{Z}}_{+}$, $i=2,\ldots,s$ and $j=1,\ldots,r$. Then

[TABLE]

for $c_{1}\gg c_{2}\gg\ldots\gg c_{r}\gg 0$.

Proof.

We can assume $c_{j}>\deg(n_{j})$ for $j=2,\ldots,s$. Then it is enough to achieve

[TABLE]

which is equivalent to the inequalities

[TABLE]

known to be satisfied for $c_{1}\gg c_{2}\gg\ldots\gg c_{r}\gg 0$. ∎

3.2. Admissible configurations

Definition 3.3**.**

For an affine positive normal monoid $M$ of rank $r$ and an extremal generator $m\in M$, a triple $(\mathcal{H},\Delta_{1},\Delta_{2})$ is called an admissible configuration if $\mathcal{H}\subset{\mathbb{R}}^{r}\setminus\{0\}$ is a rational affine hyperplane, such that {\mathbb{R}}_{+}M={\mathbb{R}}_{+}\big{(}({\mathbb{R}}_{+}M)\cap\mathcal{H}\big{)}, and $\Delta_{1},\Delta_{2}\subset\mathcal{H}$ are rational pyramids with apex $\Phi(m)$, satisfying the conditions:

1. (a)

$\Phi(M)\subset\Delta_{1}\subset\Delta_{2}$, 2. (b)

$\Delta_{1}$ an $\Delta_{2}$ are tangent to $\Phi(M)$ at $\Phi(m)$, 3. (c)

$M(\Delta_{1})={\mathbb{Z}}_{+}m+M(F_{1})$, where $F_{1}\subset\Delta_{1}$ is the facet, opposite to $\Phi(m)$, 4. (d)

$M(\Delta_{2})={\mathbb{Z}}_{+}m+M(F_{2})$, where $F_{2}\subset\Delta_{2}$ is the facet, opposite to $\Phi(m)$, 5. (e)

$F_{2}\cap\Delta_{1}=\emptyset$.

In the notation above, $M(F_{1})$ and $M(F_{2})$ are isomorphic monoids and, consequently, so are the monoids $M(\Delta_{1})$ and $M(\Delta_{2})$.

Lemma 3.4**.**

An admissible configurations exists for any affine positive normal monoid $M$ with $\operatorname{rank}(M)\geq 2$ and any extremal generator $m\in M$.

Proof.

By Lemma 2.6(a), ${\mathbb{Z}}m+M={\mathbb{Z}}m+M_{0}\cong{\mathbb{Z}}m\times M_{0}$ for an affine positive normal submonoid $M_{0}\subset\operatorname{gp}(M)$. Let $n_{1},\ldots,n_{r-1}$ be a basis of $\operatorname{gp}(M_{0})\cong{\mathbb{Z}}^{r-1}$, such that $M_{0}\subset{\mathbb{Z}}_{+}n_{1}+\cdots+{\mathbb{Z}}_{+}n_{r-1}$ (Lemma 2.5(c)). Then, for $k\gg 0$, the following triple is an admissible configuration:

[TABLE]

∎

4. Patching unimodular rows

We say that a commutative square of ring homomorphisms

[TABLE]

has the Milnor patching property for unimodular rows if for every natural number $n\geq 3$ and every element $\mathbf{f}\in\operatorname{Um}_{n}(A_{1})$, whose image $\mathbf{f}^{\prime}\in\operatorname{Um}_{n}(A^{\prime})$ satisfies $\mathbf{f}^{\prime}\underset{A^{\prime}}{\sim}\mathbf{e}$, there exists $\mathbf{g}\in\operatorname{Um}_{n}(A)$ with the image $\mathbf{g}^{\prime}\in\operatorname{Um}_{n}(A_{1})$, satisfying $\mathbf{g}^{\prime}\underset{A_{1}}{\sim}\mathbf{f}$.

A Karoubi square is a commutative square of rings of the following type

[TABLE]

where $S\subset A$ is a multiplicative subset, $S$ acts regularly on $A$, $\rho(S)$ acts regularly on $B$, and the homomorphism $A/sA\to B/\rho(s)B$ is an isomorphism for every $s\in S$. A Karoubi square is always a pull-back diagram.

Lemma 4.1**.**

(Milnor Patching)* The following two types of commutative squares of ring homomorphisms have the Milnor patching property:*

1. (a)

\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{A^{\prime}}

• a pull-back diagram, where either $\pi$ or $\sigma$ is surjective,*

2. (b)

[TABLE] \textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{-1}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{-1}\rho}$$\textstyle{\rho(S)^{-1}B}

• a Karoubi square.*

This is proved in [9, Proposition 9.1(a)], with the details for one sketched step included in [12, Lemma 8]. The basis is the equality $\operatorname{E}_{n}(A^{\prime}{})=\operatorname{E}_{n}(A_{1})\operatorname{E}_{n}(A_{2})$, obvious for the case (a) and proved in [25, Lemma 2.4] for the case (b). (Vorst actually shows $\operatorname{E}_{n}(A^{\prime}{})=\operatorname{E}_{n}(A_{2})\operatorname{E}_{n}(A_{1})$, but the two equalities are equivalent.)

Let $A=A_{0}\oplus A_{1}\oplus\cdots$ be a graded ring and $\mathbf{f}\in\operatorname{Um}_{n}(A)$. Let $\mathbf{f}(0)$ denote the image of $\mathbf{f}$ in $\operatorname{Um}_{n}(A_{0})$ under the augmentation $A\to A_{0}$. In this notation we have

Proposition 4.2**.**

(Quillen Patching)* Assume $\mathbf{f}(0)=\mathbf{e}$. Then $\mathbf{f}\underset{A}{\sim}\mathbf{e}$ if and only if $\mathbf{f}_{\mu}\underset{A_{\mu}}{\sim}\mathbf{e}$ for every maximal ideal $\mu\subset A_{0}$.*

(Here $A_{\mu}=(A_{0}\setminus\mu)^{-1}A$.)

This is [8, Corollary 7.4]. It is based on Suslin’s $K_{1}$-analogue [20, Theorem 3.1] of Quillen’s well-known local-global patching for projective modules [18, Theorem 1].

5. Integral and subintegral extensions

5.1. Integral extensions

Let $M=M(\Delta)\cup M(\Gamma)$ be a pyramidal decomposition of an affine positive normal monoid $M$ with vertex $m$ and $\deg:M\to{\mathbb{Z}}$ be an associated pyramidal degree. For an element $f\in R[M]$, we define $\deg(f)$ as the maximum degree of the terms of $f$. Call an element $f=\sum_{j=1}^{k}r_{j}m_{j}\in R[M]$ monic (with respect to $\deg$) if $r_{1}m_{1}=um^{c}$ for some $u\in U(R)$, $c\in{\mathbb{N}}$, and $\deg(m_{1})>\deg(m_{j})$ for $j=2,\ldots,k$.

Call a ring homomorphism $A\to B$ integral if $B$ is integral over $\operatorname{Im}(A)$.

Lemma 5.1**.**

Let $M=M(\Delta)\cup M(\Gamma)$ be a pyramidal decomposition, $\deg:M\to{\mathbb{Z}}$ be an associated pyramidal degree, and $f\in R[M]$ be a monic element. Then the ring homomorphism $R[M(\Gamma)]\to R[M]/fR[M]$ is integral.

Proof.

Since $M$ is normal, every element $g\in R[M]$ admits a representation $g=fg_{1}+f_{1}$, where $g_{1},f_{1}\in R[M]$ with $\deg(f_{1})<\deg(f)$. In particular, as an $R[M(\Gamma)]$-module, $R[M]/fR[M]$ is generated by the image of the subset $M_{<\deg(f)}:=\{m\in M\ |\ \deg(m)<\deg f\}$ and, therefore, by the image of $M_{\leq\deg(f)}:=\{m\in M\ |\ \deg(m)\leq\deg f\}$. But, according to [4, Theorem 2.12], $RM_{\leq\deg(f)}$ is a finitely generated $R[M(\Gamma)]$-module. ∎

5.2. Subintegral extensions

Call an extension of rings $A\subset B$ elementary subintegral if $B=A[b]$ for some $b\in B$ with $b^{2},b^{3}\in A$. A ring extension $A\subset B$ is subintegral if it is a filtered union of elementary subintegral extensions.

Theorem 5.2**.**

For a subintegral extension of rings $A\subset B$ and an element $\mathbf{f}\in\operatorname{Um}_{n}(A)$, where $n\geq 3$, we have $\mathbf{f}\underset{A}{\sim}\mathbf{e}$ if and only if $\mathbf{f}\underset{B}{\sim}\mathbf{e}$.

This is the main result of [12]. It represents the ‘unimodular row’ counterpart of [22, Theorem 14.1]. Although the statement is about arbitrary rings, the proof in [12] uses monoid rings. More precisely, it uses the main result of [8]. A stronger result in the context of the Euler class groups under subintegral extensions was later derived in [5].

6. Reduction to the interior of normal monoids

Lemma 6.1**.**

1. (a)

In order to prove Theorem 1.1 for torsion free monoids, it is sufficient to prove it in the special case when $R$ is a local Noetherian ring and $M=N_{*}$ for an affine positive normal monoid $N$. 2. (b)

In order to prove Theorem 1.1 for a ring $R$ and an affine positive monoid $M$, it is enough to prove it for the monoid rings $R[M(F)_{*}]$, where $F$ runs over the faces of $\Phi(M)$ (including $\Phi(M)$ itself).

Proof.

(a) Let $L$ be a torsion free monoid. Since $L$ is the inductive limit of its affine submonoids, we can assume that $L$ is itself affine. Consider the pullback diagram

[TABLE]

Since $U(L)$ is a free abelian group, Theorem 2.2 yields the transitivity of the elementary actions on $\operatorname{Um}_{n}(R[U(L)])$. Applying Lemma 4.1(a) and using that $\left(L\setminus U(L)\right)\cup\{1\}$ is the union of its affine submonoids, we can also assume that $L$ is an affine positive monoid. Since $\operatorname{sn}(L)$ is a filtered union of subintegral extensions of monoids, by Theorem 5.2 we can further assume $L$ is seminormal.

Since the elementary action on $\operatorname{Um}_{n}(R)$ is transitive for $n\geq d+2$, Lemma 2.4(b) and Proposition 4.2 make it possible to reduce the general case to $R$ local.

Let $F_{1},\ldots,F_{k}$ be the set of non-empty faces of the polytope $\Phi(L)$, including $\Phi(L)$ itself, indexed in such a way that $i\leq j$ implies $\dim F_{i}\leq\dim F_{j}$. In particular, $F_{k}=\Phi(L)$. We have the pull-back diagrams of $R$-algebras with the natural horizontal injective maps:

[TABLE]

(Above, we assume $\sum_{j>k}R\operatorname{int}(L(F_{j}))=0$.)

By Lemma 2.6(b), for every $i$, the ring at the upper-left corner of $(\mathbb{D}_{i})$ is of the type $R[N_{*}]$, where $N$ is an affine, positive, and normal monoid. Moreover, the ring at the upper-right corner of $(\mathbb{D}_{k})$ is $R[L]$ and that at the lower-right corner of $(\mathbb{D}_{1})$ is $R$. Consequently, Lemma 4.1(a) allows induction on $i$.

(b) The claim follows from the same inductive process, based on the corresponding pull-back diagrams. ∎

7. Quasi-monic elements

The following lemma and its proof, suggested by the referee, is a simplification of the original overly complicated version.

Lemma 7.1**.**

Let $B$ be a Noetherian ring of dimension $d$, $\mathbf{a}=(b_{1},\ldots,b_{n})\in\operatorname{Um}_{n}(B)$ for some $n\geq 2$, and $Bb_{1}+By=B$. Then, for any finite family of ideals $\nu_{1},\ldots,\nu_{p}\subset B$, there is an element $b\in Bb_{2}+\cdots+Bb_{n}$ such that, for all natural numbers $c$, one has

[TABLE]

Proof.

Enlarging the $\nu_{i}$ one may assume they are maximal ideals. We may assume that they are distinct, hence comaximal. If $b_{1}\notin\nu_{i}$ for all $i$, take $b=0$. Else reorder and choose $q$ so that $b_{1}\in\nu_{i}$ if and only if $i\leq q$. Using the Chinese Remainder Theorem, choose $z\in B$ so that $z\notin\nu_{i}$ if and only if $i\leq q$. The restriction to $Bz$ of $B\to\prod_{i\leq q}B/\nu_{i}$ is still a surjective map and it has $Bzb_{1}$ in its kernel. So $Bzb_{2}+\cdots+Bzb_{n}$ maps onto $\prod_{i\leq q}B/\nu_{i}$. Choose $b\in Bzb_{2}+\cdots+Bzb_{n}$ so that $b\notin\nu_{i}$ if $i\leq q$. Notice that $by^{c}\notin\nu_{i}$ if and only if $b_{1}\in\nu_{i}$. ∎

Definition 7.2**.**

Let $R$ be a ring. An $R$-subalgebra (not necessarily Noetherian or monomial) $A\subset R[{\mathbb{Z}}_{+}^{r}]$ is called $t_{1}$-tilted if there is a neighborhood $\Phi(t_{1})\in\mathcal{U}\subset\Phi({\mathbb{Z}}_{+}^{r})$ such that $m\in{\mathbb{Z}}_{+}^{r}\setminus\{0\}$ and $\varphi(m)\in\mathcal{U}$ imply $m\in A$.

For an element $\mathbf{g}\in\operatorname{Um}_{n}(R[t_{1},\ldots,t_{r}])$ we denote by $\mathbf{g}|_{t_{1}=0}$ the image of $\mathbf{g}$ in $\operatorname{Um}_{n}(R[t_{2},\ldots,t_{r}])$ after substituting [math] for $t_{1}$.

Lemma 7.3**.**

Let $R$ be a local Noetherian ring of dimension $d$. Assume $A\subset R[{\mathbb{Z}}_{+}^{r}]$ is a $t_{1}$-tilted subalgebra and $\mathbf{g}=(g_{1},\ldots,g_{n})\in\operatorname{Um}_{n}(A)$ for some $n\geq 2$, such that $\mathbf{g}|_{t_{1}=0}\in\operatorname{Um}_{n}(R)$. Then there exists $\mathbf{h}=(h_{1},\ldots,h_{n})\in\operatorname{Um}_{n}(A)$, for which $\mathbf{g}\underset{A}{\sim}\mathbf{h}$ and $\operatorname{ht}_{R[{\mathbb{Z}}_{+}^{r}]}\left(R[{\mathbb{Z}}_{+}^{r}]h_{1}+\cdots+R[{\mathbb{Z}}_{+}^{r}]h_{i}\right)\geq i$ for all $i$.

Proof.

For an element $\mathbf{l}=(l_{1},\ldots,l_{n})\in\operatorname{Um}_{n}(A)$ we introduce the condition

[TABLE]

Since $R$ is local and $\mathbf{g}|_{t_{1}=0}\in\operatorname{Um}_{n}(R)$, by a suiatable elementary transformation over $R$ we can achieve $g_{1}|_{t_{1}=0}=\cdots=g_{n-1}|_{t_{1}=0}=1$. Consequently, without loss of generality, we can assume that $\mathbf{g}$ satisfies (1).

Let $0\leq k<n$. Assume there exists $\mathbf{h}^{\prime}=(h^{\prime}_{1},\ldots,h^{\prime}_{n})\in\operatorname{Um}_{n}(A)$, for which

1. $\centerdot$

$\mathbf{g}\underset{A}{\sim}\mathbf{h}^{\prime}$, 2. $\centerdot$

the condition (1) is satisfied, 3. $\centerdot$

the components satisfy the inequalities in Lemma 7.3 for $i<k$ (when $k>1$).

We will induct on $k$ to achieve the height inequalities for all $i\leq n-1$; there is nothing to prove for $i=n$.

Without loss of generality, $R[{\mathbb{Z}}_{+}^{r}]h_{1}^{\prime}+\cdots+R[{\mathbb{Z}}_{+}^{r}]h_{k-1}^{\prime}\not=R[{\mathbb{Z}}_{+}^{r}]$. Put

[TABLE]

Since $\mathbf{h}^{\prime}$ satisfies (1), Lemma 7.1 implies the existence of an element

[TABLE]

such that for all natural numbers $c$ one has

[TABLE]

Since $h^{\prime}_{1},\ldots,h^{\prime}_{k-1}\in\nu_{s}$ for every $s$, the element $h^{\prime}=f_{k+1}h^{\prime}_{k+1}+\cdots+f_{n}h^{\prime}_{n}$ satisfies

[TABLE]

Since $A$ is $t_{1}$-tilted, for $c\gg 0$ we also have

[TABLE]

The last inclusion implies that, for $c\gg 0$, we have

[TABLE]

In view of (2), the minimal primes over the ideal

[TABLE]

are not among the $\nu_{s}$. In particular, the unimodular row on the right of (3) satisfies (1) and the height inequalities for $i=1,\ldots,k$. ∎

For an element $f\in R[{\mathbb{Z}}_{+}^{r}]$, its leading term in the lexicographical order with respect to $t_{1}>\ldots>t_{r}$ will be denoted by $L(f)$. Call an element $f\in R[{\mathbb{Z}}_{+}^{r}]$ quasi-monic is $L(f)=um$ for some $u\in U(R)$ and $m\in{\mathbb{Z}}_{+}^{r}$.

Corollary 7.4**.**

Let $R$ be a local Noetherian ring of dimension $d$. Assume $A\subset R[{\mathbb{Z}}_{+}^{r}]$ is a $t_{1}$-tilted $R$-subalgebra and $\mathbf{g}=(g_{1},\ldots,g_{n})\in\operatorname{Um}_{n}(A)$ for some $n\geq d+2$. Assume $\mathbf{g}|_{t_{1}=0}\in\operatorname{Um}_{n}(R)$. Then there exists $\mathbf{h}=(h_{1},\ldots,h_{n})$, such that $h_{n}$ is quasi-monic and $\mathbf{g}\underset{A}{\sim}\mathbf{h}$.

Proof.

By Lemma 7.3, without loss of generality we can assume $\operatorname{ht}(I)\geq d+1$ for $I=R[{\mathbb{Z}}_{+}^{r}]g_{1}+\cdots+R[{\mathbb{Z}}_{+}^{r}]g_{d+1}$. Then Lemma 2.3(b), applied to the decreasing sequence of rings $R[t_{1},t_{2},\ldots,t_{r}]\supset R[t_{2},\ldots,t_{r}]\supset\ldots\supset R[t_{r}]$, implies that $I$ contains a quasi-monic element $g$. Assume $L(g)=um$ for some $u\in U(R)$ and $m\in{\mathbb{Z}}_{+}^{r}$. Then, using that $A$ is $t_{1}$-tilted, we can take $\mathbf{h}=(g_{1},\ldots,g_{n-1},g_{n}+t_{1}^{c}g)$ with $c\gg 0$. In fact, for $c\gg 0$ we have $L(g_{n}+t_{1}^{c}g)=t_{1}^{c}L(g)$ and $t_{1}^{c}g\in Ag_{1}+\cdots+Ag_{d+1}$. ∎

8. Pyramidal descent

8.1. Local pyramidal descent

For the rest of Section 8.1, we assume that:

1. $\centerdot$

$(R,\mu)$ is a local Noetherian ring of dimension $d$, 2. $\centerdot$

$n\geq\max(d+2,3)$, 3. $\centerdot$

$M$ is a positive, affine, normal monoid with $\operatorname{rank}(M)\geq 2$, 4. $\centerdot$

$M=M(\Delta)\cup M(\Gamma)$ is a pyramidal decomposition with vertex $m$, 5. $\centerdot$

$(\mathcal{H},\Delta_{1},\Delta_{2})$ is an admissible configuration for the pair $(M,m)$, 6. $\centerdot$

$\mathfrak{m}\subset R[M(\Gamma)]$ is the maximal ideal generated by $\left(M(\Gamma)\setminus\{1\}\right)\cup\mu$.

Consider the following commutative diagram of rings, where the vertical arrows refer to the identity embeddings:

[TABLE]

where:

1. $\centerdot$

$r=\#\operatorname{Hilb}(M(\Delta_{1})=\#\operatorname{Hilb}(M(\Delta_{2})$ (see Section 2.3 for Hilbert bases), 2. $\centerdot$

the $R$-algebra homomorphisms $\pi_{i}$ ($i=1,2$) are induced by surjective monoid homomorphisms $h_{i}:{\mathbb{Z}}_{+}^{r}\to M(\Delta_{i})$, satisfying the conditions: $h_{1}(t_{1})=h_{2}(t_{1})=m$ and the following triples of points are collinear

[TABLE] 3. $\centerdot$

the inner and outer squares are pull-back diagrams, 4. $\centerdot$

the slanted arrows are the induced rings embeddings.

We will keep the identification $R[{\mathbb{Z}}_{+}^{r}]=R[t_{1},\ldots,t_{r}]$ at the lower-left corner of the outer square, and think of ${\mathbb{Z}}_{+}^{r}$ at the lower-left corner of the inner square as the multiplicative monoid, generated by $t_{1},t_{1}^{k_{2}}t_{2},\ldots,t_{1}^{k_{r}}t_{r}$ for appropriate $k_{2},\ldots,k_{r}\in{\mathbb{N}}$ so that the slanted arrows in $(\ref{two-pullback})$ become the identity embeddings. In particular, the upper maps in the squares are the restrictions of the respective bottom maps and so we use the same notation $\pi_{i}$ for them.

Denote ${\mathbb{Z}}_{+}^{r}(M)=h_{2}^{-1}(M)$. Then $A_{2}=R\left[{\mathbb{Z}}_{+}^{r}(M)+\ker\pi_{2}\right]\subset R[{\mathbb{Z}}_{+}^{r}]$, the $R$-subalgebra generated by ${\mathbb{Z}}_{+}^{r}(M)\cup\ker\pi_{2}$.

Observe that the subalgebra $A_{2}\subset R[{\mathbb{Z}}_{+}^{r}]$ is $t_{1}$-tilted. It is neither a finitely generated nor a monomial $R$-algebra as soon as $\ker\pi_{2}\not=0$, i.e., when $M(\Delta_{2})$ or, equivalently, $M(\Delta_{1})$ is not a free monoid.

Theorem 8.1**.**

Assume the elementary action on $\operatorname{Um}_{n}(R[M(\Delta_{1})])$ is transitive. Then $\mathbf{f}_{\mathfrak{m}}\underset{R[M]_{\mathfrak{m}}}{\sim}\mathbf{e}$ for every $\mathbf{f}\in\operatorname{Um}_{n}(R[M])$.

(Here $R[M]_{\mathfrak{m}}=(R[M(\Gamma)]\setminus\mathfrak{m})^{-1}R[M]$.)

Proof.

Let $\mathbf{f}=(f_{1},\ldots,f_{n})\in\operatorname{Um}_{n}(R[M])$. Since $\mathbf{f}\underset{R[M(\Delta_{1})]}{\sim}\mathbf{e}$, Lemma 4.1 implies the existence of $\mathbf{g}=(g_{1},\ldots,g_{n})\in\operatorname{Um}_{n}(A_{1})$ such that $\pi_{1}(\mathbf{g})\underset{R[M]}{\sim}\mathbf{f}$. We can assume $\pi_{1}(\mathbf{g})=\mathbf{f}$.

Considering $\mathbf{g}$ as an element of $\operatorname{Um}_{n}(A_{2})$, we have $\mathbf{g}|_{t_{1}=0}\in\operatorname{Um}_{n}(R)$. (This is where we use admissible configurations: the corresponding condition may not be satisfied over $A_{1}$.) By Corollary 7.4, without loss of generality we can assume that

[TABLE]

Fix a rational subsimplex $\Delta_{0}\subset\Phi({\mathbb{Z}}_{+}(M))$ such that $\Phi({\mathbb{Z}}_{+}^{d})$ is tangent to $\Delta_{0}$ at $\Phi(t_{1})$ and ${\mathbb{Z}}_{+}^{r}(\Delta_{0})\cong{\mathbb{Z}}_{+}^{r}$. This can be done by choosing

[TABLE]

for some $k\gg 0$. We have ${\mathbb{Z}}_{+}^{r}(\Delta_{0})\subset{\mathbb{Z}}_{+}^{r}(M)$.

By Theorem 2.2, there exists $\varepsilon\in\operatorname{E}_{n}(R[{\mathbb{Z}}_{+}^{r}])$, whose first row is $\mathbf{g}$.

For an $r$-tuple of natural numbers $(c_{1},\ldots,c_{r})$, consider the $R$-algebra endomorphism

[TABLE]

The crucial observation is that the non-unit monomials in the reduced forms of the entries of the matrix $\varepsilon^{-1}\tau(\varepsilon)$ align in the direction of $t_{1}\in{\mathbb{Z}}_{+}^{d}\subset{\mathbb{R}}_{+}^{d}$ as $c_{1},\ldots,c_{d}\to\infty$. This implies

[TABLE]

By Lemma 2.1 and Theorem 2.2, there exist $\varepsilon_{0}\in\operatorname{E}_{n}(R[{\mathbb{Z}}_{+}^{r}(\Delta_{0})]$ and $\alpha_{0}\in\operatorname{GL}_{n-1}(R[{\mathbb{Z}}_{+}^{r}(\Delta_{0})])$, such that

[TABLE]

Thus, for $c_{1},\ldots,c_{r}\gg 0$, we have

[TABLE]

Consequently, for every $c_{1},\ldots,c_{r}\gg 0$, there exist $f_{1}^{\prime},\ldots,f_{n-1}^{\prime}\in R[M]$ such that

[TABLE]

($\pi_{2}$ is the map from (4)). Pick a pyramidal degree $\deg:M\to{\mathbb{Z}}$, associated to the decomposition $M=M(\Delta)\cup M(\Gamma)$. Because of (5), Lemma 3.2 implies that $\pi_{2}(\tau(g_{n}))$ is monic for $c_{1}\gg\ldots\gg c_{1}\gg 0$. Then, by Lemma 5.1, the homomorphism $R[M(\Gamma)]_{\mathfrak{m}}\to R[M]_{\mathfrak{m}}/(\pi_{2}(\tau(g_{n}))$ is integral. Since $R[M]$ is finitely generated over $R$ and $R[M(\Gamma)]_{\mathfrak{m}}$ is local, the quotient ring $R[M]_{\mathfrak{m}}/(\pi(\tau(g_{n}))$ is semi-local. Consequently, [16, Corollary 7.5] implies the equivalence $\bar{\mathbf{f}}_{\mathfrak{m}}\sim\mathbf{e}$ over the ring $R[M]_{\mathfrak{m}}/(\pi_{2}(\tau(g_{n}))$, where the ‘bar’ refers to the reduction $\mod(\pi_{2}(\tau(g_{n}))$, i.e., \bar{\mathbf{f}}_{\mathfrak{m}}\in\operatorname{Um}_{n-1}\big{(}R[M]_{\mathfrak{m}}/(\pi_{2}(\tau(g_{n}))\big{)}. By lifting the involved elementary transformation to $R[M]_{\mathfrak{m}}$, one derives the desired equivalence $\mathbf{f}_{\mathfrak{m}}\underset{R[M]_{\mathfrak{m}}}{\sim}\mathbf{e}$; see Proposition 5.6 [16, Ch. I]. ∎

8.2. Pyramidal descent and Theorem 1.1 for torsion free monoids

We need further facts on unimodular rows and polytopes.

The following lemma is derived in the proof of [22, Proposition 10.3]. It represents the second use of non-finitely generated algebras (after Section 7).

Lemma 8.2**.**

Let $M$ be an affine positive normal monoid and $M=M(\Delta)\cup M(\Gamma)$ be a pyramidal decomposition. Then for a local ring $(R,\mu)$ and the maximal ideal $\mathfrak{m}=R\operatorname{int}(M(\Gamma))+\mu\subset R[M(\Gamma)_{*}]$, the following diagram is a Karoubi square:

[TABLE]

A sequence of rational polytopes $\{P_{i}\}_{i=1}^{\infty}$ is called admissible if, for every $i$, either $P_{i+1}=\Gamma_{i}$, where $P_{i}=\Delta_{i}\cup\Gamma_{i}$ as for the $\Phi$-images of pyramidal decompositions, or $P_{i}\subset P_{i+1}\subset P_{1}$. The following is [6, Lemma 2.8]:

Lemma 8.3**.**

For a rational polytope $P_{1}$ and a neighborhood $\mathcal{U}\subset P_{1}$, there exists an admissible sequence of polytopes $\{P_{i}\}_{i=1}^{\infty}$ with $P_{i}\subset\mathcal{U}$ for $i\gg 0$.

For an affine positive normal monoid $M$ of rank $r$ we define its complexity as the smallest integer ${\mathsf{k}}(M)$ for which there exists a polytope $Q\subset\Phi(M)$ of dimension ${\mathsf{k}}(M)-1$ and points $v_{1},\ldots,v_{r-{\mathsf{k}}(M)}\in\Phi(M)$, satisfying the equality

[TABLE]

The condition (6) is equivalent to the existence of a sequence of polytopes

[TABLE]

where $Q_{i}$ is a pyramid over $Q_{i-1}$ for each $i\geq 1$.

Observe that, for $M$ as above, ${\mathsf{k}}(M)=0$ if and only if $\Phi(M)$ is a simplex. In particular, by the main result of [8], Theorem 1.1 is true for an affine positive normal monoid of complexity [math].

For the rest of Section 8.2 we assume that $R$, $d$, and $n$ are as in Theorem 1.1 and that $R=(R,\mu)$ is local. We also assume that $M$ is an affine positive normal monoid with $\operatorname{rank}(M)=r$.

Let $\Phi(M)=\operatorname{conv}(v_{1},\ldots,v_{r-\mathsf{k}(M)},Q)$, as in $(\ref{multipyramid})$. For a rational polytope $P\subset Q$, we will use the notation

[TABLE]

Observe that every pyramidal decomposition $M(P)=M(\Delta)\cup M(\Gamma)$ gives rise to the pyramidal decomposition $M(\tilde{P})=M(\tilde{\Delta})\cap M(\tilde{\Gamma})$.

The proof of the following lemma is straightforward.

Lemma 8.4**.**

Let $P\subset Q$ be a rational polytope and $m\in M(\tilde{P})$ be the extremal generator, corresponding to a vertex of $P$. Assume $(\mathcal{H},\Delta_{1},\Delta_{2})$ is an admissible configuration for the monoid $M(\tilde{P})$ with respect to $m$. Then {\mathsf{k}}\big{(}M(\Delta_{1})\big{)}={\mathsf{k}}(M(\tilde{\Delta}_{1}))<{\mathsf{k}}(M).

Proposition 8.5**.**

(Pyramidal descent)* Assume Theorem 1.1 has been shown for the monoid rings of the form $R[N]$, where $N$ is an affine positive normal monoid with ${\mathsf{k}}(N)<{\mathsf{k}}(M)$. Then for a rational polytope $P\subset Q$, a pyramidal decomposition $M(P)=M(\Delta)\cup M(\Gamma)$, and a unimodular row $\mathbf{f}\in\operatorname{Um}_{n}(R[M(\tilde{P})_{*}]$, there exists $\mathbf{g}\in\operatorname{Um}_{n}(R[M(\tilde{\Gamma})_{*}])$, such that $\mathbf{f}\sim_{R[M(\tilde{P})_{*}]}\mathbf{g}$.*

Proof.

For a point $z\in\operatorname{Aff}(\Phi(M))$ and a real number $c$, the homothetic transformation of $\operatorname{Aff}(\Phi(M))$ with coefficient $c$ and centered at $z$ will be denoted by $c^{z}$.

Pick any rational point $z\in\operatorname{int}(\tilde{\Gamma})$ and a rational number $c\in(0,1)$, sufficiently close to $1$, such that \mathbf{f}\in\operatorname{Um}_{n}\big{(}R[M(c^{z}(\tilde{P}))]\big{)}. We have the pyramidal decomposition

[TABLE]

By Lemma 8.4 and the assumption on monoids of complexity $<{\mathsf{k}}(M)$, Theorem 8.1 applies to this pyramidal decomposition, yielding the equivalence

[TABLE]

where $\mathfrak{n}=R(M(c^{z}(\tilde{\Gamma}))\setminus\{1\})+\mu$. We have the inclusion

[TABLE]

where $\mathfrak{m}=R(M(\tilde{\Gamma})\setminus\{1\})+\mu$. In particular, Lemma 8.2 implies $\mathbf{f}\underset{R[M_{*}]}{\sim}\mathbf{g}$ for some $\mathbf{g}\in\operatorname{Um}_{n}(R[M(\tilde{\Gamma})_{*}])$. ∎

Proof of Theorem 1.1 for torsion free monoids.

By Lemma 6.1(a), it is enough to show the desired transitivity for the monoid ring $R[M_{*}]$. We will induct on $\mathsf{k}(M)$.

If ${\mathsf{k}}(M)=0$ then $\Phi(M)$ is a simplex. Consequently, $M_{*}$ is a filtered union of affine positive normal monoids $L$, for which the polytopes $\Phi(L)$ are simplices, and we are done by [8]. Assume ${\mathsf{k}}(M)>0$ and the transitivity has been shown for the rings of the form $R[N_{*}]$, where $N$ is an affine positive normal monoid with ${\mathsf{k}}(N)<{\mathsf{k}}(M)$. Because, for every face $F\subset\Phi(N)$, ${\mathsf{k}}(N(F))\leq{\mathsf{k}}(N)$ and $N(F)$ is normal if $N$ is, Lemma 6.1(b) implies the transitivity also for the rings $R[N]$ with $N$ affine, positive, normal, and ${\mathsf{k}}(N)<{\mathsf{k}}(M)$. By Lemma 8.3(b), there exists an admissible sequence of rational polytopes $\{P_{i}\}_{i=1}^{\infty}$, such that $P_{1}=Q$ and $P_{j}$ is a rational simplex for some $j$. By Proposition 8.5, there exists an element $\mathbf{h}\in\operatorname{Um}_{n}(R[M(\tilde{P}_{j})]_{*})$ with $\mathbf{h}\underset{R[M_{*}]}{\sim}\mathbf{f}$. But, by the base case, $\mathbf{h}\sim_{R[M(\tilde{P}_{j})_{*}]}\mathbf{e}$. ∎

Remark 8.6**.**

Let $M$ be an affine positive normal monoid and $(\mathcal{H},\Delta_{1},\Delta_{2})$ an admissible configuration with respect to an extremal generator $m\in M$. Not only is the polytope $\Delta_{1}$ closer to a simplex compared to the polytope $\Phi(M)$, making the induction in the concluding part of the proof of Theorem 1.1 possible, but even the monoid $M(\Delta_{1})$ itself is closer to a free monoid – it splits off a free monoid summand. Based on this observation, one can define a subtler induction process where the base case is the case of polynomial algebras. This way one can avoid the use of [8] and base the whole argument on Suslin’s work [20]. The referee found a mistake in our original attempt at inducting on the complexity of monoid structures and explained how to fix it. This would also require an extension of Lemma 8.2. Proposition 8.5 and the induction at the end of the proof of Theorem 1.1 is a variation: it requires no changes in Lemma 8.2 but, on the other hand, uses [8]. This approach is also similar to the proof of [13, Proposition 3.2].

9. Monoids with torsion

Unless specified otherwise, a monoid in this section means a commutative and cancellative monoid, possibly with torsion, i.e., no longer is the group $\operatorname{gp}(M)$ assumed to be torsion free.

For a monoid $L$, let $\operatorname{t}(L)$ denote the the torsion subgroup of $\operatorname{gp}(L)$. For a monoid $L$, denote \bar{L}=\operatorname{Im}\big{(}L\to\operatorname{gp}(L)/\operatorname{t}(L)\big{)}. The correspondence $L\mapsto\bar{L}$ is a left adjoint functor for the embedding functor of the category of monoids without torsion into that of monoids.

The normalization $\operatorname{n}(L)$ and seminormalization $\operatorname{sn}(L)$ of a monoid $L$ is defined in the same way as for the class of torsion free monoids in Section 2.4.

For a finitely generated monoid $L$, we will identify $\operatorname{gp}(L)$ with $\operatorname{gp}(\bar{L})\times\operatorname{t}(L)$ along some isomorphism $\operatorname{gp}(L)\cong\operatorname{gp}(\bar{L})\times\operatorname{t}(L)$. (There is no canonical choice for such an isomorphism.) As before, we think of $\operatorname{gp}(\bar{L})$ as ${\mathbb{Z}}^{r}$, where $r=\operatorname{rank}(\operatorname{gp}(L))$.

We need several facts, starting with an extension of Lemma 2.6(b) to monoids with torsion.

Lemma 9.1**.**

Let $L$ be a finitely generated monoid with trivial $U(L)$ and ${\mathbb{F}}(L)$ be the set of faces of the cone ${\mathbb{R}}_{+}\bar{L}$, including [math] and ${\mathbb{R}}_{+}\bar{L}$. For every $F\in{\mathbb{F}}(L)$, there is a subgroup $T_{F}\subset\operatorname{t}(L)$, such that:

1. (a)

\operatorname{sn}(L)=\bigcup_{{\mathbb{F}}(L)}\big{(}\operatorname{int}(\operatorname{n}(\bar{L}\cap F))\times T_{F}\big{)}; 2. (b)

$T_{{\mathbb{R}}_{+}\bar{L}}=\operatorname{t}(L)$; 3. (c)

$T_{F_{1}}\subset T_{F_{2}}$* whenever $F_{1}\subset F_{2}$.*

Proof.

The extension of monoids $L\subset\operatorname{n}(\bar{L})\times\operatorname{t}(L)$ is integral, i.e., every element of $\operatorname{n}(\bar{L})\times\operatorname{t}(L)$ has a positive multiple in $L$. In particular, $\operatorname{n}(L)=\operatorname{n}(\bar{L})\times\operatorname{t}(L)$. In fact, for any element $(l,t)\in\operatorname{n}(\bar{L})\times\operatorname{t}(L)$, there exists $c_{1}\in{\mathbb{N}}$ such that $c_{1}l\in\bar{L}$. Assume $(c_{1}l,s)\in L$ for some $s\in\operatorname{t}(L)$. Then $(c_{1}c_{2})\cdot(l,t)=(c_{1}c_{2}l,0)\in L$ for any multiple $c_{2}$ of the orders of the elements $c_{1}t,s\in\operatorname{t}(L)$.

Since $\operatorname{n}(L)$ is a finitely generated monoid, ${\mathbb{Q}}[\operatorname{n}(L)]$ is a finitely genarated ${\mathbb{Q}}[L]$-algebra. (Despite the use of monoid rings here, we continue using additive notation for the monoid operation.) But ${\mathbb{Q}}[L]\subset{\mathbb{Q}}[\operatorname{n}(L)]$ is also an integral extension. Therefore, ${\mathbb{Q}}[\operatorname{n}(L)]$ is module-finite over ${\mathbb{Q}}[L]$. There is a finite generating subset of this module

[TABLE]

We have $(\lambda,\tau)+\operatorname{n}(L)\subset L$, where $(\lambda,\tau)=\sum_{i=1}^{k}(l_{i2},t_{i2})\in L$. (In the torsion free case, we have just recovered Lemma 2.5(a).) Consequently, all sufficiently high multiples of every element $(t,l)\in\operatorname{int}(\operatorname{n}(\bar{L}))\times\operatorname{t}(L)$ are in $(\lambda,\tau)+\operatorname{n}(L)$. Hence, $\operatorname{int}(\operatorname{n}(\bar{L}))\times\operatorname{t}(L)\subset\operatorname{sn}(L)$.

The same argument, applied to any face $F\subset{\mathbb{R}}_{+}\bar{L}$, yields

[TABLE]

for the subgroup

[TABLE]

This implies (a,b), and the part (c) follows from the inclusion $(l+l^{\prime}{},t)\in\operatorname{int}(\operatorname{n}(\bar{L}\cap F_{2}))\times T_{F_{2}}$ for any two elements $(l,t)\in\operatorname{int}(\operatorname{n}(\bar{L}\cap F_{1}))\times T_{F_{1}}$ and $l^{\prime}{}\in\operatorname{int}(\operatorname{n}(\bar{L}\cap F_{2}))$. ∎

Lemma 9.2**.**

In order to prove Theorem 1.1, it is enough to prove it when $U(M)$ is trivial.

Proof.

We can assume $M$ is finitely generated. Let $U(M)={\mathbb{Z}}^{s}\times H$ for a finite group $H$. Since $\dim R=\dim R[H]$, the elementary action on the unimodular $n$-rows over $R[U(M)]=R[H][{\mathbb{Z}}^{s}]$ is transitive by Theorem 2.2. The pullback diagram, similar to the first square in the proof of Lemma 6.1, reduces the transitivity question to the submonoid $(M\setminus U(M))\cup\{1\}\subset M$. ∎

Lemma 9.3**.**

Assume $R$ is a local Noetherian ring of dimension $d$, $n\geq\max(d+2,3)$, and $B\subset R[t]$ is an $R$-subalgebra, containing $t^{m}$ for some $m$. Then the elementary action on $\operatorname{Um}_{n}(B)$ is transitive.

Proof.

For an integral extension of Noetherian rings $R_{1}\subset R_{2}$ and an ideal $I\subset R_{2}$, a standard commutative algebra argument yields $\operatorname{ht}_{R_{1}}(I\cap R_{1})\geq\operatorname{ht}_{R_{2}}I$. (See, for instance, Step 2 in the proof of [8, Lemma 6.5].)

Let $\mathbf{f}=(f_{1},\ldots,f_{n})\in\operatorname{Um}_{n}(B)$. By Lemma 2.3(a), we can assume $\operatorname{ht}(Bf_{1}+\cdots+Bf_{d+1})\geq d+1$. Because of the integral extension $R[t^{m}]\subset B$, the height inequality above and Lemma 2.3(b) imply L\big{(}(Bf_{1}+\cdots+Bf_{d+1})\cap R[t^{m}]\big{)}=R. In particular, $(Bf_{1}+\cdots+Bf_{d+1})\cap R[t^{m}]$ contains a monic polynomial $g$. Without loss of generality, $\deg(g)>0$. Since $\mathbf{f}\underset{B}{\sim}(f_{1},\ldots,f_{n-1},f_{n}+g^{c})$ and $f_{n}+g^{c}$ is monic for $c\gg 0$, we can further assume that $f_{n}$ is monic.

Since $R$ is Noetherian and $R[t]$ is module finite over $R[t^{m}]$, the algebra $B$ is also module finite over $R[t^{m}]$. In particular, $B$ is a finitely generated $R$-algebra and, in particular, Noetherian. The extension $B\subset R[t]$ is integral. So, every maximal ideal in $B$ lifts to $R[t]$. The extension $R\to R[t]/(f_{n})$ is integral and $R$ is local, implying $R[t]/(f_{n})$ is semilocal. Since there are only finitely many maximal ideals in $R[t]$, containing $f_{n}$, the number of maximal ideals in $B$, containing $f_{n}$, is also finite, i.e., $B/(f_{n})$ is semi-local. By Proposition 5.6 and Corollary 7.5 in [16, Ch. I], $\mathbf{f}\underset{B}{\sim}\mathbf{e}$. ∎

For two monoids $L_{1},L_{2}$ with $U(L_{1})=\{0\}$, we introduce the following submonoid

[TABLE]

Proof of Theorem 1.1.

By Lemma 9.2, we can assume that $U(M)$ is trivial.

Using Lemma 9.1 and the obvious adjustment of the argument in the proof of Lemma 6.1, the problem reduces to the transitivity of the action

[TABLE]

where $R$ and $n$ are as in Theorem 1.1, $M$ is a torsion free (not necessarily affine) monoid with trivial $U(M)$, and $n_{1},\ldots,n_{p}\in{\mathbb{N}}$.

We will prove the transitivity of the action (7) by induction on $p$.

The base case $p=0$ corresponds to torsion free monoids, already considered in Section 8.2. Assume the transitivity holds for $p-1$. At this points we can also assume $M$ is an affine positive monoid. In this case, pulling back a grading $R[M]=R\oplus R_{1}\oplus\cdots$ as in Lemma 2.4(b) along the projection M\leftthreetimes\big{(}{\mathbb{Z}}_{n_{1}}\times\cdots\times{\mathbb{Z}}_{n_{p}}\big{)}\to M, we obtain a grading R\left[M\leftthreetimes\big{(}{\mathbb{Z}}_{n_{1}}\times\cdots\times{\mathbb{Z}}_{n_{p}}\big{)}\right]=R\oplus S_{1}\oplus\cdots and so, by Proposition 4.2, $R$ can be assumed to be local.

Consider the following pull-back diagram with the vertical identity embeddings:

[TABLE]

where:

1. $\centerdot$

$\Lambda=A+B$, 2. $\centerdot$

$A=R\left[(M\leftthreetimes{\mathbb{Z}}_{+})\leftthreetimes\left({\mathbb{Z}}_{n_{1}}\times\cdots\times{\mathbb{Z}}_{n_{p-1}}\right)\right]$, 3. $\centerdot$

$B$ is the subalgebra $R[t^{n_{p}},t^{n_{p}+1}-t,\ldots,t^{2n_{p}-1}-t^{n_{p}-1}]\subset R[{\mathbb{Z}}_{+}]$, 4. $\centerdot$

$\pi$ is induced by $t\mapsto x$ for some generator $x\in{\mathbb{Z}}_{n_{p}}$.

Since $\dim R[{\mathbb{Z}}_{n_{p}}]=d$, the induction hypothesis implies that the elementary action on the unimodular $n$-rows over the ring at the lower-right corner of the diagram above is transitive. Therefore, by Lemma 4.1(a), it is enough to show that the elementary action on $\operatorname{Um}_{n}(\Lambda)$ is also transitive.

The very last pull-back diagram to be used by us is the following

[TABLE]

with the vertical surjective $R$-homomorphisms, induced by mapping all nontrivial monoid elements to $0\in R$, and the natural injective horizontal maps. Since $R$ local, the elementary action on $\operatorname{Um}_{n}(B)$ is transitive by Lemma 9.3. But the elementary action on $\operatorname{Um}_{n}(A)$ is also transitive by the induction hypothesis because $M\leftthreetimes{\mathbb{Z}}_{+}$ is torsion free. Thus, Lemma 4.1(a) completes the argument. ∎

Question. Does Theorem 1.1 extend to all commutative monoids?