#  Quantum toric degeneration of quantum flag and Schubert varieties

L. Rigal, P. Zadunaisky

###### Abstract

We show that certain homological regularity properties of graded connected
algebras, such as being AS-Gorenstein or AS-Cohen-Macaulay, can be tested by
passing to associated graded rings. In the spirit of noncommutative algebraic
geometry, this can be seen as an analogue of the classical result that, in a
flat family of varieties over the affine line, regularity properties of the
exceptional fiber extend to all fibers. We then show that quantized coordinate
rings of flag varieties and Schubert varieties can be filtered so that the
associated graded rings are twisted semigroup rings in the sense of [RZ]. This
is a noncommutative version of the result due to Caldero [C] stating that flag
and Schubert varieties degenerate into toric varieties, and implies that
quantized coordinate rings of flag and Schubert varieties are AS-Cohen-
Macaulay.

##  1 Introduction

Let $\Bbbk$ be a field, and let $A$ be a noetherian commutative algebra over
$\Bbbk$. If we put an ascending filtration on $A$ then we can build the Rees
ring of the filtration, which is a free $\Bbbk[t]$-algebra $\mathcal{R}$ such
that $A\cong\mathcal{R}/(t-\lambda)\mathcal{R}$ for all
$\lambda\in\Bbbk^{\times}$, while $\mathcal{R}/t\mathcal{R}$ is isomorphic to
the associated graded ring. In geometric terms, if $\Bbbk$ is algebraically
closed then the variety associated to $\mathcal{R}$ is a flat family over the
affine line, whose generic fiber is isomorphic to $\operatorname{Spec}A$ and
whose fiber over $0$ is isomorphic to $\operatorname{Spec}\leavevmode\nobreak\
\operatorname{\mathsf{gr}}\leavevmode\nobreak\ A$; in this context the fiber
over $0$ is called a degeneration of $\operatorname{Spec}A$. A standard result
from algebraic geometry states that if the fiber over $0$ is regular (resp.
Gorenstein, Cohen-Macaulay, or any other of a long list of properties) then
_all_ fibers are regular (resp. Gorenstein, Cohen-Macaulay, etc.)

Of course, the idea of studying a ring by imposing a filtration and passing to
the associated graded ring is a basic tool in an algebraist’s toolbox, and can
be applied outside of a geometric context. In particular the hypothesis of
commutativity is not necessary for filtered-to-graded methods to work.
However, in the spirit of noncommutative algebraic geometry, we should look at
the case where $A$ is noetherian, $\mathbb{N}$-graded and connected (i.e.
$A_{0}=\Bbbk$) with an eye on the geometric case. Although in this case there
are no varieties associated to our algebras as in the commutative setting, we
have suitable analogues of the notions of being regular, or Gorenstein, or
Cohen-Macaulay, defined in purely homological terms (see paragraph LABEL:AS-
reg). Hence it makes sense to ask whether these properties are “stable by flat
deformation”, i.e. if the fact that $\operatorname{\mathsf{gr}}A$ has any of
these properties implies that $A$ also has that property. The objective of
this paper is to develop these ideas in order to study natural classes of
noncommutative algebraic varieties, and its first main result 2 shows that
indeed, good geometric properties are stable by degeneration in this context.

In the commutative setting, the usual flag variety and its Schubert
subvarieties are examples where the degeneration method is successful. The
same holds for the more general flag and Schubert varieties that one may
associate to a semisimple Lie group. On the other hand, the theory of quantum
groups provides natural quantum analogues of flag and Schubert varieties,
whose classical counterparts can be recovered as semiclassical limits when the
deformation parameter tends to $1$. This is the class of noncommutative
varieties we intend to study.

The degeneration approach to the study of flag and Schubert varieties was
pioneered by de Concini, Eisenbud and Procesi [DCEP82], and followed by
Gonciulea and Lakshmibai [GL], and others. Caldero was the first to prove that
any Schubert variety of an arbitrary flag variety degenerates to an _affine
toric_ variety in [C]. A degeneration to a toric variety is particularly
convenient, since the geometric properties of toric varieties are easily
tractable, being encoded in the combinatorial properties of a semigroup
naturally attached to them in the affine case. For a general survey on the
subject of toric degenerations, including recent results, we refer the reader
to [FFL].

It is striking that even though he is interested in the classical objects, in
[C] Caldero uses the theory of quantum groups (more precisely the global bases
of Luzstig and Kashiwara) to produce bases of the coordinate rings of the
classical objects. His main argument then relies on Littelmann’s string
parametrization of the crystal basis of the negative part of the quantized
enveloping algebra, which allows him to show that his bases have good
multiplicative properties. It follows that one can build out of them adequate
filtrations of the coordinate rings, leading to a toric degeneration.

In contrast, we wish to produce degenerations of quantum Schubert varieties at
the noncommutative level. This led us to introduce noncommutative analogues of
affine toric varieties as a suitable target for degeneration and to study
their geometric properties in [RZ2], where we show that just as in the
classical case, these properties are encoded in the associated semigroup.
Inspired by Caldero’s work, in this article we show that quantum Schubert
varieties degenerate into quantum toric varieties and explore the consequences
of this fact.

We now discuss the contents of each section.

Section 2 begins with a recollection of general material on connected
$\mathbb{N}^{r}$-graded algebras, their homological regularity properties, and
the behavior of these properties with respect to a change of grading. In the
last subsection we prove our main transfer result: if a connected
$\mathbb{N}^{r}$-graded algebra has a filtration by finite dimensional graded
subspaces, then (under a mild technical condition known as property $\chi$)
the original algebra inherits the regularity properties of the associated
graded algebra, see Theorem LABEL:transfer. The key point is the existence of
a spectral sequence relating the $\operatorname{\mathsf{Ext}}$ spaces between
an algebra and its associated graded algebra; property $\chi$ guarantees that
these spaces are finite dimensional, which is crucial for the transfer of
properties.

The aim of section 3 is to introduce a general framework allowing to establish
the existence of a noncommutative toric degeneration. We first review the
basic theory of affine semigroups and the regularity properties of quantum
affine toric varieties from [RZ2]. We then introduce the notion of an algebra
of $(S,\varphi)$-type, where $S$ is an affine semigroup
$S\subset\mathbb{N}^{r}$ and $\varphi:S\longrightarrow\mathbb{N}$ is a
semigroup morphism. Any positive affine semigroup has a unique minimal
presentation as a quotient of a finitely generated free noncommutative monoid.
An algebra of $(S,\varphi)$-type is an algebra with a presentation modeled
after the presentation of $S$, with as many generators as $S$ and whose
product reflects that of $S$ up to terms of smaller order with respect to the
order induced on $S$ by $\varphi$. We then show that an algebra of
$(S,\varphi)$-type degenerates to a quantum toric variety if and only if it
has a basis of monomials indexed by $S$. At this stage it is useful to recall
the notion of an Algebra with a Straightening Law (see [DCEP82] in the
commutative case and [RZ] in the quantum case). It turns out that, under a
mild assumption, this structure provides a natural toric degeneration.

Schubert varieties in the grassmannian enjoy such an ASL structure, both in he
classical and quantum case. However, in more general Schubert varieties such a
structure no longer exists. The notion of an algebra of $(S,\varphi)$-type is
a generalization of the ASL structure which does apply in this more general
context, and the aforementioned basis indexed by $S$ plays the role of the
standard monomial basis. For more details regarding these two notions see
LABEL:s-phi-type. We finish this section by introducing the notion of an
$(S,<)$-basis, where $S$ is a semigroup and $<$ is a total order on $S$. It
turns out that the existence of such a structure associated to an affine
semigroup $S$, with the order obtained by pulling back the lexicographic order
of $\mathbb{N}^{t}$ through an embedding $S\hookrightarrow\mathbb{N}^{t}$, is
enough to obtain an $(S,\varphi)$-type structure and to prove the existence of
a toric degeneration.

Section 4 contains a review of the definitions of quantum flag varieties and
their Schubert subvarieties. We then introduce bases for these objects
originally defined by Caldero, who showed that they are parametrized by affine
semigroups using a variant of Littelmann’s string parametrization, and that
they have the multiplicative properties needed to produce a toric
degeneration. More precisely, we show that these bases are $(S,<)$-bases for
an adequate semigroup $S$. We stress that, since Caldero was interested only
in classical varieties, he worked over the field $\mathbb{C}(q)$, where $q$ is
a transcendental variable, and with an enlarged version of the quantum group
$U_{q}(\mathfrak{g})$ (which Janzen calls the adjoint type). Since we are
interested in the quantum setting, we work over the ring
$\mathbb{Z}[v,v^{-1}]$ and eventually specialize to an arbitrary field
containing a deformation parameter $q$ which is not a root of unity.
Furthermore we use the small (simply connected) version of
$U_{q}(\mathfrak{g})$.

While technical, we felt that these differences merited a detailed exposition
of the arguments, which we present in paragraphs LABEL:T:A-crystal-basis to
LABEL:Schubert-deg. Our proof that Caldero’s bases have the desired
multiplicative properties is different that the one found in [C]. While he
uses multiplicative properties of the dual global basis, we analize the
coproduct of $U^{-}_{q}(\mathfrak{g})$ in terms of the global basis. To do
this we are led to study a second basis introduced by Littelman in [Lit],
consisting of certain monomials of divided differences. This monomial basis
seems to be better adapted for computations, and has been used to obtain more
general degeneration results in the classical case by Fang, Fourier and
Littelmann [FFL2]. However, the global basis, or rather its string
parametrization, is crucial in order to prove that one obtains a _toric_
degeneration.

Acknowledgements: The second author would like to thank Xin Fang for an
illuminating dicussion on the relation between monomial and canonical bases,
and Köln University for its hospitality.

##  2 Degeneration of graded connected algebras

Let $r$ be a positive integer. Throughout this section $A$ denotes a
noetherian connected $\mathbb{N}^{r}$-graded $\Bbbk$-algebra. Here connected
means that the homogeneous component of $A$ of degree $(0,\ldots,0)$ is
isomorphic to $\Bbbk$ as a ring, so the ideal generated by all homogeneous
elements of non-zero degree is the unique maximal graded ideal of $A$; we
denote this ideal by $\mathfrak{m}$. Clearly $A/\mathfrak{m}\cong\Bbbk$ as
vector spaces, and whenever we consider $\Bbbk$ as $A$-bimodule, it will be
with this structure.

### Graded modules

####

grmod-generalities We denote by $\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A$
the category of $\mathbb{Z}^{r}$-graded $A$-modules with homogeneous morphisms
of degree $0$. We review some general properties of this category; the reader
is referred to [NV]*chapter 2 for proofs and details.

The category $\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A$ has enough
projectives and injectives, so we may speak of the graded projective and
injective dimensions of an object $M$, which we denote by
$\operatorname{pdim}_{A}^{\mathbb{Z}^{r}}M$ and
$\operatorname{injdim}^{\mathbb{Z}^{r}}_{A}M$, respectively. We denote by
$\mathbb{Z}^{r}\operatorname{\mathsf{mod}}A$ the subcategory of finitely
generated $\mathbb{Z}^{r}$-graded $A$-modules. Since $A$ is noetherian
$\mathbb{Z}^{r}\operatorname{\mathsf{mod}}A$ is an abelian category with
enough projectives.

For every object $M$ of $\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A$ and every
$\xi\in\mathbb{Z}^{r}$ we denote by $M_{\xi}$ the homogeneous component of $M$
of degree $\xi$. Also, we denote by $M[\xi]$ the object of
$\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A$ with the same underlying
$A$-module as $M$ and with homogeneous components
$M[\xi]_{\zeta}=M_{\zeta+\xi}$ for all $\zeta\in\mathbb{Z}^{r}$. If
$f:N\longrightarrow M$ is a morphism in
$\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A$ then the same function defines a
morphism $f[\xi]:N[\xi]\longrightarrow M[\xi]$. In this way we get an
endofunctor
$[\xi]:\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A\longrightarrow\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A$,
called the $\xi$-suspension functor; it is an autoequivalence, with inverse
$[-\xi]$.

Given $N,M$ objects of $\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A$ we set

$\displaystyle\operatorname{\underline{\mathsf{Hom}}}_{A}(N,M)=\bigoplus_{\xi\in\mathbb{Z}^{r}}\operatorname{\mathsf{Hom}}_{\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A}(N,M[\xi]).$

This is a $\mathbb{Z}^{r}$-graded vector space, with its component of degree
$\xi\in\mathbb{Z}^{r}$ equal to the space of homogeneous $A$-linear maps of
degree $\xi$ from $N$ to $M$. For every $i\geq 0$ we denote by
$\operatorname{\underline{\mathsf{Ext}}}_{A}^{i}$ the $i$-th right derived
functor of $\operatorname{\underline{\mathsf{Hom}}}_{A}$. We point out that

$\operatorname{\underline{\mathsf{Hom}}}_{A}(N[\xi],M)\cong\operatorname{\underline{\mathsf{Hom}}}_{A}(N,M[-\xi])\cong\operatorname{\underline{\mathsf{Hom}}}_{A}(N,M)[-\xi],$

as $\mathbb{Z}^{r}$-graded vector spaces, and that these isomorphisms induce
analogous ones for the corresponding right derived functors.

####

ext-change-of-grading Given $\xi=(\xi_{1},\ldots,\xi_{r})\in\mathbb{Z}^{r}$ we
set $|\xi|=\xi_{1}+\cdots+\xi_{r}$. Given a $\mathbb{Z}^{r}$-graded vector
space $V$, we denote by $|V|$ the $\mathbb{Z}$-graded vector space whose
$n$-th homogeneous component is

$\displaystyle|V|_{n}=\bigoplus_{|\xi|=n}V_{\xi}.$

In particular $|A|$ is a connected $\mathbb{N}$-graded algebra. Also, if $M$
is a $\mathbb{Z}^{r}$-graded $A$-module then $|M|$ is a $\mathbb{Z}$-graded
$|A|$-module, and this assignation is functorial. Since $A$ is noetherian,
[RZ2]*Proposition 1.3.7 implies that for every $i\geq 0$ and any pair of
$\mathbb{Z}^{r}$-graded modules $N,M$, with $N$ finitely generated, there is
an isomorphism of $\mathbb{Z}$-graded modules

$|\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(N,M)|\cong\operatorname{\underline{\mathsf{Ext}}}^{i}_{|A|}(|N|,|M|),$

natural in both variables.

### Homological regularity properties

In this subsection we discuss some homological properties a connected
$\mathbb{N}^{r}$-graded algebra may posses. Most of the material found in this
section is standard for connected $\mathbb{N}$-graded algebras.

####

chi Let $M$ be a $\mathbb{Z}^{r}$-graded $A$-module. We say that $\chi(M)$
holds if for each $i\geq 0$ the graded vector space
$\operatorname{\underline{\mathsf{Ext}}}_{A}^{i}(\Bbbk,M)$ is finite
dimensional, and say that the algebra $A$ has property $\chi$ if $\chi(M)$
holds for every finitely generated $\mathbb{Z}^{r}$-graded $A$-module $M$.
Property $\chi$ was originally introduced in [AZ]*section 3 and plays a
fundamental role in noncommutative algebraic geometry.

####

torsion-functor Associated to $A$ and $\mathfrak{m}$ there is a _torsion
functor_

$\displaystyle\Gamma_{\mathfrak{m}}:\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A$
$\displaystyle\longrightarrow\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A$
$\displaystyle M$ $\displaystyle\longmapsto\\{x\in
M\mid\mathfrak{m}^{n}x=0\mbox{ for }n\gg 0\\},$

which acts on morphisms by restriction and correstriction. The torsion functor
is left exact, and for each $i\geq 0$ its $i$-th right derived functor is
denoted by $H_{\mathfrak{m}}^{i}$ and called the _$i$ -th local cohomology
functor_ of $A$.

There exists a natural isomorphism

$\displaystyle\Gamma_{\mathfrak{m}}\cong\varinjlim_{n}\operatorname{\underline{\mathsf{Hom}}}_{A}(A/\mathfrak{m}^{n},-)$

which by standard homological algebra extends to natural isomorphisms

$\displaystyle
H^{i}_{\mathfrak{m}}\cong\varinjlim_{n}\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(A/\mathfrak{m}^{n},-)$

for all $i\geq 1$. The proof of this fact is completely analogous to the one
found in [BS]*Theorem 1.3.8 for commutative ungraded algebras.

We denote by $A^{\circ}$ the opposite algebra of $A$, which is also a
connected $\mathbb{N}^{r}$-graded algebra, and by $\mathfrak{m}^{\circ}$ its
maximal graded ideal. We write $\Gamma_{\mathfrak{m}^{\circ}}$ and
$H^{i}_{\mathfrak{m}^{\circ}}$ for the corresponding torsion and local
cohomology functors, respectively.

####

depth-and-ldim Given an object $M$ of
$\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}A$, its _depth_ and _local
dimension_ are defined as

$\displaystyle\operatorname{depth}_{\mathfrak{m}}M$
$\displaystyle=\inf\\{i\in\mathbb{N}\mid\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(\Bbbk,M)\neq
0\\},$ $\displaystyle\operatorname{ldim}_{\mathfrak{m}}M$
$\displaystyle=\sup\\{i\in\mathbb{N}\mid H^{i}_{\mathfrak{m}}(M)\neq 0\\},$

respectively. The _local cohomological dimension_ of $A$, denoted by
$\operatorname{lcd}_{\mathfrak{m}}A$, is the supremum of the
$\operatorname{ldim}_{\mathfrak{m}}M$ with $M$ finitely generated.

####

AS-reg The following definition is taken from [RZ2]*Definition 2.1.1. It is an
$\mathbb{N}^{r}$-graded analogue of the definition of the AS-Cohen-Macaulay,
AS-Gorenstein and AS-regular properties for connected $\mathbb{N}$-graded
algebras found in the literature, see for example the introduction to [JZ].

######  Definition 1.

Let $A$ be a connected noetherian $\mathbb{N}^{r}$-graded algebra.

  1. 1.

$A$ is called _AS-Cohen-Macaulay_ if there exists $n\in\mathbb{N}$ such that
$H^{i}_{\mathfrak{m}}(A)=0$ and $H_{\mathfrak{m}^{\circ}}^{i}(A)=0$ for all
$i\neq n$.

  2. 2.

$A$ is called _left AS-Gorenstein_ if it has finite graded injective dimension
$n$ and there exists $\ell\in\mathbb{Z}^{r}$, called the _Gorenstein shift_ of
$A$, such that

$\operatorname{\underline{\mathsf{Ext}}}_{A}^{i}(\Bbbk,A)\cong\begin{cases}\Bbbk[\ell]&\mbox{for
}i=n,\\\ 0&\mbox{for }i\neq n,\end{cases}$

as $\mathbb{Z}^{r}$-graded $A^{\circ}$-modules. We say $A$ is _right_ AS-
Gorenstein if $A^{\circ}$ is left AS-Gorenstein. Finally $A$ is _AS-
Gorenstein_ if $A$ and $A^{\circ}$ are left AS-Gorenstein, with the same
injective dimensions and Gorenstein shifts.

  3. 3.

$A$ is called _AS-regular_ if it is AS-Gorenstein, and its left and right
graded global dimensions are finite and equal.

####

invariance-by-grading The properties discussed in paragraphs LABEL:chi to
LABEL:AS-reg are defined in terms of the category of $\mathbb{Z}^{r}$-graded
$A$-modules. In LABEL:ext-change-of-grading we defined the algebra $|A|$,
which is equal to $A$ as algebra but is endowed with a connected
$\mathbb{N}$-grading induced by the grading on $A$ and the group morphism
$|\cdot|$. The maximal graded ideals of $A$ and $|A|$ coincide as vector
spaces, so we may ask whether the fact that $A$ has property $\chi$, or finite
local dimension, or the AS-Cohen-Macaulay property, etc., implies that $|A|$
has the corresponding property.

Let us say that a property $P$ does not depend on the grading of $A$ if the
following holds: for every connected $\mathbb{N}^{t}$-graded algebra $B$ with
maximal ideal $\mathfrak{n}$, which is isomorphic to $A$ as algebra through an
isomorphism that sends $\mathfrak{n}$ to $\mathfrak{m}$, $A$ has property $P$
if and only if $B$ has property $P$. As shown in [RZ2]*Corollary 1.3.9, the
local dimension of $A$ does not depend on the grading of $A$. An analogous
result is proved in [RZ2]*Remark 2.1.7 for the properties defined in LABEL:AS-
reg. However, the situation is more delicate for property $\chi$. The
following lemma shows that property $\chi$ is independent of the grading of
$A$ under the hypothesis that $A$ has finite local dimension; we do not know
whether this hypothesis can be eliminated.

######  Lemma 1.

Suppose $\operatorname{lcd}_{\mathfrak{m}}A<\infty$. Then the algebra $A$ has
property $\chi$ if and only if $\chi(A)$ holds.

  * Proof.

If $A$ has property $\chi$ then clearly $\chi(A)$ holds. To prove the opposite
implication, assume $\chi(A)$ holds. Recall from LABEL:ext-change-of-grading
that for every $\mathbb{Z}^{r}$-graded $A$-module $M$ and for every $i\geq 0$
there exists a graded vector space isomorphism

$|\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(\Bbbk,M)|\cong\operatorname{\underline{\mathsf{Ext}}}_{|A|}^{i}(|\Bbbk|,|M|),$

so $\chi(M)$ holds if and only if $\chi(|M|)$ holds; in view of this, the
hypothesis implies $\chi(|A|)$ holds.

Since
$\operatorname{ldim}_{|\mathfrak{m}|}|A|=\operatorname{ldim}_{\mathfrak{m}}A<\infty$,
we may apply [RZ2]*Proposition 2.2.6 and conclude that $|A|$ has property
$\chi$. From this it follows that $\chi(|M|)$ holds for every
$\mathbb{Z}^{r}$-graded $A$-module $M$, and hence so does $\chi(M)$. ∎

####

chi-and-local-cohomology We finish this subsection with a technical result on
the relation between property $\chi$ and local cohomology.

######  Lemma 2.

For every $n\in\mathbb{N}$, let $A_{\geq n}$ be the ideal generated by all
homogeneous elements of degree $\xi$ with $|\xi|\geq n$. Let $M$ be a finitely
generated $\mathbb{Z}^{r}$-graded $A$-module such that $\chi(M)$ holds. Then
for every $i\geq 0$ and every $t\in\mathbb{Z}$ there exists
$n_{0}\in\mathbb{Z}$ such that

$\displaystyle\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(A/A_{\geq
n},M)_{\xi}$ $\displaystyle\cong H^{i}_{\mathfrak{m}}(M)_{\xi}$

for all $n\geq n_{0}$ and all $\xi\in\mathbb{Z}^{r}$ such that $|\xi|\geq t$.

  * Proof.

Since $\mathfrak{m}$ is finitely generated, say by elements
$x_{1},\ldots,x_{r}$ with degrees $\xi_{1},\ldots,\xi_{r}$ such that
$|\xi_{i}|\geq 1$, clearly $\mathfrak{m}^{n}\subset A_{\geq n}$. Setting
$l=\max\\{|\xi_{i}|:1\leq i\leq r\\}$ we obtain $A_{\geq
ln}\subset\mathfrak{m}^{n}$. Knowing this, the proof of [BS]*Proposition 3.1.1
easily adapts to show that for every $i\geq 0$ there exist natural
isomorphisms

$\displaystyle\varinjlim_{n}\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(A/A_{\geq
n},-)\cong\varinjlim_{n}\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(A/\mathfrak{m}^{n},-)\cong
H^{i}_{\mathfrak{m}}.$

The statement of the lemma will follow if we show that for all $\xi$ as in the
statement, the homogeneous component of degree $\xi$ of the natural map

$\displaystyle\pi^{n}:\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(A/A_{\geq
n},M)\longrightarrow H^{i}_{\mathfrak{m}}(M)$

is an isomorphism for $n\gg 0$.

Fixing $t$ as in the statement, [AZ]*Proposition 3.5 (1) implies that the
natural map
$\pi^{n}_{d}:\operatorname{\underline{\mathsf{Ext}}}^{i}_{|A|}(|A|/|A_{\geq
n}|,|M|)_{d}\longrightarrow H^{i}_{|\mathfrak{m}|}(|M|)_{d}$ is an isomorphism
for all $d\geq t$ if $n$ is large enough. Now by [RZ2]*Propositions 1.3.7 and
1.3.8 there exist isomorphisms

$\displaystyle\operatorname{\underline{\mathsf{Ext}}}^{i}_{|A|}(|A|/|A_{\geq
n}|,|M|)_{d}$
$\displaystyle\cong\bigoplus_{|\xi|=d}\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(A/A_{\geq
n},M)_{\xi};$ $\displaystyle H^{i}_{|\mathfrak{m}|}(|M|)_{d}$
$\displaystyle\cong\bigoplus_{|\xi|=d}H^{i}_{\mathfrak{m}}(M)_{\xi}.$

Since the assignation $M\mapsto|M|$ is functorial, we also get that
$\pi^{n}_{d}=\bigoplus_{|\xi|=d}\pi^{n}_{\xi}$. Thus for all $\xi$ such that
$|\xi|\geq t$, the map $\pi^{n}_{\xi}$ is an isomorphism if $n$ is large
enough. ∎

### Transfer of regularity properties by degeneration

In this subsection we prove that if $A$ has a filtration compatible with its
grading, and the associated graded algebra has property $\chi$, then the
regularity properties discussed in the previous subsection transfer from
$\operatorname{\mathsf{gr}}A$ to $A$. All undefined terms regarding
filtrations can be found in [VO]*chapter I.

####

regular-filtration Recall that $A$ denotes a noetherian
$\mathbb{N}^{r}$-graded algebra. The general setup for the subsection is as
follows: we assume that $A$ has a _connected_ filtration, that is an
exhaustive filtration $\mathcal{F}=\\{F_{n}A\\}_{n\geq 0}$, with
$\Bbbk=F_{0}A\subset F_{1}A\subset\cdots\subset
F_{n}A\subset\cdots\bigcup_{n\geq 0}F_{n}A=A$, such that each layer $F_{n}A$
is a finite dimensional graded vector space, and $F_{n}A\cdot F_{m}A\subset
F_{n+m}A$ for all $n,m\in\mathbb{N}$. For each $\xi\in\mathbb{N}^{r}$ the
homogeneous component $A_{\xi}$ has an induced filtration
$\\{F_{n}A_{\xi}\\}_{n\geq 0}$, where $F_{n}A_{\xi}=F_{n}A\cap A_{\xi}$. Since
$A_{\xi}$ is finite dimensional this filtration is finite, so the associated
graded ring $\operatorname{\mathsf{gr}}A$ is a connected and locally finite
$\mathbb{N}^{r+1}$-graded algebra.

Given any $\mathbb{Z}^{r}$-graded $A$-module $M$ with a filtration whose
layers are $\mathbb{Z}^{r}$-graded subspaces, we can construct the
$\mathbb{Z}^{r+1}$-graded $\operatorname{\mathsf{gr}}A$-module
$\operatorname{\mathsf{gr}}M$. If $M$ is any $\mathbb{Z}^{r}$-graded
$A$-module then it can be endowed with such a filtration as follows: fix a
graded subspace $N\subset M$ that generates $M$ over $A$, and for each $n\geq
0$ set $F_{n}M=(F_{n}A)N$. Any such filtration is called _standard_ , and is
an exhaustive and discrete filtration by graded subspaces. If $M$ is finitely
generated and $N$ is finite dimensional then the layers of this filtration are
also finite dimensional.

####

P:ext-ss The main tool used to transfer homological information from
$\operatorname{\mathsf{gr}}A$ to $A$ is a spectral sequence that we associate
to any pair of $\mathbb{Z}^{r}$-graded $A$-modules $N,M$, which converges to
$\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(N,M)$ and whose first page
consists of the homogeneous components of
$\operatorname{\underline{\mathsf{Ext}}}^{i}_{\operatorname{\mathsf{gr}}A}(\operatorname{\mathsf{gr}}N,\operatorname{\mathsf{gr}}M)$.
The proof is straightforward, but relies on several graded analogues of
classical constructions for filtered rings. These constructions can be found
in [VO]*Chapter I and [MR]*Section 7.6, and the proofs found in the references
easily adapt to the graded context, so we use them without further comment.

In order to keep track of the extra component in the grading when passing to
associated graded objects, we make a slight abuse of notation: given a
$\mathbb{Z}^{r+1}$-graded vector space $V$, we denote by $V_{(\xi,p)}$ its
homogeneous component of degree $(\xi_{1},\ldots,\xi_{r},p)$.

######  Proposition 1.

Let $A$ be an $\mathbb{N}^{r}$-graded algebra with a connected filtration, and
assume $\operatorname{\mathsf{gr}}A$ is noetherian. Let $M,N$ be filtered
$\mathbb{Z}^{r}$-graded $A$-modules, with $N$ finitely generated, and suppose
that the filtration on $N$ is standard and the filtration on $M$ is discrete.
Then for every $\xi\in\mathbb{Z}^{r}$ there exists a convergent spectral
sequence

$\displaystyle
E(N,M)_{\xi}:E_{p,q}^{1}=\operatorname{\underline{\mathsf{Ext}}}_{\operatorname{\mathsf{gr}}A}^{-p-q}(\operatorname{\mathsf{gr}}N,\operatorname{\mathsf{gr}}M)_{(\xi,p)}$
$\displaystyle\Rightarrow\operatorname{\underline{\mathsf{Ext}}}_{A}^{-p-q}(N,M)_{\xi}$
$\displaystyle p,q\in\mathbb{Z},$

such that the filtration of the vector spaces on the right hand side is
finite.

  * Proof.

By the $\mathbb{Z}^{r}$-graded version of [MR]*Theorem 6.17, there exists a
projective resolution $P^{\bullet}\longrightarrow N$ by filtered projective
$\mathbb{Z}^{r}$-graded $A$-modules with filtered differentials, such that the
associated graded complex
$\operatorname{\mathsf{gr}}P^{\bullet}\longrightarrow\operatorname{\mathsf{gr}}N$
is a $\mathbb{Z}^{r+1}$-graded projective resolution of the
$\operatorname{\mathsf{gr}}A$-module $\operatorname{\mathsf{gr}}N$. Using the
filtration for the $\operatorname{\underline{\mathsf{Hom}}}$ spaces defined in
[VO]*section I.2, the complex
$\operatorname{\underline{\mathsf{Hom}}}_{A}(P^{\bullet},M)$ is a graded
complex with a filtration by graded subcomplexes, whose differentials are
filtered maps.

If we fix $\xi\in\mathbb{Z}^{r}$, the homogeneous component
$\operatorname{\underline{\mathsf{Hom}}}_{A}(P^{\bullet},M)_{\xi}$ is a
complex of filtered finite dimensional vector spaces. By [W]*5.5.1.2 there
exists a spectral sequence with page one equal to

$\displaystyle E_{p,q}^{1}$
$\displaystyle=H_{p+q}\left(\frac{F_{p}\operatorname{\underline{\mathsf{Hom}}}_{A}(P^{\bullet},M)_{\xi}}{F_{p-1}\operatorname{\underline{\mathsf{Hom}}}_{A}(P^{\bullet},M)_{\xi}}\right)$
$\displaystyle p,q\in\mathbb{Z},$

that converges to

$\displaystyle
H_{p+q}(\operatorname{\underline{\mathsf{Hom}}}_{A}(P^{\bullet},M)_{\xi})\cong\operatorname{\underline{\mathsf{Ext}}}^{-p-q}_{A}(N,M)_{\xi}.$

This last space is finite dimensional, and hence the filtration on it is
finite. Thus we only need to prove that for each $p,q\in\mathbb{Z}$ there
exists an isomorphism

$\displaystyle
H_{p+q}\left(\frac{F_{p}\operatorname{\underline{\mathsf{Hom}}}_{A}(P^{\bullet},M)_{\xi}}{F_{p-1}\operatorname{\underline{\mathsf{Hom}}}_{A}(P^{\bullet},M)_{\xi}}\right)$
$\displaystyle\cong\operatorname{\underline{\mathsf{Ext}}}_{\operatorname{\mathsf{gr}}A}^{-p-q}(\operatorname{\mathsf{gr}}N,\operatorname{\mathsf{gr}}M)_{(\xi,p)}.$

By [VO]*Lemma 6.4, there exists an isomorphism of complexes

$\displaystyle\varphi(P^{\bullet},M):\operatorname{\mathsf{gr}}(\operatorname{\underline{\mathsf{Hom}}}_{A}(P^{\bullet},M))\longrightarrow\operatorname{\underline{\mathsf{Hom}}}_{\operatorname{\mathsf{gr}}A}(\operatorname{\mathsf{gr}}P^{\bullet},\operatorname{\mathsf{gr}}M),$

which is defined explicitly in the reference. Direct inspection shows that the
map $\varphi(P^{\bullet},M)$ is homogeneous, so looking at its component of
degree $\xi$ we obtain an isomorphism

$\displaystyle\frac{F_{p}\operatorname{\underline{\mathsf{Hom}}}_{A}(P^{\bullet},M)_{\xi}}{F_{p-1}\operatorname{\underline{\mathsf{Hom}}}_{A}(P^{\bullet},M)}_{\xi}\cong\operatorname{\underline{\mathsf{Hom}}}_{\operatorname{\mathsf{gr}}A}(\operatorname{\mathsf{gr}}P^{\bullet},\operatorname{\mathsf{gr}}M)_{(\xi,p)}.$

Since $\operatorname{\mathsf{gr}}P^{\bullet}$ is a $\mathbb{Z}^{r+1}$-graded
projective resolution of $\operatorname{\mathsf{gr}}N$, we obtain the desired
isomorphism by applying $H_{p+q}$ to both sides of the isomorphism. ∎

The following Corollary is an immediate consequence of the previous
Proposition.

######  Corollary 1.

Let $A$ be an $\mathbb{N}^{r}$-graded algebra with a connected filtration, and
assume $\operatorname{\mathsf{gr}}A$ is noetherian. Let $M,N$ be filtered
$\mathbb{Z}^{r}$-graded $A$-modules, with $N$ finitely generated, and suppose
that the filtration on $N$ is standard and the filtration on $M$ is discrete.
Then the following hold.

  1. (a)

For each $i\geq 0$ and each $\xi\in\mathbb{Z}^{r}$

$\dim_{\Bbbk}\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(N,M)_{\xi}\leq\sum_{p=-\infty}^{\infty}\dim_{\Bbbk}\operatorname{\underline{\mathsf{Ext}}}^{i}_{\operatorname{\mathsf{gr}}A}(\operatorname{\mathsf{gr}}N,\operatorname{\mathsf{gr}}M)_{(\xi,p)}.$

  2. (b)

$\operatorname{pdim}_{A}^{\mathbb{Z}^{r}}N\leq\operatorname{pdim}_{\operatorname{\mathsf{gr}}A}^{\mathbb{Z}^{r+1}}\operatorname{\mathsf{gr}}N$
and
$\operatorname{injdim}_{A}^{\mathbb{Z}^{r}}M\leq\operatorname{injdim}_{\operatorname{\mathsf{gr}}A}^{\mathbb{Z}^{r+1}}\operatorname{\mathsf{gr}}M$.

  3. (c)

If $\chi(\operatorname{\mathsf{gr}}M)$ holds then $\chi(M)$ holds.

####

ldim We now prove a result that relates the local cohomology of a
$\mathbb{Z}^{r}$-graded $A$-module $M$ with that of its associated graded
module $\operatorname{\mathsf{gr}}M$. We will do this by combining Proposition
LABEL:P:ext-ss with Lemma LABEL:chi-and-local-cohomology and the formalism of
the change of grading functors introduced in [RZ]*Section 1.3. We recall the
relevant details. Given a group morphism
$\varphi:\mathbb{Z}^{r}\longrightarrow\mathbb{Z}^{t}$, with $t\in\mathbb{N}$,
there exists a functor
$\varphi_{!}:\mathbb{Z}^{r}\operatorname{\mathsf{Mod}}\Bbbk\longrightarrow\mathbb{Z}^{t}\operatorname{\mathsf{Mod}}\Bbbk$
that sends a $\mathbb{Z}^{r}$-graded vector space $V$ to the
$\mathbb{Z}^{t}$-graded vector space $\varphi_{!}(V)$, whose homogeneous
component of degree $\zeta\in\mathbb{Z}^{t}$ is

$V_{\zeta}=\bigoplus_{\varphi(\xi)=\zeta}V_{\xi}.$

Notice that $\varphi_{!}$ does not change the underlying vector space of its
argument, only its grading. If $R$ is a $\mathbb{Z}^{r}$-graded algebra then
$\varphi_{!}(R)$ is a $\mathbb{Z}^{t}$-graded algebra, and if $M$ is a
$\mathbb{Z}^{r}$-graded $R$-module then $\varphi_{!}(M)$ is a
$\mathbb{Z}^{t}$-graded $\varphi_{!}(R)$-module with the same underlying
$R$-module structure as $M$.

######  Corollary 2.

Let $A$ be an $\mathbb{N}^{r}$-graded algebra with a connected filtration, and
assume $\operatorname{\mathsf{gr}}A$ is noetherian. Let $M$ be a filtered
$\mathbb{Z}^{r}$-graded $A$-module with a discrete filtration, and assume
$\chi(\operatorname{\mathsf{gr}}M)$ holds. Then for each $i\geq 0$ and each
$\xi\in\mathbb{Z}^{r}$

$\dim_{\Bbbk}H^{i}_{\mathfrak{m}}(M)_{\xi}\leq\sum_{p=-\infty}^{\infty}\dim_{\Bbbk}H^{i}_{\operatorname{\mathsf{gr}}\mathfrak{m}}(\operatorname{\mathsf{gr}}M)_{(\xi,p)}.$

  * Proof.

Let $\pi:\mathbb{Z}^{r+1}\longrightarrow\mathbb{Z}^{r}$ be the projection to
the first $r$-coordinates, and set $B=\pi_{!}(\operatorname{\mathsf{gr}}A)$,
so for every $\xi\in\mathbb{Z}^{r}$

$\displaystyle
B_{\xi}=\bigoplus_{p\in\mathbb{Z}}\operatorname{\mathsf{gr}}A_{(\xi,p)}=\operatorname{\mathsf{gr}}(A_{\xi}).$

Thus $B$ is a connected $\mathbb{N}^{r}$-graded algebra. Set
$\mathfrak{n}=\pi_{!}(\operatorname{\mathsf{gr}}\mathfrak{m})$, which is the
maximal graded ideal of $B$, and set
$\tilde{M}=\pi_{!}(\operatorname{\mathsf{gr}}M)$.

By [RZ2]*Proposition 1.3.7 $\chi(\operatorname{\mathsf{gr}}M)$ implies
$\chi(\tilde{M})$ and by item (c) of Corollary LABEL:P:ext-ss it also implies
$\chi(M)$, so we may apply Lemma LABEL:chi-and-local-cohomology to $M$ and
$\tilde{M}$, and deduce that for given $\xi\in\mathbb{Z}^{r}$ and $i\geq 0$
the natural maps

$\displaystyle\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(A/A_{\geq
n},M)_{\xi}$ $\displaystyle\longrightarrow H^{i}_{\mathfrak{m}}(M)_{\xi}$
$\displaystyle\operatorname{\underline{\mathsf{Ext}}}^{i}_{B}(B/B_{\geq
n},\tilde{M})_{\xi}$ $\displaystyle\longrightarrow
H^{i}_{\mathfrak{n}}(\tilde{M})_{\xi}$

are isomorphisms for $n\gg 0$.

Combining this with [RZ2]*Propositions 1.3.7 and 1.3.8, we obtain a chain of
isomorphisms

$\displaystyle\bigoplus_{p\in\mathbb{Z}}\operatorname{\underline{\mathsf{Ext}}}^{i}_{\operatorname{\mathsf{gr}}A}$
$\displaystyle\left(\frac{\operatorname{\mathsf{gr}}A}{\operatorname{\mathsf{gr}}(A_{\geq
n})},\operatorname{\mathsf{gr}}M\right)_{(\xi,p)}$
$\displaystyle\cong\operatorname{\underline{\mathsf{Ext}}}^{i}_{B}\left(B/B_{\geq
n},\tilde{M}\right)_{\xi}\cong
H^{i}_{\mathfrak{n}}(\tilde{M})_{\xi}\cong\bigoplus_{p\in\mathbb{Z}}H^{i}_{\operatorname{\mathsf{gr}}\mathfrak{m}}(\operatorname{\mathsf{gr}}M)_{(\xi,p)}$

By definition $\operatorname{\mathsf{gr}}(A/A_{\geq
n})\cong\operatorname{\mathsf{gr}}A/\operatorname{\mathsf{gr}}(A_{\geq n})$,
so applying item (a) of Corollary LABEL:P:ext-ss and taking $n\gg 0$ we obtain

$\displaystyle\dim_{\Bbbk}H^{i}_{\mathfrak{m}}(M)_{\xi}$
$\displaystyle=\dim_{\Bbbk}\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(A/A_{\geq
n},M)_{\xi}$
$\displaystyle\leq\sum_{p=-\infty}^{\infty}\dim_{\Bbbk}\operatorname{\underline{\mathsf{Ext}}}^{i}_{\operatorname{\mathsf{gr}}A}(\operatorname{\mathsf{gr}}A/\operatorname{\mathsf{gr}}(A_{\geq
n}),M)_{(\xi,p)}$
$\displaystyle=\sum_{p=-\infty}^{\infty}\dim_{\Bbbk}H^{i}_{\operatorname{\mathsf{gr}}\mathfrak{m}}(\operatorname{\mathsf{gr}}M)_{(\xi,p)}.$

∎

####

transfer We are now ready to prove the main result of this section.

######  Theorem 1.

Suppose $A$ is a connected $\mathbb{N}^{r}$-graded algebra endowed with a
connected filtration, and that $\operatorname{\mathsf{gr}}A$ is noetherian and
has property $\chi$. Then the following hold.

  1. (a)

$A$ has property $\chi$.

  2. (b)

$\operatorname{lcd}_{\mathfrak{m}}A\leq\operatorname{lcd}_{\operatorname{\mathsf{gr}}\mathfrak{m}}\operatorname{\mathsf{gr}}A$.

  3. (c)

If $\operatorname{\mathsf{gr}}A$ is AS-Cohen-Macaulay, AS-Gorenstein or AS-
Regular, so is $A$.

  * Proof.

The hypothesis that $\operatorname{\mathsf{gr}}A$ is noetherian implies that
$A$ is noetherian [MR]*1.6.9. If $M$ is any finitely generated
$\mathbb{Z}^{r}$-graded $A$-module, we may filter it using the procedure
described in LABEL:regular-filtration. Since
$\chi(\operatorname{\mathsf{gr}}M)$ holds by hypothesis, item (c) of Corollary
LABEL:P:ext-ss implies $\chi(M)$ holds, which proves that $A$ has property
$\chi$.

Item 2 follows from Corollary LABEL:ldim, as does the fact that if
$\operatorname{\mathsf{gr}}A$ is AS-Cohen-Macaulay so is $A$. If
$\operatorname{\mathsf{gr}}A$ is AS-Gorenstein of injective dimension $n$ and
Gorenstein shift $(\xi,p)\in\mathbb{Z}^{r+1}$, the algebra $A$ has injective
dimension at most $n$ by item (b) of Corollary LABEL:P:ext-ss. Also, for each
$\zeta\in\mathbb{Z}^{r}$ the spectral sequence $E(\Bbbk,A)_{\zeta}$ of
Proposition LABEL:P:ext-ss degenerates at page $1$: it is zero if
$\zeta\neq-\xi$, while for $\zeta=-\xi$ there is a single, one dimensional
non-zero entry in the diagonal $-p-q=n$. Hence we obtain vector space
isomorphisms

$\displaystyle\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(\Bbbk,A)_{\zeta}$
$\displaystyle\cong\begin{cases}\Bbbk&\mbox{ if $\zeta=-\xi$ and $i=n$;}\\\
0&\mbox{otherwise.}\end{cases}$

Thus $\operatorname{\underline{\mathsf{Ext}}}^{i}_{A}(\Bbbk,A)\cong\Bbbk[\xi]$
as $\mathbb{Z}^{r}$-graded vector spaces. Any such isomorphism is also
$A^{\circ}$-linear, so $A$ is left AS-Gorenstein of injective dimension $n$
and Gorenstein shift $\xi$. The same proof applies to show that $A$ is right
AS-Gorenstein with the same injective dimension and Gorenstein shift, so $A$
is AS-Gorenstein.

Finally, assume that $\operatorname{\mathsf{gr}}A$ is AS-regular. Then $A$ is
AS-Gorenstein, and item (b) of Corollary LABEL:P:ext-ss implies that it has
finite left and right global dimensions. These dimensions are equal to the
left and right projective dimensions of $\Bbbk$ as $A$-module [RZ2]*Lemma
2.1.5, which in turn equal the left and right injective dimensions of $A$ and
hence coincide, so $A$ is AS-regular. ∎

##  3 Quantum affine toric degenerations

In this section we recall the homological properties of quantum affine toric
varieties proved in [RZ2], and give necessary and sufficient conditions for a
connected $\mathbb{N}^{r}$-graded algebra $A$ to have a connected filtration
such that the associated graded ring is a quantum positive affine toric
variety.

### Quantum affine toric varieties

####

affine-semigroups Recall that an affine semigroup is a finitely generated
monoid isomorphic to a subsemigroup of $\mathbb{Z}^{r}$ for some
$r\in\mathbb{N}$. An embedding of $S$ is an injective semigroup morphism
$i:S\hookrightarrow\mathbb{Z}^{r}$ for some $r\in\mathbb{N}$. By definition,
every affine semigroup $S$ is commutative and cancellative, so it has a group
of fractions, which we denote by $G(S)$, and the natural map from $S$ to
$G(S)$ is injective. The group $G(S)$ is a finitely generated and torsion-free
commutative group, so there exists a natural number $r$ such that
$G(S)\cong\mathbb{Z}^{r}$ as groups; we refer to $r$ as the _rank_ of $S$ and
denote it $\operatorname{rk}S$. We say that $i$ is a _full embedding_ if the
image of $S$ generates $\mathbb{Z}^{r}$ as a group, in which case
$r=\operatorname{rk}S$. Fixing an isomorphism
$G(S)\cong\mathbb{Z}^{\operatorname{rk}(S)}$ we obtain a full embedding of $S$
in an obvious way.

####

interior Let $S$ be an affine semigroup. Fixing an embedding
$i:S\hookrightarrow\mathbb{Z}^{r}$ we identify $S$ with its image and see
$\mathbb{Z}^{r}$ as a subgroup of $\mathbb{R}^{r}$ in the obvious way. The
_rational cone_ generated by $S$ in $\mathbb{R}^{r}$ is the set

$\mathbb{R}_{+}S=\left\\{\sum_{i=1}^{n}r_{i}s_{i}\mid r_{i}\in\mathbb{R}_{\geq
0},s_{i}\in S,n\in\mathbb{N}\right\\}.$

The _relative interior_ of $S$ is defined as
$\operatorname{\mathsf{relint}}S=S\cap\mathbb{R}_{+}S^{\circ}$, where
$\mathbb{R}_{+}S^{\circ}$ is the topological interior of $\mathbb{R}_{+}S$ as
a subset of the vector space $\mathbb{R}S$ endowed with the subspace topology
induced from $\mathbb{R}^{r}$. The relative interior is intrinsic to $S$ and
does not depend on the chosen embedding [BG]*Remark 2.6.

An affine semigroup $S$ of rank $r$ is called _normal_ if it verifies the
following property: given $z\in G(S)$, if there exists $m\in\mathbb{N}^{*}$
such that $mz\in S$, then $z\in S$. If we identify $S$ with a subset of
$\mathbb{Z}^{r}$ through a full embedding and consider the real cone
$\mathbb{R}_{+}S$ of $S$ inside $\mathbb{R}^{r}$ in the obvious way, then
Gordan’s lemma [BH]*Proposition 6.1.2 states that $S$ is normal if and only if
$S=\mathbb{Z}^{r}\cap\mathbb{R}_{+}S$.

######  Example 1.

We now introduce an example that will recur through the rest of this section.
Recall that a lattice is a poset $\mathcal{L}$ such that any two elements
$x,y$ have an infimum, called the _meet_ of $x$ and $y$ and denoted $x\wedge
y$, and a supremum, called the _join_ of $x$ and $y$ and denoted $x\vee y$.
For example, given $u\in\mathbb{N}$ the poset $\mathbb{N}^{u}$ with the
product order is a lattice, with join (resp. meet) given by taking the maximum
(resp. minimum) at each coordinate. A lattice is said to be _finite_ if its
underlying set is finite, and _distributive_ if the binary operation $\wedge$
is distributive over $\vee$, and vice versa. Clearly $\mathbb{N}^{u}$ is a
distributive lattice, although of course not a finite one. Given
$u,v\in\mathbb{N}$ with $u\leq v$, the set $\Pi_{u,v}$ consisting of all
$(a_{1},\ldots a_{u})\in\mathbb{N}^{u}$ such that $1\leq
a_{1}<\cdots<a_{u}\leq v$ is a finite distributive sublattice of
$\mathbb{N}^{u}$.

An element $z$ of a distributive lattice is said to be _join-irreducible_ if
it is not the minimum, and whenever $z=x\vee y$ then $z=x$ or $z=y$. Let
$\mathcal{L}$ be a distributive lattice, and let
$J(\mathcal{L})=\\{x_{1},\ldots,x_{r}\\}$ be the set of its join irreducible
elements, enumerated so that $x_{i}<x_{j}$ implies $i<j$. Following
[RZ2]*subsection 3.3 set $\mathsf{str}(\mathcal{L})$ to be the subset of
$\mathbb{N}^{r+1}$ consisting of tuples $(a_{0},a_{1},\ldots,a_{r})$ such that
$a_{0}\geq a_{i}$ for all $i=1,\ldots,r$, and $a_{i}\geq a_{j}$ whenever
$x_{i}<x_{j}$ as elements of $\mathcal{L}$. It turns out that
$\mathsf{str}(\mathcal{L})$ is a normal affine semigroup, which we will call
the affine semigroup associate to $\mathcal{L}$. Later on we will give an
equivalent definition of $S(\mathcal{L})$ in terms of the lattice, which will
justify its name. Below we give an example with $\mathcal{L}=\Pi_{2,4}$.

$(1,2)$$(1,3)$$(1,4)$$(2,3)$$(2,4)$$(3,4)$$J(\Pi_{2,4})=\\{(1,3),(1,4),(2,3),(3,4)\\}$$\mathsf{str}(\Pi_{2,4})=\left\\{(a,b,c,d,e)\in\mathbb{N}^{5}\
\Bigg{\lvert}\begin{array}[]{rl}a&\geq b,c,d,e;\\\ b&\geq c,d,e;\\\ c,d&\geq
e\end{array}\right\\}$The lattice $\Pi_{2,4}$ and its affine semigroup. Framed
elements are join-irreducibles.

####

hilbert-basis Let $S$ be an affine semigroup. An element $s\in S$ is called
_irreducible_ if whenever $s=x+y$ with $x,y\in S$, then either $x$ or $y$ is
invertible. On the other hand $S$ is called _positive_ if its only invertible
element is $0$. If $S$ is positive then the set of its irreducible elements is
finite and generates $S$; for this reason it is called the _Hilbert basis_ of
$S$. The fact that $S$ is positive also implies that there exists a full
embedding $S\hookrightarrow\mathbb{Z}^{r}$ such that the image of $S$ is
contained in $\mathbb{N}^{r}$. It follows that a semigroup is positive if and
only if there exists $r\geq 0$ such that $S$ is isomorphic to a finitely
generated subsemigroup of $\mathbb{N}^{r}$. Proofs of these results can be
found in [BG]*pp. 54–56.

####

presentation Let $S$ be a positive affine semigroup and let
$\\{s_{1},\ldots,s_{n}\\}$ be its Hilbert Basis. We denote by
$\pi:\mathbb{N}^{n}\longrightarrow S$ the semigroup morphism defined by the
assignation $e_{i}\mapsto s_{i}$ for each $1\leq i\leq n$, where $e_{i}$ is
the $i$-th element in the canonical basis of $\mathbb{N}^{n}$. This map
determines an equivalence relation in $\mathbb{N}^{n}$ where $p\sim
p^{\prime}$ if and only if $\pi(p)=\pi(p^{\prime})$; we denote this relation
by $L(\pi)$. Clearly $L(\pi)$ is compatible with the additive structure of
$\mathbb{N}^{n}$, and hence the quotient $\mathbb{N}^{n}/L(\pi)$ is a
commutative monoid with the operation induced by addition in $\mathbb{N}^{n}$,
and there is an isomorphism of monoids $\mathbb{N}^{n}/L(\pi)\cong S$.

In general, an equivalence relation on $\mathbb{N}^{n}$ closed under addition
is called a _congruence_. If $\rho$ is a subset of
$\mathbb{N}^{n}\times\mathbb{N}^{n}$ then the congruence generated by $\rho$
is the smallest congruence containing the set $\rho$. Redei’s theorem
[RGS]*Theorem 5.12 states that every congruence in $\mathbb{N}^{n}$ is
finitely generated, i.e. there exists a finite set
$\rho\subset\mathbb{N}^{n}\times\mathbb{N}^{n}$ that generates it; in
particular there exists a finite set
$P=\\{(p_{1},p^{\prime}_{1}),\ldots,(p_{m},p^{\prime}_{m})\\}\subset L(\pi)$
which generates $L(\pi)$. A _presentation_ of $S$ will be for us a pair
$(\pi,P)$, where $\pi:\mathbb{N}^{n}\longrightarrow S$ is the map described
above and $P$ is a finite generating set of the congruence $L(\pi)$. By the
previous discussion every positive affine semigroup has a presentation.

######  Example 2.

Let $\mathcal{L}$ be a finite distributive lattice, and let $F(\mathcal{L})$
be the free commutative monoid on $\mathcal{L}$, which is clearly isomorphic
to $\mathbb{N}^{|\mathcal{L}|}$. We will give a presentation of the group
$\mathsf{str}(\mathcal{L})$ introduced in LABEL:interior as a quotient of
$F(\mathcal{L})$.

As before, we denote by $\\{x_{1},\ldots,x_{r}\\}$ the set of join-irreducible
elements of $\mathcal{L}$. We denote by $[l]$ the image of an element
$l\in\mathcal{L}$ in $F(\mathcal{L})$ and define a map
$\pi:F(\mathcal{L})\longrightarrow\mathsf{str}(\mathcal{L})$ given by
$\pi([l])=e_{0}+\sum_{x_{i}\leq l}e_{i}$. As shown in [RZ2]*Proposition 3.3.3
this map is surjective. Furthermore $L(\pi)$ is the equivalence relation
generated by the set $\\{([l]+[l^{\prime}],[l\vee l^{\prime}]+[l\wedge
l^{\prime}])\mid l,l^{\prime}\in\mathcal{L}\\}$. Notice also that the Hilbert
basis of $\mathsf{str}(\mathcal{L})$ is precisely the image of $\mathcal{L}$,
and the map $\pi$ restricted to $\mathcal{L}$ is a lattice morphism. Thus
every distributive lattice $\mathcal{L}$ can be realized as a sublattice of
$\\{0,1\\}^{|J(\mathcal{L})|+1}$.

$(1,0,0,0,0)$$(1,1,0,0,0)$$(1,1,1,0,0)$$(1,1,0,1,0)$$(1,1,1,1,0)$$(1,1,1,1,1)$The
lattice $\Pi_{2,4}$, with each element replaced by its image through $\pi$.

####

twisted-semigroup-algebras Let $S$ be a commutative semigroup with identity. A
$2$-cocycle over $S$ is a function $\alpha:S\times
S\longrightarrow\Bbbk^{\times}$ such that
$\alpha(s,s^{\prime})\alpha(s+s^{\prime},s^{\prime\prime})=\alpha(s,s^{\prime}+s^{\prime\prime})\alpha(s^{\prime},s^{\prime\prime})$
for all $s,s^{\prime},s^{\prime\prime}\in S$. Given a $2$-cocycle $\alpha$
over $S$, the _$\alpha$ -twisted semigroup algebra_ $\Bbbk^{\alpha}[S]$ is the
associative $\Bbbk$-algebra whose underlying vector space has basis
$\\{X^{s}\mid s\in S\\}$ and whose product over these generators is given by
$X^{s}X^{t}=\alpha(s,t)X^{s+t}$. This is a noncommutative deformation of the
classical semigroup algebra $\Bbbk[S]$. For more details see [RZ2]*section 3;
in this reference the group is assumed to be cancellative, but this is not
relevant for the definition of the twisted algebra.

######  Definition 2.

Let $A$ be a connected $\mathbb{N}^{r}$-graded algebra. We say that $A$ is a
_quantum positive affine toric variety_ if there exist a positive affine
semigroup $S$ and a $2$-cocycle $\alpha$ over $S$ such that $A$ is isomorphic
to the twisted semigroup algebra $\Bbbk^{\alpha}[S]$, and for each $s\in
S\setminus\\{0\\}$ the element $X^{s}$ is homogeneous of nonzero degree with
respect to the $\mathbb{N}^{r}$-grading of $\Bbbk^{\alpha}[S]$ induced by this
isomorphism. In that case we refer to $S$ as the underlying semigroup of $A$.

Let $r\geq 1$ and let $\psi:S\longrightarrow\mathbb{N}^{r}$ be a monoid
morphism such that $\psi^{-1}(0)=\\{0\\}$. The twisted semigroup algebra
$\Bbbk^{\alpha}[S]$ can be endowed with a connected $\mathbb{N}^{r}$-grading
setting $\deg X^{s}=\psi(s)$ for each $s\in S$. Conversely, any connected
grading such that the elements of the form $X^{s}$ are homogeneous arises in
this manner. In particular, if $A$ is a quantum positive toric variety with
underlying semigroup $S$ then there is a corresponding monoid morphism
$\psi:S\longrightarrow\mathbb{N}^{r}$, to which we will refer as the _grading_
morphism.

######  Example 3.

Let $q\in\Bbbk^{\times}$ and set $\mathbf{q}=\begin{pmatrix}1&q&1&1&q\\\
q^{-1}&1&q&q&q^{-2}\\\ 1&q^{-1}&1&1&q\\\ 1&q^{-1}&1&1&q\\\
q^{-1}&q^{2}&q^{-1}&q^{-1}&1\end{pmatrix}$.

Let $\Bbbk_{\mathbf{q}}[X]$ be the twisted polynomial ring generated by
variables $X_{1},\ldots,X_{5}$ subject to the relations
$X_{i}X_{j}=\mathbf{q}_{i,j}X_{j}X_{i}$. Given
$t=(a,b,c,d,e)\in\mathbb{N}^{5}$ we set
$X^{t}=X_{1}^{a}X_{2}^{b}X_{3}^{c}X_{4}^{d}X_{5}^{e}$. The formula
$X^{t}X^{s}=\alpha(s,t)X^{t+s}$ induces a map
$\alpha:\mathbb{N}^{5}\times\mathbb{N}^{5}\longrightarrow\Bbbk^{\times}$, and
associativity of the product of $\Bbbk_{\mathbf{q}}[X]$ implies that $\alpha$
is a $2$-cocycle. Recall from the examples in LABEL:interior and
LABEL:presentation that $S=\mathsf{str}(\Pi_{2,4})\subset\mathbb{N}^{5}$ is an
affine semigroup. The restriction of $\alpha$ to $S$ is also a $2$-cocycle,
and $\Bbbk^{\alpha}[S]$ is isomorphic to the subalgebra of
$\Bbbk_{\mathbf{q}}[X]$ generated by the elements $X^{t}$ with $t$ running
over the Hilbert basis of $S$. Setting

$\displaystyle Y_{1}$ $\displaystyle=X_{1}$ $\displaystyle Y_{2}$
$\displaystyle=X_{1}X_{2}$ $\displaystyle Y_{3}$
$\displaystyle=X_{1}X_{2}X_{3}$ $\displaystyle Y_{4}$
$\displaystyle=X_{1}X_{2}X_{4}$ $\displaystyle Y_{5}$
$\displaystyle=X_{1}X_{2}X_{3}X_{4}$ $\displaystyle Y_{6}$
$\displaystyle=X_{1}X_{2}X_{3}X_{4}X_{5}$

it is routine to check that $Y_{j}Y_{i}=q^{-2}Y_{i}Y_{j}$ for
$(i,j)\in\\{(1,5),(2,4)\\}$, that $Y_{2}$ and $Y_{3}$ commute, and that
$Y_{j}Y_{i}=q^{-1}Y_{i}Y_{j}$ for all other possible pairs $i<j$. There is one
more relation among these monomials, namely $Y_{2}Y_{3}=q^{-1}Y_{1}Y_{4}$.
Thus $\Bbbk^{\alpha}[S]$ has as many generators as $S$, and can be presented
by the commutation relations between them plus one extra relation arising from
the fact that there is exactly one pair of incomparable elements in
$\Pi_{2,4}$. We will see below in Proposition LABEL:P:presentation that all
quantum positive affine toric varieties and several related algebras have
similar presentations.

####

properties-of-qatv Quantum positive affine toric varieties were studied from
the point of view of noncommutative geometry in [RZ2]*section 3. The following
is a summary of the results proved there.

######  Proposition 2.

Let $S$ be a positive affine semigroup and let $A$ be a positive quantum toric
variety with underlying semigroup $S$. Then the following hold.

  1. (a)

$A$ is noetherian and integral.

  2. (b)

$A$ has property $\chi$ and finite local dimension equal to the rank of $S$.

  3. (c)

Suppose $S$ is normal. Then $A$ is AS-Cohen-Macaulay and a maximal order in
its division ring of fractions. Furthermore, $A$ is AS-Gorenstein if and only
if there exists $s\in S$ such that $\operatorname{\mathsf{relint}}S=s+S$.

  * Proof.

This is proved in [RZ2]*section 3.2 in the case where the grading morphism
$\psi$ is given by a full embedding of $S$ in $\mathbb{N}^{r}$ for some $r\geq
0$. The proposition follows from the fact that all the properties mentioned in
it are independent of the grading, see paragraph LABEL:invariance-by-grading.
∎

### Algebras with a quantum toric degeneration

Classically, a toric degeneration of an algebraic variety $V$ is a flat
deformation of $V$ into a toric variety $T$. Since varieties inherit many good
properties from their flat deformations, and positive affine toric varieties
are well studied, toric degeneration is a standard method to study algebraic
varieties. With this in mind we introduce a noncommutative analogue of toric
degeneration.

######  Definition 3.1.

Let $A$ be a connected $\mathbb{N}^{r}$-graded algebra. We will say that $A$
has a _quantum positive affine toric degeneration_ if it has a connected
filtration such that its associated graded ring is a quantum positive affine
toric variety. We refer to the underlying semigroup of this quantum positive
affine toric variety as the semigroup associated to the degeneration.

For the sake of brevity we will write “quantum toric degeneration”, omitting
the adjectives “positive” and “affine”. In view of Theorem LABEL:transfer and
Proposition LABEL:twisted-semigroup-algebras, an algebra with a quantum toric
degeneration is noetherian, integral, has property $\chi$ and finite local
dimension. Furthermore, we can determine whether $A$ is AS-Cohen-Macaulay or
AS-Gorenstein by studying the semigroup associated to the degeneration.

####

s-phi-type A standard technique for proving toric degeneration of a variety is
to find the structure of a Hodge algebra in the coordinate ring of the
variety. We now introduce a noncommutative notion, inspired in the definition
of Hodge algebras and its descendants such as classical and quantum algebras
with a straightening law, which will play a similar role for the rest of this
article.

######  Definition 3.

Let $A$ be a noetherian connected $\mathbb{N}^{r}$-graded algebra. Let $S$ be
a positive affine semigroup and let $(\pi,P)$ be a presentation of $S$ as
defined in LABEL:presentation, with $P=\\{(p_{i},p^{\prime}_{i})\mid 1\leq
i\leq m\\}$. Let $\varphi:\leavevmode\nobreak\ S\leavevmode\nobreak\
\longrightarrow\leavevmode\nobreak\ \mathbb{N}$ be a semigroup morphism such
that $\varphi^{-1}(0)=\\{0\\}$, and set $\tilde{\varphi}=\varphi\circ\pi$.

We say that the algebra $A$ is of $(S,\varphi)$-type with respect to $(\pi,P)$
if the following hold.

  1. 1.

$A$ is generated as algebra by a finite set of homogeneous elements
$\\{b_{1},\ldots,b_{n}\\}$ of the same cardinality as the Hilbert basis of
$S$. We set $\mathbf{b}^{\xi}=\displaystyle\prod_{i=1}^{n}b_{i}^{\xi_{i}}$ for
each $\xi\in\mathbb{N}^{n}$.

  2. 2.

For each $1\leq i<j\leq n$ and each $\xi\in\mathbb{N}^{n}$ such that
$\tilde{\varphi}(\xi)<\varphi(s_{i}+s_{j})$, there exist
$c_{i,j}\in\Bbbk^{\times}$ and $c^{i,j}_{\xi}\in\Bbbk$ such that

$\displaystyle
b_{j}b_{i}=c_{i,j}b_{i}b_{j}+\sum_{\tilde{\varphi}(\xi)<\varphi(s_{i}+s_{j})}c^{i,j}_{\xi}\mathbf{b}^{\xi}.$

  3. 3.

For each $1\leq i\leq m$ and each $\xi\in\mathbb{N}^{n}$ such that
$\tilde{\varphi}(\xi)<\tilde{\varphi}(p_{i})$ there exist
$d_{i}\in\Bbbk^{\times}$ and $d^{i}_{\xi}\in\Bbbk$ such that

$\displaystyle\mathbf{b}^{p^{\prime}_{i}}=d_{i}\mathbf{b}^{p_{i}}+\sum_{\tilde{\varphi}(\xi)<\tilde{\varphi}(p_{i})}d^{i}_{\xi}\mathbf{b}^{\xi}.$

We say that $A$ is of $(S,\varphi)$-type if there exists a presentation of $S$
such that $A$ is of $(S,\varphi)$-type with respect to it.

######  Remark 1.

Since $\tilde{\varphi}(e_{i})=\varphi(s_{i})>0$ for all $i$, given
$l\in\mathbb{N}$ there exist finitely many $\xi\in\mathbb{N}^{n}$ such that
$\tilde{\varphi}(\xi)\leq l$ and so the sums on the right hand side of the
formulas displayed in 2 and 3 are finite.

The reader familiar with Hodge algebras will notice that there is a condition
missing in the definition, namely the existence of a set of linearly
independent monomials on the generators. Thus the trivial algebra is of
$(S,\varphi)$-type for any $S$ and $\varphi$. This omission will be revised in
LABEL:P:equivalent-qatd, where the existence of a linearly independent set
will be shown to be equivalent to the existence of a quantum toric
degeneration for the algebra.

######  Example 4.

As stated above, this definition is inspired by that of quantum graded
algebras with a straightening law (quantum graded ASL for short) as defined in
[LR1]*Definition 1.1.1. A quantum graded ASL is not necessarily an algebra of
$(S,\varphi)$-type, but the results from [RZ]*section 5 show that the
quantized coordinate rings of grassmannians and their Richardson subvarieties
have this structure. Our paradigmatic example is the quantum grassmannian.

Fix a field $\Bbbk$ and let $q\in\Bbbk^{\times}$. The quantum grassmannian
$\mathcal{G}_{q}(2,4)$ is the algebra generated by elements
$\\{[ij],(i,j)\in\Pi_{2,4}\\}$, subject to the commutation relations

$\displaystyle[34][12]$ $\displaystyle=q^{-2}[12][34]$ $\displaystyle[24][13]$
$\displaystyle=q^{-2}[13][24]+q^{-2}(q-q^{-1})[12][34]$

while $YX=q^{-1}XY$ for any other pair $X<Y$ (here we identify the generating
set with $\Pi_{2,4}$ in the obvious way). There is also a quantum Plücker
relation, given by $[14][23]=q^{-1}[13][24]-q^{-2}[12][34]$. Notice that the
set of generators can be identified with $\Pi_{2,4}$, and inherits the
structure of a distributive lattice.

Setting $\mathsf{wt}$ as in [RZ]*Definition 4.3 we obtain an assignation

$\displaystyle[12]$ $\displaystyle\mapsto 21$ $\displaystyle[13]$
$\displaystyle\mapsto 12$ $\displaystyle[14]\mapsto 3$ $\displaystyle[23]$
$\displaystyle\mapsto 11$ $\displaystyle[24]$ $\displaystyle\mapsto 2$
$\displaystyle[34]\mapsto 1$

which extends to a semigroup morphism
$\varphi:\mathsf{str}(\Pi_{2,4})\longrightarrow\mathbb{N}$. It is now a matter
of routine computations to check that $\mathcal{G}_{q}(2,4)$ is of
$(\mathsf{str}(\Pi_{2,4}),\varphi)$-type.

####

type-filtration If $A$ is an algebra of $(S,\varphi)$-type we write
$F_{l}A=\langle\mathbf{b}^{\xi}\mid\tilde{\varphi}(\xi)\leq l\rangle$ for each
$l\in\mathbb{N}$. The fact that $\tilde{\varphi}(e_{i})>0$ for all $i$ implies
that this is a finite dimensional vector space, and that $F_{0}A=\Bbbk$. The
following lemma shows that $\mathcal{F}=\\{F_{l}A\\}_{l\geq 0}$ is a connected
filtration on $A$.

######  Lemma 3.

Let $S$ be a positive affine semigroup. Let $A$ be a noetherian connected
$\mathbb{N}^{r}$-graded algebra, and assume that it is of $(S,\varphi)$-type
with respect to a presentation $(\pi,P)$. Then the following hold.

  1. 1.

Given $\xi,\nu\in\mathbb{N}^{n}$ there exists $c_{\xi,\nu}\in\Bbbk^{\times}$
such that $\mathbf{b}^{\xi}\mathbf{b}^{\nu}\equiv
c_{\xi,\nu}\mathbf{b}^{\xi+\nu}\mod F_{\tilde{\varphi}(\xi+\nu)-1}A$. In
particular $\mathcal{F}$ is a filtration on $A$.

  2. 2.

Given $(\xi,\nu)\in L(\pi)$ there exists $d_{\xi,\nu}\in\Bbbk^{\times}$ such
that $\mathbf{b}^{\xi}\equiv d_{\xi,\nu}\mathbf{b}^{\nu}\mod
F_{\tilde{\varphi}(\xi)-1}A$.

  * Proof.

To prove item 1 we proceed by induction on $\tilde{\varphi}(\xi+\nu)$, with
the $0$-th step being obvious since $F_{0}A=\Bbbk$ is a subalgebra of $A$.
Suppose that the result holds for all $l<\tilde{\varphi}(\xi+\nu)$ and let $i$
be the least integer such that $\xi_{i}\neq 0$, so
$\mathbf{b}^{\xi}\mathbf{b}^{\nu}=b_{i}(\mathbf{b}^{\xi-
e_{i}}\mathbf{b}^{\nu})$. Using the inductive hypothesis we obtain

$\displaystyle b_{i}(\mathbf{b}^{\xi-e_{i}}\mathbf{b}^{\nu})$
$\displaystyle=b_{i}\bigg{(}c_{\xi-e_{i},\nu}\mathbf{b}^{\xi-
e_{i}+\nu}+\sum_{\tilde{\varphi}(\rho)<\tilde{\varphi}(\xi-
e_{i}+\nu)}c_{\rho}\mathbf{b}^{\rho}\bigg{)}$ $\displaystyle=c_{\xi-
e_{i},\nu}b_{i}\mathbf{b}^{\xi-
e_{i}+\nu}+\sum_{\tilde{\varphi}(\rho)<\tilde{\varphi}(\xi-
e_{i}+\nu)}c_{\rho}b_{i}\mathbf{b}^{\rho}$

where the $c_{\rho}\in\Bbbk$ and $c_{\xi-e_{i},\nu}\in\Bbbk^{\times}$. The
inductive hypothesis also implies each product $b_{i}\mathbf{b}^{\rho}$ lies
in $F_{\tilde{\varphi}(\xi+\nu)-1}A$, so

$b_{i}\mathbf{b}^{\xi-e_{i}}\mathbf{b}^{\nu}\equiv c_{\xi-
e_{i},\nu}b_{i}\mathbf{b}^{\xi-e_{i}+\nu}\mod
F_{\tilde{\varphi}(\xi+\nu)-1}A.$

Now let $j$ be the least integer such that $\nu_{j}\neq 0$. If $i\leq j$ then
$b_{i}\mathbf{b}^{\xi-e_{i}+\nu}=\mathbf{b}^{\xi+\nu}$ and we are finished;
otherwise, using item 2 of Definition LABEL:s-phi-type and a similar argument
as before, we obtain

$\displaystyle b_{i}\mathbf{b}^{\xi-e_{i}+\nu}=(b_{i}b_{j})(\mathbf{b}^{\xi-
e_{i}+\nu-e_{j}})\equiv c_{j,i}(b_{j}b_{i})(\mathbf{b}^{\xi-e_{i}+\nu-
e_{j}})\mod F_{\tilde{\varphi}(\xi+\nu)-1}A$

The same reasoning applied to the product $b_{j}(b_{i}\mathbf{b}^{\xi-
e_{i}+\nu-e_{j}})$ shows that

$\displaystyle b_{j}(b_{i}\mathbf{b}^{\xi-e_{i}+\nu-e_{j}})$
$\displaystyle\equiv c_{e_{i},\xi-e_{i}+\nu-e_{j}}b_{j}(\mathbf{b}^{\xi+\nu-
e_{j}})\mod F_{\tilde{\varphi}(\xi+\nu)-1}A.$

Since $j<i$, the definition of $j$ implies $b_{j}(\mathbf{b}^{\xi+\nu-
e_{j}})=\mathbf{b}^{\xi+\nu}$, so the proof of item 1 is complete.

We now prove item 2. Set

$T=\\{(\xi,\nu)\in L(\pi)\mid\mathbf{b}^{\xi}\equiv c\mathbf{b}^{\nu}\mod
F_{\tilde{\varphi}(\xi)-1}\mbox{ for some }c\in\Bbbk^{\times}\\}.$

We will show that $L(\pi)\subset T$, which clearly implies the desired result.
By definition $T$ is an equivalence relation, and item 1 implies it is a
congruence on $\mathbb{N}^{n}$. By item 3 of Definition LABEL:s-phi-type,
every pair $(p_{i},p^{\prime}_{i})\in P$ lies in $T$. Since $L(\pi)$ is the
smallest congruence containing $P$, we deduce that $L(\pi)\subset T$. ∎

####

P:equivalent-qatd Let $S$ be a positive affine semigroup and let $(\pi,P)$ be
a presentation of $S$. A _section_ of $\pi$ is a function
$\tau:S\longrightarrow\mathbb{N}^{n}$ such that
$\pi\circ\tau=\operatorname{Id}_{S}$, that is $s=\sum_{i}\tau(s)_{i}s_{i}$ for
every $s\in S$. If $A$ is an algebra of $(S,\varphi)$-type then Lemma
LABEL:type-filtration implies that for any section $\tau$ of $\pi$ the set
$\\{\mathbf{b}^{\tau(s)}\mid s\in S\\}$ spans $A$. We now show that an algebra
has a quantum toric degeneration with associated semigroup $S$ if and only if
it is of $(S,\varphi)$-type for an adequate morphism $\varphi$ and the
spanning set determined by any section is linearly independent.

######  Proposition 3.

Let $S$ be a positive affine semigroup, and let $A$ be a noetherian connected
$\mathbb{N}^{r}$-graded algebra. The following statements are equivalent.

  1. 1.

The algebra $A$ has a quantum toric degeneration with associated semigroup
$S$.

  2. 2.

For every presentation $(\pi,P)$ of $S$ there exists a semigroup morphism
$\varphi:S\longrightarrow\mathbb{N}$ such that $A$ is of $(S,\varphi)$-type
with respect to $(\pi,P)$, and for every section
$\tau:S\longrightarrow\mathbb{N}^{n}$ of $\pi$ the set
$\\{\mathbf{b}^{\tau(s)}\mid s\in S\\}$ is linearly independent.

  3. 3.

There exist a presentation $(\pi,P)$, a semigroup morphism
$\varphi:S\longrightarrow\mathbb{N}$ and a section
$\tau:S\longrightarrow\mathbb{N}^{n}$ of $\pi$ such that $A$ is of
$(S,\varphi)$-type with respect to $(\pi,P)$ and the set
$\\{\mathbf{b}^{\tau(s)}\mid s\in S\\}$ is linearly independent.

  * Proof.

We first show that 1 implies 2. By hypohtesis there exists a filtration by
graded subspaces $\mathcal{F}=\\{F_{l}A\\}_{l\geq 0}$ such that
$\operatorname{\mathsf{gr}}_{\mathcal{F}}A\cong\Bbbk^{\alpha}[S]$ as
$\mathbb{N}^{r+1}$-graded algebras for some $2$-cocycle $\alpha:S\times
S\longrightarrow\Bbbk^{\times}$, with the grading on the twisted semigroup
algebra given by a semigroup morphism $\psi:S\longrightarrow\mathbb{N}^{r+1}$
such that $\psi^{-1}(0)=\\{0\\}$. We identify
$\operatorname{\mathsf{gr}}_{\mathcal{F}}A$ with $\Bbbk^{\alpha}[S]$ through
this isomorphism to simplify notation.

Fix a presentation $(\pi,P)$ of $S$. For each $1\leq i\leq n$ we choose
homogeneous elements $b_{i}\in A$ such that
$\operatorname{\mathsf{gr}}b_{i}=X^{s_{i}}$. By definition of the product of
an associated graded ring, for each $\xi\in\mathbb{N}^{n}$ the element
$\prod_{i=1}^{n}(\operatorname{\mathsf{gr}}b_{i})^{\xi_{i}}$ equals either
$\operatorname{\mathsf{gr}}\left(\prod_{i=1}^{n}b_{i}^{\xi_{i}}\right)$ or
zero. Since $\Bbbk^{\alpha}[S]$ is an integral ring the last possibility
cannot occur, so $\operatorname{\mathsf{gr}}(\mathbf{b}^{\xi})$ equals a
nonzero multiple of $X^{\pi(\xi)}$. Thus if
$\tau:S\longrightarrow\mathbb{N}^{n}$ is a section of $\pi$ then for each
$s\in S$ there exists a nonzero constant $c_{s}$ such that
$\operatorname{\mathsf{gr}}b^{\tau(s)}\equiv c_{s}X^{s}$, and so the set
$\\{\operatorname{\mathsf{gr}}\mathbf{b}^{\tau(s)}\mid s\in S\\}$ is a basis
of $\Bbbk^{\alpha}[S]$, which implies that $\\{\mathbf{b}^{\tau(s)}\mid s\in
S\\}$ is a basis of $A$. This also proves that $A$ satisfies item 1 of
Definition LABEL:s-phi-type.

Let $\varphi:S\longrightarrow\mathbb{N}$ be the additive map given by
$s\mapsto\psi(s)_{r+1}$; equivalently $\varphi(s)$ is the minimal $l$ such
that $\mathbf{b}^{\xi}\in F_{l}A$ for all $\xi\in\pi^{-1}(s)$. In particular
$\varphi(s_{i})>0$ since $F_{0}A=\Bbbk$. Also
$F_{l}A=\left\langle\mathbf{b}^{\xi}\mid\tilde{\varphi}(\xi)\leq
l\right\rangle$, and since for each $\xi\in\mathbb{N}^{n}$ there exists a
nonzero constant $c_{\xi}$ such that $\operatorname{\mathsf{gr}}b^{\xi}\equiv
c_{\xi}\operatorname{\mathsf{gr}}b^{\tau(\pi(\xi))}$ we actually have
$F_{l}A=\left\langle\mathbf{b}^{\tau(s)}\mid\varphi(s)\leq l\right\rangle$.
Finally, for each $1\leq i,j\leq n$ and each $1\leq k\leq m$ there exist
$c_{i,j},d_{k}\in\Bbbk^{\times}$ such that

$\displaystyle\operatorname{\mathsf{gr}}(b_{j}b_{i})$
$\displaystyle=c_{i,j}\operatorname{\mathsf{gr}}(b_{i}b_{j}),$
$\displaystyle\operatorname{\mathsf{gr}}\mathbf{b}^{p^{\prime}_{k}}$
$\displaystyle=d_{k}\operatorname{\mathsf{gr}}\mathbf{b}^{p_{k}}$

hold in $\operatorname{\mathsf{gr}}_{\mathcal{F}}A$, which implies that items
2 and 3 of Definition LABEL:s-phi-type hold in $A$ for the morphism $\varphi$
we have just defined. Thus $A$ is of $(S,\varphi)$-type, and we have proved 1
implies 2.

We said in LABEL:presentation that every positive affine semigroup has a
presentation so clearly 2 implies 3. Let us see that 3 implies 1. Define the
filtration $\mathcal{F}=\\{F_{l}A\\}_{l\geq 0}$ as in LABEL:type-filtration.
By item 2 of Lemma LABEL:type-filtration, the set
$\\{\mathbf{b}^{\tau(s)}\mid\varphi(s)\leq l\\}$ generates $F_{l}A$ for each
$l\in\mathbb{N}$, and since by hypothesis it is linearly independent, it is a
basis of $F_{l}A$. Hence $\operatorname{\mathsf{gr}}_{\mathcal{F}}A$ is
generated by $\\{\operatorname{\mathsf{gr}}\mathbf{b}^{\tau(s)}\mid s\in
S\\}$. Once again by Lemma LABEL:type-filtration for each $s,s^{\prime}\in S$
there exist $\beta(s,s^{\prime}),\alpha(s,s^{\prime})\in\Bbbk^{\times}$ such
that

$\displaystyle(\operatorname{\mathsf{gr}}\mathbf{b}^{\tau(s)})(\operatorname{\mathsf{gr}}\mathbf{b}^{\tau(s^{\prime})})=\beta(s,s^{\prime})\operatorname{\mathsf{gr}}\mathbf{b}^{\tau(s)+\tau(s^{\prime})}=\alpha(s,s^{\prime})\operatorname{\mathsf{gr}}\mathbf{b}^{\tau(s+s^{\prime})}.$

Associativity of the product of $\operatorname{\mathsf{gr}}_{\mathcal{F}}A$
implies that $\alpha:S\times S\longrightarrow\Bbbk^{\times}$ is a $2$-coycle,
so we may consider the $\Bbbk$-linear map
$\Bbbk^{\alpha}[S]\longrightarrow\operatorname{\mathsf{gr}}_{\mathcal{F}}A$
induced by the assignation
$X^{s}\mapsto\operatorname{\mathsf{gr}}\mathbf{b}^{\tau(s)}$, which is a
multiplicative map. Since $S$ is positive we must have $\tau(0)=0$, and hence
$\alpha(s,0)=\alpha(0,s)=1$ for all $s\in S$ which implies that our
multiplicative map is unitaty and hence an isomorphism of $\Bbbk$-algebras.
Furthermore, the elements $\operatorname{\mathsf{gr}}\mathbf{b}^{\tau(s)}$ are
homogeneous, so this algebra is indeed a quantum positive affine toric
variety. ∎

######  Example 5.

We return one last time to the example of the quantum grassmannian
$\mathcal{G}_{q}(2,4)$ over an arbitrary field discussed in LABEL:s-phi-type.
We extend the order of $\Pi_{2,4}$ to a total order so we can identify the
free abelian semigroup over $\Pi_{2,4}$ with $\mathbb{N}^{6}$, and let
$\pi:\mathbb{N}^{6}\longrightarrow\mathsf{str}(\Pi_{2,4})$ be the presentation
morphism described in LABEL:presentation. For each
$s\in\mathsf{str}(\Pi_{2,4})$ the fiber $\pi^{-1}(s)$ is finite and has a
unique maximal element with respect to the total lexicographic order of
$\mathbb{N}^{6}$, which we denote by $\tau(s)$. The monomials corresponding to
this section are precisely the standard monomials introduced in
[LR1]*Definition 3.2.1. Using the map $\varphi$ from LABEL:s-phi-type the
quantum toric variety obtained by degeneration is the twisted semigroup
algebra presented in LABEL:twisted-semigroup-algebras (this explains the
rather odd choice of commutation coefficients in that example).

A similar argument holds not just for quantum grassmannians, but for the large
class of quantum graded ASL satisfying condition $(C)$ introduced in
[RZ]*Definition 4.1. This includes all quantum grassmannians in type $A$,
along with their Schubert and Richardson subvariaties. In the following
section we will show that quantized coordinate rings of Schubert subvarieties
of arbitrary flag varieties also have a quantum toric degeneration.

####

lex-degeneration We now introduce a second notion related to quantum toric
degenerations. Recall that a commutative semigroup $S$ is said to be well-
ordered if there exists a well-order $<$ on $S$ compatible with the additive
structure, i.e. such that for all $s,s^{\prime},s^{\prime\prime}\in S$ the
inequality $s<s^{\prime}$ implies
$s+s^{\prime\prime}<s^{\prime}+s^{\prime\prime}$.

######  Definition 4.

Let $S$ be a commutative semigroup, and let $<$ be a well-order on $S$
compatible with the semigroup structure. Let $A$ be a connected
$\mathbb{N}^{l}$-graded algebra for some $l\in\mathbb{N}$. An _$(S, <)$-basis_
for $A$ is an ordered basis $\\{b_{s}\mid s\in S\\}$ consisting of homogeneous
elements, such that for all $s,s^{\prime},s^{\prime\prime}\in S$ with
$s^{\prime\prime}<s+s^{\prime}$ there exist
$c_{s,s^{\prime}}\in\Bbbk^{\times}$ and
$c_{s,s^{\prime}}^{s^{\prime\prime}}\in\Bbbk$ such that

$b_{s}b_{s^{\prime}}=c_{s,s^{\prime}}b_{s+s^{\prime}}+\sum_{s^{\prime\prime}<s+s^{\prime}}c_{s,s^{\prime}}^{s^{\prime\prime}}b_{s^{\prime\prime}}.$

####

S-order-bases Let $S$ be a commutative semigroup with neural element $0$.
Assume $<$ is a well-order on $S$ compatible with its additive structure, and
let $A$ be an algebra. An $(S,<)$-filtration on $A$ is a collection of vector
spaces $\mathcal{F}=\\{F_{s}A\mid s\in S\\}$, such that $F_{s}A\cdot
F_{s^{\prime}}A\subset F_{s+s^{\prime}}A$ for all $s,s^{\prime}\in S$, and
such that $F_{s}A\subseteq F_{s^{\prime}}A$ whenever $s\leq s^{\prime}$. The
standard notions related to $\mathbb{N}$-filtrations translate easily to the
context of $(S,<)$-filtrations. We will assume that our filtrations are always
_exhaustive_ , so $A=\bigcup_{s\in S}F_{s}A$, and _discrete_ , so $F_{s}A=0$
for all $s<0$.

Let $A$ be an $S$-filtered algebra. The associated graded algebra
$\operatorname{\mathsf{gr}}_{\mathcal{F}}A$ is defined setting
$F_{<s}A=\sum_{t<s}F_{t}A$ and taking

$\operatorname{\mathsf{gr}}_{\mathcal{F}}A=\bigoplus_{s\in
S}\frac{F_{s}A}{F_{<s}A}.$

As usual, for each element $a\in A$ we may define
$\operatorname{\mathsf{gr}}a$ as the image of $a$ in the quotient
$F_{s}A/F_{<s}A$ where $s$ is the first element of $S$ such that $a\in
F_{s}A$; notice that this element exists because $<$ is a well-order. The
product can then be defined as in the $\mathbb{N}$-filtered case, namely if
$a,b\in A$ and $s,t$ are minimal elements such that $a\in F_{s}A$ and $b\in
F_{t}A$, then $(\operatorname{\mathsf{gr}}a)(\operatorname{\mathsf{gr}}b)$
equals the image of $ab$ in $F_{s+t}A/F_{<s+t}A$, which equals
$\operatorname{\mathsf{gr}}(ab)$ if $t+s$ is minimal with respect to the
property that $ab\in F_{t+s}A$ and zero otherwise.

######  Lemma 4.

Let $S$ be a positive affine semigroup and let $<$ be a well-order of $S$
compatible with the semigroup structure. Let $A$ be a noetherian
$\mathbb{N}^{l}$-graded connected algebra for some $l\in\mathbb{N}$, and
assume $A$ has an $(S,<)$-basis $\mathcal{B}=\\{b_{s}\mid s\in S\\}$. Set
$F_{s}A=\langle b_{t}\mid t\leq s\in S\rangle$ and $\mathcal{F}=\\{F_{s}A\mid
s\in S\\}$. Also let $\\{s_{1},\ldots,s_{n}\\}$ be the Hilbert basis of $S$
and set $b_{i}=b_{s_{i}}$ for all $1\leq i\leq n$. The following hold.

  1. 1.

The family $\mathcal{F}$ is an $(S,<)$-filtration. Furthermore, each quotient
$F_{s}A/F_{<s}A$ is of dimension $1$ and $F_{0}A=\langle 1\rangle$.

  2. 2.

There exists a $2$-cocycle $\alpha$ over $S$ such that
$\operatorname{\mathsf{gr}}_{\mathcal{F}}A$ is isomorphic as $S$-graded
algebra to $\Bbbk^{\alpha}[S]$ for the obvious $S$-grading on this.

  3. 3.

The algebra $A$ is generated by the set $\\{b_{i}\mid 1\leq i\leq n\\}$.

  * Proof.

The fact that $\mathcal{F}$ is an exhaustive and discrete $(S,<)$-filtration
is an immediate consequence of the definition of an $(S,<)$-basis. Also
$F_{<s}A=\langle b_{t}\mid t<s\rangle$, so $F_{s}A/F_{<s}A$ is generated by
the image of $b_{s}$ in the quotient, which is nonzero. Finally, writing $1$
as a linear combination of the $b_{s}$ and using a leading term argument, it
is easy to see that $1\in F_{0}A$ and hence it must generate it. Notice that
this implies that $b_{0}$ is a scalar so without loss of generality we may
assume that $b_{0}=1$. This proves item $1$.

Set $\alpha(s,s^{\prime})=c_{s,s^{\prime}}$ for each $s,s^{\prime}\in S$. By
the first item the set $\\{\operatorname{\mathsf{gr}}b_{s}\mid s\in S\\}$ is a
basis of $\operatorname{\mathsf{gr}}_{\mathcal{F}}A$, and
$(\operatorname{\mathsf{gr}}b_{s})(\operatorname{\mathsf{gr}}b_{s^{\prime}})=c_{s,s^{\prime}}\operatorname{\mathsf{gr}}b_{s+s^{\prime}}$
by definition of the product on the associated graded ring. Associativity of
the product in $\operatorname{\mathsf{gr}}_{\mathcal{F}}A$ implies then that
$\alpha$ is a $2$-cocycle over $S$, and furthermore the map
$\operatorname{\mathsf{gr}}_{\mathcal{F}}A\longrightarrow\Bbbk^{\alpha}[S]$
sending $\operatorname{\mathsf{gr}}b_{s}$ to $X^{s}$ is a multiplicative
$S$-graded vector-space isomorphism. Since we are assuming that $b_{0}=1$, it
follows that $\alpha(1,s)=\alpha(s,1)=1$ for all $s\in S$ and hence our
isomorphism preserves the unit, and is thus a ring isomorphism. This proves
item 2.

Finally, in order to prove that the $b_{i}$’s generate $A$ it is enough to
show that each $b_{s}$ is in the algebra generated by these elements. Suppose
this is not the case. Then, since $S$ is well-ordered by $<$, there exists a
minimal $s$ such that $b_{s}$ is not in the algebra generated by the
$b_{i}$’s. Take $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{N}^{n}$ such that
$\sum_{i}\xi_{i}s_{i}=s$. By the definition of the product in the associated
graded ring, $\prod_{i}(\operatorname{\mathsf{gr}}b_{i})^{\xi_{i}}$ equals
either $\operatorname{\mathsf{gr}}(\mathbf{b}^{\xi})$ or zero, and since
$\operatorname{\mathsf{gr}}_{\mathcal{F}}A\cong\Bbbk^{\alpha}[S]$ is integral
(see [RZ2]*Lemma 3.2.3) the second possibility can not occur. Thus
$\operatorname{\mathsf{gr}}(\mathbf{b}^{\xi})$ is a nonzero element of
$\operatorname{\mathsf{gr}}_{\mathcal{F}}A$ of degree $\pi(\xi)$, so item 1 of
this lemma implies that
$\operatorname{\mathsf{gr}}b_{s}=c\operatorname{\mathsf{gr}}(\mathbf{b}^{\xi})$
for some $c\in\Bbbk^{\times}$, and hence
$b_{s}=c\mathbf{b}^{\xi}+\sum_{t<s}c_{t}b_{t}$. By the minimality of $s$ all
the $b_{t}$’s appearing in the sum on the right hand side of the equation lie
in the algebra generated by the $b_{i}$’s, and clearly so does
$\mathbf{b}^{\xi}$, a contradiction. ∎

####

S-ordered-basis-degeneration There is an obvious way to obtain well-orderings
on positive affine semigroups. Since $S$ is a positive affine semigroup it can
be embedded in $\mathbb{N}^{r}$ for some $r\geq 0$ through a monoid morphism
$\iota:S\longrightarrow\mathbb{N}^{r}$. Now $\mathbb{N}^{r}$ is a well-ordered
semigroup with the lexicographic order, which is compatible with its additive
structure, so we may pull-back the lexicographic order through $\iota$ and
thus obtain a well-order $<^{\iota}$ over $S$, which is also compatible with
its additive structure. Notice that in this case $0$ is always the minimal
element of $S$.

######  Proposition 4.

Let $S$ be a positive affine semigroup, and let $A$ be a noetherian connected
$\mathbb{N}^{l}$-graded algebra for some $l\in\mathbb{N}$. The algebra $A$ has
a quantum affine toric degeneration with underlying semigroup $S$ if and only
if there exists an embedding $\iota:S\longrightarrow\mathbb{N}^{t}$ such that
$A$ has an ordered $(S,<^{\iota})$-basis.

  * Proof.

Suppose $A$ has a quantum affine toric degeneration with underlying semigroup
$S$. Then by Proposition LABEL:P:equivalent-qatd there exists a semigroup
morphism $\varphi:S\longrightarrow\mathbb{N}$ such that $A$ is of
$(S,\varphi)$-type, and we may choose any section
$\tau:S\longrightarrow\mathbb{N}^{n}$ to obtain a basis
$\mathcal{B}=\\{\mathbf{b}^{\tau(s)}\mid s\in S\\}$. Let
$\rho:S\longrightarrow\mathbb{N}^{t-1}$ be an embedding ($t>1$) and let
$\iota:S\longrightarrow\mathbb{N}^{t}$ be defined as
$\iota(s)=(\varphi(s),\rho(s))$, which is an embedding of $S$ since $\rho$ is
an embedding. Write $<$ for $<^{\iota}$, and notice that
$\varphi(s)<\varphi(s^{\prime})$ implies $s<s^{\prime}$. By Lemma LABEL:type-
filtration, for all $s,s^{\prime}\in S$ and all $s^{\prime\prime}$ such that
$\varphi(s^{\prime\prime})<\varphi(s+s^{\prime})$ there exist
$c_{s,s^{\prime}}\in\Bbbk^{\times}$ and
$c_{s,s^{\prime}}^{s^{\prime\prime}}\in\Bbbk$ such that

$\mathbf{b}^{\tau(s)}\mathbf{b}^{\tau(s^{\prime})}=c_{s,s^{\prime}}\mathbf{b}^{\tau(s+s^{\prime})}+\sum_{\varphi(s^{\prime\prime})<\varphi(s+s^{\prime})}c_{s,s^{\prime}}^{s^{\prime\prime}}\mathbf{b}^{\tau(s^{\prime\prime})},$

which implies $\mathcal{B}$ is an ordered $S$-basis with respect to $<$.

Now assume $A$ has an ordered $S$-basis with respect to some total order $<$
induced by an embedding $\iota:S\longrightarrow\mathbb{N}^{t}$. Since $<$ is
the pull-back of the lexicographic order through an embedding, we might as
well assume $S\subset\mathbb{N}^{t}$ and that $<$ is the lexicographic order.
By the previous lemma, we already know that the $b_{i}$’s generate $A$, so all
that is left to do is to prove the existence of an additive map
$\varphi:S\longrightarrow\mathbb{N}$ and that the desired relations hold.

As before, we denote by $\pi:\mathbb{N}^{n}\longrightarrow S$ the map
$(\xi_{1},\ldots,\xi_{n})\mapsto\sum_{i}\xi_{i}s_{i}$. Recall that using the
$(S,<)$-filtration $\mathcal{F}$ defined in the previous lemma, we proved that
$\operatorname{\mathsf{gr}}_{\mathcal{F}}A\cong\Bbbk^{\alpha}[S]$ for some
$2$-cocycle $S$. This implies that for all $1\leq i<j\leq n$ and all
$\xi\in\mathbb{N}^{n}$ such that $\pi(\xi)<s_{i}+s_{j}$ there exist
$c_{i,j}\in\Bbbk^{\times}$ and $c^{i,j}_{\xi}\in\Bbbk$ such that

$\displaystyle
b_{j}b_{i}=c_{i,j}b_{i}b_{j}+\sum_{\pi(\xi)<\pi(s_{i}+s_{j})}c^{i,j}_{\xi}\mathbf{b}^{\xi},$

and for each $1\leq i\leq m$ and each $\xi\in\mathbb{N}^{n}$ such that
$\pi(\xi)<p_{i}$ there exist $d_{i}\in\Bbbk^{\times}$ and
$d^{i}_{\xi}\in\Bbbk$ such that

$\displaystyle\mathbf{b}^{p^{\prime}_{i}}=d_{i}\mathbf{b}^{p_{i}}+\sum_{\pi(\xi)<p_{i}}d^{i}_{\xi}\mathbf{b}^{\xi}.$

Let $C\subset S$ be the set consisting of the following elements:

    * –

all $s_{i}+s_{j}$ with $1\leq i<j\leq n$;

    * –

all $\pi(p_{i})$ with $1\leq i\leq m$;

    * –

all $\pi(\xi)$ such that $c^{i,j}_{\xi}\neq 0$ for some $1\leq i<j\leq n$;

    * –

and all $\pi(\xi)$ such that $d^{i}_{\xi}\neq 0$ for some $1\leq i\leq m$.

The set $C$ is finite and hence is contained in a cube $[0,N]^{t}$ for $N$
large enough. Set $\varphi:\mathbb{N}^{t}\longrightarrow\mathbb{N}$ to be the
semigroup morphism defined by
$(c_{1},\ldots,c_{t})\mapsto\sum_{i=1}^{t}c_{i}(N+1)^{t-i}$. If $c\in C$ then
$\varphi(c)$ is the unique natural number such that its $N+1$-adic expansion
has $c_{i}$ as its $i$-th digit. This implies that $\varphi$ respects the
restriction of the lexicographic order to $C$, and thus $A$ is of
$(S,\varphi)$-type.

Let $\tau:S\longrightarrow\mathbb{N}^{n}$ be any section of $\pi$. The algebra
$\operatorname{\mathsf{gr}}_{\mathcal{F}}A$ has a natural $S$-grading, and for
each $s\in S$ the element $\operatorname{\mathsf{gr}}\mathbf{b}^{\tau(s)}$ is
of degree $s$. As we have already observed, this is a non-zero element so the
set $\\{\operatorname{\mathsf{gr}}\mathbf{b}^{\tau(s)}\mid s\in S\\}$ is a
basis of $\operatorname{\mathsf{gr}}_{\mathcal{F}}A$, which implies that
$\\{\mathbf{b}^{\tau(s)}\mid s\in S\\}$ is a basis of $A$. Thus by Proposition
LABEL:P:equivalent-qatd $A$ has a quantum affine toric degeneration with
underlying semigroup $S$. ∎

######  Remark 2.

The trick of turning the $S$-filtration into an $\mathbb{N}$-filtration using
$N+1$-adic expansions is due to Caldero [C]*Lemma 3.2. A similar though less
general version of this idea appears in [GL] and [RZ].

####

P:presentation We finish this section with an easy consequence of Lemma
LABEL:type-filtration. It will not be used in the sequel, but we include it
for completeness.

######  Proposition 5.

Let $S$ be a positive affine semigroup and $A$ a noetherian connected
$\mathbb{N}^{r}$\- graded algebra. If $A$ is of $(S,\varphi)$-type for some
monoid morphism $\varphi:S\longrightarrow\mathbb{N}$, and there exists some
section $\tau$ of $\pi$ such that the set $\\{\mathbf{b}^{\tau(s)}\mid s\in
S\\}$ is linearly independent, then the relations given in items 2 and 3 of
Definition LABEL:s-phi-type give a presentation of $A$.

  * Proof.

Since $A$ is of $(S,\varphi)$-type, it is generated as algebra by homogeneous
elements $b_{1},\ldots,b_{n}$, and there exist constants
$c_{i,j},c^{i,j}_{\xi},d_{i},d^{i}_{\xi}$ such that $A$ complies with
Definition LABEL:s-phi-type. Furthermore, the relations described in items 2
and 3 of this definition are homogeneous.

Let $B$ be the free algebra generated by $X_{1},\ldots,X_{n}$ and let $I$ be
the ideal of $B$ generated by the elements

$\displaystyle X_{j}X_{i}$ $\displaystyle-
c_{i,j}X_{i}X_{j}-\sum_{\tilde{\varphi}(\xi)<\varphi(s_{i}+s_{j})}c^{i,j}_{\xi}X^{\xi},$
$\displaystyle\mbox{for }1\leq i<j\leq n;$ $\displaystyle X^{p^{\prime}_{i}}$
$\displaystyle-
d_{i}X^{p_{i}}-\sum_{\tilde{\varphi}(\xi)<\tilde{\varphi}(p_{i})}d^{i}_{\xi}X^{\xi},$
$\displaystyle\mbox{for }1\leq i\leq m;$

where $X^{\xi}=X_{1}^{\xi_{1}}X_{2}^{\xi_{2}}\cdots X_{n}^{\xi_{n}}$ for each
$\xi\in\mathbb{N}^{n}$. We put an $\mathbb{N}^{r}$ grading on $B$ by setting
$\deg X_{i}=\deg b_{i}$, and this induces an $\mathbb{N}^{r}$ grading on
$B/I$. Since $B$ is a free algebra the assignation $X_{i}\mapsto b_{i}$
induces a morphism of graded algebras $B\longrightarrow A$, which factors
through $B/I$. We thus obtain a morphism of $\mathbb{N}^{r}$-graded algebras
$f:B/I\longrightarrow A$.

We denote by $Y_{i}$ the image of $X_{i}$ in $B/I$. Clearly $B/I$ is an
$(S,\varphi)$\- algebra, and the algebra map $f:B/I\longrightarrow A$ sends
$Y_{i}$ to $b_{i}$ for all $i$. Since $f(Y^{\tau(s)})=\mathbf{b}^{\tau(s)}$,
the set $\\{Y^{\tau(s)}\mid s\in S\\}$ is linearly independent and hence a
basis of $B/I$. Thus $f$ maps a basis onto a basis, so it is an isomorphism. ∎

##  4 Quantum affine toric degeneration of quantum Schubert varieties

We apply the results in the previous section to study Schubert varieties of
quantum flag varieties. We recall the definitions of quantum flag and Schubert
varieties with some detail in order to establish notation. We then adapt an
argument due to P. Caldero to show that these algebras have $(S,<)$-bases for
adequate semigroups $S$. The main ingredient in the construction of these
bases is the canonical or global basis of $U_{q}^{-}(\mathfrak{g})$ discovered
independently by Lusztig and Kashiwara. The semigroup arises out of the string
parametrization of this basis.

### Quantum flag and Schubert varieties

####

lie-algebra-notation Let $\mathfrak{g}$ be a complex semisimple Lie algebra.
We denote by $\Phi$ the root system of $\mathfrak{g}$ with respect to a fixed
Cartan subalgebra, and by $\mathbb{Z}\Phi$ its root lattice. We also fix a
basis $\Pi\subset\Phi$ of positive roots, and write
$P=\\{\varpi_{\alpha}\mid\alpha\in\Pi\\}$ for the set of corresponding
fundamental weights. We denote by $\Lambda$ the weight lattice
$\sum_{\alpha\in\Pi}\mathbb{Z}\varpi_{\alpha}$ and by $\Lambda^{+}$ the set of
dominant integral weights $\sum_{\alpha\in\Pi}\mathbb{N}\varpi_{\alpha}$.

Let $W$ be the Weyl group of $\mathfrak{g}$, and $s_{\alpha}\in W$ the
reflection corresponding to $\alpha\in\Pi$. Given an element $w\in W$ we
denote its length by $\ell(w)$, and set $N$ to be the length of $w_{0}$, the
longest element of $W$. A _decomposition_ of $w\in W$ is a word on the
generators $s_{\alpha}$ that equals $w$ in $W$. The decomposition is _reduced_
if it is of minimal length, i.e. its length equals $\ell(w)$. We denote by
$(-,-)$ the standard $W$-invariant pairing between $\mathbb{Z}\Phi$ and
$\Lambda$, and write
$\langle\lambda,\alpha\rangle=\frac{2(\lambda,\alpha)}{(\alpha,\alpha)}$ for
all $\lambda\in\Lambda$ and $\alpha\in\Phi$, so if
$\lambda=\sum_{\alpha\in\Pi}r_{\alpha}\varpi_{\alpha}$ then
$\langle\lambda,\alpha\rangle=r_{\alpha}$.

####

quantized-enveloping-algebra Fix $q\in\Bbbk^{\times}$. Let
$U_{q}(\mathfrak{g})$ be the quantum enveloping algebra of $\mathfrak{g}$;
this is an algebra generated by elements
$F_{\alpha},E_{\alpha},K_{\alpha}^{\pm 1}$ for $\alpha\in\Pi$, with the
relations given in [Jan]*Definition 4.3. We denote by
$U_{q}^{+}(\mathfrak{g}),U_{q}^{-}(\mathfrak{g})$ the subalgebras generated by
the $E_{\alpha}$’s and the $F_{\alpha}$’s, which are respectively called the
positive and negative parts of $U_{q}(\mathfrak{g})$ [Jan]*4.4. As shown in
[Jan]*Proposition 4.11, $U_{q}(\mathfrak{g})$ is a Hopf algebra.

If $q$ is not a root of unity and $\mathsf{char}\Bbbk>3$ then by [Jan]*chapter
5 for each $\lambda\in\Lambda^{+}$ there is an irreducible highest-weight
representation of $U_{q}(\mathfrak{g})$ of type $\mathbf{1}$, which we denote
by $V_{q}(\lambda)$. Each $V_{q}(\lambda)$ decomposes as the direct sum of
weight spaces $\bigoplus_{\mu\in\Lambda}V_{q}(\lambda)_{\mu}$; the dimensions
of the weight spaces are the same as the corresponding representation over
$\mathfrak{g}$, so the Weyl character formula holds for these representations,
see [Jan]*5.15.

#### L

et $G$ be the simply connected, connected algebraic group with Lie algebra
$\mathfrak{g}$. Since $U_{q}(\mathfrak{g})$ is a Hopf algebra, its dual
$U_{q}(\mathfrak{g})^{*}$ is an algebra with convolution product induced by
the coproduct of $U_{q}(\mathfrak{g})$. There is a map
$V_{q}(\lambda)^{*}\otimes V_{q}(\lambda)\longrightarrow
U_{q}(\mathfrak{g})^{*}$ defined by sending $\varphi\otimes v\in
V_{q}(\lambda)^{*}\otimes V_{q}(\lambda)$ to the linear functional
$c^{\lambda}_{\varphi,v}$, which assigns to each $u\in U_{q}(\mathfrak{g})$
the scalar $c^{\lambda}_{\varphi,v}(u)=\varphi(uv)$. Functionals of the form
$c^{\lambda}_{\varphi,v}$ are called _matrix coefficients_. The $\Bbbk$-linear
span of the matrix coefficients is a subalgebra of $U_{q}(\mathfrak{g})^{*}$
denoted by $\mathcal{O}_{q}[G]$, called the quantized algebra of coordinate
functions over the group $G$ [Jan]*7.11.

####

q-full-flag-varieties Quantum analogues of flag varieties and their Schubert
subvarieties were introduced by Soibelman in [S] and by Lakshmibai and
Reshetikhin in [qLR]; we review their definition. We assume that $q$ is not a
root of unity. Fix a maximal Borel subgroup $B$ of $G$. The _full flag
variety_ associated to $G$ is $G/B$. Let $C^{+}_{q}(\lambda)$ be the vector
space of matrix coefficients of the form $c^{\lambda}_{\varphi,v_{\lambda}}$
in $U_{q}(\mathfrak{g})^{*}$, where $v_{\lambda}$ is a highest weight vector
in $V_{q}(\lambda)$, and set

$\mathcal{O}_{q}[G/B]=\bigoplus_{\lambda\in\Lambda^{+}}C^{+}_{q}(\lambda)\subset\mathcal{O}_{q}[G]\subset
U_{q}(\mathfrak{g})^{*}.$

This is called the _quantum full flag variety_ of $G$. The product of two
matrix coefficients in $\mathcal{O}_{q}[G/B]$ is again in
$\mathcal{O}_{q}[G/B]$, and its decomposition as a direct sum gives
$\mathcal{O}_{q}[G/B]$ the structure of a $\Lambda^{+}$-graded algebra.

Let $I$ be a subset of the set of fundamental weights and set
$\mathcal{J}(I)=\sum_{\varpi\notin I}\mathbb{N}\varpi$. Denote by
$W_{I}\subset W$ the subgroup generated by the reflections $s_{\alpha}$ with
$\varpi_{\alpha}\in I$, and for each class in $W/W_{I}$ pick a representative
of minimal length. We denote by $W^{I}$ the set of these representatives.
Since the Weyl character formula holds, for each $w\in W$ and each
$\lambda\in\mathcal{J}(I)$ the vector space $V_{q}(\lambda)_{w\lambda}$ has
dimension $1$. The _Demazure module_ $V_{q}(\lambda)_{w}$ is the
$U_{q}^{+}(\mathfrak{g})$-submodule of $V_{q}(\lambda)$ generated by a vector
of weight $w\lambda$ in $V_{q}(\lambda)$.

The set $I$ determines a Lie subalgebra $\mathfrak{p}\subset\mathfrak{g}$, and
a parabolic subgroup $P_{I}\subset G$. The variety $G/P_{I}$ is the
corresponding generalized flag variety. To these data we associate the
$\Lambda^{+}$-graded subalgebra of $\mathcal{O}_{q}[G/B]$

$\mathcal{O}_{q}[G/P_{I}]=\bigoplus_{\lambda\in\mathcal{J}(I)}C_{q}^{+}(\lambda)$

called the _quantum partial flag variety associated to $I$_.

Given vector spaces $V_{2}\subset V_{1}$, we denote by $V_{2}^{\perp}$ the set
of linear functionals over $V_{1}$ which are zero on $V_{2}$. For every $w\in
W^{I}$ the vector space

$J_{w}^{I}=\bigoplus_{\lambda\in\mathcal{J}(I)}\left\langle
c_{\varphi,v_{\lambda}}^{\lambda}\in C_{q}^{+}(\lambda)\mid\varphi\in
V_{q}(\lambda)_{w}^{\perp}\right\rangle\subset\mathcal{O}_{q}[G/P_{I}]$

is an ideal of $\mathcal{O}_{q}[G/P_{I}]$ called the _Schubert ideal_
associated to $w$. The quotient algebra
$\mathcal{O}_{q}[G/P_{I}]_{w}=\mathcal{O}_{q}[G/P_{I}]/J^{I}_{w}$ is called
the _quantum Schubert variety_ associated to $w$.

### Degeneration of quantum Schubert varieties

Our aim is to show that quantum Schubert varieties have quantum affine toric
degenerations. In order to do so we work for a moment over the field
$\mathbb{Q}(v)$, where $v$ is an indeterminate over $\mathbb{Q}$, and consider
the $\mathbb{Q}(v)$-algebra $U=U_{v}(\mathfrak{g})$. We now review Caldero’s
proof of the existence of an $(S,<_{\mathsf{lex}})$-basis of
$\mathcal{O}_{v}[G/B]$, and its natural extension to arbitrary partial flag
and Schubert varieties [C]. Since Caldero is interested in classical flag
varieties, he works with a large base field $\mathbb{C}(q)$ that allows him to
specialize at $q=1$ and still get algebraic varieties over the complex
numbers. We give a different version of his argument which works over
$\mathbb{Q}(v)$.

#### F

ix as our base field $\mathbb{Q}(v)$, and set $U=U_{v}(\mathfrak{g})$. We
denote by $U^{+}$ and $U^{-}$ the algebras $U^{+}_{v}(\mathfrak{g})$ and
$U^{-}_{v}(\mathfrak{g})$, respectively.

Let $\mathcal{A}=\mathbb{Z}[v,v^{-1}]\subset\mathbb{Q}(v)$. For each
$n\in\mathbb{N}$ and each $\alpha\in\Pi$ we write $v_{\alpha}$ for
$v^{(\alpha,\alpha)/2}$. We also use the notation
$[n]_{\alpha}=\frac{v_{\alpha}^{n}-v_{\alpha}^{-n}}{v_{\alpha}-v_{\alpha}^{-1}}$,
and $[n]_{\alpha}!=[1]_{\alpha}[2]_{\alpha}\cdots[n]_{\alpha}$. Finally, we
set $F_{\alpha}^{(n)}=\frac{1}{[n]_{\alpha}!}F_{\alpha}^{n}$ and
$E_{\alpha}^{(n)}=\frac{1}{[n]_{\alpha}!}E_{\alpha}^{n}$. The algebra $U$ has
an $\mathcal{A}$-form which we denote by $U_{\mathcal{A}}$; it is the
$\mathcal{A}$-subalgebra of $U$ generated by the elements of the form
$F_{\alpha}^{(n)},E_{\alpha}^{(n)},K_{\alpha}^{\pm 1}$ for all $\alpha\in\Pi$
and all $n\geq 0$ [Jan]*11.1. The algebra $U^{+}$, resp. $U^{-}$, also has an
$\mathcal{A}$-form which we denote by $U_{\mathcal{A}}^{+}$, resp.
$U_{\mathcal{A}}^{-}$; it is generated by all the $E_{\alpha}^{(n)}$, resp.
$F^{(n)}_{\alpha}$, with $n\geq 0$. These $\mathcal{A}$-forms are compatible
with the weight decomposition of $U$. By construction
$U\cong\mathbb{Q}(v)\otimes_{\mathcal{A}}U_{\mathcal{A}}$, and analogous
results hold for $U_{\mathcal{A}}^{+}$ and $U_{\mathcal{A}}^{-}$.

For the rest of this section we fix a nonzero highest weight vector
$v_{\lambda}\in V_{v}(\lambda)$. Setting
$V_{\mathcal{A}}(\lambda)=U^{-}_{\mathcal{A}}v_{\lambda}\subset
V_{v}(\lambda)$ we obtain $\mathcal{A}$-forms of $U$-modules. These
$\mathcal{A}$-forms are compatible with the weight decompositions of the
original objects, see [Jan]*chapter 11. Now let $\lambda$ be a dominant
integral weight and fix $w\in W$. The $\mathcal{A}$-form of the Demazure
module $V_{v}(\lambda)_{w}$ is defined as
$V_{\mathcal{A}}(\lambda)_{w}=U_{\mathcal{A}}^{+}V_{\mathcal{A}}(\lambda)_{w\lambda}$.

####

T:A-crystal-basis The algebra $U_{\mathcal{A}}^{-}$ has a homogeneous
$\mathcal{A}$-basis, called the _canonical_ or _global_ basis of $U^{-}$,
discovered independently by Lusztig and Kashiwara. Its construction is the
subject of [Jan]*chapters 9 - 11, and we will use the notation from this
source to recapitulate some relevant facts.

Set $A_{0}\subset\mathbb{Q}(v)$ to be the ring of rational functions without a
pole at $0$. For each $\alpha\in\Pi$ define the operators
$\tilde{E}_{\alpha},\tilde{F}_{\alpha}:U^{-}\longrightarrow U^{-}$ as in
[Jan]*10.2, and let $\mathcal{L}(\infty)$ be the $A_{0}$-lattice generated by
all elements of the form
$\tilde{F}_{\alpha_{1}}\tilde{F}_{\alpha_{2}}\cdots\tilde{F}_{\alpha_{r}}(1)$;
by definition these are weight elements, so setting
$\mathcal{L}(\infty)_{-\nu}=\mathcal{L}(\infty)\cap U^{-}_{-\nu}$ for each
$\nu\in\mathbb{Z}\Phi$ with $\nu\geq 0$, we get
$\mathcal{L}(\infty)=\bigoplus_{\nu\geq 0}\mathcal{L}(\infty)_{-\nu}$, and
furthermore each $\mathcal{L}(\infty)_{-\nu}$ is a finitely generated
$A_{0}$-module that generates $U^{-}_{\nu}$ over $\mathbb{Q}(v)$.

Set

$\displaystyle\mathcal{B}(\infty)_{-\nu}$
$\displaystyle=\left\\{\tilde{F}_{\alpha_{1}}\tilde{F}_{\alpha_{2}}\cdots\tilde{F}_{\alpha_{r}}(1)+v\mathcal{L}(\infty)\mid\sum_{i}\alpha_{i}=\nu\right\\}\subset\mathcal{L}(\infty)_{-\nu}/v\mathcal{L}(\infty)_{-\nu},$

and set $\mathcal{B}(\infty)=\bigsqcup_{\nu\geq 0}\mathcal{B}(\infty)_{-\nu}$.
Although it is not obvious, $\mathcal{B}(\infty)_{-\nu}$ is a basis of
$\mathcal{L}(\infty)_{-\nu}/v\mathcal{L}(\infty)_{-\nu}$ [Jan]*Proposition
10.11; this is the _crystal_ basis of $U^{-}_{-\nu}$ at $v=0$. It turns out
that each $b\in\mathcal{B}(\infty)_{-\nu}$ has a unique lift
$G(b)\in\mathcal{L}(\infty)_{-\nu}\cap U_{\mathcal{A}}^{-}$, which is
invariant under the action of certain automorphisms of $U$ [Jan]*Theorem 11.10
a). The set $G(\mathcal{B})$ of all $G(b)$ with $b\in\mathcal{B}(\infty)$ is
the global basis of $U_{\mathcal{A}}^{-}$.

Let $w\in W$, and let $w=s_{\alpha_{1}}\cdots s_{\alpha_{r}}$ be a reduced
decomposition of $w$. Set $\mathcal{B}_{w}(\infty)$ as the set of elements in
$\mathcal{B}(\infty)$ of the form
$\tilde{F}_{\alpha_{1}}^{k_{1}}\tilde{F}_{\alpha_{2}}^{k_{2}}\cdots\tilde{F}_{\alpha_{r}}^{k_{r}}(1)+v\mathcal{L}(\infty)$
with $k_{j}\geq 0$. This set does not depend on the decomposition of $w$
[K1]*Proposition 3.2.5.

######  Theorem 2.

Let $\nu\in\mathbb{Z}\Phi$ with $\nu\geq 0$.

  1. (a)

The set $\\{G(b)\mid b\in\mathcal{B}(\infty)_{-\nu}\\}$ is an
$\mathcal{A}$-basis of $(U_{\mathcal{A}}^{-})_{-\nu}$.

  2. (b)

Let $\lambda$ be a dominant integral weight. The set $\\{G(b)v_{\lambda}\mid
b\in\mathcal{B}(\infty)_{-\nu}\\}\setminus\\{0\\}$ is an $\mathcal{A}$-basis
of $V_{\mathcal{A}}(\lambda)_{\lambda-\nu}$. Furthermore, if
$G(b)v_{\lambda}=G(b^{\prime})v_{\lambda}\neq 0$ then $b=b^{\prime}$.

  3. (c)

Let $w\in W$. The set $B_{w}(\lambda)=\\{G(b)v_{\lambda}\mid
b\in\mathcal{B}_{w}(\infty)\\}\setminus\\{0\\}$ is an $\mathcal{A}$-basis of
$V_{\mathcal{A}}(\lambda)_{w}$.

  * Proof.

The first two items are part of [Jan]*Theorem 11.10. The third is
[K1]*Proposition 3.2.5 (vi). ∎

####

q-var-bases We now use the global basis to produce bases for quantum Schubert
varieties. For each dominant integral weight $\lambda$ we write
$\mathcal{B}(\lambda)=\\{b\in\mathcal{B}(\infty)\mid G(b)v_{\lambda}\neq
0\\}$. By the previous theorem the set $\\{G(b)v_{\lambda}\mid
b\in\mathcal{B}(\lambda)\\}$ is a basis of $V_{v}(\lambda)$, so we can take
its dual basis. Given $b\in\mathcal{B}(\lambda)$, we denote by
$b_{\lambda}^{*}$ the unique element of $V_{v}(\lambda)^{*}$ such that
$b^{*}_{\lambda}(G(b^{\prime})v_{\lambda})=\delta_{b,b^{\prime}}$ for all
$b^{\prime}\in\mathcal{B}(\lambda)$. Thus to each element
$b\in\mathcal{B}(\lambda)$ we can associate the matrix coefficient
$b_{\lambda}=c^{\lambda}_{b^{*}_{\lambda},v_{\lambda}}\in C^{+}_{v}(\lambda)$,
and the set $\\{b_{\lambda}\mid b\in\mathcal{B}(\lambda)\\}$ is a basis of
$C^{+}_{v}(\lambda)$. Since the quantum flag variety is the direct sum of all
these spaces with $\lambda$ running over all dominant integral weights, we
obtain a basis of $\mathcal{O}_{v}[G/B]$ as defined in LABEL:q-full-flag-
varieties by taking

$FB=\\{b_{\lambda}\mid b\in\mathcal{B}(\lambda),\lambda\in\Lambda^{+}\\}.$

If $I$ is a subset of the fundamental weights, then we obtain a basis of the
partial flag variety $\mathcal{O}_{v}[G/P_{I}]$ by taking

$FB_{I}=\\{b_{\lambda}\mid
b\in\mathcal{B}(\lambda),\lambda\in\mathcal{J}(I)\\}.$

Finally, the third item of the previous theorem implies that the ideal
$J^{I}_{w}$ defined in LABEL:q-full-flag-varieties is spanned over
$\mathbb{Q}(v)$ by all elements of the form $b_{\lambda}$ with
$b\in\mathcal{B}(\lambda)\setminus\mathcal{B}_{w}(\infty)$. Setting
$\mathcal{B}_{w}(\lambda)=\mathcal{B}_{w}(\infty)\cap\mathcal{B}(\lambda)$ we
obtain a basis for the quantum Schubert variety $\mathcal{O}_{v}[G/P_{I}]_{w}$
by taking the image of

$SB_{I}(w)=\\{b_{\lambda}\mid
b\in\mathcal{B}_{w}(\lambda),\lambda\in\mathcal{J}(I)\\}.$

in the quotient.

####

Littelman-parametrizations Recall that the Kashiwara operators
$\tilde{E}_{\alpha},\tilde{F}_{\alpha}$ induce operators
$\tilde{E}_{\alpha},\tilde{F}_{\alpha}:\mathcal{B}(\infty)\longrightarrow\mathcal{B}(\infty)\cup\\{0\\}$
[Jan]*10.12. By definition the operators $\tilde{E}_{\alpha}$ are locally
nilpotent as operators on $U$ [Jan]*10.2 and hence as operators on
$\mathcal{B}(\infty)$, so it makes sense to set
$e_{\alpha}(b)=\max\\{k\in\mathbb{N}\mid\tilde{E}_{\alpha}^{k}(b)\neq 0\\}$
for each $b\in\mathcal{B}(\infty)$. We write
$\overline{E}_{\alpha}(b)=\tilde{E}_{\alpha}^{e_{\alpha}(b)}(b)$.

To each reduced decomposition of $w\in W$ we can associate a parametrization
of $\mathcal{B}_{w}(\infty)$, known as a string parametrization, introduced by
Littelmann [Lit] and Berenstein and Zelevinsky [BZ]. If
$\tilde{w}=s_{\alpha_{1}}\cdots s_{\alpha_{r}}$ is the chosen decomposition
then we define
$\Lambda_{\tilde{w}}:\mathcal{B}(\infty)\longrightarrow\mathbb{N}^{r}$ by the
formula

$\Lambda_{\tilde{w}}(b)=(e_{\alpha_{1}}(b),e_{\alpha_{2}}(\overline{E}_{\alpha_{1}}(b)),\ldots,e_{\alpha_{r}}(\overline{E}_{\alpha_{r-1}}\cdots\overline{E}_{\alpha_{1}}(b))).$

If $a=\Lambda_{\tilde{b}}$ then $\tilde{F}_{\alpha_{1}}^{a_{1}}\cdots
F_{\alpha_{N}}^{a_{N}}1=b$, so this map is injective. Now according to
[BZ]*Proposition 3.5, the set
$S_{\tilde{w}}=\Lambda_{\tilde{w}}(\mathcal{B}_{w}(\infty))$ is the set of
integral points of a convex polyhedral cone, and hence by Gordan’s lemma a
normal affine semigroup.

A decomposition of $w_{0}$, the longest word of $W$, is said to be _adapted_
to $w$ if it is of the form $s_{\alpha_{1}}\cdots s_{\alpha_{N}}$ with
$s_{\alpha_{1}}\cdots s_{\alpha_{\ell(w)}}=w$. For every element $w\in W$
there exists a decomposition of the longest word of $W$ adapted to $w$, or in
other words the longest word of $W$ is the maximum for the weak right Bruhat
order on $W$, see [BB]*Proposition 3.1.2.

######  Definition 5.

Set $\Pi=\\{\alpha_{1},\ldots,\alpha_{n}\\}$ and set
$\varpi_{i}=\varpi_{\alpha_{i}}$. Fix $w\in W$ and fix $\tilde{w}_{0}$ a
decomposition of $w_{0}$ adapted to $w$. We define

$\displaystyle\Gamma_{\tilde{w}_{0}}:FB$
$\displaystyle\longrightarrow\mathbb{N}^{N}\times\mathbb{N}^{n}$
$\displaystyle b_{\lambda}$
$\displaystyle\longmapsto\Lambda_{\tilde{w}_{0}}(b)\times(\langle\lambda,\alpha_{1}\rangle,\ldots,\langle\lambda,\alpha_{n}\rangle).$

and set
$\tilde{S}_{\tilde{w}_{0}}=\Gamma_{\tilde{w}_{0}}(FB),\tilde{S}_{\tilde{w}_{0}}^{I}=\Gamma_{\tilde{w}_{0}}(FB_{I})$
and
$\tilde{S}_{\tilde{w}_{0},\tilde{w}}^{I}=\Gamma_{\tilde{w}_{0}}(SB_{I}(w))$.

If
$\Gamma_{\tilde{w}_{0}}(b_{\lambda})=\Gamma_{\tilde{w}_{0}}(b^{\prime}_{\lambda^{\prime}})$
then $\lambda=\lambda^{\prime}$ and
$\Lambda_{\tilde{w}_{0}}(b_{\lambda})=\Lambda_{\tilde{w}_{0}}(b^{\prime}_{\lambda^{\prime}})$,
which implies $b_{\lambda}=b^{\prime}_{\lambda^{\prime}}$. Thus
$\Gamma_{\tilde{w}_{0}}$ is injective and the sets
$\tilde{S}_{\tilde{w}_{0}},\tilde{S}_{\tilde{w}_{0}}^{I}$ and
$\tilde{S}_{\tilde{w}_{0},\tilde{w}}^{I}$ parametrize bases of the quantized
coordinate rings of the full flag variety, the partial flag variety associated
to the set $I$, and of the Schubert variety associated to $I$ and $w$,
respectively.

######  Lemma 5.

The sets $\tilde{S}_{\tilde{w}_{0}},\tilde{S}_{\tilde{w}_{0}}^{I}$ and
$\tilde{S}_{\tilde{w}_{0},\tilde{w}}^{I}$ are normal affine semigroups.

  * Proof.

According to [Lit]*Proposition 1.5 $\tilde{S}_{\tilde{w}_{0}}$ is the set of
all $a\times(r_{1},\ldots,r_{n})\in S_{\tilde{w}_{0}}\times\mathbb{N}^{n}$
such that

$\displaystyle a_{l}$
$\displaystyle\leq\left\langle\lambda-\sum_{j=1}^{N-l}a_{N-j+1}\alpha_{i_{N-j+1}},\alpha_{i_{l}}\right\rangle$
$\displaystyle(1\leq l\leq N)$

where $\lambda=\sum_{i}r_{i}\varpi_{i}$. Thus $\tilde{S}_{\tilde{w}_{0}}$ is
the set of points of $S_{\tilde{w}_{0}}\times\mathbb{N}^{n}$ that comply with
these inequalities, and hence it is also a normal affine semigroup.

Furthermore, $\tilde{S}_{\tilde{w}_{0}}^{I}=\Gamma_{\tilde{w}_{0}}(FB_{I})$ is
the intersection of $\tilde{S}_{\tilde{w}_{0}}$ with the hyperplanes defined
by the equations $x_{N+i}=0$ for all $i$ such that $\varpi_{i}\in I$, and
hence is also a normal affine semigroup. Finally, the fact that the
decomposition $\tilde{w}_{0}$ is adapted to $w$ implies that
$\tilde{S}_{\tilde{w}_{0},\tilde{w}}^{I}=\Gamma_{\tilde{w}_{0}}(SB_{I}(w))$ is
the intersection of $\tilde{S}_{\tilde{w}_{0}}^{I}$ with the hyperplanes
defined by $x_{i}=0$ for all $\ell(w)<i\leq N$, and hence it is also a normal
affine semigroup. ∎

####

coproduct-coefficients We have just shown that the bases of quantum flag
varieties and quantum Schubert varieties defined in LABEL:q-var-bases are
parametrized by normal semigroups. All that is left to check is that they have
the multiplicative property of $(S,<_{\mathsf{lex}})$-bases, where
$<_{\mathsf{lex}}$ is the lexicographic order of
$\mathbb{N}^{N}\times\mathbb{N}^{n}$.

Let $\lambda,\lambda^{\prime}\in\Lambda^{+}$ and
$b\in\mathcal{B}(\lambda),b^{\prime}\in\mathcal{B}(\lambda^{\prime})$. Recall
that $b_{\lambda}^{*}$ denotes the element in the dual basis of
$V_{v}(\lambda)^{*}$ as defined in LABEL:q-var-bases. The product
$b_{\lambda}b^{\prime}_{\lambda^{\prime}}$ is by definition the matrix
coefficient corresponding to the functional $b_{\lambda}^{*}\otimes
b_{\lambda^{\prime}}^{\prime*}$ and the vector $v_{\lambda}\otimes
v_{\lambda^{\prime}}$ over $V(\lambda)\otimes V(\lambda^{\prime})$. Now the
$U$-module generated by $v_{\lambda}\otimes v_{\lambda^{\prime}}$ is
isomorphic to $V_{v}(\lambda+\lambda^{\prime})$, so $b_{\lambda}^{*}\otimes
b_{\lambda^{\prime}}^{\prime*}$ naturally induces an element in
$C_{v}^{+}(\lambda+\lambda^{\prime})$, and the product
$b_{\lambda}b^{\prime}_{\lambda^{\prime}}$ is a linear combination of matrix
coefficients in $C^{+}_{v}(\lambda+\lambda^{\prime})$

$b_{\lambda}b^{\prime}_{\lambda^{\prime}}=\sum_{b^{\prime\prime}\in\mathcal{B}(\lambda+\lambda^{\prime})}c^{b^{\prime\prime}}_{b,b^{\prime}}b^{\prime\prime}_{\lambda+\lambda^{\prime}}$

with $c^{b^{\prime\prime}}_{b,b^{\prime}}\in\mathbb{Q}(v)$. In order to show
that $FB$ is an $(\tilde{S}_{\tilde{\omega}_{0}},<_{\mathsf{lex}})$-basis we
must show that $c^{b^{\prime\prime}}_{b,b^{\prime}}\neq 0$ implies that
$\Gamma_{\tilde{\omega}_{0}}(b^{\prime\prime})\leq\Gamma_{\tilde{\omega}_{0}}(b)+\Gamma_{\tilde{\omega}_{0}}(b^{\prime})$,
and that if equality holds then $c_{b,b^{\prime}}^{b^{\prime\prime}}$ is
nonzero. Notice that this would also imply that $FB_{I}$ is an
$(\tilde{S}^{I}_{\tilde{\omega}_{0}},<_{\mathsf{lex}})$-basis and that
$SB_{I}(\tilde{w})$ is a
$(\tilde{S}^{I}_{\tilde{\omega}_{0},\tilde{w}},<_{\mathsf{lex}})$-basis.

By definition of a dual basis the scalar $c_{b,b^{\prime}}^{b^{\prime\prime}}$
is the value $b_{\lambda}b^{\prime}_{\lambda^{\prime}}(G(b^{\prime\prime}))$.
On the other hand, by the definition of the product of matrix coefficients

$b_{\lambda}b^{\prime}_{\lambda^{\prime}}(G(b^{\prime\prime}))=b_{\lambda}^{*}\otimes
b_{\lambda^{\prime}}^{\prime*}(\Delta(G(b^{\prime\prime}))\cdot
v_{\lambda}\otimes v_{\lambda^{\prime}}),$

so we need to study the coproduct $\Delta(G(b^{\prime\prime}))$. It follows
from [Jan]*4.9 (4) that
$\Delta(F_{\alpha}^{(r)})=\sum_{i+j=r}v_{\alpha}^{ji}F^{(i)}_{\alpha}\otimes
F^{(j)}_{\alpha}K_{\alpha}^{-i}$, where $v_{\alpha}=v^{(\alpha,\alpha)/2}$.
Thus by an argument similar to that of [Jan]*Lemma 4.12 we see that

$\displaystyle\Delta(U^{-}_{\mathcal{A}})_{-\nu}\subset\bigoplus_{0\leq\mu\leq\nu}(U^{-}_{\mathcal{A}})_{-\mu}\otimes(U^{-}_{\mathcal{A}})_{-(\nu-\mu)}K_{-\mu}$

It follows that $\Delta(G(b^{\prime\prime}))$ is an $\mathcal{A}$-linear
combination of terms $G(b_{(1)})\otimes G(b_{(2)})K_{\mu}$, with
$b_{(1)},b_{(2)}\in\mathcal{B}(\infty)$, and $\mu$ the weight of $b_{(1)}$.
Among all these terms there is one of the form
$d_{b,b^{\prime}}^{b^{\prime\prime}}(G(b)\otimes
G(b^{\prime})K_{\operatorname{\mathsf{wt}}(b)})$ with
$d_{b,b^{\prime}}^{b^{\prime\prime}}\in\mathcal{A}$, and thus
$c_{b,b^{\prime}}^{b^{\prime\prime}}=v^{(\operatorname{\mathsf{wt}}(b),\lambda^{\prime})}d_{b,b^{\prime}}^{b^{\prime\prime}}$.
Notice that unlike before, the element $d_{b,b^{\prime}}^{b^{\prime\prime}}$
is independent of $\lambda$ and $\lambda^{\prime}$.

The problem of showing that $FB$ is indeed an
$(\tilde{S}_{\tilde{w}_{0}},<_{\mathsf{lex}})$-basis thus reduces to showing
that if $d_{b,b^{\prime}}^{b^{\prime\prime}}\neq 0$ then
$\Lambda_{\tilde{w}_{0}}(b^{\prime\prime})\leq_{\mathsf{lex}}\Lambda_{\tilde{w}_{0}}(b)+\Lambda_{\tilde{w}_{0}}(b^{\prime})$;
this also implies that $FB_{I}$ and $SB_{I}(\tilde{w})$ are
$(\tilde{S}_{\tilde{w}_{0}},<_{\mathsf{lex}})$-bases. Caldero shows that this
is indeed the case in [C]*Theorem 2.3, under the hypothesis that $q$ is
transcendental over $\mathbb{C}$. We give an alternative proof in the
following paragraphs.

####

coproduct-coefficients-convex We fix some notation. Given a decomposition of
the longest word of $W$ $\tilde{w}_{0}=s_{\alpha_{1}}\cdots s_{\alpha_{N}}$,
for each $a\in\mathbb{N}^{N}$ we write $F^{(a)}=F_{\alpha_{1}}^{(a_{1})}\cdots
F_{\alpha_{N}}^{(a_{N})}$ and
$\tilde{F}^{a}=\tilde{F}_{\alpha_{1}}^{a_{1}}\cdots\tilde{F}_{\alpha_{N}}^{a_{N}}$.
If $a$ lies in the image of $\Lambda_{\tilde{w}_{0}}$ then we write
$b^{a}=\Lambda_{\tilde{w}_{0}}^{-1}(a)$.

According to [Lit]*Proposition 10.3 the monomials $F^{(a)}$ with $a$ in the
image of $\Lambda_{\tilde{\omega}_{0}}$ form a weight basis of
$U_{\mathcal{A}}^{-}$. In fact, if we fix $\nu\in\mathbb{Z}\Phi,\nu\geq 0$,
the change of basis matrix between Littelman’s monomial basis and the global
basis of $(U_{\mathcal{A}}^{-})_{-\nu}$ is unipotent if we order the bases
according to the lexicographic order of the corresponding
$a\in\operatorname{Im}\Lambda_{\tilde{\omega}_{0}}$. The following lemma
records this fact. On the other hand, we can consider a monomial of the form
$F^{(a)}$ with $a$ outside the image of $\Lambda_{\tilde{\omega}_{0}}$; we
show that in this case the monomial is a linear combination of monomials whose
exponents are strictly larger than $a$ in the lexicographic order, or
equivalently, elements $G(b^{a^{\prime}})$ with $a^{\prime}>_{\mathsf{lex}}a$.

######  Lemma 6.

Fix $\tilde{w}_{0}=s_{\alpha_{1}}\cdots s_{\alpha_{N}}$ a decomposition of the
longest word of $W$. Let $a\in\mathbb{N}^{N}$, and let
$\nu\in\mathbb{Z}\Phi,\nu\geq 0$ be such that $F^{(a)}\in U^{-}_{-\nu}$.

  1. (a)

Let $\alpha\in\Pi$ and $r>0$. Let $b\in\mathcal{B}(\infty)_{-\nu}$. Then for
each $b^{\prime}\in\mathcal{B}(\infty)_{-\nu-r\alpha}$ with
$e_{\alpha}(b^{\prime})>r+e_{\alpha}(b)$ there exists
$x_{b^{\prime}}\in\mathcal{A}$ such that

$\displaystyle F^{(r)}_{\alpha}G(b)$
$\displaystyle=\genfrac{[}{]}{0.0pt}{0}{r+e_{\alpha}(b)}{r}G(\tilde{F}_{\alpha}^{r}b)+\sum_{e_{\alpha}(b^{\prime})>r+e_{\alpha}(b)}x_{b^{\prime}}G(b^{\prime}).$

  2. (b)

Suppose $a$ lies in the image of $\Lambda_{\tilde{w}_{0}}$. Then for each
$a^{\prime}\in\Lambda_{\tilde{w}_{0}}(\mathcal{B}(\infty)_{-\nu})$ with
$a^{\prime}>_{\mathsf{lex}}a$ there exist
$x_{a,a^{\prime}},y_{a,a^{\prime}}\in\mathcal{A}$ such that

$\displaystyle F^{(a)}$
$\displaystyle=G(b^{a})+\sum_{a^{\prime}>_{\mathsf{lex}}a}x_{a,a^{\prime}}G(b^{a^{\prime}}),$
$\displaystyle G(b^{a})$
$\displaystyle=F^{(a)}+\sum_{a^{\prime}>_{\mathsf{lex}}a}y_{a,a^{\prime}}F^{(a^{\prime})}.$

  3. (c)

Suppose $a$ does not lie in the image of $\Lambda_{\tilde{w}_{0}}$. Let
$b=\tilde{F}^{a}1+v\mathcal{L}(\infty)\in\mathcal{B}(\infty)_{-\nu}$ and let
$s=\Lambda_{\tilde{w}_{0}}(b)$. Then $s>_{\mathsf{lex}}a$, and for each
$a^{\prime}\in\Lambda_{\tilde{w}_{0}}(\mathcal{B}(\infty)_{-\nu})$ with
$a^{\prime}>_{\mathsf{lex}}a$ there exists $z_{a,a^{\prime}}\in\mathcal{A}$
such that

$\displaystyle F^{(a)}$
$\displaystyle=\sum_{a^{\prime}>_{\mathsf{lex}}a}z_{a,a^{\prime}}G(b^{a^{\prime}}).$

Furthermore $z_{a,s}\neq 0$.

  * Proof.

Recall from [Jan]*Lemma 11.3 that
$\tilde{F}_{\alpha}^{r}(U_{\mathcal{A}}^{-})_{-\nu}=\sum_{s\geq
0}F_{\alpha}^{(r+s)}(U_{\mathcal{A}}^{-})_{-\nu+s\alpha}$, and that for each
$u\in(U_{\mathcal{A}}^{-})_{-\nu}$ we have

$\displaystyle\tilde{F}_{\alpha}^{r}u=F_{\alpha}^{(r)}u+\tilde{F}_{\alpha}^{r+1}u^{\prime}=F_{\alpha}^{(r)}u+\sum_{s\geq
1}F_{\alpha}^{(r+s)}u_{s}$

where $u^{\prime}\in(U_{\mathcal{A}}^{-})_{\nu+\alpha}$ and
$u_{s}\in(U_{\mathcal{A}}^{-})_{\nu+s\alpha}$. Set $e=e_{\alpha}(b)$. As shown
in the proof of [Jan]*Lemma 11.12, in p. 249 below equation (1),

$\displaystyle G(b)$
$\displaystyle=\tilde{F}_{\alpha}^{e}(G(\tilde{E}_{\alpha}^{e}b))+\tilde{F}_{\alpha}^{e+1}u^{\prime}$
($\dagger$)

with $u^{\prime}\in(U_{\mathcal{A}}^{-})_{\nu+\alpha}$, so

$\displaystyle G(b)$
$\displaystyle=F_{\alpha}^{(e)}(G(\tilde{E}_{\alpha}^{e}b))+\sum_{s\geq
1}F_{\alpha}^{(e+s)}u_{s}.$

Multiplying this by $F_{\alpha}^{(r)}$ we get

$\displaystyle F_{\alpha}^{(r)}G(b)$
$\displaystyle=\genfrac{[}{]}{0.0pt}{0}{r+e}{r}F_{\alpha}^{(r+e)}(G(\tilde{E}_{\alpha}^{e}b))+\sum_{s\geq
1}F_{\alpha}^{(r+e+s)}u^{\prime}_{s}.$

Using $\dagger$ ‣ Proof. we get

$\displaystyle\tilde{F}_{\alpha}^{r+e}(G(\tilde{E}_{\alpha}^{e}b))=\tilde{F}_{\alpha}^{r+e}(G(\tilde{E}_{\alpha}^{e+r}\tilde{F}_{\alpha}^{r}b))=G(\tilde{F}_{\alpha}^{r}b)+\tilde{F}_{\alpha}^{r+e+1}u^{\prime\prime}$

and finally

$\displaystyle F_{\alpha}^{(r)}G(b)$
$\displaystyle\equiv\genfrac{[}{]}{0.0pt}{0}{r+e}{r}G(\tilde{F}_{\alpha}^{r}b)\mod\tilde{F}_{\alpha}^{r+e+1}(U_{\mathcal{A}}^{-})_{-\nu}.$

By [Jan]*Lemma 11.12 the $G(b^{\prime})$ with $e_{\alpha}(b^{\prime})>r+e$
form an $\mathcal{A}$-basis of
$\tilde{F}_{\alpha}^{r+e+1}(U_{\mathcal{A}}^{-})_{-\nu}$, so we are done with
item (a).

As mentioned above, item (b) is a consequence of [Lit]*Proposition 10.3, which
states that given a dominant integral weight $\lambda$ and a highest weight
vector $v_{\lambda}\in V_{\mathcal{A}}(\lambda)$, then if
$G(b^{a})v_{\lambda}\neq 0$ there exist $x_{a,a^{\prime}}$ as in the statement
such that

$\displaystyle F^{(a)}v_{\lambda}$
$\displaystyle=G(b^{a})v_{\lambda}+\sum_{a^{\prime}>_{\mathsf{lex}}a}x_{a,a^{\prime}}G(b^{a^{\prime}})v_{\lambda}.$

As shown in the proof of [Jan]*Theorem 10.10, there exists a dominant integral
weight $\lambda$ such that the map $U^{-}_{-\nu}\longrightarrow
V_{v}(\lambda)_{\lambda-\nu}$ given by $G(b)\mapsto G(b)v_{\lambda}$ is an
isomorphism, so the first formula is proved. This means that the (finite)
matrix of the coefficients of the $F^{(a)}\in(U_{\mathcal{A}}^{-})_{-\nu}$ in
the global basis of $(U_{\mathcal{A}}^{-})_{-\nu}$ with the order induced by
the lexicographic order, is lower triangular with ones in the diagonal. The
second formula follows by taking the inverse of this matrix.

Let us prove the last item. Suppose that $s<_{\mathsf{lex}}a$. Then by
definition $s_{1}=a_{1},\ldots,s_{j}=a_{j},s_{j+1}<a_{j+1}$ for some $1\leq
j\leq N$. This implies that

$\displaystyle\tilde{E}_{\alpha_{j+1}}^{s_{j+1}+1}\tilde{E}_{\alpha_{j}}^{s_{j}}\cdots\tilde{E}_{\alpha_{1}}^{s_{1}}(b)=\tilde{F}_{\alpha_{j+1}}^{a_{j}-s_{j}-1}\cdots\tilde{F}_{\alpha_{N}}^{a_{N}}1+v\mathcal{L}(\infty)\neq
0,$

since the maps $\tilde{F}_{\alpha}$ are injective. This contradicts the
definition of $\Lambda_{\tilde{w}_{0}}(b)$, so $s\geq_{\mathsf{lex}}a$ and
since $a$ is not in the image of $\Lambda_{\tilde{w}_{0}}$ the inequality is
strict.

We now prove the following intermediate result: for each $a\in\mathbb{N}^{N}$
we have
$F^{(a)}=z^{\prime}_{a,s}G(b^{s})+\sum_{a^{\prime}>_{\mathsf{lex}}a}z^{\prime}_{a,a^{\prime}}F^{(a^{\prime})}$,
where
$s=\Lambda_{\tilde{\omega}_{0}}^{-1}(\tilde{F}^{a}(1+v\mathcal{L}(\infty)))$
and $z^{\prime}_{a,a^{\prime}},z^{\prime}_{a,s}$ lie in $\mathcal{A}$ for all
$a^{\prime}$. We prove this by descending induction on $j=\min\\{i\mid
a_{i}\neq 0\\}$, starting with the case $j=N$. In that case item (a) implies
that $\tilde{F}^{a}=G(b^{a})$ and we are done.

Now let $\overline{a}\in\mathbb{N}^{N}$ be the $N$-tuple given by
$\overline{a}_{j}=0$ and $\overline{a}_{i}=a_{i}$ for all $i\neq j$. By the
inductive hypothesis we have

$\displaystyle F^{(a)}$
$\displaystyle=F_{\alpha_{j}}^{(a_{j})}F^{(\overline{a})}=z^{\prime}_{\overline{a},\overline{s}}F^{a_{j}}_{\alpha_{j}}G(b^{\overline{s}})+\sum_{\overline{a}^{\prime}>\overline{a}}z^{\prime}_{\overline{a},\overline{a}^{\prime}}F^{(a_{j})}_{\alpha_{j}}F^{(\overline{a}^{\prime})}.$

Now $F^{(a_{j})}_{\alpha_{j}}F^{(\overline{a}^{\prime})}$ is a scalar multiple
of $F^{(a^{\prime})}$ where $a^{\prime}_{j}=a_{j}$ and
$a^{\prime}_{i}=\overline{a}_{i}$, and clearly $a^{\prime}>_{\mathsf{lex}}a$.
On the other hand by item (a)

$\displaystyle F^{a_{j}}_{\alpha_{j}}G(b^{\overline{s}})$
$\displaystyle=\genfrac{[}{]}{0.0pt}{0}{a_{j}+e}{a_{j}}G(\tilde{F}^{a_{j}}_{\alpha_{j}}b^{\overline{s}})+\sum_{e_{\alpha_{j}}(b^{\prime})>a_{j}+e_{\alpha_{j}}(b)}x_{b^{\prime}}G(b^{\prime})$

where $e=e_{\alpha_{j}}(b^{\overline{s}})$. Since
$b^{\overline{s}}=\tilde{F}^{\overline{a}}$ we have
$\tilde{F}_{\alpha_{j}}^{a_{j}}b^{s}=\tilde{F}^{a}(1+v\mathcal{L}(\infty))=b^{s}$.
On the other hand, the condition
$e_{\alpha_{j}}(b^{\prime})>a_{j}+e_{\alpha_{j}}(b)$ guarantees that
$\Lambda_{\tilde{\omega}_{0}}(b^{\prime})>_{\mathsf{lex}}a$ and by item (b)
each $G(b^{\prime})$ is a linear combination of monomials $F^{(a^{\prime})}$
with $a^{\prime}>_{\mathsf{lex}}a$. This completes the proof of the
intermediate result.

Now consider the set $D$ of all
$a\notin\operatorname{Im}\Lambda_{\tilde{\omega}_{0}}$ such that
$F^{(a)}\in(U^{-}_{\mathcal{A}})_{-\nu}$ for which the statement of (c) fails.
This set is finite and totally ordered by the lexicographic order, so if it is
not empty then it has a maximal element $d$. Now by the intermediate statement
$F^{(d)}=z^{\prime}_{d,s}G(b^{s})+\sum_{d^{\prime}>_{\mathsf{lex}}d}z^{\prime}_{d,d^{\prime}}F^{(d^{\prime})}$,
and since no $d^{\prime}$ can be in $D$, this sum is an $\mathcal{A}$-linear
combination of elements $G(b^{s^{\prime}})$ with
$s^{\prime}>_{\mathsf{lex}}d^{\prime}>_{\mathsf{lex}}d$. This contradicts the
fact that $d\in D$, and the contradiction arose from supposing $D$ was
nonempty. Thus (c) holds in all cases. ∎

#### R

ecall that we have defined $c_{b,b^{\prime}}^{b^{\prime\prime}}\in\mathcal{A}$
as the coefficient of $b^{\prime\prime}_{\lambda+\lambda^{\prime}}$ in
$b_{\lambda}b_{\lambda^{\prime}}$, and that we have shown that it equals a
power of $v$ times $d_{b,b^{\prime}}^{b^{\prime\prime}}$, the coefficient of
$G(b)\otimes G(b^{\prime})K_{-\mu}$ in $\Delta(G(b^{\prime\prime}))$, where
$-\mu$ is the weight of $G(b)$. We have also shown above that the statement of
the following proposition is equivalent to $FB$ being a
$(\tilde{S}_{\tilde{\omega}_{0}},<_{\mathsf{lex}})$-basis.

######  Proposition 6.

Fix $\tilde{w}_{0}=s_{\alpha_{1}}\cdots s_{\alpha_{N}}$ a decomposition of the
longest word of $W$. Let
$b,b^{\prime},b^{\prime\prime}\in\mathcal{B}(\infty)$. If
$d_{b,b^{\prime}}^{b^{\prime\prime}}\neq 0$ then
$\Lambda_{\tilde{w}_{0}}(b^{\prime\prime})\leq_{\mathsf{lex}}\Lambda_{\tilde{w}_{0}}(b)+\Lambda_{\tilde{w}_{0}}(b^{\prime})$,
and if equality holds then $d_{b,b^{\prime}}^{b^{\prime\prime}}$ is a power of
$v$.

  * Proof.

We have already observed that
$\Delta(F_{\alpha}^{(r)})=\sum_{i+j=r}v_{\alpha}^{ij}F_{\alpha}^{(i)}\otimes
F_{\alpha}^{(j)}K^{-i}_{\alpha}$, and it follows that for each
$a\in\mathbb{N}^{N}$ we get
$\Delta(F^{(a)})=\sum_{t+u=a}v^{z(t,u)}F^{(t)}\otimes
F^{(u)}K_{\operatorname{\mathsf{wt}}(F^{(t)})}$, with $z(t,u)\in\mathbb{Z}$.
By item (b) of the previous lemma

$\displaystyle\Delta(G(b^{a}))$
$\displaystyle=\Delta(F^{(a)})+\sum_{a^{\prime}>_{\mathsf{lex}}a}y_{a,a^{\prime}}\Delta(F^{(a^{\prime})})$
$\displaystyle=\sum_{t+u=a}v^{z(t,u)}F^{(t)}\otimes
F^{(u)}K_{\operatorname{\mathsf{wt}}(F^{(t)})}+\sum_{t+u>_{\mathsf{lex}}a}y_{a,t+u}v^{z(t,u)}F^{(t)}\otimes
F^{(u)}K_{\operatorname{\mathsf{wt}}(F^{(t)})},$

and using item (c) of the lemma we get that the element in the last display
equals

$\displaystyle\sum_{t+u=a}v^{z(t,u)}G(b^{t})\otimes
G(b^{u})K_{\operatorname{\mathsf{wt}}(b^{t})}+\sum_{t+u>_{\mathsf{lex}}a}y^{\prime}_{t,u}G(b^{t})\otimes
G(b^{u})K_{\operatorname{\mathsf{wt}}(b^{t})},$

where $t,u$ run over the image of $\Lambda_{\tilde{w}_{0}}$, and
$z(t,u)\in\mathbb{Z}$ and $y^{\prime}_{t,u}\in\mathcal{A}$ for each such pair
$(t,u)$. The result follows by taking $b^{a}=b^{\prime\prime}$. ∎

######  Remark 3.

It is possible to take an alternative approach, and define bases for quantum
flag and Schubert varieties as in paragraph LABEL:q-var-bases starting with
the monomial basis instead of the global basis, and in this case we also
obtain an $(S,<_{\mathsf{lex}})$-basis. This is the approach taken by Fang,
Fourier and Littelmann in [FFL2], where they obtain many different monomial
bases for (classical) enveloping algebras and hence many different
degenerations for a larger class of varieties (in the commutative case). The
price to pay is that one loses control over the semigroup parameterizing the
basis i.e. the exponents of the monomial basis. In general it is not known
whether this semigroup is affine (though in some cases it is known that it is
not normal, see the aforementioned article). In our case this is guaranteed by
the fact that this semigroup is the same as that arising from the string
parametrization, which is known to be affine. Thus even in the alternative
approach the relation between the monomial basis and the canonical basis is
essential.

####

Schubert-deg Now let $\Bbbk$ be an arbitrary field with
$\operatorname{char}\Bbbk>2$, or $\operatorname{char}\Bbbk>3$ if the root
system of $\mathfrak{g}$ has an irreducible component of type $G_{2}$, and let
$q\in\Bbbk^{\times}$ be a nonroot of unity. There is a morphism
$\mathcal{A}\longrightarrow\Bbbk$ induced by the assignation $v\mapsto q$,
which makes $\Bbbk$ into an $\mathcal{A}$-bimodule. There is an algebra map
$\kappa:U^{-}_{q}(\mathfrak{g})\longrightarrow\Bbbk\otimes_{\mathcal{A}}U_{\mathcal{A}}^{-}$,
given by sending $F_{\alpha_{i}}\in U_{q}^{-}(\mathfrak{g})$ to
$1\otimes_{\mathcal{A}}F_{\alpha_{i}}$. This map is obviously surjective, and
it respects the weight decomposition of both algebras. By [Jan]*8.24 Remark
(3), the dimension of the weight components of both algebras are given by the
Kostant partition function, and hence they are equal. Thus $\kappa$ is an
isomorphism.

The map $\Bbbk\otimes_{\mathcal{A}}V_{\mathcal{A}}(\lambda)\longrightarrow
V_{q}(\lambda)$ is $U^{-}_{q}(\mathfrak{g})$-linear and sends highest weight
vectors to highest weight vectors, so it is an isomorphism and the global
basis of $V_{\mathcal{A}}(\lambda)$ maps to a basis of $V_{q}(\lambda)$, which
we also call the global basis of $V_{q}(\lambda)$. Also, if we set
$C_{\mathcal{A}}^{+}(\lambda)$ as the $\mathcal{A}$-span of $b_{\lambda}$ for
$b\in\mathcal{B}(\lambda)$ we get an isomorphism
$\Bbbk\otimes_{\mathcal{A}}C_{\mathcal{A}}^{+}(\lambda)\mapsto
C_{q}^{+}(\lambda)$, with the image of the $b_{\lambda}$ forming the dual
basis of the global basis of $V_{q}(\lambda)$. Thus if we set
$\mathcal{O}_{\mathcal{A}}[G/P_{I}]_{w}$ to be the $\mathcal{A}$-span of
$SB_{I}(w)$ inside $\mathcal{O}_{v}[G/P_{I}]_{w}$, we get that the natural map
$\Bbbk\otimes_{\mathcal{A}}\mathcal{O}_{\mathcal{A}}[G/P_{I}]_{w}\longrightarrow\mathcal{O}_{q}[G/P_{I}]_{w}$
is an isomorphism.

######  Theorem 3.

Let $\Bbbk$ be any field and let $q\in\Bbbk^{\times}$ be a nonroot of unity.
Let $I\subset P$ be a set of fundamental weights, let $w\in W$, and let
$\tilde{w}_{0}$ be a reduced decomposition of $w_{0}$ adapted to $w$.

The quantum Schubert variety $\mathcal{O}_{q}[G/P_{I}]_{w}$ degenerates to a
quantum affine toric variety with associated semigroup
$\tilde{S}^{I}_{\tilde{w}_{0},\tilde{w}}$. In particular any quantum Schubert
variety has property $\chi$, finite local dimension, the AS-Cohen-Macaulay
property, and is a maximal order in its skew-field of fractions.

  * Proof.

For each dominant integral weight $\lambda$ and each
$a\in\Lambda_{\tilde{w}_{0}}(\mathcal{B}(\lambda))$, we denote by
$b^{a}_{\lambda}$ the element
$1\otimes_{\mathcal{A}}b_{\lambda}\in\Bbbk\otimes_{\mathcal{A}}\mathcal{O}_{v}[G/P_{I}]_{w}$
where $b=\Gamma_{\tilde{w}_{0}}^{-1}(a)$. With this notation, it follows from
LABEL:coproduct-coefficients-convex that for each pair of dominant integral
weights $\lambda,\lambda^{\prime}$ and each
$a\in\Lambda_{\tilde{w}_{0}}(\mathcal{B}(\lambda)),a^{\prime}\in\Lambda_{\tilde{w}_{0}}(\mathcal{B}(\lambda^{\prime})),a^{\prime\prime}\in\Lambda_{\tilde{w}_{0}}(\mathcal{B}(\lambda+\lambda^{\prime}))$
such that $a^{\prime\prime}\geq_{\mathsf{lex}}a+a^{\prime}$, there exists
$x_{a^{\prime\prime}}\in\mathcal{A}$ such that

$\displaystyle
b^{a}_{\lambda}b^{a^{\prime}}_{\lambda^{\prime}}=\sum_{a^{\prime\prime}\geq
a+a^{\prime}}x_{a^{\prime\prime}}b^{a^{\prime\prime}}_{\lambda+\lambda^{\prime}},$

with $x_{a+a^{\prime}}$ a power of $q$. This implies that the basis
$\\{1\otimes b\mid b\in SB_{I}(w)\\}$ is a
$(\tilde{S}^{I}_{\tilde{w}_{0},\tilde{w}},\leq_{\mathsf{lex}}\leavevmode\nobreak\
)$ basis. Thus by Proposition LABEL:S-ordered-basis-degeneration we get the
degeneration result.

Since $\tilde{S}^{w}_{w_{0},I}$ is a normal semigroup, we know by Proposition
LABEL:properties-of-qatv that the associated graded ring of the quantum
Schubert variety has property $\chi$, finite local dimension and the AS-Cohen-
Macaulay property, which $\mathcal{O}_{q}[G/P_{I}]_{w}$ inherits by Theorem
LABEL:transfer. Also by Proposition LABEL:properties-of-qatv, a quantum affine
toric variety whose underlying semigroup is normal is a maximal order in its
ring of fractions, and it follows from [Mau]*Chapitre IV, Proposition 2.1 and
Chapitre V, Corollaire 2.6 that $\mathcal{O}_{q}[G/P_{I}]_{w}$ is also a
maximal order. ∎

## References

Laurent RIGAL,
Université Paris 13, Sorbonne Paris Cité, LAGA, UMR CNRS 7539, 99 avenue J.-B.
Clément, 93430 Villetaneuse, France; e-mail<EMAIL_ADDRESS>

Pablo ZADUNAISKY,
Universidad CAECE, Departamento de Matemáticas. Av. de Mayo 866 - Buenos
Aires, Argentina. e-mail<EMAIL_ADDRESS>