# The symmetry of finite group schemes, Watanabe type theorem, and the
$a$-invariant of the ring of invariants

Mitsuyasu Hashimoto Partially supported by JSPS KAKENHI Grant number 20K03538
and MEXT Promotion of Distinctive Joint Research Center Program
JPMXP0723833165.

(Dedicated to Professor Mel Hochster and Professor Craig Huneke)

###### Abstract

Let $k$ be a field, and $G$ be a $k$-group scheme of finite type. Let
$G_{\mathrm{ad}}$ be the $k$-scheme $G$ with the adjoint action of $G$. We
call $\lambda_{G,G}=H^{0}(\mathop{\operator@font Spec}\nolimits
k,e^{*}(\omega_{G_{\mathrm{ad}}}))$ the Knop character of $G$, where
$e:\mathop{\operator@font Spec}\nolimits k\rightarrow G_{\mathrm{ad}}$ is the
unit element, and $\omega_{G_{\mathrm{ad}}}$ is the $G$-canonical module. We
prove that $\lambda_{G,G}$ is trivial in the following cases: (1) $G$ is
finite, and $k[G]^{*}$ is a symmetric algebra; (2) $G$ is finite and étale;
(3) $G$ is finite and constant; (4) $G$ is smooth and connected reductive; (5)
$G$ is abelian; (6) $G$ is finite, and the identity component $G^{\circ}$ of
$G$ is linearly reductive; (7) $G$ is finite and linearly reductive. Let $V$
be a small $G$-module of dimension $n<\infty$. We assume that $\lambda_{G,G}$
is trivial. Let $H=\mathbb{G}_{m}$ be the one-dimensional torus, and let $V$
be of degree one as an $H$-module so that $S=\mathop{\operator@font
Sym}\nolimits V^{*}$ is a $\tilde{G}$-algebra generated by degree one
elements, where $\tilde{G}=G\times H$. We set $A=S^{G}$. Then we have (i)
$\omega_{A}\cong\omega_{S}^{G}$ as $(H,A)$-modules; (ii) $a(A)\leq-n$ in
general, where $a(A)$ denotes the $a$-invariant. Moreover, the following are
equivalent: (1) The action $G\rightarrow\text{\sl{GL}}(V)$ factors through
$\text{\sl{SL}}(V)$; (2) $\omega_{S}\cong S(-n)$ as $(\tilde{G},S)$-modules;
(3) $\omega_{S}\cong S$ as $(G,S)$-modules; (4) $\omega_{A}\cong A(-n)$ as
$(H,A)$-modules; (5) $A$ is quasi-Gorenstein; (6) $A$ is quasi-Gorenstein and
$a(A)=-n$; (7) $a(A)=-n$. This partly generalizes recent results of
Liedtke–Yasuda arXiv:2304.14711v2 and Goel–Jeffries–Singh arXiv:2306.14279v1.

0002020 Mathematics Subject Classification. Primary 13A50; Secondary 14L15,
16T05. Key Words and Phrases. unimodular Hopf algebra, $a$-invariant,
canonical module, invariant subring, finite group scheme, small action.

##  1 Introduction

Let $k$ be a field, $V$ a finite-dimensional $k$-vector space, and $G$ a
finite subgroup of $\text{\sl{GL}}(V)$. Let $S=\mathop{\operator@font
Sym}\nolimits V^{*}=k[V]$, and $A=S^{G}$. Hochster and Eagon proved that in
non-modular case (that is, the case that the order $|G|$ of $G$ is not
divisible by the characteristic of $k$), $A$ is Cohen–Macaulay. K.-i. Watanabe
proved that in non-modular case, $G\subset\text{\sl{SL}}(V)$ if and only if
$G$ does not have a pseudo-reflection and $A$ is Gorenstein [Wat1, Wat2].
Since then, his result has been generalized by several authors. Fleischmann
and Woodcock [FW] and Braun [Bra] proved that if $G\subset\text{\sl{GL}}(V)$
is a finite subgroup without pseudo-reflection, then $A$ is quasi-Gorenstein
(or equivalently, $\omega_{A}\cong A$) if and only if
$G\subset\text{\sl{SL}}(V)$.

It has been known that the condition that the finite group $G$ does not have a
pseudo-reflection sometimes can be generalized to more general $G$. The
condition is replaced by the condition that $\pi:V=\mathop{\operator@font
Spec}\nolimits S\rightarrow\mathop{\operator@font Spec}\nolimits A=V/\\!/G$ is
a principal $G$-bundle off codimension two or more, and called an almost
principal bundle or quasi-torsor [Has4, C-R], and we call this condition ‘$V$
is small.’ Namely, we say that $V$ is small if there exist some open subset
$W$ of $\mathop{\operator@font Spec}\nolimits A$ and $G$-stable open subset
$U$ of $\pi^{-1}(W)$ such that $\mathop{\operator@font
codim}\nolimits(V\setminus U,V)\geq 2$, $\mathop{\operator@font
codim}\nolimits(V/\\!/G\setminus W,V/\\!/G)\geq 2$, and $\pi:U\rightarrow W$
is a principal $G$-bundle (or a $G$-torsor) in the sense that $\pi$ is
faithfully flat, and $\Phi:G\times U\rightarrow U\times_{W}U$ given by
$\Phi(g,u)=(gu,u)$ is an isomorphism. Note that if $G$ is a finite constant
group, then $V$ is small if and only if $G\subset\text{\sl{GL}}(V)$, and $G$
does not have a pseudo-reflection.

Knop [Kno] pointed out that the equivalence $\omega_{A}\cong A\iff
G\subset\text{\sl{SL}}(V)$ is not true any more even if $G$ is a
(disconnected) reductive group over an algebraically closed field of
characteristic zero, and the action is small. Letting $\lambda_{\mathrm{ad}}$
be the top exterior power of $\mathop{\operator@font Lie}\nolimits(G)^{*}$,
the dual of the adjoint representation, the triviality of
$\det_{V}\otimes\lambda_{\mathrm{ad}}^{*}$ was important [Kno, Satz 2]. Note
that $\lambda_{\mathrm{ad}}$ is trivial if $G$ is finite, and we can recover
Watanabe’s original result.

We define $\lambda_{G,G}=H^{0}(\mathop{\operator@font Spec}\nolimits
k,e^{*}(\omega_{G_{\mathrm{ad}}}))$, and call it the Knop character of $G$,
where $e:\mathop{\operator@font Spec}\nolimits k\rightarrow G_{\mathrm{ad}}$
is the unit element, $\omega_{G_{\mathrm{ad}}}$ is the $G$-equivariant
canonical module of $G_{\mathrm{ad}}$. If, moreover, $G$ is a normal closed
subgroup scheme of another affine $k$-group scheme $\tilde{G}$ of finite type,
then $\lambda_{G,G}$ is a character of $\tilde{G}$, and we denote it by
$\lambda_{\tilde{G},G}$. Note that
$\lambda_{G,G}\cong\lambda_{\mathrm{ad}}^{*}$, if $G$ is $k$-smooth. By [Has4,
(11.22)], it is easy to see that if $V$ is small, then $\omega_{A}\cong A(a)$
if and only if $\omega_{S}=S\otimes\det_{V^{*}}\cong
S(a)\otimes_{k}\lambda_{\tilde{G},G}$ if and only if
$\det_{V}\cong\lambda_{G,G}$ as $G$-modules, and $a=-n$. If, moreover,
$\lambda_{G,G}\cong k$, then $A$ is quasi-Gorenstein if and only if
$G\subset\text{\sl{SL}}(V)$, and if these conditions are satisfied, then
$a(A)=-n$. So it is natural to ask, when $\lambda_{G,G}$ is trivial. In [Has4,
(11.21)], it is pointed out that if $G$ is finite and linearly reductive,
étale, or connected reductive, then $\lambda_{G,G}$ is trivial, but
$\lambda_{G,G}$ is nontrivial if $k$ is a field of characteristic not two and
$G=O(2)$.

In this paper, we discuss when $\lambda_{G,G}$ is trivial, assuming that $G$
is finite (but not étale). It is well-known that the group algebra $kG$ is
symmetric [SY, Example IV.2.6]. A finite dimensional $k$-Hopf algebra is
Frobenius in general [SY, Theorem VI.3.6]. In general, a finite dimensional
$k$-Hopf algebra $H$ is not symmetric even if $H$ is cocommutative, or
equivalently, $H=k[G]^{*}$ for some finite $k$-group scheme $G$, see [LS, p.
85]. We prove that $\lambda_{G,G}$ is trivial if and only if the notions of
the left integral and the right integral agree in $H=k[G]^{*}$. The latter
condition is called the unimodular property of $H$. As the square $s_{H}^{2}$
of the antipode $s_{H}$ of $H$ is the identity, $H$ is unimodular if and only
if $H$ is a symmetric algebra, see [Hum, Rad].

As an application of the $G$-triviality of $\lambda_{G,G}$, we prove the
following.

######  Theorem 3.5.

Let $k$ be a field, $G$ be an affine $k$-group scheme of finite type, and $V$
be a small $G$-module of dimension $n<\infty$. We assume that $\lambda_{G,G}$
is trivial. Let $H=\mathbb{G}_{m}$ be the one-dimensional torus, and let $V$
be of degree one as an $H$-module so that $S=\mathop{\operator@font
Sym}\nolimits V^{*}$ is a $\tilde{G}$-algebra generated by degree one
elements, where $\tilde{G}=G\times H$. We set $A=S^{G}$. Then we have

  1. (i)

$\omega_{A}\cong\omega_{S}^{G}$ as $(H,A)$-modules;

  2. (ii)

$a(A)\leq-n$ in general, where $a(A)$ denotes the $a$-invariant.

Moreover, the following are equivalent:

  1. (1)

The action $G\rightarrow\text{\sl{GL}}(V)$ factors through
$\text{\sl{SL}}(V)$;

  2. (2)

$\omega_{S}\cong S(-n)$ as $(\tilde{G},S)$-modules;

  3. (3)

$\omega_{S}\cong S$ as $(G,S)$-modules;

  4. (4)

$\omega_{A}\cong A(-n)$ as $(H,A)$-modules;

  5. (5)

$A$ is quasi-Gorenstein;

  6. (6)

$A$ is quasi-Gorenstein and $a(A)=-n$;

  7. (7)

$a(A)=-n$.

For the case that $G$ is finite and constant, the theorem was proved (in a
stronger form) by Goel, Jeffries, and Singh [GJS]. Note that they do not
require that the action of $G$ on $V$ is small. They proved that $a(A)\leq
a(S)=-n$ in general. They also proved that the equality $a(A)=-n$ holds if and
only if the image of $G\rightarrow\text{\sl{GL}}(V)$ is a subgroup of
$\text{\sl{SL}}(V)$ without pseudo-reflections for the case that $G$ is finite
and constant [GJS, Proposition 4.1, Theorem 4.4]. It is interesting to ask if
these are true for any finite group scheme $G$. Note also that the equivalence
(1)$\Leftrightarrow$(5) for the case that $G$ is finite linearly reductive
(but not necessarily constant) was proved recently by Liedtke and Yasuda [LY].

Acknowledgment. The author thanks K. Goel, A. Singh, K.-i. Watanabe, and T.
Yasuda for valuable communications. He is also grateful to A. Masuoka for
valuable advice.

##  2 Preliminaries

### L

et $k$ be a field, $f:\tilde{G}\rightarrow H$ be a homomorphism between affine
$k$-group schemes of finite type with $G=\mathop{\operator@font Ker}\nolimits
f$. Let $\mathcal{F}(\tilde{G})$ be the category of $\tilde{G}$-schemes
separated of finite type over $k$. For
$(h_{Z}:Z\rightarrow\mathop{\operator@font Spec}\nolimits
k)\in\mathcal{F}(\tilde{G})$, the $\tilde{G}$-dualizing complex of $Z$ (or
better, of $h_{Z}$) is $h_{Z}^{!}(k)$ by definition, and we denote it by
${\mathbb{I}}_{Z}={\mathbb{I}}_{Z}(G)$, where $(-)^{!}$ denotes the twisted
inverse [Has2]. The $\tilde{G}$-canonical module $\omega_{Z}$ is the lowest
nonzero cohomology group of ${\mathbb{I}}_{Z}$. It is a coherent
$\tilde{G}$-module. If $Z=\mathop{\operator@font Spec}\nolimits B$ is affine,
$H^{0}(Z,\omega_{Z})$ is denoted by $\omega_{B}$, and is called the
$\tilde{G}$-equivariant canonical module of $B$. When we forget the
$\tilde{G}$-structure, ${\mathbb{I}}_{Z}$ is the dualizing complex of the
scheme $Z$ without the $\tilde{G}$-action [Has2, (31.17)].

### A

morphism $\varphi:X\rightarrow Y$ of $\tilde{G}$-schemes of finite type over
$k$ is called a $\tilde{G}$-enriched principal $G$-bundle if $G$ acts
trivially on $Y$, $\varphi$ is faithfully flat, and the morphism $\Phi:G\times
X\rightarrow X\times_{Y}X$ given by $\Phi(g,x)=(gx,x)$ is an isomorphism. As
$G$ is affine, flat, and Gorenstein over $\mathop{\operator@font
Spec}\nolimits k$, $\varphi$ is affine, flat, and Gorenstein.

### L

et $X$ be a scheme and $U$ its open subset. We say that $U$ is $n$-large if
$\mathop{\operator@font codim}\nolimits(X\setminus U,X)\geq n+1$, where we
regard that the codimension of the empty set in $X$ is $\infty\geq n+1$.

######  Definition 2.1 (cf. [Has4, (10.2)]).

A diagram of $\tilde{G}$-schemes of finite type

(1)
$\textstyle{X}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{\rho}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{Y}$

is called a $\tilde{G}$-enriched $n$-almost rational principal $G$-bundle if
(1) $G$ acts trivially on $Y$; (2) $j$ is an open immersion, and $j(V)$ is
$n$-large in $Y$; (3) $i$ is an open immersion, and $i(U)$ is $n$-large in
$X$; (4) $\rho:U\rightarrow V$ is a principal $G$-bundle. That is, $\rho$ is
faithfully flat, and $\Phi:G\times U\rightarrow U\times_{V}U$ given by
$\Phi(n,u)=(nu,u)$ is an isomorphism.

### I

n what follows, $1$-large and $1$-almost will simply be called large and
almost, respectively. A $\tilde{G}$-morphism $\varphi:X\rightarrow Y$ is said
to be a $\tilde{G}$-enriched $n$-almost principal $G$-bundle with respect to
$U$ and $V$ if $U$ is a $\tilde{G}$-stable open subset of $X$, $V$ is an
$H$-stable open subset of $Y$, and the diagram (1) is a $\tilde{G}$-enriched
$n$-almost rational principal $G$-bundle. We say that a $\tilde{G}$-morphism
$\varphi:X\rightarrow Y$ is a $\tilde{G}$-enriched $n$-almost principal
$G$-bundle if it is so with respect to $U$ and $V$ for some $U$ and $V$.

######  Lemma 2.2.

Let $\varphi:X\rightarrow Y$ be a $\tilde{G}$-enriched almost principal
$G$-bundle between $\tilde{G}$-schemes of finite over $k$. Assume that $X$ is
normal, and that $\mathcal{O}_{Y}\rightarrow(\varphi_{*}\mathcal{O}_{X})^{G}$
is an isomorphism. y Let $\mathop{\operator@font Ref}\nolimits(\tilde{G},X)$
be the category of coherent $(\tilde{G},\mathcal{O}_{X})$-modules which are
reflexive as $\mathcal{O}_{X}$-modules, and Let $\mathop{\operator@font
Ref}\nolimits(H,Y)$ be the category of coherent $(H,\mathcal{O}_{Y})$-modules
which are reflexive as $\mathcal{O}_{Y}$-modules. Then we have: The functor
$\mathcal{G}:\mathop{\operator@font
Ref}\nolimits(\tilde{G},X)\rightarrow\mathop{\operator@font
Ref}\nolimits(H,Y)$ given by
$\mathcal{G}(\mathcal{M})=(\varphi_{*}\mathcal{M})^{G}$ is an equivalence, and
$\mathcal{F}:\mathop{\operator@font
Ref}\nolimits(H,Y)\rightarrow\mathop{\operator@font
Ref}\nolimits(\tilde{G},X)$ given by
$\mathcal{F}(\mathcal{N})=(\varphi^{*}\mathcal{N})^{**}$ is its quasi-inverse,
where $(-)^{**}$ is the double dual.

###### Proof.

Follows immediately from [Has4, (10.13),(11.3)]. A short proof for the case
that everything is affine can be found in [HK, (2.4)]. ∎

### L

et $X$ be a $\tilde{G}$-scheme, and $L$ be a $\tilde{G}$-module. Let
$h_{X}:X\rightarrow\mathop{\operator@font Spec}\nolimits k$ be the structure
map. Then for a quasi-coherent $(\tilde{G},\mathcal{O}_{X})$-module
$\mathcal{M}$, we denote $\mathcal{M}\otimes_{\mathcal{O}_{X}}h_{X}^{*}L$ by
$\mathcal{M}\otimes_{k}L$. Note that $G$ is a normal closed subgroup scheme of
$\tilde{G}$. So $\tilde{G}$ acts on $G$ by the adjoint action. We denote this
scheme by $G_{\mathrm{ad}}$. Let $e:\mathop{\operator@font Spec}\nolimits
k\rightarrow G_{\mathrm{ad}}$ be the unit element. It is a $\tilde{G}$-stable
closed immersion. We denote the $\tilde{G}$-module
$H^{0}(\mathop{\operator@font Spec}\nolimits
k,e^{*}(\omega_{G_{\mathrm{ad}}}))$ by $\lambda_{\tilde{G},G}$, and we call it
the Knop character of $G$ (enriched by $\tilde{G}$). If $G$ is $k$-smooth,
then $\lambda_{G,G}\cong\det_{\mathfrak{g}}^{*}$ by [Has2, (28.11)], where
$\mathfrak{g}=\mathop{\operator@font Lie}\nolimits G$ is the adjoint
representation of $G$, and $\det$ denotes the top exterior power. Its dual
$\det_{\mathfrak{g}}=\lambda_{G,G}^{*}$ is denoted by $\lambda_{\mathrm{ad}}$
in [Kno], and played an important role in studying Gorenstein property of
invariant subrings [Kno, Satz 2].

######  Lemma 2.3.

Let $\varphi:X=\mathop{\operator@font Spec}\nolimits T\rightarrow
Y=\mathop{\operator@font Spec}\nolimits B$ be a $\tilde{G}$-enriched almost
principal $G$-bundle which is also a morphism in $\mathcal{F}(\tilde{G})$ with
$X$ and $Y$ affine normal. Then
$\omega_{T}=(T\otimes_{B}\omega_{B})^{**}\otimes_{k}\lambda_{\tilde{G},G}$,
where $(-)^{**}=\mathop{\operator@font
Hom}\nolimits_{T}(\mathop{\operator@font Hom}\nolimits_{T}(-,T),T)$ is the
double dual. We also have that
$\omega_{B}=(\omega_{T}\otimes_{k}\lambda_{\tilde{G},G}^{*})^{G}$. In
particular, if moreover, $\lambda_{\tilde{G},G}\cong k$, then
$\omega_{T}\cong(T\otimes_{B}\omega_{B})^{**}$ and
$\omega_{B}\cong\omega_{T}^{G}$.

###### Proof.

This is a special case of [Has4, (11.22)]. ∎

### L

et $\Lambda$ be a finite-dimensional $k$-algebra. We say that $\Lambda$ is
Frobenius if ${}_{\Lambda}\Lambda\cong D(\Lambda_{\Lambda})$ as left
$\Lambda$-modules, where $D=\mathop{\operator@font Hom}\nolimits_{k}(-,k)$.
This is equivalent to say that there is a nondegenerate bilinear form
$\beta:\Lambda\times\Lambda\rightarrow k$ such that $\beta(ac,b)=\beta(a,cb)$.
If, moreover, such a $\beta$ is symmetric, we say that $\Lambda$ is symmetric.
This is equivalent to say that the bimodule ${}_{\Lambda}\Lambda_{\Lambda}$ is
isomorphic to $D({}_{\Lambda}\Lambda_{\Lambda})$.

### L

et $\Gamma$ be a finite dimensional $k$-Hopf algebra. We define

$\textstyle\int^{l}_{\Gamma}:=\\{x\in\Gamma^{*}\mid\forall
y\in\Gamma^{*}\;yx=\epsilon(y)x\\},$

where $\epsilon:\Gamma^{*}\rightarrow k$ is given by
$\epsilon(y)=y(1_{\Gamma})$. An element of $\int^{l}_{\Gamma}$ is called a
left integral of $\Gamma$.

### N

ote that

$\textstyle\int^{l}_{\Gamma}=(\Gamma^{*})^{\Gamma}=\\{\psi\in\Gamma^{*}\mid\omega_{\gamma^{*}}(\psi)=\psi\otimes
1\\}=\mathop{\operator@font Hom}\nolimits_{\Gamma}(\Gamma,k).$

Indeed, if $\gamma_{1},\ldots,\gamma_{n}$ is a $k$-basis of $\Gamma$ and
$\gamma^{*}_{1},\ldots,\gamma^{*}_{n}$ is its dual basis, then the comodule
structure $\omega_{\Gamma^{*}}$ of $\Gamma^{*}$ is given by
$\omega_{\Gamma^{*}}(\alpha)=\sum_{i=1}^{n}\gamma^{*}_{i}\alpha\otimes\gamma_{i}$
for $\alpha\in\Gamma^{*}$. In other words,
$\omega(\alpha)=\sum_{(\alpha)}\alpha_{(0)}\otimes\alpha_{(1)}$ is given by
$\sum_{(\alpha)}\langle\beta,\alpha_{(1)}\rangle\alpha_{(0)}=\beta\alpha$ for
$\beta\in\Gamma^{*}$. So $\omega(\psi)=\psi\otimes 1$ is equivalent to say
that $\rho\psi=\epsilon(\rho)\psi$ for $\rho\in\Gamma^{*}$, as desired.

### W

e also define $\int^{r}_{\Gamma}=\\{x\in\Gamma^{*}\mid\forall
y\in\Gamma^{*}\;xy=x\varepsilon(y)\\}$, and an element of $\int^{r}_{\Gamma}$
is called a right integral of $\Gamma$. It is known that
$\dim_{k}\int^{l}_{\Gamma}=\dim_{k}(\int^{r}_{\Gamma})^{*}=1$ [Swe1, Corollary
5.1.6]. If $\int^{l}_{\Gamma}=\int^{r}_{\Gamma}$, then we say that
$\Gamma^{*}$ is unimodular. Radford proved [Rad] that $\Gamma^{*}$ is a
symmetric algebra if and only if $\Gamma^{*}$ is unimodular and
$\mathcal{S}^{2}$ is an inner automorphism of $\Gamma^{*}$, where
$\mathcal{S}$ is the antipode of $\Gamma$. Suzuki [Suz] constructed an example
of a finite dimensional unimodular $k$-Hopf algebra which is not symmetric.

######  Lemma 2.4.

Let $G$ be a finite $k$-group scheme, $H$ an affine $k$-group scheme of finite
type, and $\tilde{G}=G\times H$ is the direct product. Let $\Gamma=k[G]$ be
the coordinate ring of $G$. Then the following are equivalent.

  1. (1)

$\Gamma^{*}$ is symmetric.

  2. (2)

$\Gamma^{*}$ is unimodular.

  3. (3)

$\lambda_{\tilde{G},G}\cong k$ as $\tilde{G}$-modules.

  4. (4)

$\lambda_{G,G}\cong k$ as $G$-modules.

###### Proof.

As $\Gamma^{*}$ is cocommutative,
$s^{2}=\mathord{\operator@font{id}}_{\Gamma^{*}}$, where $s$ is the antipode
of $\Gamma^{*}$. By [Hum, Theorem 1,2] (see also [Rad]),
(1)$\Leftrightarrow$(2) holds.

We prove (2)$\Rightarrow$(3). Note that $\Gamma^{*}$ is a $(G,k[G])$-module.
In other words, $\Gamma^{*}$ is a $\Gamma$-Hopf module. Let
$\zeta:\Gamma^{*}\rightarrow\Gamma^{*}\otimes_{k}\Gamma$ be the map given by
$\zeta(\gamma)=\sum_{(\gamma)}\gamma_{(0)}(\mathcal{S}\gamma_{(1)})\otimes\gamma_{(2)}$,
where $\mathcal{S}$ is the antipode of $\Gamma$. By the proof of [Swe1,
Theorem 4.1.1], $\zeta$ is injective and $\mathop{\operator@font
Im}\nolimits\zeta=\int^{l}_{\Gamma}\otimes k[G]$. Now let us consider the same
map $\zeta:k[G_{\mathrm{ad}}]^{*}\rightarrow k[G_{\mathrm{ad}}]^{*}\otimes
k[G_{\mathrm{ad}}]$. It is easy to see that this is a
$(G,k[G_{\mathrm{ad}}])$-homomorphism. Note also that $\int^{l}_{\Gamma}$ is a
$G$-submodule of $k[G_{\mathrm{ad}}]$ and $k[G_{r}]$. As we have
$k[G_{\mathrm{ad}}]^{*}\cong\omega_{k[G_{\mathrm{ad}}]}$ by the duality of
finite morphisms, see [Has2, (27.8)]. As
$\int^{l}_{\Gamma}=\int^{r}_{\Gamma}$, we have that $\int^{l}_{\Gamma}$ as the
submodule of $k[G_{\mathrm{ad}}]$ is also $G$-trivial (isomorphic to $k$).
Hence $\zeta$ induces an isomorphism $k[G_{\mathrm{ad}}]^{*}\cong
k[G_{\mathrm{ad}}]$ of $(G,k[G_{\mathrm{ad}}])$-modules. As $H$ acts trivially
on $G_{\mathrm{ad}}$, the isomorphism is that of
$(\tilde{G},k[G_{\mathrm{ad}}])$-modules. Pulling back this isomorphism by the
unit element $e:\mathop{\operator@font Spec}\nolimits k\rightarrow
G_{\mathrm{ad}}$, we get $\lambda_{\tilde{G},G}\cong k$, as desired.

(3)$\Rightarrow$(4) is trivial.

(4)$\Rightarrow$(2). The argument above shows that
$k[G_{\mathrm{ad}}]^{*}\cong\lambda^{\prime}\otimes k[G_{\mathrm{ad}}]$, where
$\lambda^{\prime}$ is $\int^{l}_{\Gamma}$ as a $G$-submodule of
$k[G_{\mathrm{ad}}]^{*}$. As $\int^{l}_{\Gamma}$ is trivial as a $G$-submodule
of $k[G_{l}]$, we have that $\lambda^{\prime}$ agrees with $\int^{l}_{\Gamma}$
as a $G$-submodule of $k[G_{r}]$. The assumption (4) means
$k\cong\lambda\cong\lambda^{\prime}$. So
$\int^{l}_{\Gamma}\subset\int^{r}_{\Gamma}$. As we know that both
$\int^{l}_{\Gamma}$ and $\int^{r}_{\Gamma}$ are one-dimensional, we have that
$\Gamma^{*}$ is unimodular. ∎

##  3 Main theorem

### L

et $S$ be a $k$-algebra of finite type on which $G$ acts. Let $A=S^{G}$ be the
ring of invariants. If the canonical map $\mathop{\operator@font
Spec}\nolimits S\rightarrow\mathop{\operator@font Spec}\nolimits A$ is an
almost principal $G$-bundle, then we say that the $G$-action on $S$ is small.
If $V$ is a $G$-module and $S=k[V]=\mathop{\operator@font Sym}\nolimits V^{*}$
is small, then we say that the representation $V$ of $G$ is small. If $G$ is a
finite (constant) group, then $V$ is small if and only if the action is
faithful, and $G\subset\text{\sl{GL}}(V)$ does not have a pseudo-reflection.
Letting each element of $V^{*}$ of degree one, $S=\mathop{\operator@font
Sym}\nolimits V^{*}$ is a graded $G$-algebra. So letting $H=\mathbb{G}_{m}$
and $\tilde{G}=G\times H$, we have that $S$ is a $\tilde{G}$-algebra.

######  Lemma 3.1 (cf. [Has4, Remark 11.21]).

In the following cases, we have that the Knop character
$\lambda_{\tilde{G},G}$ is trivial as $\tilde{G}$-modules.

  1. (1)

$G$ is finite, and $k[G]^{*}$ is a symmetric algebra;

  2. (2)

$G$ is finite and étale;

  3. (3)

$G$ is finite and constant;

  4. (4)

$G$ is smooth and connected reductive;

  5. (5)

$G$ is abelian;

  6. (6)

$G$ is finite, and the identity component $G^{\circ}$ of $G$ is linearly
reductive;

  7. (7)

$G$ is finite and linearly reductive.

###### Proof.

By Lemma 2.4, (1) is already proved, and it suffices to show that
$\lambda_{G,G}\cong k$ for (2)–(7).

For the case that (2), (3), or (4) is assumed, $G$ is $k$-smooth, and hence
$\lambda_{G,G}=\det_{\mathfrak{g}^{*}}=\bigwedge^{\mathrm{top}}\mathfrak{g}^{*}$.
As the $0^{\mathrm{th}}$ exterior power is always trivial, (4) has been
proved. The assertion (3) is a special case of (2) (also, direct proofs are
well-known, see for example, [SY, Example IV.2.6]).

We prove (4). We may assume that $k$ is algebraically closed. Let $T$ be a
maximal torus of $G$. As $\lambda_{G,G}=\det_{\mathfrak{g}}^{*}$ is one-
dimensional, it suffices to show that $\det_{\mathfrak{g}}$ is trivial as a
$T$-module. We have
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha}$,
where $\Phi$ is the set of roots (that is, nonzero weights of $\mathfrak{g}$).
It is known that $\dim\mathfrak{g}_{\alpha}=1$ for each $\alpha\in\Phi$, and
$\Phi=-\Phi$. Thus $\det_{\mathfrak{g}}$ has weight
$0+\sum_{\alpha\in\Phi}\alpha=0$.

We prove (5). If $G$ is abelian, then the action of $G$ on $G_{\mathrm{ad}}$
is trivial. So $\omega_{G_{\mathrm{ad}}}$ and $\lambda_{G,G}$ are $G$-trivial.

We prove (6). We may assume that $k$ is algebraically closed of characteristic
$p>0$. By [Swe2, (3.11)], $G^{\circ}=\mathop{\operator@font Spec}\nolimits kM$
for some abelian $p$-group $M$. Note that $\pi^{0}(G)=G_{\operator@font{red}}$
is a closed subgroup scheme, and is a constant finite group. Note that
$G=G^{\circ}\rtimes G_{\operator@font{red}}$ is a semidirect product. As
$G_{\operator@font{red}}$ acts on $G^{\circ}$ by the adjoint action, it acts
on the character group $\chi(G^{\circ})=M$. As the action is that of groups,
it fixes the unit element $1_{M}$ of $M$. As a $G^{\circ}$-module,
$k[G^{\circ}_{r}]=k[G^{\circ}_{l}]$ is decomposed into the sum of one-
dimensional $G^{\circ}$-modules as $\bigoplus_{m\in M}k\cdot m$. Note that
$k\cdot m$ is isomorphic to $k$ if and only if $m=1_{M}$, and that
$\int^{r}_{k[G^{\circ}]}=\int^{l}_{k[G^{\circ}]}$ is generated by the
projection $\pi:k[G]\rightarrow k$ given by $\pi(1_{M})=1$ and $\pi(m)=0$ for
$m\in M\setminus\\{1_{M}\\}$. As $gm\neq 1_{M}$ if $m\neq 1_{M}$ and
$g1_{M}=1_{M}$ for any $g\in G_{\operator@font{red}}$, we have that $g\pi=\pi$
for any $g\in G_{\operator@font{red}}$, where $(g\pi)(m)=\pi(g^{-1}(m))$. This
shows that $\lambda_{G,G^{\circ}}\cong k$. As $G^{\circ}$ is a $G$-stable
(closed and) open neighborhood of the unit element $e$ in $G_{\mathrm{ad}}$,
we have that $\lambda_{G,G}=\lambda_{G,G^{\circ}}\cong k$, as desired.

We prove (7). By [Has3, Lemma 2.2], $G^{\circ}$ is linearly reductive. By (6),
the assertion is clear now. ∎

######  Example 3.2 (cf. [Kno, p. 51]).

$\lambda_{G,G}$ is not $G$-trivial in general, even if $k$ is an algebraically
closed field of characteristic zero, and $G$ is $k$-smooth. Let
$k=\mathbb{C}$, and consider

$G=O_{2}=\\{A\in\text{\sl{GL}}_{2}(\mathbb{C})\mid{}^{t}AA=E_{2}\\},$

where $E_{2}$ is the identity matrix. Then the Lie algebra $\mathfrak{g}$ of
$G$ is

$\\{B\in\mathop{\mathfrak{gl}}\nolimits_{2}(\mathbb{C})=\mathop{\operator@font
Mat}\nolimits_{2}(\mathbb{C})\mid{}^{t}B+B=O\\},$

on which $G$ acts by the action $(A,B)\mapsto AB{}^{t}A$. It is easy to see
that the action is nontrivial, and hence $\lambda_{G,G}=\mathfrak{g}^{*}$ is
also nontrivial.

######  Example 3.3.

$\lambda_{G,G}$ is not $G$-trivial in general, even if $G$ is finite. Consider
the restricted Lie algebra (see [Jac, (V.7)] for definition) $L$ over a field
$k$ of characteristic $p>0$ with the basis $e,f$ with the relations $[f,e]=e$,
$f^{p}=f$, and $e^{p}=0$. Take the restricted universal enveloping algebra $V$
of $L$. Letting each element of $x\in L$ primitive (i.e., $\Delta(x)=x\otimes
1+1\otimes x$), $V$ is a $p^{2}$-dimensional cocommutative Hopf algebra which
is not unimodular, see [LS, p. 85]. Letting $G=\mathop{\operator@font
Spec}\nolimits V^{*}$, we have that $\lambda_{G,G}$ is not trivial by Lemma
2.4.

######  Lemma 3.4.

Let $k$ be a field, $G$ and $H$ be affine $k$-group schemes of finite type,
and $\tilde{G}=G\times H$. Let $S$ be a $\tilde{G}$-algebra, and assume that
the action of $G$ on $S$ is small. We assume that $S$ is normal, and
$\lambda_{G,G}$ is trivial. Let $L$ be an $(H,A)$-module which is projective
as an $A$-module. Then the following are equivalent:

  1. (1)

$\omega_{S}\cong S\otimes_{A}L$ as $(\tilde{G},S)$-modules;

  2. (2)

$\omega_{A}\cong L$ as $(H,A)$-modules,

where the action of $G$ on $L$ is trivial.

###### Proof.

(1)$\Rightarrow$(2). By Lemma 2.3,
$\omega_{A}\cong\omega_{S}^{G}\cong(S\otimes_{A}L)^{G}\cong A\otimes_{A}L\cong
L$, since $L$ is $G$-trivial and $A$-flat.

(2)$\Rightarrow$(1). We have $\omega_{S}^{G}\cong\omega_{A}\cong L$. Applying
the functor $(S\otimes_{A}-)^{**}$, which is the quasi-inverse of
$(-)^{G}:\mathop{\operator@font
Ref}\nolimits(\tilde{G},S)\rightarrow\mathop{\operator@font
Ref}\nolimits(H,A)$, we get isomorphisms

$\omega_{S}\cong(S\otimes_{A}\omega_{S}^{G})^{**}\cong(S\otimes_{A}L)^{**}\cong
S\otimes_{A}L$

of $(\tilde{G},S)$-modules. ∎

######  Theorem 3.5.

Let $k$ be a field, $G$ be an affine $k$-group scheme of finite type, and $V$
be a small $G$-module of dimension $n<\infty$. We assume that $\lambda_{G,G}$
is trivial. Let $H=\mathbb{G}_{m}$ be the one-dimensional torus, and let $V$
be of degree one as an $H$-module so that $S=\mathop{\operator@font
Sym}\nolimits V^{*}$ is a $\tilde{G}$-algebra generated by degree one
elements, where $\tilde{G}=G\times H$. We set $A=S^{G}$. Then we have

  1. (i)

$\omega_{A}\cong\omega_{S}^{G}$ as $(H,A)$-modules;

  2. (ii)

$a(A)\leq-n$ in general, where $a(A)$ denotes the $a$-invariant.

Moreover, the following are equivalent:

  1. (1)

The action $G\rightarrow\text{\sl{GL}}(V)$ factors through
$\text{\sl{SL}}(V)$;

  2. (2)

$\omega_{S}\cong S(-n)$ as $(\tilde{G},S)$-modules;

  3. (3)

$\omega_{S}\cong S$ as $(G,S)$-modules;

  4. (4)

$\omega_{A}\cong A(-n)$ as $(H,A)$-modules;

  5. (5)

$A$ is quasi-Gorenstein;

  6. (6)

$A$ is quasi-Gorenstein and $a(A)=-n$;

  7. (7)

$a(A)=-n$.

###### Proof.

The assertion (i) is clear by Corollary 2.3. We prove (ii). We have an
$(H,A)$-linear isomorphism $\omega_{A}\rightarrow\omega_{S}^{G}$ by
assumption, and $\omega_{S}^{G}\subset\omega_{S}=S\otimes_{k}\det_{V}$. So
$a(A)\leq a(S)=-n$ in general. The equality holds only if $\det_{V}$ is
trivial. Namely, we have (7)$\Rightarrow$(1).

(1) is equivalent to say that $\det_{V}\cong k(-n)$. Combining this with the
fact $\omega_{S}\cong S\otimes_{k}\det_{V}$, we get (1)$\Rightarrow$(2).

(2)$\Rightarrow$(3) is trivial.

(3)$\Rightarrow$(1). $S\otimes_{k}\det_{V}\cong\omega_{S}\cong S$ as
$(G,S)$-modules. So

$\textstyle\det_{V}\cong S/S_{+}\otimes_{S}(S\otimes_{k}\det_{V})\cong
S/S_{+}\otimes_{S}\omega_{S}\cong S/S_{+}\otimes_{S}S\cong S/S_{+}\cong k$

as $G$-modules. This shows (1).

(2)$\Rightarrow$(4). $\omega_{A}\cong\omega_{S}^{G}\cong S(-n)^{G}\cong A(-n)$
as $(H,A)$-modules.

(4)$\Rightarrow$(6)$\Rightarrow$(5) is trivial.

(5)$\Rightarrow$(3). By assumption, $\omega_{A}$ is projective. As $A$ is
positively graded and $\omega_{A}$ is a graded finitely generated module of
rank one, we have that $\omega_{A}\cong A(a)$ for some $a\in{\mathbb{Z}}$. By
Lemma 3.4, we have that $\omega_{S}\cong S\otimes_{A}A(a)\cong S(a)$ as
$(\tilde{G},S)$-modules. Forgetting the grading, we have that $\omega_{S}\cong
S$ as $(G,S)$-modules, as desired.

(6)$\Rightarrow$(7) is trivial. ∎

######  Remark 3.6.

Goel–Jeffries–Singh [GJS] proved better theorems than Theorem 3.5 for the case
that $G$ is finite and constant. They proved the inequality $a(A)\leq-n$
without assuming that the action is small. They also prove there that
$a(A)=-n$ implies that the action is small (and hence
$G\subset\text{\sl{SL}}(V)$), see [GJS, Proposition 4.1, Theorem 4.4]. The
author does not know if these are true for a general finite group scheme $G$.

The equivalence (1)$\Leftrightarrow$(5) for the case that $G$ is finite and
constant was first proved by Fleischmann and Woodcock [FW] and Braun [Bra].
The author proved that $\omega_{S}^{G}\cong\omega_{A}$ if $G$ is finite
linearly reductive, without assuming that the action is small [Has2, (32.4)].
The equivalence (1)$\Leftrightarrow$(5) for the case that $G$ is finite
linearly reductive was proved by Liedtke–Yasuda [LY, Proposition 4.7] ($A$ is
strongly $F$-regular this case, and hence quasi-Gorenstein is equivalent to
Gorenstein there).

######  Example 3.7.

We give an example of higher-dimensional $G$. Let $m,n,t$ be positive integers
such that $2\leq t\leq m\leq n$. Let $W_{1}=k^{n}$, $W_{2}=k^{m}$,
$E=k^{t-1}$, and $G=\text{\sl{GL}}(E)$. We consider that $G$ acts on $E$ as a
vector representation, while the actions of $G$ on $W_{1}$ and $W_{2}$ are
trivial. We set $V=(\mathop{\operator@font
Hom}\nolimits_{k}(E,W_{2})\oplus\mathop{\operator@font
Hom}\nolimits_{k}(W_{1},E))^{*}$, $S=\mathop{\operator@font Sym}\nolimits V$,
and $A=S^{G}$. We define $X=V^{*}=\mathop{\operator@font
Hom}\nolimits(E,W_{2})\times\mathop{\operator@font
Hom}\nolimits(W_{1},E)=\mathop{\operator@font Spec}\nolimits
S=E^{n}\times(E^{*})^{m}$, and $Y=\mathop{\operator@font Spec}\nolimits
A=X/\\!/G$. The quotient map $\pi:X\rightarrow Y$ is identified with the map
$\Pi:X\rightarrow Y_{t}$ given by $(\varphi,\psi)\mapsto\varphi\circ\psi$,
where $Y_{t}=\\{\rho\in\mathop{\operator@font
Hom}\nolimits(W_{1},W_{2})\mid\mathop{\operator@font rank}\nolimits\rho<t\\}$
is the determinantal variety, see [DP]. Note that $\Pi$ is a
$\text{\sl{GL}}(W_{1})\times G\times\text{\sl{GL}}(W_{2})$-enriched almost
principal $G$-bundle, see [Has1]. So by the theorem, we have that $a(A)\leq
a(S)=-(m+n)(t-1)$, and the equality holds if and only if $A$ is Gorenstein.
Note also that the usual grading of $A=k[\mathop{\operator@font
Hom}\nolimits(W_{1},W_{2})^{*}]/I_{t}$, where $I_{t}$ is the determinantal
ideal, is the one such that each element of $\mathop{\operator@font
Hom}\nolimits(W_{1},W_{2})^{*}$ is of degree one. However, the grading used
here is the one which is inherited from the grading of $S$, and each element
of $\mathop{\operator@font Hom}\nolimits(W_{1},W_{2})^{*}$ is of degree two.
For the case that $k$ is of characteristic zero, Lascoux’s resolution [Las]
tells us that $a(A)=2(-mn+n(m-t+1))=-2n(t-1)\leq-(m+n)(t-1)=a(S)$, doubling
the degree to adopt our grading inherited from $S$. Being a graded ASL over a
distributive lattice, $A$ is Cohen–Macaulay, and the Hilbert series of $A$ is
independent of $k$, see [BH]. So $a(A)$ is also independent of $k$, and we
always have $a(A)=-2n(t-1)$. So $a(A)=a(S)$ if and only if $m=n$. This shows
that $A$ is Gorenstein if and only if $m=n$, and this is the well-known
theorem by Svanes [Sva].

######  Example 3.8.

Let $k$ be an algebraically closed field of characteristic $p>0$, and $\ell$
be an odd prime number which does not divide $p-1$. Let

$G=\left.\left\\{\begin{bmatrix}t&\alpha\\\
0&1\end{bmatrix}\,\right|\,t\in\boldsymbol{\mu}_{\ell},\;\alpha\in\boldsymbol{\alpha}_{p}\right\\},$

where $\boldsymbol{\alpha}_{p}=\mathop{\operator@font Spec}\nolimits
k[a]/(a^{p})$ is the first Frobenius kernel of the additive group
$\mathbb{G}_{a}=\mathop{\operator@font Spec}\nolimits k[a]=\mathbb{A}^{1}$,
and $\boldsymbol{\mu}_{\ell}=\mathop{\operator@font Spec}\nolimits
k[T]/(T^{\ell}-1)\subset\text{\sl{GL}}_{1}$. Note that $G$ acts on the vector
representation $W=k^{2}$ in a natural way. Let $V=W\oplus W^{*}$. It is easy
to see that $V$ is small and $G\subset\text{\sl{SL}}(V)$. It is also easy to
see that $\lambda_{G,G}=\mathop{\operator@font soc}\nolimits
k[(\boldsymbol{\alpha}_{p})_{\mathrm{ad}}]^{*}=(\mathop{\operator@font
soc}\nolimits k[a]/(a^{p}))^{*}=(ka^{p-1})^{*}$. With the adjoint action, we
have $\sigma_{t}\cdot a=t^{-2}a$, where $\sigma=\begin{bmatrix}t&0\\\
0&1\end{bmatrix}$. By assumption, $\lambda_{G,G}$ is nontrivial. By [Has4,
Corollary 11.22], $A=S^{G}$ is not quasi-Gorenstein, where
$A=k[V]=\mathop{\operator@font Sym}\nolimits V^{*}$.

### L

et $G$ be a finite $k$-group scheme, $H$ a $k$-group scheme of finite type,
and set $\tilde{G}=G\times H$. Let $S$ be a $\tilde{G}$-algebra, and
$A=S^{G}$. In [C-R], the trace map $\mathop{\operator@font
Tr}\nolimits_{S/A}:S\rightarrow A$ is defined. Let $\delta_{G}:k[G]\rightarrow
k$ be a non-zero left integral (that is,
$\delta_{G}\in\int^{l}_{k[G]}\setminus\\{0\\}$). This is equivalent to say
that $\delta_{G}\in\mathop{\operator@font Hom}\nolimits_{G}(k[G_{l}],k)$. For
any $G$-algebra $S$, let $\mathop{\operator@font
Tr}\nolimits_{S/A}:S\rightarrow S^{\prime}$ be the composite

$S\xrightarrow{\omega_{S}}S^{\prime}\otimes_{k}k[G]\xrightarrow{1_{S^{\prime}}\otimes\delta_{G}}S^{\prime}\otimes_{k}k=S^{\prime},$

where $S^{\prime}$ is the $A$-module $S$ with the trivial $G$-action. By [C-R,
Definition-Proposition 3.6], the image of $\mathop{\operator@font
Tr}\nolimits_{S/A}$ is contained in $A=S^{G}$, and hence the map
$\mathop{\operator@font Tr}\nolimits_{S/A}:S\rightarrow A$ is induced. It is
easy to see that $\mathop{\operator@font Tr}\nolimits_{S/A}$ is $A$-linear.

### A

ssume that the Hopf algebra $k[G]^{*}$ is unimodular. Then $\delta_{G}$ is
also a right integral. That is, $\delta_{G}:k[G_{r}]\rightarrow k$ is
$G$-linear (note that $k[G_{r}]$ is a $k[G]$-comodule algebra letting the
coproduct $\Delta:k[G_{r}]\rightarrow k[G_{r}]\otimes k[G_{r}]$ the coaction).
As $\omega_{S}:S\rightarrow S^{\prime}\otimes_{k}k[G_{r}]$ is also $G$-linear,
we have that $\mathop{\operator@font Tr}\nolimits_{S/A}:S\rightarrow A$ is
$(G,A)$-linear. Moreover, $\delta_{G}$ is $H$-linear, since $H$ acts trivially
on $G_{r}$. Letting the action of $H$ on $S^{\prime}$ be the same as that on
$S$, we have that $\omega_{S}:S\rightarrow S^{\prime}\otimes_{k}k[G_{r}]$ is
also $H$-linear. Thus $\mathop{\operator@font Tr}\nolimits_{S/A}$ is
$(\tilde{G},A)$-linear.

######  Theorem 3.9.

Let $S$ be a $k$-algebra of finite type. Let $G$ be a finite $k$-group scheme,
$H$ be a $k$-group scheme of finite type, and $\tilde{G}=G\times H$. Assume
that $\tilde{G}$ acts on $S$. If $k[G]^{*}$ is symmetric and either

  1. (1)

The map $\mathop{\operator@font Spec}\nolimits
S\rightarrow\mathop{\operator@font Spec}\nolimits A$ is a $\tilde{G}$-enriched
principal $G$-bundle; or

  2. (2)

The action of $G$ on $S$ is small, and $S$ satisfies the $(S_{2})$-condition,

then $\zeta:S\rightarrow\mathop{\operator@font Hom}\nolimits_{A}(S,A)$
($s\mapsto(t\mapsto\mathop{\operator@font Tr}\nolimits_{S/A}(st))$) is an
isomorphism of $(\tilde{G},S)$-modules.

###### Proof.

This is [C-R, Corollary 3.13] except that we need to prove that the map
$\zeta$ is $\tilde{G}$-linear. This is done in the discussion above. ∎

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Mitsuyasu Hashimoto

Department of Mathematics

Osaka Metropolitan University

Sumiyoshi-ku, Osaka 558–8585, JAPAN

e-mail<EMAIL_ADDRESS>