# Indecomposability of graded modules
over a graded ring

Mitsuyasu Hashimoto Partially supported by JSPS KAKENHI Grant number 20K03538
and MEXT Promotion of Distinctive Joint Research Center Program
JPMXP0619217849. Yuntian Yang

###### Abstract

Let $R=\bigoplus_{i\geq 0}R_{i}$ be a Noetherian commutative non-negatively
graded ring such that $(R_{0},\mathfrak{m}_{0})$ is a Henselian local ring.
Let $\mathfrak{m}$ be its unique graded maximal ideal
$\mathfrak{m}_{0}+\bigoplus_{i>0}R_{i}$. Let $T$ be a module-finite (non-
commutative) graded $R$-algebra. Let $T\mathop{\operator@font grmod}\nolimits$
denote the category of finite graded left $T$-modules, and $M\in
T\mathop{\operator@font grmod}\nolimits$. Then the following are equivalent:
(1) $\hat{M}$ is an indecomposable $\hat{T}$-module, where $\widehat{(-)}$
denotes the $\mathfrak{m}$-adic completion; (2) $M_{\mathfrak{m}}$ is an
indecomposable $T_{\mathfrak{m}}$-module; (3) $M$ is an indecomposable
$T$-module; (4) $M$ is indecomposable as a graded $T$-module. As a corollary
we prove that for two finite graded left $T$-modules $M$ and $N$, the
following are equivalent: (1) If $M=M_{1}\oplus\cdots\oplus M_{s}$ and
$N=N_{1}\oplus\cdots\oplus N_{t}$ are decompositions into indecomposable
objects in $T\mathop{\operator@font grmod}\nolimits$, then $s=t$, and there
exist some permutation $\sigma\in\mathfrak{S}_{s}$ and integers
$d_{1},\ldots,d_{s}$ such that $N_{i}\cong M_{\sigma i}(d_{i})$, where
$-(d_{i})$ denotes the shift of degree; (2) $M\cong N$ as $T$-modules; (3)
$M_{\mathfrak{m}}\cong N_{\mathfrak{m}}$ as $T_{\mathfrak{m}}$-modules; (4)
$\hat{M}\cong\hat{N}$ as $\hat{T}$-modules. As an application, we compare the
FFRT property of rings of characteristic $p$ in the graded sense and in the
local sense.

0002020 Mathematics Subject Classification. Primary 16W50; Secondary 13A02.
Key Words and Phrases. graded ring, indecomposable module

##  1 Introduction

In many cases, a theory in Noetherian local rings has its graded version or
∗version. Moreover, such a graded version often determines the ring-theoretic
property of the graded ring $A$ as a non-graded ring.

We give an easy illustrative example. Let $A=\bigoplus_{i\geq 0}$ be a
finitely generated positively graded (that is, $A_{0}=k$) commutative algebra
over a field $k$. Then we can define a homogeneous system of parameters of $A$
to be a sequence $x_{1},\ldots,x_{d}$ of homogeneous elements of positive
degrees such that $\dim_{k}A/(x_{1},\ldots,x_{d})<\infty$, and the length of
the sequence $d$ is small as possible among such sequences. It is well-known
that a sequence $x_{1},\ldots,x_{r}$ of homogeneous elements of positive
degree is a homogeneous system of parameters if and only if their images in
the local ring $A_{\mathfrak{m}}$ is a system of parameters in the local
sense. In particular, if $x_{1},\ldots,x_{d}$ is a homogeneous system of
parameters, then we have $d=\dim A_{\mathfrak{m}}$. What is interesting is
$\dim A=\dim A_{\mathfrak{m}}$ in fact [BH, Section 1.5]. Moreover, there is a
homogeneous system of parameters which is a regular sequence if and only if
$A_{\mathfrak{m}}$ is Cohen–Macaulay if and only if $A$ is Cohen–Macaulay
[MR].

Some important results in this direction can be found in the textbook [BH].

In this paper, we compare the indecomposability of finitely generated module
$M$ over a module-finite algebra over a graded ring along this line, and we
get the following. Namely, we have

######  Theorem 3.6.

Let $R=\bigoplus_{i\geq 0}R_{i}$ be a Noetherian commutative graded ring such
that $(R_{0},\mathfrak{m}_{0})$ is Henselian local,
$R_{+}=\bigoplus_{i>0}R_{i}$, $\mathfrak{m}=R_{+}+\mathfrak{m}_{0}$, and $T$ a
$\mathbb{Z}$-graded module-finite (non-commutative) $R$-algebra. Let $M$ be a
finitely generated graded left $T$-module. Then the following are equivalent.

  1. (1)

$\hat{M}$ is indecomposable as a $\hat{T}$-module, where $\widehat{(-)}$
denotes the $\mathfrak{m}$-adic completion.

  2. (2)

$M_{\mathfrak{m}}$ is indecomposable as a $T_{\mathfrak{m}}$-module.

  3. (3)

$M$ is indecomposable as a $T$-module.

  4. (4)

$M$ is indecomposable as a graded $T$-module.

As a corollary, we prove

######  Corollary 3.7.

Let $M$ and $N$ be objects in $T\mathop{\operator@font grmod}\nolimits$, the
category of finitely generated graded left $T$-modules. Then the following are
equivalent.

  1. (1)

If $M=M_{1}\oplus\cdots\oplus M_{s}$ and $N=N_{1}\oplus\cdots\oplus N_{t}$ are
decompositions into indecomposable objects in $T\mathop{\operator@font
grmod}\nolimits$, then $s=t$, and there exist some permutation
$\sigma\in\mathfrak{S}_{s}$ and integers $d_{1},\ldots,d_{s}$ such that
$N_{i}\cong M_{\sigma i}(d_{i})$, where $-(d_{i})$ denotes the shift of
degree.

  2. (2)

$M\cong N$ as $T$-modules.

  3. (3)

$M_{\mathfrak{m}}\cong N_{\mathfrak{m}}$ as $T_{\mathfrak{m}}$-modules.

  4. (4)

$\hat{M}\cong\hat{N}$ as $\hat{T}$-modules.

As an application, we prove a comparison theorem for the finite
$F$-representation type (FFRT for short) property of an $F$-finite non-
negatively graded ring $R=\bigoplus_{i\geq 0}R_{i}$ such that
$(R_{0},\mathfrak{m}_{0})$ is an $F$-finite Henselian local ring of a prime
characteristic $p$ in the graded sense and the FFRT property of the
$\mathfrak{m}$-adic completion of $R$, where
$\mathfrak{m}=\mathfrak{m}_{0}+\bigoplus_{i>0}R_{i}$ (Corollary 3.8). See
Corollary 3.8 for the definition of FFRT. This property for rings of
characteristic $p$ was defined by K. E. Smith and M. Van den Bergh [SVdB], and
has been studied extensively [AK, DQ, HB, HO, Shi1, Shi2].

Acknowledgment. The essential part of this joint work was done in 2019, and is
recorded as [Y]. In [Y], Theorem 3.6 and Corollary 3.7 for the case that $R=T$
was proved. The authors are grateful to Professor Osamu Iyama for valuable
discussions.

##  2 Preliminaries

### F

or a ring $A$, we denote the set of units of $A$ by $A^{\times}$. We say that
$A$ is local if $A\setminus A^{\times}$ is an additive subgroup of $A$. This
is equivalent to say that $A\neq 0$, and $A\setminus A^{\times}$ is closed
under addition. If so, $A\setminus A^{\times}$ is the unique maximal left
ideal of $A$. It is also the unique maximal right ideal of $A$, and agrees
with the radical $\mathop{\operator@font rad}\nolimits A$. Note that $A$ is
local if and only if $A/\mathop{\operator@font rad}\nolimits A$ is a division
ring, see [L, (19.1)].

### L

et $B=\bigoplus_{i\in{\mathbb{Z}}}B_{i}$ be a graded ring, and
$M=\bigoplus_{i\in{\mathbb{Z}}}M_{i}$ be a graded left module. A left graded
submodule $N$ is said to be ∗maximal if $N$ is a maximal element of the set
$\\{N\subset M\mid\text{$N$ is a graded submodule of $M$ and $N\neq M$}\\}$.
The intersection of all the ∗maximal graded submodules is denoted by
${}^{*}\mathop{\operator@font rad}\nolimits M$. If $f:M\rightarrow M^{\prime}$
is a homomorphism of graded left $B$-modules, then
$f({}^{*}\mathop{\operator@font rad}\nolimits
M)\subset{}^{*}\mathop{\operator@font rad}\nolimits M^{\prime}$. In
particular, if $b\in B_{i}$, then $({}^{*}\mathop{\operator@font rad}\nolimits
B)b\subset{}^{*}\mathop{\operator@font
rad}\nolimits(B(i))=({}^{*}\mathop{\operator@font rad}\nolimits B)(i)$, and
${}^{*}\mathop{\operator@font rad}\nolimits B$ is a two-sided ideal.

######  Lemma 2.1.

Let $B$ be as above. Then any nonzero finitely generated left graded module
$M$ has a ∗maximal submodule. In particular,
$M\neq{}^{*}\mathop{\operator@font rad}\nolimits M$.

###### Proof.

Let $\Gamma$ be the set of graded proper submodules of $M$. As $M$ is nonzero,
$\Gamma$ is nonempty. If $\Omega$ is a non-empty chain of $\Gamma$, then
$\sum_{N\in\Omega}N\neq M$, and $\sum_{N\in\Omega}N\in\Gamma$. Indeed, if
$\sum_{N\in\Omega}N=M$, then there exists some $N$ that contains all the
generators of $M$. This implies $N=M\in\Omega\subset\Gamma$ and this is a
contradiction. So by Zorn’s lemma, $\Gamma$ has a maximal element, and this is
what we wanted to prove. ∎

######  Corollary 2.2.

Let $B$ be as above, and assume that $B\neq 0$. Then $B$ has a left ∗maximal
ideal. In particular, $B\neq{}^{*}\mathop{\operator@font rad}\nolimits B$. ∎

######  Lemma 2.3.

Let $B$ be as above, $i\in{\mathbb{Z}}$, and $b\in B_{i}$. Then the following
are equivalent.

  1. (1)

For any $c\in B_{-i}$, $1+cb\in B_{0}^{\times}$.

  2. (2)

For any $c\in B_{-i}$, $1+cb\in B^{\times}$.

  3. (3)

$b\in{}^{*}\mathop{\operator@font rad}\nolimits B$.

  4. (4)

$b\in{}^{*}\mathop{\operator@font rad}\nolimits B^{\mathop{\operator@font
op}\nolimits}$, where $B^{\mathop{\operator@font op}\nolimits}$ is the
opposite ring of $B$.

  5. (5)

For any $c\in B_{-i}$, $1+bc\in B^{\times}$.

  6. (6)

For any $c\in B_{-i}$, $1+bc\in B_{0}^{\times}$.

###### Proof.

It is obvious that $B_{0}^{\times}\subset B_{0}\cap B^{\times}$. Conversely,
if $x\in B_{0}\cap B^{\times}$ and $xy=1$ with $y=\sum y_{j}$ ($y_{j}\in
B_{j}$), then $xy_{0}=1$, and hence $x\in B_{0}^{\times}$. So
(1)$\Leftrightarrow$(2) and (5)$\Leftrightarrow$(6) are obvious.

If $b\notin{}^{*}\mathop{\operator@font rad}\nolimits B$, then
$b\notin\mathfrak{m}$ for some ∗maximal left ideal $\mathfrak{m}$ of $B$. So
$B=Bb+\mathfrak{m}$, and $1+cb\in\mathfrak{m}$ for some $c\in B_{-i}$. Thus we
have (2)$\Rightarrow$(3). Similarly, (5)$\Rightarrow$(4) is proved.

(3)$\Rightarrow$(2). If $b\in{}^{*}\mathop{\operator@font rad}\nolimits B$ and
$c\in B_{-i}$, then $cb\in(\mathop{\operator@font rad}\nolimits B)_{0}$. So
$1+cb\in B_{0}$ is not contained in any ∗maximal ideal of $B$, and hence
$B(1+cb)=B$. So there exists some $d\in B_{0}$ such that $d(1+cb)=1$. If
$d\in\mathfrak{m}$ for some ∗maximal ideal $\mathfrak{m}$ of $B$, then
$d+dcb\in\mathfrak{m}$, since $dcb\in{}^{*}\mathop{\operator@font
rad}\nolimits B$. This shows that $1=d(1+cb)\in\mathfrak{m}$, which is absurd.
So $d\in B^{\times}$, and hence $1+cb\in B^{\times}$.

(3)$\Rightarrow$(5). As $b\in{}^{*}\mathop{\operator@font rad}\nolimits B$ and
${}^{*}\mathop{\operator@font rad}\nolimits B$ is a two-sided ideal,
$bc\in{}^{*}\mathop{\operator@font rad}\nolimits B$. So by the assertion
(3)$\Rightarrow$(2), which we have already proved, we have that $1+bc\in
B^{\times}$.

(4)$\Rightarrow$(2) follows from (3)$\Rightarrow$(5) above, applied to the
graded ring $B^{\mathop{\operator@font op}\nolimits}$. ∎

######  Lemma 2.4.

Let $B$ be as above. Then the following are equivalent.

  1. (1)

$B_{0}$ is local.

  2. (2)

$B$ has a unique ∗maximal left ideal.

  3. (3)

$B$ has a unique ∗maximal right ideal.

  4. (4)

For each $i\in{\mathbb{Z}}$, $B_{i}\setminus B^{\times}$ is an additive
subgroup of $B_{i}$.

###### Proof.

(1)$\Rightarrow$(2). If $B=0$, then $B_{0}=0$, and $B_{0}$ is not local. So
$B\neq 0$, and $B$ has a ∗maximal left ideal by Lemma 2.1. If $B$ has two
∗maximal left ideals $\mathfrak{m}_{1}$ and $\mathfrak{m}_{2}$ with
$\mathfrak{m}_{1}\neq\mathfrak{m}_{2}$, then there exists some
$a_{1}\in\mathfrak{m}_{1}\cap B_{0}$ and $a_{2}\in\mathfrak{m}_{2}\cap B_{0}$
such that $a_{1}+a_{2}=1$, and $B_{0}$ is not local.

(2)$\Rightarrow$(4). Let $\mathfrak{m}$ be the unique ∗maximal left ideal. If
$b\in B_{i}\setminus B^{\times}$, then $b\in Bb\subset\mathfrak{m}$, and
$B_{i}\setminus B^{\times}=B_{i}\cap\mathfrak{m}$.

(4)$\Rightarrow$(1). This is trivial.

(1)$\Leftrightarrow$(3) follows from (1)$\Leftrightarrow$(2), which already
has been proved, applied to $B^{\mathop{\operator@font op}\nolimits}$. ∎

### W

e say that $B$ is ∗local if $B$ satisfies the equivalent conditions in Lemma
2.4.

######  Lemma 2.5 (graded Nakayama’s lemma).

Let $B$ be as above, and $J={}^{*}\mathop{\operator@font rad}\nolimits B$. If
$M$ is a finitely generated graded left $B$-module and $JM=M$, then $M=0$.

###### Proof.

For any homogeneous element $m$ of $M$, $x\mapsto xm$ is a graded homomorphism
of left $B$-modules $B\rightarrow M(-\deg m)$, and
${}^{*}\mathop{\operator@font rad}\nolimits B$ is mapped to
${}^{*}\mathop{\operator@font rad}\nolimits M$ by this map. This shows that
$JM\subset{}^{*}\mathop{\operator@font rad}\nolimits M\subset M$. By
assumption, we have that ${}^{*}\mathop{\operator@font rad}\nolimits M=M$. By
Lemma 2.1, we have that $M=0$. ∎

##  3 Main results

######  Lemma 3.1.

Let $(A,\mathfrak{m})$ be a Henselian local ring, and $\Lambda$ a module-
finite (non-commutative) $A$-algebra. Then $\Lambda$ is semi-perfect, and the
category of finite left $\Lambda$-modules is Krull–Schmidt.

For Henselian local rings, see [LW, Section A.3] and [M].

###### Proof.

Let $V$ be a simple left $\Lambda$-module. Let
$\bar{\Lambda}=\Lambda/\mathop{\operator@font rad}\nolimits\Lambda$. As
$\mathfrak{m}\Lambda\subset\mathop{\operator@font rad}\nolimits\Lambda$ (by
Nakayama’s lemma applied to the finite $A$-module $\Lambda$), we have that
$\bar{\Lambda}$ is a finite-dimensional $A/\mathfrak{m}$-algebra with the
trivial radical. So $\bar{\Lambda}$ is semi-simple. As $V$ is a simple
$\bar{\Lambda}$-module, there exists some idempotent $\bar{e}$ of
$\bar{\Lambda}$ such that $V=\bar{\Lambda}\bar{e}$. By [LW, Theorem A.30],
$\bar{e}$ lifts to an idempotent $e$ of $\Lambda$. Let $P=\Lambda e$. Then
$P/JP=(\Lambda/J)e=\bar{\Lambda}\bar{e}=V$. By Nakayama’s lemma, $JP$ is the
unique maximal submodule of $P$, and hence the canonical surjective map
$P\rightarrow P/JP\cong V$ is a projective cover by [Kra, Lemma 3.6]. So
$\Lambda$ is semi-perfect, and the category of left $\Lambda$-modules is
Krull–Schmidt, see [Kra, Proposition 4.1]. ∎

### L

et $R=\bigoplus_{i>0}R_{i}$ be a Noetherian ${\mathbb{Z}}_{\geq 0}$-graded
commutative ring such that $(R_{0},\mathfrak{m}_{0})$ is Henselian local. We
set $\mathfrak{m}=R_{+}+\mathfrak{m}_{0}$, where $R_{+}=\bigoplus_{i>0}R_{i}$.

######  Lemma 3.2.

Let $M$ be a finite graded $R$-module. Then $\bigcap_{r\geq
1}\mathfrak{m}^{r}M=0$.

###### Proof.

Let $N=\bigcap_{r\geq 1}\mathfrak{m}^{r}M$. Note that $N$ is a finite graded
$R$-submodule of $M$. By Artin–Rees lemma, there exists some $c\geq 1$ such
that

$N=N\cap\mathfrak{m}^{c+1}M=\mathfrak{m}(N\cap\mathfrak{m}^{c}M)\subset\mathfrak{m}N\subset
N.$

As $\mathfrak{m}N=N$, $N=0$ by graded Nakayama’s lemma. ∎

Let $T=\bigoplus_{i\in{\mathbb{Z}}}T_{i}$ be a graded $R$-algebra which is a
finite $R$-module. Let $J$ be the ∗radical ${}^{*}\mathop{\operator@font
rad}\nolimits T$ of $T$.

######  Lemma 3.3.

$J^{r}\subset\mathfrak{m}T\subset J$ for some $r\geq 1$.

###### Proof.

If $\mathfrak{m}T\not\subset J$, then there exists some ∗maximal left ideal
$\mathfrak{n}$ of $T$ such that $\mathfrak{n}+\mathfrak{m}T=T$. By graded
Nakayama’s lemma, $\mathfrak{n}=T$, and this is absurd. So
$\mathfrak{m}T\subset J$.

To prove that $J^{r}\subset\mathfrak{m}T$ for some $r\geq 1$, we may assume
that $\mathfrak{m}=0$. Then $T$ is a finite-dimensional
$R_{0}/\mathfrak{m}_{0}$-algebra. As $J^{r}=J^{r+1}$ for some $r\geq 1$,
$J^{r}=0$ by graded Nakayama’s lemma again. ∎

######  Lemma 3.4.

Let $e$ be an idempotent of $\hat{T}$, where $\hat{T}$ is the
$\mathfrak{m}$-adic completion of $T$. If $e\in J\hat{T}=\hat{J}$, then $e=0$.
In particular, if $e$ is an idempotent of $T$ such that $e\in J$, then $e=0$.

###### Proof.

By Lemma 3.3, we can take $r\geq 1$ such that $J^{r}\subset\mathfrak{m}T$. So
$e=e^{r}\in\mathfrak{m}\hat{T}$. As $e=e^{n}\in\mathfrak{m}^{n}\hat{T}$ for
any $n\geq 1$, so $e\in\bigcap_{n\geq 1}\mathfrak{m}^{n}\hat{T}=0$. Since
$T\rightarrow\hat{T}$ is injective, the last assertion follows immediately. ∎

######  Lemma 3.5.

If $T$ is ∗local, then $J=\mathop{\operator@font rad}\nolimits
T_{0}+\bigoplus_{i\neq 0}T_{i}$, and $T/J\cong T_{0}/\mathop{\operator@font
rad}\nolimits T_{0}$ is a division ring.

###### Proof.

Replacing $T$ by $T/J$, we may assume that $J=0$. Then $\mathfrak{m}T\subset
J=0$. Replacing $R$ by $R/\mathfrak{m}$, we may assume that $R=R_{0}$ is a
field concentrated in degree zero. Then $T$ is a finite-dimensional
$R$-algebra. If $i\neq 0$, $a\in R_{i}$, and $a$ is a unit, then $R_{ir}\neq
0$ for $r\in{\mathbb{Z}}$, and $R$ cannot be finite-dimensional. So
$T_{i}\subset{}^{*}\mathop{\operator@font rad}\nolimits T$ for $i\neq 0$. On
the other hand, we have

$({}^{*}\mathop{\operator@font rad}\nolimits T)_{0}=T_{0}\setminus
T^{\times}=T_{0}\setminus T_{0}^{\times}=\mathop{\operator@font rad}\nolimits
T_{0}.$

So ${}^{*}\mathop{\operator@font rad}\nolimits T=\mathop{\operator@font
rad}\nolimits T_{0}+\bigoplus_{i\neq 0}T_{i}$. Hence
$T/{}^{*}\mathop{\operator@font rad}\nolimits T\cong
T_{0}/\mathop{\operator@font rad}\nolimits T_{0}$. As $T_{0}$ is local, this
is a division ring. ∎

### L

et $T\mathop{\operator@font grmod}\nolimits$ denote the category of finitely
generated graded left $T$-modules. We say that $M\in T\mathop{\operator@font
grmod}\nolimits$ is ∗indecomposable if it is an indecomposable object of
$T\mathop{\operator@font grmod}\nolimits$. For $M\in\mathop{\operator@font
grmod}\nolimits T$, the endomorphism ring of $M$ as an object of
$T\mathop{\operator@font grmod}\nolimits$ is $E_{0}=(\mathop{\operator@font
End}\nolimits_{T}M)_{0}$, the degree zero component of
$E=\mathop{\operator@font End}\nolimits_{T}M$, the endomorphism ring of $M$ as
a (non-graded) $T$-module. Note that $E_{0}$ is finite as an $R_{0}$-module.
As we assume that $R_{0}$ is Henselian local, $E_{0}$ is semi-perfect, and
hence the additive category $T\mathop{\operator@font grmod}\nolimits$ is
Krull–Schmidt by Lemma 3.1. In particular, $M$ is ∗indecomposable if and only
if $E$ is ∗local, that is, $E_{0}$ is local.

######  Theorem 3.6.

Let $R=\bigoplus_{i\geq 0}R_{i}$ be a Noetherian commutative graded ring such
that $(R_{0},\mathfrak{m}_{0})$ is Henselian local,
$R_{+}=\bigoplus_{i>0}R_{i}$, $\mathfrak{m}=R_{+}+\mathfrak{m}_{0}$, and $T$ a
$\mathbb{Z}$-graded module-finite (non-commutative) $R$-algebra. Let $M$ be a
finitely generated graded left $T$-module. Then the following are equivalent.

  1. (1)

$\hat{M}$ is indecomposable as a $\hat{T}$-module, where $\widehat{(-)}$
denotes the $\mathfrak{m}$-adic completion.

  2. (2)

$M_{\mathfrak{m}}$ is indecomposable as a $T_{\mathfrak{m}}$-module.

  3. (3)

$M$ is indecomposable as a $T$-module.

  4. (4)

$M$ is indecomposable as a graded $T$-module.

###### Proof.

(1)$\Rightarrow$(2). If $M_{\mathfrak{m}}\cong N_{1}\oplus N_{2}$ as a
$T_{\mathfrak{m}}$ module with $N_{1}\neq 0$ and $N_{2}\neq 0$, then taking
the completion, $\hat{M}\cong\hat{N}_{1}\oplus\hat{N}_{2}$, and
$\hat{N}_{1}\neq 0$ and $\hat{N}_{2}\neq 0$, and $\hat{M}$ is not
indecomposable. If $M_{\mathfrak{m}}=0$, then $\hat{M}=0$, and $\hat{M}$ is
not indecomposable.

(2)$\Rightarrow$(3). If $M=0$, then $M_{\mathfrak{m}}=0$. Assume that there is
a decomposition $M=M_{1}\oplus M_{2}$ with $M_{1}\neq 0$ and $M_{2}\neq 0$ in
the category of $T$-modules. Let $m_{1}$ and $m_{2}$ be non-zero elements of
$M_{1}$ and $M_{2}$, respectively. Then it is easy to see that there exists
some $r\geq 1$ such that both $m_{1}$ and $m_{2}$ are nonzero in
$M/\mathfrak{m}^{r}M$ by Lemma 3.2. Then $M_{1}/\mathfrak{m}^{r}M_{1}\neq 0$
and $M_{2}/\mathfrak{m}^{r}M_{2}\neq 0$. This shows
$(M_{1})_{\mathfrak{m}}\neq 0$ and $(M_{2})_{\mathfrak{m}}\neq 0$. This
contradicts the indecomposability of $M_{\mathfrak{m}}$, since
$M_{\mathfrak{m}}=(M_{1})_{\mathfrak{m}}\oplus(M_{2})_{\mathfrak{m}}$.

(3)$\Rightarrow$(4). Set $E=\mathop{\operator@font End}\nolimits_{T}M$. Then
$E\neq 0$, and $E$ does not have a non-trivial idempotent. So $E_{0}\neq 0$,
and $E_{0}$ does not have a non-trivial idempotent. As the endomorphism ring
of $M$ as an object of $T\mathop{\operator@font grmod}\nolimits$ is $E_{0}$,
we have that $M$ is indecomposable as an object of $T\mathop{\operator@font
grmod}\nolimits$.

(4)$\Rightarrow$(1). Let $\hat{e}$ be an idempotent of
$\hat{E}=\mathop{\operator@font End}\nolimits_{\hat{T}}\hat{M}$. If
$\hat{e}\in\hat{J}$, then $\hat{e}=0$ by Lemma 3.4. If $1-\hat{e}\in\hat{J}$,
then $1-\hat{e}=0$. So if $\hat{e}$ is nontrivial, then the image of $\hat{e}$
in $\hat{E}/\hat{J}\cong E_{0}/J_{0}$ must be still nontrivial, but this is
absurd, since $E_{0}/J_{0}$ is a division ring where there is no nontrivial
idempotent. ∎

######  Corollary 3.7.

Let $M$ and $N$ be objects in $T\mathop{\operator@font grmod}\nolimits$, the
category of finitely generated graded left $T$-modules. Then the following are
equivalent.

  1. (1)

If $M=M_{1}\oplus\cdots\oplus M_{s}$ and $N=N_{1}\oplus\cdots\oplus N_{t}$ are
decompositions into indecomposable objects in $T\mathop{\operator@font
grmod}\nolimits$, then $s=t$, and there exists some permutation
$\sigma\in\mathfrak{S}_{s}$ and integers $d_{1},\ldots,d_{s}$ such that
$N_{i}\cong M_{\sigma i}(d_{i})$, where $-(d_{i})$ denotes the shift of
degree.

  2. (2)

$M\cong N$ as $T$-modules.

  3. (3)

$M_{\mathfrak{m}}\cong N_{\mathfrak{m}}$ as $T_{\mathfrak{m}}$-modules.

  4. (4)

$\hat{M}\cong\hat{N}$ as $\hat{T}$-modules.

###### Proof.

(1)$\Rightarrow$(2)$\Rightarrow$(3)$\Rightarrow$(4) is trivial. We prove
(4)$\Rightarrow$(1). As $\hat{R}$ is a Noetherian complete local ring and
$\hat{T}$ is its module-finite algebra, the category of finite
$\hat{T}$-modules is Krull–Schmidt. As
$\hat{M}=\hat{M}_{1}\oplus\cdots\oplus\hat{M}_{s}$ and
$\hat{N}=\hat{N}_{1}\oplus\cdots\oplus\hat{N}_{t}$ are decompositions into
indecomposable $\hat{T}$-modules by Theorem 3.6, we have that $s=t$, and after
change of indices (if necessary), there are isomorphisms
$\hat{M}_{i}\cong\hat{N}_{i}$ for $i=1,\ldots,s$. So we may assume that
$s=t=1$.

Let $\hat{\phi}:\hat{M}\rightarrow\hat{N}$ be a $\hat{T}$-isomorphism, and let
$\hat{\psi}:\hat{N}\rightarrow\hat{M}$ be its inverse. We can write
$\hat{\phi}=\sum_{i}\hat{a}_{i}\phi_{i}$ and
$\hat{\psi}=\sum_{j}\hat{b}_{j}\psi_{j}$ with
$\phi_{i}\in\mathop{\operator@font Hom}\nolimits_{T}(M,N)_{u_{i}}$,
$u_{i}\in{\mathbb{Z}}$, $\hat{a}_{i}\in\hat{R}$,
$\psi_{j}\in\mathop{\operator@font Hom}\nolimits_{T}(N,M)_{v_{j}}$,
$v_{j}\in{\mathbb{Z}}$, and $\hat{b}_{j}\in\hat{R}$. So
$1_{\hat{M}}=\hat{\psi}\hat{\phi}=\sum_{ij}\hat{a}_{i}\hat{b}_{j}\psi_{j}\phi_{i}$.
As $1_{\hat{M}}\notin\hat{J}$, there exists some $(i,j)$ such that
$\psi_{j}\phi_{i}\notin J$, where $J={}^{*}\mathop{\operator@font
rad}\nolimits\mathop{\operator@font End}\nolimits_{T}M$. As $\psi_{j}\phi_{i}$
is a homogeneous element of $E=\mathop{\operator@font End}\nolimits_{T}M$,
which is ∗local, we have that $\psi_{j}\phi_{i}$ is a unit of $E$. In
particular, $\psi_{j}$ is a split epimorphism. As $N$ is also indecomposable,
we have that $\psi_{j}$ is a $T$-isomorphism. So we have that
$\psi_{j}:N\rightarrow M(v_{j})$ is an isomorphism in $T\mathop{\operator@font
grmod}\nolimits$. ∎

######  Corollary 3.8.

Let $R=\bigoplus_{i\geq 0}R_{i}$ be a Noetherian ${\mathbb{Z}}_{\geq
0}$-graded commutative ring such that $(R_{0},\mathfrak{m}_{0})$ is an
$F$-finite Henselian local ring of prime characteristic $p$. Let
$\mathfrak{m}=\mathfrak{m}_{0}+R_{+}$, where $R_{+}=\bigoplus_{i>0}R_{i}$. Let
$\hat{R}$ be the $\mathfrak{m}$-adic completion of $R$. Let
$M_{1},\ldots,M_{r}$ be finitely generated ${\mathbb{Q}}$-graded $R$-modules.
Then the following are equivalent.

  1. (1)

$R$ has finite $F$-representation type (FFRT for short) in the graded sense
with $M_{1},\ldots,M_{r}$. That is,

    1. (1-a)

For each $i$, $M_{i}$ is indecomposable;

    2. (1-b)

For each $i$, there exists some $e\geq 0$ and $c\in{\mathbb{Q}}$ such that
$M_{i}(c)$ is a direct summand of ${}^{e}R$;

    3. (1-c)

For each $e\geq 0$, any indecomposable direct summand of ${}^{e}R$ is
isomorphic to $M_{i}(c)$ for some $1\leq i\leq r$ and $c\in{\mathbb{Q}}$.

  2. (2)

The local ring $\hat{R}$ has FFRT with $\hat{M}_{1},\ldots,\hat{M}_{r}$. That
is,

    1. (2-a)

For each $i$, $\hat{M}_{i}$ is indecomposable;

    2. (2-b)

For each $i$, there exists some $e\geq 0$ such that $\hat{M}_{i}$ is a direct
summand of ${}^{e}\hat{R}$;

    3. (2-c)

For each $e\geq 0$, any indecomposable direct summand of ${}^{e}\hat{R}$ is
isomorphic to $\hat{M}_{i}$ for some $1\leq i\leq r$.

In particular, $R$ has FFRT in the graded sense if and only if $\hat{R}$ has
FFRT.

###### Proof.

By Theorem 3.6, (1-a) and (2-a) are equivalent.

Note that for each $e$,

${}^{e}\hat{R}\cong\varprojlim{}^{e}(R/\mathfrak{m}^{n})=\varprojlim{}^{e}R/{}^{e}(\mathfrak{m}^{n})=\varprojlim{}^{e}R/{}^{e}((\mathfrak{m}^{n})^{[p^{e}]})=\varprojlim{}^{e}R/(\mathfrak{m}^{n}){}^{e}R=\widehat{{}^{e}R},$

where $\mathfrak{m}^{[p^{r}]}$ is the ideal of $R$ generated by
$\\{a^{p^{r}}\mid a\in\mathfrak{m}\\}$.

(1-a)$\Rightarrow$(2-a). If $M_{i}(c)$ is a direct summand of ${}^{e}R$, then
$\hat{M}_{i}$ is a direct summand of $\widehat{{}^{e}R}={}^{e}\hat{R}$.

(2-a)$\Rightarrow$(1-a). Assume that $\hat{M}_{i}$ is a direct summand of
${}^{e}\hat{R}$. If we have a decomposition

(1) ${}^{e}R\cong N_{1}\oplus\cdots\oplus N_{r},$

where each $N_{j}$ is an indecomposable ${\mathbb{Q}}$-graded $R$-modules,
then

(2)
${}^{e}\hat{R}\cong\widehat{{}^{e}R}\cong\hat{N}_{1}\oplus\cdots\oplus\hat{N}_{r}.$

This is the decomposition into indecomposable modules by Theorem 3.6. So by
the Krull–Schmidt property of the category of $\hat{R}$-modules,
$\hat{M}_{i}\cong\hat{N}_{j}$ for some $j$. By Corollary 3.7, $N_{j}\cong
M_{i}(c)$ for some $c\in{\mathbb{Q}}$.

(1-c)$\Rightarrow$(2-c). Let $\hat{N}$ be a direct summand of ${}^{e}\hat{R}$.
By assumption, there is a decomposition (1) such that each $N_{j}$ is
isomorphic to $M_{i(j)}(c_{j})$ for some $1\leq i(j)\leq r$ and
$c_{j}\in{\mathbb{Q}}$. Then the isomorphism (2) holds, and by the
Krull–Schmidt property, $\hat{N}\cong\hat{N}_{j}\cong\hat{M}_{i(j)}$.

(2-c)$\Rightarrow$(1-c). Let $L$ be a direct summand of ${}^{e}R$. Then
$\hat{L}$ is a direct summand of ${}^{e}\hat{R}$. So $\hat{L}\cong\hat{M}_{i}$
for some $i$. Hence by Corollary 3.7, $L\cong M_{i}(c)$ for some
$c\in{\mathbb{Q}}$.

We prove the last assertion. The ‘only if’ part is clear from what we have
proved above. We prove the ‘if’ part. Let $\hat{R}$ have FFRT with the finite
indecomposable $\hat{R}$-modules $\hat{L}_{1},\ldots,\hat{L}_{r}$. So for each
$i$, $\hat{L}_{i}$ is a direct summand of ${}^{e}\hat{R}$ for some $e$. Now
let (1) be the decomposition of ${}^{e}R$ into indecomposable
${\mathbb{Q}}$-graded $R$-modules. Then we have an isomorphism (2), which is a
decomposition into indecomposables by Theorem 3.6. By the Krull–Schmidt,
$\hat{L}_{i}\cong\hat{N}_{j}$ for some $j$. In particular, there is a finite
indecomposable ${\mathbb{Q}}$-graded module $M_{i}$ such that
$\hat{M}_{i}\cong\hat{L}_{i}$. By what we have proved above, $R$ has FFRT with
$M_{1},\ldots,M_{r}$ in the graded sense, as required. ∎

##  References

  * [AK] K. Alhazmy and M. Katzman, FFRT properties of hypersurfaces and their $F$-signatures, J. Algebra 18 (2019), 1950215, 15 pp.
  * [BH] W. Bruns and J. Herzog, Cohen–Macaulay Rings. 2nd ed., Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, 1998.
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  * [HO] N. Hara and R. Ohkawa, The FFRT property of two-dimensional normal graded rings and orbifold curves, Adv. Math. 30 (2020), 107215, 37 pp.
  * [HB] M. Hochster and L. Núñez-Betancourt, Support of local cohomology modules over hypersurfaces and rings with FFRT, Math. Res. Lett. 24 (2017), 401–420.
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  * [L] T. Y. Lam, A First Course in Noncommutative Rings, Springer, 1991.
  * [LW] G. J. Leuschke and R. Wiegand, Cohen–Macaulay Representations, Mathematical Surveys and Monographs 181, AMS, 2012.
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Mitsuyasu Hashimoto

Department of Mathematics

Osaka Metropolitan University

Sumiyoshi-ku, Osaka 558–8585, JAPAN

e-mail<EMAIL_ADDRESS>
Yuntian Yang

Saitama, JAPAN

e-mail<EMAIL_ADDRESS>


# Social Robot Scenarios for Real-World Child and Family Care Settings through
Participatory Design

Anouk Neerincx1 1 Department of Information and Computing Sciences, Utrecht
University<EMAIL_ADDRESS>
###### Abstract

This paper discusses a 5-year PhD project, focused upon the implementation of
social robots for general child and family care settings in the Netherlands.
The project is a collaboration with general Dutch family care organisations as
well as specialized child mental health care organisations. The project adapts
a bottom-up, participatory design approach, where end users are included in
all stages of the project. End users consist of children, parents, and family
care professionals, who all have different needs, regarding the social robot
behaviors as well as the participatory design methods. This paper provides
suggestions to deal with these differences in designing social robots for
child mental support in real-world settings.

##  I PROBLEM STATEMENT AND COMMUNITY

Social robots show potential in the field of child education [1] and child
health care [2]. Previous research shows that children generally like to work
with robots. Robots can increase engagement in children during treatment,
reduce stress, and robots can be used as a multimedia tool. Social robots can
therefore make healthcare treatments as well as education more enjoyable for
children. Additionally, a social robot can help with establishing a connection
between the child and the therapist and/or teacher, for example by creating a
safe environment for the child, stimulating self-disclosure. Mental as well as
general health care organizations in the Netherlands express the need to make
use of the potential of technological solutions that exist (such as social
robots) for child mental support in various care and educational settings [3].
Before implementing such solutions, it is important to first design those
solutions user-centered, before studying the effects and risks thoroughly
(long-term) in real-world settings. The discussed 5-year PhD project (started
November 2019), focuses on designing and testing social robot behaviors for
child mental support, together with several Dutch health care organizations,
primary schools, and science festivals

###  I-A Organisations

Several organisations are involved in this PhD project. First of all, the
Dutch Child and Family Center (CJG), which focuses on providing general health
care (e.g., vaccinations, eye tests) as well as family coaching. Second,
Levvel, which provides specialized mental health care, such as improving
social-emotional skills. Third, several primary schools are involved by
organizing robot co-design workshops for their students. Finally, we
collaborate with several science festivals, to co-design and test social
robots in public spaces.

###  I-B Research Questions

The two main research questions studied in this PhD project are:

  * •

How can a social robot evoke appropriate self-disclosures (e.g., to facilitate
the connection between child and professional, to support social-emotional
skills)?

  * •

How can a social robot reduce stress during treatments?

The main target group of our research is children aged 8 to 12 years old,
since this age group is suitable for working with robots, and it is the
biggest client group of our collaborators.

##  II METHODS & PRELIMINARY RESULTS

###  II-A Dutch Child and Family Center

####  II-A1 Exploration

In an exploration phase, the Dutch Child and Family Center included
perspectives of child professionals (N=12) and parents (N=5), to explore
different scenarios and user requirements. Four focus groups were carried out
of 1,5 hours each. The focus group methodology was based on grounded theory,
and included open-ended questions as well as card sorting exercises.

Main examples of user requirements are:

  * •

Robot shall adapt its behavior to the child’s needs

  * •

Robot shall be safe to use (data storage, system quality)

  * •

Robot shall enrich the interaction by means of positive reflections (e.g.,
feedback), “small talk”, and games

Main examples of scenarios are:

  * •

Robot as icebreaker

  * •

Robot to assist in communicating

  * •

Robot to make the treatment more fun

We also carried out a child robot co-design workshop trial at the Dutch Child
and Family Center (N=4), including creative activities such as theatre play,
drawing, and playing with different robots. After this trial, several
workshops were organized at different primary schools as well as at our
university.

####  II-A2 Testing

The designs found in the exploration phase are currently being tested and
evaluated in (low-risk) real-world settings, for example at vaccination days.
Standardized questionnaires, thematic analysis of open-ended questions, as
well as video analysis are being used in analyzing the effects. First results
of our studies show for example that robots can reduce stress by increasing
engagement (but can hinder parent participation) [4], the children evaluate
the use of social robots positively, girls trust the robot more than boys, and
children can learn from the robot.

###  II-B Primary Schools

Four different robot co-design workshops have been organized at primary
schools, with children age 8-12 (N=50). The activities consisted of drawing or
writing, theatre play, and robot programming. Results include user
requirements and scenarios. The most popular scenarios were:

  * •

Social robot at vaccination day

  * •

Social robot to assist at school / with homework

  * •

Social robot at an eye test

The workshop was evaluated by a homework assignment from the teacher, where
she asked: What did you learn? What did you like? The children stated that
they mostly learned about how robots work and about programming. They stated
that they liked the theatre activity the most.

####  II-B1 Summer School

Currently, an adapted version of the robot co-design workshop is being carried
out at a summer school at Utrecht University (expected N=500). In this
workshop, the children are asked to make a robot drawing in small groups based
on one scenario (from a set of 7 based on our previous research), online
programming of a robot, and meeting a robot in our lab. Features of the
drawings as well as online programming activity will be analyzed based on a
coding scheme.

###  II-C Levvel

In the collaboration with Levvel, focus groups were done with several child
mental health care professionals (N=8), to explore the potentials of social
robot exercises to improve social-emotional skills. Based on standardized
social-emotional exercises provided by the experts, social robot exercises are
currently under development, which will be tested in real-world settings.

###  II-D Science Festivals

Collaborations with science festivals study the topic of self-disclosure in a
first real-world robot encounter. At Betweter science festival, we studied the
effect of self-disclosure category, human personality and robot identity on
adults (N=80) self-disclosure by means of standardized questionnaires and
audio analysis (full paper at RO-MAN 2022). At expeditie NEXT science festival
for children (N=83), we did a similar study, but we added a co-design
activity, where children could fill out a comic strip of a child-robot
interaction in either an education or health care setting. Results are
currently being analyzed.

##  III DISCUSSION: WHY PARTICIPATORY DESIGN, AND EXPERIENCE

Especially when designing technological solutions for in practice, it is
important to incorporate the views and needs of the end users. Since the
researchers and designers are no experts of the specific field in which the
technology will be used, they will probably have unexpected views and needs,
which need to be taken into account for a satisfactory product. Also, in the
field of social robots, often new robots are being designed which are only
used for a certain period of time (novelty effect), then cast aside, after
which a new model is developed. Designing together with the end users might
encourage more long-term, sustainable use. By using participatory design
methods, you are making sure that the right product is being designed. It is
important though to include all end users in the process, and to adapt the
participatory design methods to the target group. For example, children can
benefit from multiple ways of expressing themselves (besides verbally). Self-
expession might also be culture-related [5]. Therefore, including more
creative methods such as drawing and theatre play might be beneficial.
Creative methods might stimulate adults to think ”out-of-the-box” as well. By
first using participatory design, and then use more ’standardized’ methods to
test those designs, you use the best of both worlds. The discussed PhD project
was started at the end of 2019 as a bottom-up, participatory design project. I
started my participatory design experience in the beginning of 2020 by
organizing focus groups.

## References

  * [1] T. Belpaeme, J. Kennedy, A. Ramachandran, B. Scassellati, and F. Tanaka, “Social robots for education: A review,” _Science robotics_ , vol. 3, no. 21, 2018.
  * [2] J. Dawe, C. Sutherland, A. Barco, and E. Broadbent, “Can social robots help children in healthcare contexts? a scoping review,” _BMJ paediatrics open_ , vol. 3, no. 1, 2019.
  * [3] A. Neerincx, D. L. Rodenburg, M. M. de Graaf, and J. F. Masthoff, “Social robots to support child and family care: A dutch use case,” in _Companion of the 2021 ACM/IEEE International Conference on Human-Robot Interaction_ , 2021, pp. 367–371.
  * [4] A. Neerincx, T. Hiwat, and M. de Graaf, “Social robot for health check and entertainment in waiting room: Child’s engagement and parent’s involvement,” in _Adjunct Proceedings of the 29th ACM Conference on User Modeling, Adaptation and Personalization_ , 2021, pp. 120–125.
  * [5] A. Neerincx, F. Sacchitelli, R. Kaptein, S. Van Der Pal, E. Oleari, and M. A. Neerincx, “Child’s culture-related experiences with a social robot at diabetes camps,” in _2016 11th ACM/IEEE International Conference on Human-Robot Interaction (HRI)_. IEEE, 2016, pp. 485–486.