###### Proof.

bibprimaryclass bibarchiveprefix

###### The index of a numerical semigroup ring

###### Generalized Loewy Length of Cohen-Macaulay Local and Graded Rings

###### Abstract.

We generalize a theorem of Ding relating the generalized Loewy length
$\text{g}\ell\ell(R)$ and index of a one-dimensional Cohen-Macaulay local ring
$(R,\mathfrak{m},k)$. Ding proved that if $R$ is Gorenstein, the associated
graded ring is Cohen-Macaulay, and $k$ is infinite, then the generalized Loewy
length and index of $R$ are equal. However, if $k$ is finite, equality may not
hold. We prove that if the index of a one-dimensional Cohen-Macaulay local
ring is finite and the associated graded ring has a homogeneous nonzerodivisor
of degree $t$, then $\text{g}\ell\ell(R)\leq\text{index}(R)+t-1$. Next we
prove that if $R$ is a one-dimensional hypersurface ring with a witness to the
generalized Loewy length that induces a regular initial form on the associated
graded ring, then the generalized Loewy length achieves this upper bound. We
then compute the generalized Loewy lengths of several families of examples of
one-dimensional hypersurface rings over finite fields. Finally, we study a
graded version of the generalized Loewy length and determine its value for
numerical semigroup rings.

###### Key words and phrases:

Auslander’s delta invariant, index, generalized Loewy length, generalized
graded length, Ding’s conjecture, hypersurface ring, numerical semigroup ring,
Cohen-Macaulay, Gorenstein

###### 2020 Mathematics Subject Classification:

11A07, 13A02, 13A30, 13C10, 13E05, 13E10, 13E15, 13H05, 13H10, 13H15, 13P05.

##  1\. Introduction

Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay local ring of Krull dimension
$d$. For a finitely-generated $R$-module $M$, Auslander’s $\delta$-invariant,
denoted $\delta_{R}(M)$, is the smallest non-negative integer $n$ such that
there exists a surjective $R$-map $X\oplus R^{n}\longrightarrow M$, where $X$
is a maximal Cohen-Macaulay $R$-module with no free direct summand. For
finitely-generated $R$-modules $M$ and $N$, $\delta_{R}$ satisfies the
following properties [11, Corollary 11.28].

(i) $\delta_{R}(M\oplus N)=\delta_{R}(M)+\delta_{R}(N)$.
(ii) $\delta_{R}(N)\leq\delta_{R}(M)$ if there is a surjective $R$-map
$M\longrightarrow N$.
(iii) $\delta_{R}(M)\leq\mu_{R}(M)$.

We see from properties (ii) and (iii) that for each $n\geq 1$,

$0\leq\delta_{R}(R/\mathfrak{m}^{n})\leq\delta_{R}(R/\mathfrak{m}^{n+1})\leq
1\,.$

Therefore, if $\delta_{R}(R/\mathfrak{m}^{n_{o}})=1$ for some $n_{o}$, then
$\delta_{R}(R/\mathfrak{m}^{n})=1$ for all $n\geq n_{o}$. By a result of
Auslander, regular local rings are precisely the Cohen-Macaulay local rings
for which $\delta_{R}(R/\mathfrak{m}^{n})=1$ for all $n\geq 1$ [11,
Proposition 11.37]. It is natural to ask when the sequence
$\\{\delta_{R}(R/\mathfrak{m}^{n})\\}_{n=1}^{\infty}$ stabilizes at one for
different classes of Cohen-Macaulay rings. The smallest positive integer $n$
for which $\delta_{R}(R/\mathfrak{m}^{n})=1$ is the following numerical
invariant defined by Auslander.

$\text{index}(R):=\text{inf}\\{n\geq
1\,|\,\delta_{R}(R/\mathfrak{m}^{n})=1\\}$

Suppose $R$ is a Cohen-Macaulay local ring with canonical module $\omega$. The
trace of $\omega$ in $R$, denoted $\tau_{\omega}(R)$, is the ideal of $R$
generated by all $R$-homomorphic images of $\omega$ in $R$. Ding proved that
if $R$ is a Cohen-Macaulay local ring with canonical module such that
$\mathfrak{m}\subseteq\tau_{\omega}(R)$, then $\text{index}(R)$ is finite and
bounded above by the generalized Loewy length of $R$ [6, Proposition 2.4].
This invariant, denoted $\operatorname{\text{g}\ell\ell}(R)$, is the smallest
positive integer $n$ for which $\mathfrak{m}^{n}$ is contained in the ideal
generated by a system of parameters of $R$. In particular,
$\text{index}(R)\leq\text{g}\ell\ell(R)$ if $R$ is Gorenstein. If, in addition
to being Gorenstein, $R$ has infinite residue field and Cohen-Macaulay
associated graded ring $\operatorname{\text{gr}_{\mathfrak{m}}}(R)$, then
$\text{index}(R)=\text{g}\ell\ell(R)$ [7, Theorem 2.1].

In general, if $R$ is a Cohen-Macaulay local ring that satisfies the above
equality, we say that $R$ satisfies Ding’s conjecture. In this paper, we study
how the finiteness of the residue field can cause Ding’s conjecture to fail.
In particular, we prove that there are infinitely-many hypersurfaces with
Cohen-Macaulay associated graded ring and finite residue field that do not
satisfy Ding’s conjecture. Each of our families of hypersurfaces generalizes
an example of Hashimoto and Shida [8, Example 3.2], who showed for
$R=\mathbb{F}_{2}\llbracket x,y\rrbracket/(xy(x+y))$ that $\text{index}(R)=3$
and $\operatorname{\text{g}\ell\ell}(R)=4$.

When $k$ is finite, the assumption that
$\operatorname{\text{gr}_{\mathfrak{m}}}(R)$ is Cohen-Macaulay does not
guarantee the existence of a homogeneous system of parameters of degree one
$x_{1}^{*},...,x_{d}^{*}$ in
$(\operatorname{\text{gr}_{\mathfrak{m}}}(R))_{1}$. If a homogeneous system of
parameters in $\operatorname{\text{gr}_{\mathfrak{m}}}(R)$ does not consist of
linear elements, it cannot be used in Ding’s argument to prove that
$\text{index}(R)=\operatorname{\text{g}\ell\ell}(R)$.

However, if $R$ is a one-dimensional Cohen-Macaulay local ring with finite
index and $\operatorname{\text{gr}_{\mathfrak{m}}}(R)$ is Cohen-Macaulay, then
we can use a homogeneous $\operatorname{\text{gr}_{\mathfrak{m}}}(R)$-regular
element of minimal degree to obtain an upper bound for
$\operatorname{\text{g}\ell\ell}(R)$ in terms of $\text{index}(R)$. In Theorem
2.3, we prove that if $R$ is one-dimensional Cohen-Macaulay and
$\operatorname{\text{gr}_{\mathfrak{m}}}(R)$ has a homogeneous nonzerodivisor
$z^{*}$, where $z\in\mathfrak{m}^{t}\setminus\mathfrak{m}^{t+1}$, then

$\operatorname{\text{g}\ell\ell}(R)\leq\text{index}(R)+t-1\,.$

If $R$ is Gorenstein, then

$\text{index}(R)\leq\operatorname{\text{g}\ell\ell}(R)\leq\text{index}(R)+t-1\,.$

When $R$ is a hypersurface ring, we have $\text{index}(R)=e(R)$, where $e(R)$
denotes the Hilbert-Samuel multiplicity of $R$ [5, Theorem 3.3]. Therefore,
the index of hypersurface rings is easy to compute: if $R=k\llbracket
x_{1},...,x_{n}\rrbracket/(f)$, $\mathfrak{m}=(x_{1},...,x_{n})R$, and
$f\in\mathfrak{m}^{r}\setminus\mathfrak{m}^{r+1}$, then
$\text{index}(R)=e(R)=r$.

In section 3, we prove that if $R$ is a one-dimensional hypersurface with a
witness $z$ to its generalized Loewy length that induces a regular initial
form on $\operatorname{\text{gr}_{\mathfrak{m}}}(R)$, then

$\operatorname{\text{g}\ell\ell}(R)=\text{ord}_{R}(z)+e(R)-1.$

We then compute the generalized Loewy lengths of families of examples of one-
dimensional hypersurface rings with finite residue field and Cohen-Macaulay
associated graded ring. These examples illustrate differences between
hypersurface rings $R$ with finite residue field and Cohen-Macaulay associated
graded ring for which $\operatorname{\text{g}\ell\ell}(R)=\text{index}(R)$ and
$\operatorname{\text{g}\ell\ell}(R)=\text{index}(R)+1$. In [3], De Stefani
gave examples of one-dimensional Gorenstein local rings with infinite residue
field for which $\operatorname{\text{g}\ell\ell}(R)=\text{index}(R)+1$.

In section 4, we let $R$ be a positively-graded Noetherian $k$-algebra, where
$k$ is an arbitrary field. We show that several families of one-dimensional
standard graded hypersurfaces attain the graded version of the upper bound for
the generalized Loewy length from Theorem 2.3. We then study a graded version
of the generalized Loewy length: the generalized graded length of $R$, denoted
$\operatorname{\bf{ggl}}(R)$. After determining bounds for
$\operatorname{\bf{ggl}}(R)$ in terms of $\operatorname{\text{g}\ell\ell}(R)$
and the minimum and maximum degrees of generators of $R$, we compute the
generalized graded length of numerical semigroup rings. For
$R=k[t^{a},t^{b}]$, where $a<b$, we prove that
$\operatorname{\bf{ggl}}(R)=ba-b+1$ and if $z$ is a witness to
$\operatorname{\bf{ggl}}(R)$, then $(z)=(t^{ia})$ for some $1\leq i\leq
1+b-a$.

##  2\. Estimating the Generalized Loewy Length of One-Dimensional Cohen-
Macaulay Rings

Throughout this section, $(R,\mathfrak{m},k)$ is a local ring. We assume that
$R$ has a nonzerodivisor $x$ of order $t$ such that multiplication by $x$ is
injective on graded components of the associated graded ring in degrees less
than $\text{index}(R)$. Generalizing [7, Lemma 2.3] to this context, we prove
that if $R$ is a one-dimensional Cohen-Macaulay local ring with finite index,
then $\operatorname{\text{g}\ell\ell}(R)\leq\text{index}(R)+t-1$.

######  Lemma 2.1.

Let $s$ and $t$ be positive integers and
$x\in\mathfrak{m}^{t}\setminus\mathfrak{m}^{t+1}$ an $R$-regular element.
Suppose the induced map
$\overline{x}:\mathfrak{m}^{i-1}/\mathfrak{m}^{i}\longrightarrow\mathfrak{m}^{i+t-1}/\mathfrak{m}^{i+t}$
is injective for $1\leq i\leq s$. Then

$(\mathfrak{m}^{s+t-1},x)/x\mathfrak{m}^{s}\cong
R/\mathfrak{m}^{s}\oplus(\mathfrak{m}^{s+t-1},x)/xR.$

###### Proof.

Let $I=xR\cap\mathfrak{m}^{s+t-1}$ and
$W=(I+\mathfrak{m}^{s+t})/\mathfrak{m}^{s+t}$. Since $W$ is a $k$-subspace of
$\mathfrak{m}^{s+t-1}/\mathfrak{m}^{s+t}$, there is a direct sum decomposition

$\mathfrak{m}^{s+t-1}/\mathfrak{m}^{s+t}=W\oplus V\,$

for some subspace $V\subseteq\mathfrak{m}^{s+t-1}/\mathfrak{m}^{s+t}$. Let
$e_{1},...,e_{n}$ be a $k$-basis for $V$. For each $i$, let
$e_{i}=\overline{y_{i}}$, where $y_{i}\in\mathfrak{m}^{s+t-1}$. Let $B$ denote
the $R$-submodule of $(\mathfrak{m}^{s+t-1},x)/x\mathfrak{m}^{s}$ generated by
$[y_{1}],...,[y_{n}]\in(\mathfrak{m}^{s+t-1},x)/x\mathfrak{m}^{s}$. We will
prove that $(\mathfrak{m}^{s+t-1},x)/x\mathfrak{m}^{s}=A\oplus B$, where
$A=xR/x\mathfrak{m}^{s}$. First we show that

$A+B=(\mathfrak{m}^{s+t-1},x)/x\mathfrak{m}^{s}.$

Choose $r_{1},...,r_{\alpha}\in R$ such that $I=(r_{1}x,...,r_{\alpha}x)$.
Then $\mathfrak{m}^{s+t-1}/\mathfrak{m}^{s+t}$ is generated as a vector space
by
$\\{\overline{r_{i}x}\\}_{i=1}^{\alpha}\cup\\{\overline{y}_{j}\\}_{j=1}^{n}$,
and by Nakayama’s lemma, $\mathfrak{m}^{s+t-1}$ is generated as an $R$-module
by $\\{r_{i}x\\}_{i=1}^{\alpha}\cup\\{y_{j}\\}_{j=1}^{n}$. Let
$[z]\in(\mathfrak{m}^{s+t-1},x)/x\mathfrak{m}^{s}$. Then
$[z]=r[x]+r^{\prime}[v]$, where $r,r^{\prime}\in R$,
$v\in\mathfrak{m}^{s+t-1}$, and
$v=r^{\prime\prime}x+\sum_{i=1}^{n}\rho_{i}y_{i}$, where
$r^{\prime\prime},\rho_{i}\in R$. So

$[z]=(r+r^{\prime}r^{\prime\prime})[x]+\sum_{i=1}^{n}r^{\prime}\rho_{i}[y_{i}]\in
A+B.$

Now we show that $A\cap B=0$. Let $[z]\in A\cap B$. Then
$[z]=a[x]=\sum_{i=1}^{n}a_{i}[y_{i}]$, where $a,a_{i}\in R$, and
$ax-\sum_{i=1}^{n}a_{i}y_{i}\in x\mathfrak{m}^{s}$. Let
$ax-\sum_{i=1}^{n}a_{i}y_{i}=xy$, where $y\in\mathfrak{m}^{s}$. Then
$\sum_{i=1}^{n}a_{i}y_{i}=(a-y)x\in I$, so

$\overline{(a-y)x}=\overline{0}\in\mathfrak{m}^{s+t-1}/\mathfrak{m}^{s+t}.$

If $a=y$ we are done, so assume $a-y\neq 0$. Then there is a nonnegative
integer $l$ such that $a-y\in\mathfrak{m}^{l}\setminus\mathfrak{m}^{l+1}$.
Suppose $0\leq l<s$. Since $\overline{(a-y)x}=\overline{0}$ in
$\mathfrak{m}^{l+t}/\mathfrak{m}^{l+t+1}$, it follows from the injectivity of
the induced map $\overline{x}$ that $a-y\in\mathfrak{m}^{l+1}$, a
contradiction. Therefore, $a-y\in\mathfrak{m}^{s}$, and $ax-xy\in
x\mathfrak{m}^{s}$. Since $xy\in x\mathfrak{m}^{s}$, $ax\in
x\mathfrak{m}^{s}$, and $[z]=a[x]=[0]$.

It follows that
$(\mathfrak{m}^{s+t-1},x)/x\mathfrak{m}^{s}=xR/x\mathfrak{m}^{s}\oplus B$ and
$B\cong(\mathfrak{m}^{s+t-1},x)/xR$. Since $x$ is $R$-regular, it follows that
$(\mathfrak{m}^{s+t-1},x)/x\mathfrak{m}^{s}\cong
R/\mathfrak{m}^{s}\oplus(\mathfrak{m}^{s+t-1},x)/xR$.∎

######  Lemma 2.2.

Let $(R,\mathfrak{m})$ be a local ring, $I\subseteq R$ an ideal, and
$x,y\in\mathfrak{m}$ such that $(x,I)=(y)$. If $I$ is not a principal ideal,
then $(x)=(y)$.

###### Proof.

Let $a,b\in R$ and $z\in I$ such that $y=ax+bz$. Let $c\in R$ such that
$x=cy$. Then $y=acy+bz$ and $(1-ac)y=bz$. Suppose $c\in\mathfrak{m}$. Then
$1-ac$ is invertible and $y=(1-ac)^{-1}bz\in I$, so $(y)=I$, which is false.
Therefore $c$ is invertible and $(x)=(y)$. ∎

######  Theorem 2.3.

Let $(R,\mathfrak{m})$ be a one-dimensional Cohen-Macaulay local ring for
which $\text{index}(R)$ is finite. Let $s=\text{index}(R)$ and
$x\in\mathfrak{m}^{t}\setminus\mathfrak{m}^{t+1}$ a nonzerodivisor, where
$t\geq 1$. If the induced map

$\overline{x}:\mathfrak{m}^{i-1}/\mathfrak{m}^{i}\longrightarrow\mathfrak{m}^{i+t-1}/\mathfrak{m}^{i+t}$

is injective for $1\leq i\leq s$, then

$\operatorname{\text{g}\ell\ell}(R)\leq\text{index}(R)+t-1.$

If $\mathfrak{m}^{s+t-1}$ is not a principal ideal, then
$\mathfrak{m}^{s+t-1}\subseteq(x).$

###### Proof.

By Lemma 2.1, $(\mathfrak{m}^{s+t-1},x)/x\mathfrak{m}^{s}\cong
R/\mathfrak{m}^{s}\oplus(\mathfrak{m}^{s+t-1},x)/xR$, so there is a surjection

$(\mathfrak{m}^{s+t-1},x)\longrightarrow R/\mathfrak{m}^{s}.$

Therefore, $\delta_{R}((\mathfrak{m}^{s+t-1},x))>0$. By [10, Lemma 2.5],
$(\mathfrak{m}^{s+t-1},x)$ is a parameter ideal of $R$. Let
$(\mathfrak{m}^{s+t-1},x)=(y)$, where $y\in\mathfrak{m}$ is a regular element.
Since $\mathfrak{m}^{s+t-1}\subseteq(y)$, we have
$\operatorname{\text{g}\ell\ell}(R)\leq s+t-1$. If $\mathfrak{m}^{s+t-1}$ is
not a principal ideal, then by Lemma 2.2 we have
$\mathfrak{m}^{s+t-1}\subseteq(x)$. ∎

######  Definition 2.4.

Let $R$ be a Cohen-Macaulay local ring with canonical module $\omega$. The
trace of $\omega$ in $R$, denoted $\tau_{\omega}(R)$, is the ideal of $R$
generated by all $R$-homomorphic images of $\omega$ in $R$.

######  Corollary 2.5.

Let $(R,\mathfrak{m})$ be a one-dimensional Cohen-Macaulay local ring with
canonical module $\omega$ such that $\mathfrak{m}\subseteq\tau_{\omega}(R)$.
Let $x\in\mathfrak{m}^{t}\setminus\mathfrak{m}^{t+1}$ such that
$x^{*}\in\operatorname{\text{gr}_{\mathfrak{m}}}(R)$ is a regular element.
Then

$\text{index}(R)\leq\operatorname{\text{g}\ell\ell}(R)\leq\text{index}(R)+t-1.$

###### Proof.

This follows from [6, Proposition 2.4] and Theorem 2.3. ∎

##  3\. Examples

In this section we derive a formula for the generalized Loewy length of one-
dimensional hypersurface rings and compute the generalized Loewy lengths of
several families of examples of one-dimensional hypersurfaces. The associated
graded ring of each of these hypersurface rings has a homogeneous
nonzerodivisor of degree one or two, so the index and generalized Loewy length
differ by at most one.

Using techniques from the proof of [8, Example 3.2], we prove that for several
families of hypersurfaces $\\{R_{n}\\}_{n=1}^{\infty}$,

$\operatorname{\text{g}\ell\ell}(R_{n})-\text{index}(R_{n})=1$

for $n\geq 1$. This difference is positive for each $n$ because of the absence
of a regular linear form in certain one-dimensional hypersurface rings over
finite fields.

Throughout this section, $S=k\llbracket x,y\rrbracket$, where $k$ is a field
and $\mathfrak{n}=(x,y)S$. We say that the order of an element $f\in S$ is $r$
if $f\in\mathfrak{n}^{r}\setminus\mathfrak{n}^{r+1}$, and write
$\text{ord}_{S}(f)=r$. Let $R=S/(f)$, where $f\in\mathfrak{n}$. Let
$\mathfrak{m}=(x,y)R$. The order of an element $z\in R$ is $r$ if
$z\in\mathfrak{m}^{r}\setminus\mathfrak{m}^{r+1}$, and we write
$\text{ord}_{R}(z)=r$. Recall that $\text{index}(R)=e(R)$. Finally, if
$(R,\mathfrak{m})$ is any local ring of embedding dimension $n$, then
$\mu_{R}(\mathfrak{m}^{r})\leq\binom{n+r-1}{r}$.

######  Lemma 3.1.

Let $R=k\llbracket x,y\rrbracket/(f)$, where $\text{ord}_{S}(f)=e$ and
$g=\operatorname{\text{g}\ell\ell}(R)$. Let $z\in\mathfrak{m}$ such that
$\mathfrak{m}^{g}\subseteq(z)$ and $i\geq 0$. If
$\operatorname{\text{g}\ell\ell}(R)\leq e+i$, then $\text{ord}_{R}(z)\leq
i+1$.

###### Proof.

Let $\text{ord}_{R}(z)=r$ and
$\zeta\in\mathfrak{n}^{r}\setminus\mathfrak{n}^{r+1}$ such that
$\overline{\zeta}=z$. Then $\mathfrak{n}^{g}\subseteq(f,\zeta)$. Let $M$ be
the $k$-vector space of leading forms of degree $g$ of elements of
$(f,\zeta)$. Since $\text{ord}_{S}(\zeta)=r$, we obtain leading forms of
degree $g$ from this element by multiplying $\zeta$ by generators of
$\mathfrak{n}^{g-r}$. Similarly, we multiply $f$ by generators of
$\mathfrak{n}^{g-e}$ to obtain leading forms of degree $g$. Therefore,

$\text{dim}_{k}\,M\leq\binom{2+(g-e)-1}{g-e}+\binom{2+(g-r)-1}{g-r}=2g-(e+r)+2.$

On the other hand, the vector space of forms of degree $g$ in
$\mathfrak{n}^{g}$ has dimension $g+1$. Therefore, $g+1\leq 2g-(e+r)+2$ and
$e+r\leq g+1$. The result follows from this inequality. ∎

######  Definition 3.2.

Let $(R,\mathfrak{m})$ be a $d$-dimensional local ring. A system of parameters
$\underline{\bf{x}}=x_{1},...,x_{d}\in\mathfrak{m}$ is a witness to
$g=\operatorname{\text{g}\ell\ell}(R)$ if
$\mathfrak{m}^{g}\subseteq(\underline{\bf{x}})$.

If $R$ is a one-dimensional hypersurface with a witness $z$ to
$\operatorname{\text{g}\ell\ell}(R)$ that induces a regular initial form on
$\operatorname{\text{gr}_{\mathfrak{m}}}(R)$, then we can compute
$\operatorname{\text{g}\ell\ell}(R)$ using the following formula. We see that
the order of $z$ is uniquely determined by
$\operatorname{\text{g}\ell\ell}(R)$ and $e(R)$.

######  Proposition 3.3.

Let $R=k\llbracket x,y\rrbracket/(f)$, where $\text{ord}_{S}(f)=e$ and
$z\in\mathfrak{m}$ such that $z^{*}$ is
$\operatorname{\text{gr}_{\mathfrak{m}}}(R)$-regular. If $z$ is a witness to
$\operatorname{\text{g}\ell\ell}(R)$, then

$\operatorname{\text{g}\ell\ell}(R)=\text{ord}_{R}(z)+e-1.$

###### Proof.

Recall that $\text{index}(R)=e$. Let $g=\operatorname{\text{g}\ell\ell}(R)$
and $n=g-e$. Then $g=e+n$ and by Lemma 3.1, $\text{ord}_{R}(z)\leq n+1$. By
Theorem 2.3, $g\leq e+\text{ord}_{R}(z)-1\leq e+n=g$. ∎

If we cannot find an element of a one-dimensional hypersurface that is a
witness to $\operatorname{\text{g}\ell\ell}(R)$ and induces a regular form on
$\operatorname{\text{gr}_{\mathfrak{m}}}(R)$, then we can use the following
lemma to estimate the generalized Loewy length.

######  Lemma 3.4.

Let $R=k\llbracket x,y\rrbracket/(f)$, where $\text{ord}_{S}(f)=e\geq 2$. If
$R$ has no nonzerodivisors of the form $\alpha x+\beta y$, where
$\alpha,\beta\in k$, then $\operatorname{\text{g}\ell\ell}(R)>e.$

###### Proof.

Since $\text{index}(R)=e$, we have $e\leq\operatorname{\text{g}\ell\ell}(R)$
[6, Proposition 2.4]. Suppose $\operatorname{\text{g}\ell\ell}(R)=e$. Let
$z\in\mathfrak{m}$ such that $\mathfrak{m}^{e}\subseteq(z)$. By Lemma 3.1, we
have $\text{ord}_{R}(z)=1$. Let
$\zeta\in\mathfrak{n}\setminus\mathfrak{n}^{2}$ be a preimage of $z$. Note
that for each invertible matrix $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\text{GL}_{2}(k)$, the map $\phi:S\longrightarrow S$
defined by $\phi(x)=ax+by$ and $\phi(y)=cx+dy$ is a $k$-algebra automorphism.
Letting an appropriate invertible matrix in $\text{GL}_{2}(k)$ act on $S$, we
may assume without loss of generality that $\zeta=x-h$ for some nonzero
element $h\in S$ with $\text{ord}_{S}(h)\geq 2$. Since $x$ is a zerodivisor on
$R$, there is an element $g\in\mathfrak{n}^{e-1}$ such that $f=xg$.

Let $R^{\prime}=S/(\zeta)$. Since $S$ is a regular local ring and
$\text{ord}_{S}(\zeta)=1$, it follows that $R^{\prime}$ is a one-dimensional
regular local ring, and thus a discrete valuation ring. Let $\overline{f}$
denote the image of $f$ in $R^{\prime}$. Then

$R/(z)\cong R^{\prime}/(\overline{f}).$

Since $\overline{g}\in(x,y)^{e-1}R^{\prime}$ and
$\overline{x}=\overline{h}\in(x,y)^{2}R^{\prime}$, it follows that
$\overline{f}\in(x,y)^{e+1}R^{\prime}$, so
$l_{R^{\prime}}(R^{\prime}/(\overline{f}))=\text{ord}_{R^{\prime}}(\overline{f})\geq
e+1$ and $l_{R}(R/(z))\geq e+1$. Now let $R_{1}:=R/(z)$ and
$\mathfrak{m}_{1}:=\mathfrak{m}/(z)$. Then

$0=\mathfrak{m}_{1}^{e}\subseteq\mathfrak{m}_{1}^{e-1}\subseteq\cdot\cdot\cdot\subseteq\mathfrak{m}_{1}\subseteq
R_{1}$

is a composition series for $R_{1}$, so $l_{R}(R/(z))=e$. This is a
contradiction.∎

If $(R,\mathfrak{m},k)$ is a one-dimensional local ring with Cohen-Macaulay
associated graded ring and infinite residue field, then
$\operatorname{\text{gr}_{\mathfrak{m}}}(R)$ has a homogeneous linear
nonzerodivisor. We now consider one-dimensional hypersurface rings with finite
residue field such that the associated graded ring does not have a homogeneous
linear nonzerodivisor. If the associated graded ring has a homogeneous
quadratic nonzerodivisor, then it follows from Theorem 2.3 and Lemma 3.4 that
the difference between the generalized Loewy length and index is one.

######  Proposition 3.5.

Let $k$ be a finite field and $R=k\llbracket
x,y\rrbracket/y(\prod\limits_{\alpha\in k}(x+\alpha y))$. Then

$\operatorname{\text{g}\ell\ell}(R)=\text{index}(R)+1=|k|+2.$

###### Proof.

We construct a homogeneous nonzerodivisor of degree $2$ in
$\operatorname{\text{gr}_{\mathfrak{m}}}(R)$. Let $f\in k[x]$ be a degree 2
irreducible polynomial. Define

$g(x,y)\in k[x,y]\,\,\,by\,\,\,g(x,y):=y^{2}f(\frac{x}{y}).$

We claim that the element
$\overline{g}=g(\overline{x},\overline{y})\in\operatorname{\text{gr}_{\mathfrak{m}}}(R)=k[x,y]/y(\prod\limits_{\alpha\in
k}(x+\alpha y))$ is $\operatorname{\text{gr}_{\mathfrak{m}}}(R)$-regular.

Let $h\in k[x,y]$ such that $\overline{g}\overline{h}=\overline{0}$. Then
there exists a polynomial $p(x,y)\in k[x,y]$ such that

$gh=py(\prod\limits_{\alpha\in k}(x+\alpha y)).$

Let $\alpha\in k$. Suppose $(x+\alpha y)\mid g$ and $q(x,y)\in k[x,y]$ such
that $(x+\alpha y)q(x,y)=g(x,y)$. Then $(x+\alpha)q(x,1)=g(x,1)=f(x)$. This
contradicts the irreducibility of $f$.

It follows that $(x+\alpha y)\mid h$. Clearly $y\nmid g$, so $y\mid h$ as
well, and $y(\prod\limits_{\alpha\in k}(x+\alpha y))\mid h$. Therefore we have
$\overline{h}=\overline{0}$, and $\overline{g}$ is
$\operatorname{\text{gr}_{\mathfrak{m}}}(R)$-regular. By Theorem 2.3 and Lemma
3.4, $\operatorname{\text{g}\ell\ell}(R)=\text{index}(R)+1$. ∎

######  Remark 3.6.

When $k=\mathbb{F}_{2}$, Proposition 3.5 is Hashimoto and Shida’s
counterexample to Ding’s conjecture: $\mathbb{F}_{2}\llbracket
x,y\rrbracket/(xy(x+y))$. In the following propositions, we compute the
generalized Loewy lengths of families of one-dimensional hypersurface rings of
the form $k\llbracket x,y\rrbracket/(xy(x^{n}+y^{n}))$, where $k$ is a finite
field and $n$ is a positive integer.

######  Proposition 3.7.

Let $n\geq 1$ and $k$ a field such that $\text{char}\,k\neq 2$ and
$\text{char}\,k\nmid 1+(-2)^{n}$. Let $R=k\llbracket
x,y\rrbracket/(xy(x^{n}+y^{n}))$. Then
$\mathfrak{m}^{n+2}=(x+2y)\mathfrak{m}^{n+1}$ and

$\operatorname{\text{g}\ell\ell}(R)=\text{index}(R)=n+2.$

###### Proof.

Since $\mathfrak{m}^{n+1}$ is generated by $\\{x^{n+1-i}y^{i}\\}_{i=0}^{n+1}$,
it follows that $(x+2y)\mathfrak{m}^{n+1}$ is generated by
$\\{x^{n+2-i}y^{i}+2x^{n+1-i}y^{i+1}\\}_{i=0}^{n+1}$. Let

$z_{i}=x^{n+2-i}y^{i}+2x^{n+1-i}y^{i+1}$

for $0\leq i\leq n+1$. Since $xy^{n+1}=-x^{n+1}y$,

$\sum_{i=1}^{n}(-2)^{i-1}z_{i}=x^{n+1}y+2(-2)^{n-1}xy^{n+1}$
$=x^{n+1}y-2(-2)^{n-1}x^{n+1}y$ $=(1+(-2)^{n})x^{n+1}y.$

Since $z_{i}\in(x+2y)\mathfrak{m}^{n+1}$ for $0\leq i\leq n+1$, we have
$x^{n+1}y\in(x+2y)\mathfrak{m}^{n+1}$ and
$\mathfrak{m}^{n+2}\subseteq(x+2y)\mathfrak{m}^{n+1}$. Therefore,
$\operatorname{\text{g}\ell\ell}(R)\leq
n+2=\text{index}(R)\leq\operatorname{\text{g}\ell\ell}(R)$. ∎

######  Corollary 3.8.

Let $k$ be a field of characteristic $p>2$ and $R=k\llbracket
x,y\rrbracket/(xy(x^{p^{n}}+y^{p^{n}}))$, where $n\geq 0$. Then
$\mathfrak{m}^{p^{n}+2}=(x+2y)\mathfrak{m}^{p^{n}+1}$, and

$\operatorname{\text{g}\ell\ell}(R)=\text{index}(R)=p^{n}+2.$

###### Proof.

Suppose $p\mid 1+(-2)^{p^{n}}$. Since $1+(-2)^{p^{n}}=1-2^{p^{n}}$, we have
$2^{p^{n}}=1\text{ mod }p$. Since $2^{p^{n}}=2\text{ mod }p$, it follows that
$2=1\text{ mod }p$, which is false. Therefore, $p\nmid 1+(-2)^{p^{n}}$. The
result now follows from Proposition 3.7.∎

If we let $p=2$ in Corollary 3.8, then the generalized Loewy length and index
of $R$ differ by one. This is a special case of Proposition 3.12. To prove
Proposition 3.12, we require the following results about the reducibility of
cyclotomic polynomials modulo prime integers and primitive roots of powers of
prime integers.

######  Lemma 3.9 (12, Theorem 2.47).

Let $K=\mathbb{F}_{q}$, where $q$ is prime and $q\nmid n$. Let $\varphi$
denote Euler’s totient function and $d$ the least positive integer such that
$q^{d}=1\text{ mod }n$. Then the $n^{\text{th}}$ cyclotomic polynomial
$\Phi_{n}$ factors into $\varphi(n)/d$ distinct monic irreducible polynomials
in $K[x]$ of degree $d$.

######  Lemma 3.10 (12, Example 2.46).

Let $p$ be prime and $m\in\mathbb{N}$. Then the $p^{m}$th cyclotomic
polynomial $\Phi_{p^{m}}$ equals

$1+x^{p^{m-1}}+x^{2p^{m-1}}+\,\cdot\cdot\cdot\,+x^{(p-1)p^{m-1}}.$

######  Lemma 3.11 (2, Proposition 3.4.1).

Let $p$ be a prime and $g$ a positive integer. Then the following three
assertions are equivalent:

  1. (1)

$g$ is a primitive root modulo $p$ and $g^{p-1}\neq 1\text{ mod }p$;

  2. (2)

$g$ is a primitive root modulo $p^{2}$;

  3. (3)

For every $i\geq 2$, $g$ is a primitive root modulo $p^{i}$.

######  Proposition 3.12.

Let $R=\mathbb{F}_{2}\llbracket
x,y\rrbracket/(xy(x^{2^{n}p^{m}}+y^{2^{n}p^{m}}))$, where $m,n\geq 0$ and
$p>3$ is a prime such that $2$ is a primitive root modulo $p^{2}$. Then

$\operatorname{\text{g}\ell\ell}(R)=\text{index}(R)+1=2^{n}p^{m}+3$

and

$\mathfrak{m}^{2^{n}p^{m}+3}\subseteq(x^{2}+xy+y^{2}).$

If $m=1$, then we need only assume that $2$ is a primitive root modulo $p$.

###### Proof.

First assume that $m>0$. We show that $x^{2}+xy+y^{2}$ is
$\operatorname{\text{gr}_{\mathfrak{m}}}(R)$-regular. Let
$S=\mathbb{F}_{2}\llbracket x,y\rrbracket$ and suppose $f,g\in S$ such that

(3.1)
$(x^{2}+xy+y^{2})f=g(xy(x^{2^{n}p^{m}}+y^{2^{n}p^{m}}))=g(xy(x^{p^{m}}+y^{p^{m}})^{2^{n}}).$

By Lemmas 3.9 through 3.11, $\Phi_{p^{i}}(x)$ is an irreducible polynomial
over $\mathbb{F}_{2}$ of degree $p^{i}-p^{i-1}$ for $1\leq i\leq m$. We obtain
the following factorization of $x^{p^{m}}+1$ into irreducible polynomials over
$\mathbb{F}_{2}$.

$x^{p^{m}}+1=(x+1)\prod\limits_{i=1}^{m}\Phi_{p^{i}}(x).$

Let $h_{i}(x,y):=y^{p^{i}-p^{i-1}}\Phi_{p^{i}}(x/y)$ for $i=1,...,m$. Then
$h_{i}(x,y)$ is a homogeneous polynomial of degree $p^{i}-p^{i-1}$, and

(3.2) $x^{p^{m}}+y^{p^{m}}=(x+y)\prod\limits_{i=1}^{m}h_{i}(x,y).$

We claim that each $h_{i}(x,y)$ is irreducible over $\mathbb{F}_{2}$. Suppose
$p,q\in\mathbb{F}_{2}[x,y]$ such that

$h_{i}(x,y)=p(x,y)q(x,y).$

Since $h_{i}$ is homogeneous, $p$ and $q$ are homogeneous. Let $y=1$ in the
above equation. Then

$\Phi_{p^{i}}(x)=h_{i}(x,1)=p(x,1)q(x,1).$

Since $\Phi_{p^{i}}$ is irreducible over $\mathbb{F}_{2}$,
$p(x,1)=\Phi_{p^{i}}(x)$ or $q(x,1)=\Phi_{p^{i}}(x)$. Assume
$p(x,1)=\Phi_{p^{i}}(x)$. Then $p(x,y)=h_{i}(x,y)$, so $h_{i}(x,y)$ is
irreducible. By equations (3.1) and (3.2), $h_{i}\mid(x^{2}+xy+y^{2})$ or
$h_{i}\mid f$. Since the degree of $h_{i}$ is

$p^{i}-p^{i-1}=p^{i-1}(p-1)\geq p^{i-1}3,$

it follows that $h_{i}\mid f$. It is clear that $x$, $y$, and $x+y$ divide $f$
as well, so $f\in(xy(x^{p^{m}}+y^{p^{m}})^{2^{n}})$, and $x^{2}+xy+y^{2}$ is a
nonzerodivisor on $\operatorname{\text{gr}_{\mathfrak{m}}}(R)$. By Theorem 2.3
and Lemma 3.4, $\operatorname{\text{g}\ell\ell}(R)=\text{index}(R)+1$ and
$\mathfrak{m}^{2^{n}p^{m}+3}\subseteq(x^{2}+xy+y^{2})$. If $m=0$, (3.1)
becomes

$(x^{2}+xy+y^{2})f=g(xy(x^{2^{n}}+y^{2^{n}}))=g(xy(x+y)^{2^{n}}).$

It follows that $f\in xy(x^{2^{n}}+y^{2^{n}})$, so $x^{2}+xy+y^{2}$ is a
nonzerodivisor on $\operatorname{\text{gr}_{\mathfrak{m}}}(R)$. Therefore,
$\mathfrak{m}^{2^{n}+3}\subseteq(x^{2}+xy+y^{2})$ and
$\operatorname{\text{g}\ell\ell}(R)=\text{index}(R)+1$. ∎

######  Remark 3.13.

Whether there are infinitely many primes $p$ such that $2$ is a primitive root
modulo $p$ is an open question. This is a special case of Artin’s conjecture
on primitive roots [2, p.66]. A list of the first primes $p$ for which $2$ is
a primitive root modulo $p$ is sequence A001122 in the OEIS.

##  4\. Generalized Loewy Length of Graded Algebras

We now consider positively-graded Noetherian $k$-algebras and a graded
analogue of the generalized Loewy length of a local ring. Throughout this
section, $k$ is an arbitrary field.

######  Definition 4.1.

Let $R=\bigoplus\limits_{i\geq 0}R_{i}$ be a positively-graded Noetherian
$k$-algebra, where $R_{0}=k$ and $\mathfrak{m}=\bigoplus\limits_{i\geq
1}R_{i}$ is the irrelevant ideal. For $n\geq 0$, let
$\mathfrak{m}_{n}:=\bigoplus\limits_{i\geq n}R_{i}$. The generalized graded
length of $R$, denoted $\operatorname{\bf{ggl}}(R)$, is the smallest positive
integer $n$ for which $\mathfrak{m}_{n}$ is contained in the ideal generated
by a homogeneous system of parameters.

In this context, the generalized Loewy length,
$\operatorname{\text{g}\ell\ell}(R)$, is the smallest positive integer $n$ for
which $\mathfrak{m}^{n}$ is contained in the ideal generated by a homogeneous
system of parameters.

With Herzog, we note that all of the above definitions can be transferred
accordingly to standard graded Gorenstein $k$-algebras [9, p.98]. For
$R=k[x,y]/(f)$, one can prove that $\text{index}(R)=\text{deg}(f)$ by using
Ding’s arguments in [5] to prove the standard graded version of [5, Theorem
3.3]. By the standard graded version of Proposition 3.3, the generalized Loewy
length of $k[x,y]/(f)$ is one less than the sum of the degree of $f$ and the
degree of a witness to $\text{g}\ell\ell(R)$.

######  Proposition 4.2.

Let $R=k[x,y]/(f)$ be standard graded, where $f\in k[x,y]$ is a form of degree
$e$. Let $z\in(x,y)R$ be a witness to $\operatorname{\text{g}\ell\ell}(R)$.
Then $\operatorname{\text{g}\ell\ell}(R)=\text{deg}_{R}(z)+e-1$.

Let $(R,\mathfrak{m})$ be a positively-graded Noetherian $k$-algebra. It is
clear that for each $n\geq 1$, we have
$\mathfrak{m}^{n}\subseteq\mathfrak{m}_{n}$, so
$\operatorname{\text{g}\ell\ell}(R)\leq\operatorname{\bf{ggl}}(R)$. We now
determine upper and lower bounds for $\operatorname{\bf{ggl}}(R)$ in terms of
$\operatorname{\text{g}\ell\ell}(R)$ and the minimum and maximum degrees of
generators of $R$.

######  Proposition 4.3.

Let $(R,\mathfrak{m})$ be a positively-graded Noetherian $k$-algebra, where
$R_{0}=k$ and $\mathfrak{m}$ is the irrelevant ideal. Suppose
$x_{1},...,x_{n}\in\mathfrak{m}$ are homogeneous elements such that
$R=k[x_{1},...,x_{n}]$. Let

$\text{min}\\{\text{deg}(x_{i})\\}_{i=1}^{n}=a\leq
b=\text{max}\\{\text{deg}(x_{i})\\}_{i=1}^{n}.$

Then

$a(\operatorname{\text{g}\ell\ell}(R))-(a-1)^{2}\leq\operatorname{\bf{ggl}}(R)\leq
b(\operatorname{\text{g}\ell\ell}(R))-b+1.$

If $a=b=1$, then
$\operatorname{\bf{ggl}}(R)=\operatorname{\text{g}\ell\ell}(R)$.

###### Proof.

We claim that for $n\geq 0$, $\mathfrak{m}_{nb+1}\subseteq\mathfrak{m}^{n+1}$.
This is trivial when $n=0$. Suppose the inclusion holds for some $n\geq 0$.
Let $x\in\mathfrak{m}_{(n+1)b+1}$ be homogeneous, and suppose
$x=\sum\limits_{i=1}^{n}s_{i}x_{i}$, where each $s_{i}\in R$ is homogeneous.
Then $\text{deg}(s_{i})\geq(n+1)b+1-\text{deg}(x_{i})\geq nb+1.$ Therefore,
$s_{i}\in\mathfrak{m}_{nb+1}\subseteq\mathfrak{m}^{n+1}$, and
$x\in\mathfrak{m}^{n+2}$. This proves the claim. Let
$n=\operatorname{\text{g}\ell\ell}(R)-1$. Then by the above inclusion,
$\operatorname{\bf{ggl}}(R)\leq b(\operatorname{\text{g}\ell\ell}(R)-1)+1$.

Let $m=\operatorname{\bf{ggl}}(R)$. There exists an integer $c\geq 0$ and an
integer $0\leq l<a$ such that $m=ac+l$. It is clear that
$\mathfrak{m}^{i}\subseteq\mathfrak{m}_{ia}$ for $i\geq 0$. We claim that
$\mathfrak{m}^{i+j}\subseteq\mathfrak{m}_{ia+j}$ for $i,j\geq 0$. Fix $i$. If
the inclusion holds for some $j\geq 0$, then

$\mathfrak{m}^{i+j+1}=\mathfrak{m}\cdot\mathfrak{m}^{i+j}\subseteq\mathfrak{m}\cdot\mathfrak{m}_{ia+j}\subseteq\mathfrak{m}_{ia+j+1}.$

It follows that $\mathfrak{m}^{c+l}\subseteq\mathfrak{m}_{m}$ and $c+l\geq
n=\operatorname{\text{g}\ell\ell}(R)$. Since $ac+al\geq an$, we have

$\operatorname{\bf{ggl}}(R)\geq a(\operatorname{\text{g}\ell\ell}(R))-(a-1)l$

and

$\operatorname{\bf{ggl}}(R)\geq
a(\operatorname{\text{g}\ell\ell}(R))-(a-1)^{2}.$

∎

Let $H=\langle a_{1},...,a_{n}\rangle$ be the numerical semigroup with unique
minimal generating set $0<a_{1}<a_{2}<\cdot\cdot\cdot<a_{n}$, where
$\text{gcd}(a_{1},...,a_{n})=1$. Let $C$ denote the conductor of $H$, the
smallest integer $n\in H$ for which every integer larger than $n$ is also in
$H$. Define $k[H]:=k[t^{a_{1}},...,t^{a_{n}}]\subseteq k[t]$, where $k[H]$ is
positively-graded via $|t^{a}|=a$.

######  Proposition 4.4.

Let $R=k[H]$, where $H=\langle a_{1},...,a_{n}\rangle$. Then
$\operatorname{\bf{ggl}}(R)=C+a_{1}$.

###### Proof.

Let $\mathfrak{m}=(t^{a_{1}},...,t^{a_{n}})$. It is clear that
$\mathfrak{m}_{C+a_{1}}\subseteq(t^{a_{1}})$, so
$\operatorname{\bf{ggl}}(R)\leq C+a_{1}$. Let $n,d\geq 0$ and suppose
$\mathfrak{m}_{C+n}\subseteq(t^{d})$. This inclusion holds if and only if
$t^{C+n+i}\in(t^{d})$ for all $i\geq 0$, which is true if and only if
$C+n+i-d\in H$ for all $i\geq 0$. This is equivalent to the inequality
$C+n-d\geq C$, or $n\geq d$. Therefore,
$\mathfrak{m}_{C+a_{1}-1}\not\subseteq(t^{d})$ for all $d\in
H\setminus\\{0\\}$. It follows that $\operatorname{\bf{ggl}}(R)=C+a_{1}$. ∎

######  Corollary 4.5.

Let $R=k[t^{a},t^{b}]$, where $a<b$. Then $\operatorname{\bf{ggl}}(R)=ba-b+1$.

###### Proof.

The conductor of $\langle a,b\rangle$ is $ba-a-b+1$ [14, p.201]. ∎

Veliche notes that for $R=k\llbracket t^{a},t^{b}\rrbracket$, where $a<b$ and
$k$ is infinite, we have
$\operatorname{\text{g}\ell\ell}(R)=\text{index}(R)=a$ [16, p.3]. She then
determines formulas for the generalized Loewy lengths of Gorenstein local
numerical semigroup rings of embedding dimension at least three over infinite
fields. [16, Corollary 2.4, Corollary 3.3, Proposition 3.9]. If we know the
conductor of the semigroup that determines one of these rings, then the
generalized graded length of the corresponding graded ring is easier to
compute than the generalized Loewy length of this local ring.

######  Proposition 4.6.

Let $R=k[H]$, where $H=\langle a,b\rangle$ and $a<b.$ Suppose $z$ is a witness
to $\operatorname{\bf{ggl}}(R)$. Then $(z)=(t^{ia})$ for some $1\leq i\leq
1+b-a$.

###### Proof.

We have $R\cong k[x,y]/(x^{b}-y^{a})=k[\overline{x},\overline{y}]$, where
$\text{deg}(\overline{x})=a$, $\text{deg}(\overline{y})=b$, and
$\mathfrak{m}=(\overline{x},\overline{y})$. Let $z\in\mathfrak{m}$ be a
witness to $\operatorname{\bf{ggl}}(R)$. Suppose $z\in(\overline{y})$. By
Proposition 4.4, $\overline{x}$ is also a witness to
$\operatorname{\bf{ggl}}(R)$, so
$\mathfrak{m}_{ab-(b-1)}\subseteq(\overline{x})\cap(\overline{y})$. Since
$a<b$ are coprime, we have $b=as+r$ for some $s>0$ and $0<r<b$, so

$ab-(b-1)=a(as+r)-(as+r-1)=a((a-1)s+r)-(r-1).$

It follows that
$\overline{x}^{(a-1)s+r}\in(\overline{x})\,\cap\,(\overline{y})=(\overline{x}\overline{y},\overline{x}^{b})$.
Since $(a-1)s+r<b$, we have
$\overline{x}^{(a-1)s+r}=\overline{f}\overline{x}\overline{y}$ for some $f\in
k[x,y]$ and $x^{(a-1)s+r}-fxy=gx^{b}-gy^{a}$ for some $g\in k[x,y]$. It
follows that $fxy-gy^{a}=x^{(a-1)s+r}-gx^{b}$. Therefore,
$y\,|\,(x^{(a-1)s+r}-gx^{b})$. Since $(a-1)s+r<b$, this is false. We conclude
that $z\not\in(\overline{y})$.

Therefore, $z=\overline{x}^{i}$ for some $1\leq i<b$. For each $n\geq 1$, we
have $\mathfrak{m}^{n}\subseteq\mathfrak{m}_{na}$, so

$\mathfrak{m}^{b}\subseteq\mathfrak{m}_{ab}\subseteq\mathfrak{m}_{ab-(b-1)}.$

It follows that
$\mathfrak{n}^{b}\subseteq(x^{i},x^{b}-y^{a})S=(x^{i},y^{a})S$, where
$\mathfrak{n}=(x,y)S$, and $S$ denotes $k[x,y]$ with the standard grading. Let
$M$ denote the $k$-vector space generated by monomials of degree $b$ in
$(x^{i},y^{a})S$. Then

$\text{dim}_{k}M\leq\binom{2+(b-i)-1}{b-i}+\binom{2+(b-a)-1}{b-a}=2+2b-(a+i).$

On the other hand, the $k$-vector space generated by monomials of degree $b$
in $\mathfrak{n}^{b}$ has dimension $b+1$. It follows that $b+1\leq
2+2b-(a+i)$ and $i\leq 1+b-a$. ∎

######  Definition 4.7.

Let $(R,\mathfrak{m})$ be a positively-graded Noetherian $k$-algebra, where
$R_{0}=k$ and $\mathfrak{m}$ is the irrelevant ideal. Let $I\subseteq R$ be a
graded ideal. We say that $I$ is a graded reduction of $\mathfrak{m}$ of
degree $d$ if there is a positive integer $i$ such that
$I\mathfrak{m}_{i}=\mathfrak{m}_{i+d}$.

It is clear that for a numerical semigroup ring $k[t^{a_{1}},...,t^{a_{n}}]$,
the ideal $(t^{a_{1}})$ is a graded reduction of $\mathfrak{m}$ of degree
$a_{1}$. We therefore ask the following questions, which parallel a question
asked by De Stefani [3, Questions 4.5 (ii)].

######  Questions 4.8.

Suppose $R$ is a positively-graded Noetherian $k$-algebra. Is there a witness
to $\operatorname{\bf{ggl}}(R)$ that generates a graded reduction of
$\mathfrak{m}$? What can be said about the degree of such a graded reduction?

## Acknowledgements

I would like to thank my thesis advisor Graham Leuschke for his support and
insight into this topic, and for many helpful conversations. I would like to
thank Eloísa Grifo for her helpful suggestions for the abstract of this paper.
Finally, I would like to thank the referee for providing helpful comments and
suggestions for this paper.

## References


# Scattering between minivalleys in a moiré material

Petar Tomić<EMAIL_ADDRESS>Peter Rickhaus  Laboratory for Solid State
Physics, ETH Zürich, CH-8093 Zürich, Switzerland  Aitor Garcia-Ruiz  National
Graphene Institute, University of Manchester, Manchester M13 9PL, United
Kingdom  Giulia Zheng  Elías Portolés  Laboratory for Solid State Physics, ETH
Zürich, CH-8093 Zürich, Switzerland  Vladimir Fal’ko  National Graphene
Institute, University of Manchester, Manchester M13 9PL, United Kingdom  Henry
Royce Institute for Advanced Materials, M13 9PL, Manchester, UK  Kenji
Watanabe  Takashi Taniguchi  National Institute for Material Science, 1-1
Namiki, Tsukuba 305-0044, Japan  Klaus Ensslin  Thomas Ihn  Folkert K. de
Vries<EMAIL_ADDRESS>Laboratory for Solid State Physics, ETH Zürich,
CH-8093 Zürich, Switzerland

###### Abstract

A unique feature of the complex band structures of moiré materials is the
presence of minivalleys, their hybridization, and scattering between them.
Here we investigate magneto-transport oscillations caused by scattering
between minivalleys - a phenomenon analogous to magneto-intersubband
oscillations - in a twisted double bilayer graphene sample with a twist angle
of $1.94\text{\,}{}^{\circ}$. We study and discuss the potential scattering
mechanisms and find an electron-phonon mechanism and valley conserving
scattering to be likely. Finally, we discuss the relevance of our findings for
different materials and twist angles.

Two-dimensional moiré materials are formed by stacking van der Waals materials
such that the layers couple and an in-plane superlattice emerges. The
superlattice formed depends on the twist (and lattice mismatch) between the
layers. Graphene is a typical van der Waals material and has a honeycomb
lattice build up from two hexagonal sublattices. The wavefunctions of the two
valleys $K$ and $K^{\prime}$ in reciprocal space are sublattice polarized at
the Dirac point [1]. Similarly, for trigonal moiré lattices, such as twisted
graphene [2, 3], the wavefunctions of the minivalleys $\kappa$ and
$\kappa^{\prime}$ are polarized on the twisted layers when the interlayer
coupling is weak. Even though this is a generic property of trigonal moiré
lattices, little is known about the scattering of charge carriers between
these minivalleys.

We focus on twisted graphene since recently a plethora of correlated states
has been observed [4, 5, 6, 7, 8, 9, 10], and in view of these correlations,
scattering in twisted graphene is a highly interesting topic. Specifically, we
choose to work with twisted double bilayer graphene (TDBG) with weak coupling
between the layers, because this offers excellent control over the minivalley
occupation, and high quality electron transport. This system resembles that of
a weakly coupled double quantum well, where the two minivalleys around
$\kappa$ and $\kappa^{\prime}$ play the roles of the two subbands. Since the
wavefunctions of the minivalleys are mostly bilayer polarized, a dual gate
geometry provides independent control over the density in the two minivalleys,
and with that, their energetic alignment [11].

A common way to obtain the leading scattering mechanism is analyzing the
temperature dependence of the electrical resistivity [12]. Measurements of the
resistivity in magic-angle twisted bilayer graphene have shown a linear
temperature dependence, suggestive of electron-phonon scattering [13, 14] or
strange metallic behavior [15, 16]. As the scattering between minivalleys is
not necessarily the leading scattering mechanism, we introduce a more targeted
approach. When applying a magnetic field, Landau levels are formed in both
minivalleys separately. The energetic (mis)alignment of the modulated
densities of states in the two minivalleys leads to oscillations in the
interminivalley scattering. This oscillating interminivalley scattering is
reflected in electrical transport through an effect analogous to magneto
intersubband oscillations (MISO) [17, 18]. In the following, we will refer to
these magneto inter-minivalley oscillations as MISO, since they encapsulate
the same physical phenomenon. The method we present is transferable to other
moiré materials with a well developed Landau level spectrum as well as control
of the energetic alignment of the minivalleys.

Here, we report measurements of MISO in TDBG with a twist angle of
$1.94^{\circ}$ and investigate the interminivalley scattering in the regime of
hole-like states ($n<0$). We introduce the bandstructure and tunability of the
TDBG device by analyzing its Shubnikov-de Haas oscillations (SdHO) as a
function of density and displacement field. Then we investigate two regions
where MISO are particularly pronounced. First, towards the Lifshitz transition
[19], we study MISO as a function of temperature and displacement field and
discuss the scattering mechanism. Second, at the onset of the second
minivalley, we discuss implications of the underlying scattering mechanism
based on the observed valley degeneracy lifting. Finally, we discuss the
impact of our findings with regard to different materials and twist angles.

Figure 1: (a) Schematic overview of the device, including setup to measure the
longitudinal and Hall voltage ($V_{\mathrm{xx}}$ and $V_{\mathrm{xy}}$), and
the alignment of magnetic field $B$. Cross-section of the device featuring the
graphite, hexagonal boron nitride (hBN), TDBG, aluminium-oxide (AlOx), and
gold (Au) layers, and the top and bottom gates ($V_{\mathrm{tg}}$ and
$V_{\mathrm{bg}}$) indicated. (b,c) Calculated band structure of TDBG
($1.94\text{\,}{}^{\circ}$) for displacement field $D=0$ and
$D=$0.1\text{\,}\mathrm{V}\mathrm{/}\mathrm{n}\mathrm{m}$$, respectively.
Minivalleys $\kappa$ and $\kappa^{\prime}$ are indicated as well as the
regions A and B. (d) $R_{\mathrm{xx}}$ as a function of density $n$ and $D$ at
temperature $T=$1.2\text{\,}\mathrm{K}$$ and $B=$2\text{\,}\mathrm{T}$$.
Regions A and B are indicated. The overlayed colors bronze and gray indicate
single minivalley regimes. The LT is highlighted by the dashed line. (e,f)
Resistance modulation $\Delta R_{\mathrm{xx}}(n,D)$ at
$B=$2\text{\,}\mathrm{T}$$ and $T=$10\text{\,}\mathrm{K}$$ and
$T=$20\text{\,}\mathrm{K}$$, respectively. The density used in Figs. 2 and 3
is indicated with a black dashed line.

We fabricate a Hall bar device [Fig. 1(a)] from TDBG sandwiched between two
hexagonal boron nitride layers. A four-terminal current-bias setup is used to
obtain the longitudinal resistance $R_{\mathrm{xx}}$ and Hall resistance
$R_{\mathrm{xy}}$, utilizing standard lock-in techniques with
$I_{\mathrm{AC}}=$100\text{\,}\mathrm{n}\mathrm{A}$$. Crucially, we are able
to control the density $n$ and displacement field $D$ separately in the device
through voltages applied to the top gate ($V_{\mathrm{tg}}$) and bottom gate
($V_{\mathrm{bg}}$) (see Supplementary Material [20]). We investigate the
device in a $\mathrm{{}^{4}He}$ cryostat with a base temperature of
$T=$1.2\text{\,}\mathrm{K}$$, and implemented temperature control. The device
is identical to the one investigated in Ref. [21]. For details on the device
fabrication and data analysis see the Supplementary Material [20].

The bandstructure of TDBG with a twist angle of $1.94^{\circ}$ is presented in
Fig. 1(b). The bands in the mini-Brioullin zone show local minima, maxima and
band gaps around the two minivalleys $\kappa$ and $\kappa^{\prime}$. Without
an applied displacement field D the minivalleys are occupied equally (region
A). In contrast, when a displacement field is applied, the energetic alignment
of the minivalleys $\Delta$ is altered [11]. Independently tuning the Fermi
energy now allows one to only occupy the bands centered around
$\kappa^{\prime}$ [Fig. 1(c)], and reach asymmetric minivalley occupations
(region B).

The tunability of our device with respect to the occupation of different
minivalleys in the hole bands is investigated using SdHO. In Fig. 1(d) we show
$R_{\mathrm{xx}}$ as a function of displacement field $D$ and total density
$n$ measured at constant magnetic field $B=$2\text{\,}\mathrm{T}$$ and
temperature $T=$1.2\text{\,}\mathrm{K}$$. We observe a single set of SdHO in
the regions masked with bronze and gray in Fig. 1(d), indicating the
occupation of a single minivalley. This corresponds to the Fermi energy being
tuned into the band gap of either minivalley [Fig. 1(c)]. The blue region in
Fig. 1(d) corresponds to the configuration where both minivalleys $\kappa$ and
$\kappa^{\prime}$ are occupied, giving rise to a pattern of two sets of SdHO.
We reproduce this pattern within a simple model by considering screening
effects of the bilayers as presented in the Supplementary Material [20].
Finally, we highlight the Lifshitz transition [19] in Fig. 1(d) by drawing the
contour (dashed line) where $R_{\mathrm{xy}}=0$. At this line, the topology of
the Fermi surface changes as the Fermi energy crosses the saddle point in the
bandstructure at the $\mu$ point [see Fig. 1(b,c)] [11].

Interestingly, apart from the SdHO pattern, density independent resistance
minima cutting through the middle of the hexagons are observed indicated with
black arrows in the region labeled A in Fig. 1(d). Unlike SdHO, which are
strongly thermally damped, these resistance minima are more pronounced at
higher temperatures as seen at $T=$10\text{\,}\mathrm{K}$$ and
$T=$20\text{\,}\mathrm{K}$$ in Fig. 1(e,f), where a smooth background has been
subtracted for improved clarity of the presentation. These oscillations are
MISO [17, 18], caused by an oscillating interminivalley scattering rate. The
displacement field tunes the energy offset between the minivalleys, $\Delta$
(indicated in Fig 1(c)) that periodically changes the energetic alignment of
the Landau levels in the minivalleys. This results in a periodic modulation of
the interminivalley scattering rate that we observe as MISO in Fig. 1(d-f).

The unprecedented tunability of our TDBG device allows us to study MISO at
various possible relative energetic alignments of the minivalleys. In
particular, we observe enhanced MISO in two distinct regions denoted A and B
in Fig. 1(d-f). Region A is located towards the Lifshitz transition for
moderate values of displacement fields, and region B at the onset of the
second minivalley.

Figure 2: (a) Line traces of the $R_{\mathrm{xx}}-R_{\mathrm{xx,0}}$ (value of
$R_{\mathrm{xx}}$ at $D=0$) versus displacement field $D$ for various
temperatures $T$ as indicated, at constant
$n=$-2.25\text{\times}{10}^{12}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$ and
$B=$2\text{\,}\mathrm{T}$$. The traces are offset by $40\text{\,}\Omega$ and
the upper four traces are multiplied by 5 for clarity. Vertical dashed lines
mark positions of MISO as guides to the eye. (b) Resistance modulation $\Delta
R_{\mathrm{xx}}$ as a function of $D$ and $B$, taken at
$n=$-2.25\text{\times}{10}^{12}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$ and
$T=$20\text{\,}\mathrm{K}$$. The black dashed lines represent the result of
the basic MISO model, while gray and bronze dashed lines represent SdHO
originating from $\kappa$ and $\kappa^{\prime}$ minivalley respectively (see
Supplementary Material [20]).

We start discussing region A showing unambiguously the difference between MISO
and SdHO. Line traces of $R_{\mathrm{xx}}$ as a function of $D$ measured at
different temperatures and constant density
$n=$-2.25\text{\times}{10}^{12}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$ are shown
in Fig. 2(a) revealing the suppression of SdHO with temperature, while MISO
persist. At $T=$20\text{\,}\mathrm{K}$$, as in Fig. 1(f), only MISO are left,
where as at $T=$1.2\text{\,}\mathrm{K}$$, as in Fig. 1(d), SdHO are dominant
and MISO are slightly visible as well.

In order to show that the oscillation spectrum of MISO also differs from SdHO
we present the resistivity modulation $\Delta R_{\mathrm{xx}}$ as a function
of $B$ and $D$ at $T=$20\text{\,}\mathrm{K}$$ in Fig. 2(b). Note, displacement
field independent Azbel-Brown-Zak oscillations (ABZO) [22, 21] are present in
the whole magnetic field range. For comparison, we plot the results of a basic
MISO model (see Supplementary Material [20]) using dashed lines that highlight
the alignment of Landau levels from different minivalleys. The condition is
fulfilled when the energy spacing of an integer number of Landau levels fits
the energy offset between the minivalleys, i.e. when $\Delta(D)=k\hbar
eB/m_{\mathrm{eff}}$ where $k$ is an integer and
$m_{\mathrm{eff}}=0.06m_{\mathrm{e}}$ the effective mass. At
$T=$20\text{\,}\mathrm{K}$$ we estimate that only a few Landau levels
participate in transport. The model fits well for moderate $D$ and deviates at
higher $D$, where we suspect the alignment condition is broken because
$m_{\mathrm{eff}}$ in the minivalleys becomes dissimilar. This is in contrast
to semiconductors with mostly parabolic bandstructure, where larger numbers of
Landau levels can overlap at the same time. Finally, we like to point out the
MISO fan in Fig. 2(b) is different from SdHO Landau fans since it has its
origin at $D=0$ [not at finite $D$ as for SdHO in Supplemental Fig. 7], and it
fans out in both directions in $D$.

Generally MISO is considered to be caused by impurity scattering [18]. One
would therefore expect to see maxima in the resistivity when the Landau levels
from both minivalleys are aligned. Here, we observed the opposite. Observing
minima at MISO resonances indicates that phonon-assisted instead of impurity
intersubband scattering is the dominant mechanism [23]. We expect the
electron-phonon scattering to be quasi-elastic, because in Fig. 2(b) the
experimentally observed phase of the oscillation matches with the phase of the
Landau level alignment in the basic MISO model. Since we observe the MISO
resonances as resistance minima in the full temperature range [Fig. 2(a)], we
envision scattering by low energy phonons (e.g. flexural phonons) that have a
flat dispersion [24, 25].

Additionally, we measured the temperature dependence of the resistivity at
$B=0$ and $D=0$, which shows a linear behavior, indicative of an electron-
phonon mechanism. However, since the amplitude of the MISO is small compared
to the smooth background [e.g. $\Delta R_{\mathrm{xx}}/R_{\mathrm{xx,0}}\sim
1\%$ at $T=$20\text{\,}\mathrm{K}$$ in Fig. 2(a)], the interminivalley
scattering is probably not the dominant scattering mechanism determining the
resistance. In the Supplementary Material [20], apart from a detailed analysis
of the temperature-dependence of the data, we present a theoretical model that
reveals that intraminivalley electron-phonon scattering is likely to be the
dominant mechanism.

Figure 3: (a) $R_{\mathrm{xx}}(D,B)$ at $T=$5\text{\,}\mathrm{K}$$ and
$n=$-2.25\text{\times}{10}^{12}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$. We
highlight an example of SdHO from $\kappa$ and $\kappa^{\prime}$ carriers with
gray and bronze dashed lines, several ABZO by purple dashed lines, and the
split MISO peaks with the dark blue arrows. (b,c,d) Possible description of
splitting of the MISO peaks. The density of states (DOS) for both minivalleys
$\kappa$ and $\kappa^{\prime}$ are shown for $D<0$, $D>0$ and $D=0$,
respectively. The valley splitting and proposed scattering is indicated. (e)
Mini-Brillouin zones of TDBG with labeled valleys ($K$ and $K^{\prime}$) and
minivalleys ($\kappa$ and $\kappa^{\prime}$).

We continue by analyzing MISO as a function of displacement and magnetic field
at $T=$5\text{\,}\mathrm{K}$$ in Fig. 3(a), focusing on the same density as in
Fig. 2(b). The hexagonal pattern is formed by two sets of SdHO originating
from respective minivalleys, as indicated in Fig. 3(a), whose discontinuous
structure is a result of screening effects. Additionally, displacement field
independent ABZO are present throughout the whole magnetic field range.
Interestingly, instead of single resistance minima, as observed at the
alignment condition of MISO in Fig. 2(b), the MISO resonances are split into
two resistance minima at higher magnetic fields, as highlighted by the dark
blue arrows in Fig. 3(a), while the SdHO are not split. We extract an
effective Landé $g$-factor of $g\approx 5$, pointing towards a splitting of
the valley ($K$ and $K^{\prime}$) degeneracy rather than the spin [26].

The possible Landau level alignments taking into account valley splitting of a
single Landau level in each minivalley are schematically shown in Fig. 3(b-d),
where three typical energetic offsets are sketched. From this picture one
would expect to observe three MISO minima. However, since we only observe two
(configuration b and c), the interminivalley scattering must have a valley
selection criterion. Note that the MISO splitting is twice that of the Landau
levels, which makes it plausible that the splitting is not apparent in the
SdHO. In Fig. 3(e) we sketch the mini-Brillouin zones with respective valleys
and minivalleys. Considering the relative distance in $k$-space it is more
likely that the scattering is valley conserving. In combination with a quasi-
elastic scattering mechanism, this implies that the valley splitting in the
two graphene bilayers should be opposite and scattering is only allowed when
the same valleys line up, as sketched in Fig. 3(b,c). Future theoretical work
is needed to confirm this possible mechanism.

Figure 4: (a) $R_{\mathrm{xx}}(n,B)$ taken at
$D=$0.24\text{\,}\mathrm{V}\mathrm{/}\mathrm{n}\mathrm{m}$$ and
$T=$1.2\text{\,}\mathrm{K}$$. The inset shows $R_{\mathrm{xx}}$ as a function
of $B$, for constant second minivalley filling factors $\nu=\pm 2$, as
indicated by the dotted lines. (b) $R_{\mathrm{xx}}-R_{\mathrm{xx,0}}$ taken
at $D=$-0.2\text{\,}\mathrm{V}\mathrm{/}\mathrm{n}\mathrm{m}$$ as a function
of perpendicular component of the magnetic field $B_{\bot}$ for $\nu=\pm 2$
and $\alpha=0^{\circ},36.5^{\circ},54.5^{\circ}$

After analyzing the region of enhanced MISO in region A of Fig. 1(d), we shift
our attention to the region at the onset of the second minivalley occupation,
denoted by B in Fig. 1(d). In Fig. 4(a) we plot $R_{\mathrm{xx}}$ as a
function of $B$ and $n$, at finite displacement field and temperature of
$T=$1.2\text{\,}\mathrm{K}$$. At densities below
$n\sim$-1.5\text{\times}{10}^{12}$cm^{-2}$, which is the density onset of the
$\kappa$ minivalley, we measure the Landau fan of $\kappa^{\prime}$ minivalley
highlighted with bronze dashed lines in Fig. 4(a). Once the Landau levels of
the second minivalley appear, the levels in the already occupied
$\kappa^{\prime}$ minivalley (dotted bronze lines) split as indicated with the
bronze arrow in Fig. 4(a), signalling a transition from 4-fold to 2-fold
degenerate levels [see also inset of Fig. 4(a)]. To find out which degeneracy
(valley or spin) is lifted, we perform additional measurements, using a
different cryostat, for two different angles of the magnetic field with
respect to the normal of device [see Fig. 4(b)]. Since the splitting depends
on $B_{\bot}$ (and not on $|B|$), we speculate that the valley ($K$ and
$K^{\prime}$) degeneracy in the already occupied $\kappa^{\prime}$ minivalley
is lifted.

The lifting of the valley degeneracy once the second minivalley gets occupied
can be a result of electron-electron interactions, as it is not expected to
occur due to single particle effects [27]. A possible electron-electron
interaction at play is the valley exchange effect. Furthermore, due to high
effective mass at the onset of the second minivalley, screening effects can
improve the mobility leading to better line-width of SdHO hence revealing the
splitting in the spectrum. However, since we observe enhanced MISO, i.e.
stronger scattering, at the onset of the second minivalley, we speculate that
the observed enhanced inter-minivalley scattering leading to MISO in region B
of Fig. 1(d) is due to an electron-electron scattering mechanism.

In summary, by using MISO as a targeted measurement approach, we have
investigated interminivalley scattering in a moiré material. We found two
regions of enhanced MISO, and discussed the likely scattering mechanisms of
electron-phonon scattering with a valley selection rule in the vicinity of the
Lifshitz transition and electron-electron scattering at the onset of the
second minivalley. The described measurement technique is transferable to
other moiré materials [28], given that they adhere to the prerequisites of a
clear Landau level spectrum and sufficient tunability. These observations may
give a handle to refine theoretical models that aim to capture interactions in
moiré systems, such as magic-angle twisted bilayer graphene.

The data used in this Letter will be made available through the ETH Research
Collection.

###### Acknowledgements.

We acknowledge the support from Peter Maerki, Thomas Baehler and the staff of
the ETH FIRST cleanroom facility. We acknowledge support from the Graphene
Flagship, the Swiss National Science Foundation via NCCR Quantum Science and
from the European Union’s Horizon 2020 research and innovation programme under
grant agreement No 862660/QUANTUM E LEAPS. E.P acknowledges support of a
fellowship from ”la Caixa” Foundation (ID 100010434) under fellowship code
LCF/BQ/EU19/11710062. K.W. and T.T. acknowledge support from the Elemental
Strategy Initiative conducted by the MEXT, Japan, Grant Number
JPMXP0112101001, JSPS KAKENHI Grant Number JP20H00354 and the
CREST(JPMJCR15F3), JST.

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