# All electrical cooling of an optically levitated nanoparticle

Oscar Kremer  Department of Electrical Engineering, Pontifical Catholic
University of Rio de Janeiro, 22451-900 Rio de Janeiro, RJ, Brazil  Igor
Califrer  Department of Physics, Pontifical Catholic University of Rio de
Janeiro, Rio de Janeiro 22451-900, Brazil  Daniel Tandeitnik  Department of
Physics, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro
22451-900, Brazil  Jean Pierre von der Weid  Department of Electrical
Engineering, Pontifical Catholic University of Rio de Janeiro, 22451-900 Rio
de Janeiro, RJ, Brazil  Guilherme Temporão  Department of Electrical
Engineering, Pontifical Catholic University of Rio de Janeiro, 22451-900 Rio
de Janeiro, RJ, Brazil  Thiago Guerreiro<EMAIL_ADDRESS>Department of
Physics, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro
22451-900, Brazil

###### Abstract

We implement an all electrical controller for 3D feedback cooling of an
optically levitated nanoparticle capable of reaching sub-Kelvin temperatures
for the center of mass motion. The controller is based on an optimal policy
where state estimation is made by delayed position measurements. The method
offers a simplified path for pre-cooling and decoupling the transverse degrees
of freedom of the nanoparticle. Numerical simulations show that in an improved
setup with quantum limited detection, 3D ground state cooling and all
electrical quantum control can be achieved.

##  I Introduction

Optical tweezers [1] have emerged as a valuable tool for isolating and
controlling the motion of micro- and nano-objects [2, 3, 4]. By clever
combinations with electric and magnetic traps and actuators [5, 6, 7, 8, 9],
optical traps can be used to design highly sensitive sensors for force,
acceleration, and torque [10, 11, 12, 13, 14], with a high degree of control
enabling cooling of the center-of-mass motion of a levitated nanoparticle to
the ground state [15, 16, 17, 18]. Moreover, tweezers provide a versatile
platform for many-body [19, 20, 21, 22, 23] and fundamental physics
experiments, with applications in diverse areas such as stochastic
thermodynamics [24, 25, 26, 27, 28], nonlinear dynamics [29, 30, 27, 31, 32],
the search for new particles and forces of nature [33, 34, 35, 36, 37, 38],
and unprecedented tests of quantum mechanics [39, 40, 41, 42, 43]. All these
applications require the levitated object to be well isolated from its
surrounding environment, which is mainly limited by the vacuum quality of the
experiment, photon recoil heating [44], and black body radiation [45].
Regarding the vacuum quality, since the nano-object is initially trapped at
atmospheric pressure, it is thermalized at room temperature, preventing stable
trapping at low pressures and rendering the trapping potential nonlinear due
to large thermal fluctuations [29]. Therefore, cooling the object’s motion is
often a prerequisite for levitation experiments.

Active feedback cooling [46, 47], in particular parametric cooling, has
emerged as the standard technique for achieving 3D cooling of the levitated
nanoparticle’s motion [48], enabling temperatures as low as sub-mK [44]. In
practice, parametric control techniques are often used as a precooling
mechanism. The performance of parametric feedback, however, comes at the cost
of employing a nonlinear control protocol which modulates a portion of the
optical trapping power according to the resonance frequencies of the
nanoparticle. In addition, expensive electro- (EOM) or acousto-optic (AOM)
modulators must be used in combination with lock-in devices capable of
modulating a signal locked to the particle’s motion. Alongside the parametric
control, once the thermal occupation number has been reduced to around
$10^{3}$, the levitated object’s charge can be exploited to further control
its motion along one direction to even lower temperatures all the way into the
quantum ground state [17, 18].

In this letter we explore an all electrical approach to pre-cool the motion of
a levitated nanoparticle from room temperature to a point where the trap’s
nonlinear features are significantly reduced and stable trapping can be
achieved in high-vacuum ($p<$
$e{-3}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$). To do so, we design
a simple electric actuator based on a custom made printed circuit board (PCB),
capable of influencing the particle’s motion via Coulomb forces. Fine
alignment of the PCB with the levitated nanoparticle is not required. After a
careful calibration of the electrical forces, we employ a delayed feedback
scheme to 3D cool the center-of-mass (CoM) motion of the particle. We
experimentally measure the effect of the delay in the feedback force and show
excellent agreement with theoretical predictions [49]. Finally, we
successfully demonstrate 3D cooling down to sub-Kelvin temperatures while
completely avoiding modulation of the trap’s power, in a first step towards
the larger effort of simplifying optomechanical cooling experiments. With
numerical simulations based on our electrical actuator we argue that, in
combination with a stiffer optical trap, quantum-limited detection for all
three axes [50] and optimal quantum state estimation [51, 52, 53], all
electrical 3D ground state cooling can be achieved in our setup.

We highlight that 3D electrical feedback cooling of levitated nanoparticles
has been recently implemented in levitated optomechanics experiments – see
[54, 55, 56] for examples using integrated chip photonics, hybrid optical Paul
trap and finely aligned electrode tips. Our setup adds a simplified solution
to that list, while still offering the possibility of 3D quantum control of a
levitated nanoparticle. This paper is organized as follows. In Sec. II we
briefly describe the equations of motion and the Linear Quadratic Regulator
(LQR), used to evaluate the optimal proportional and derivative gains used in
the control feedback. Next, Sec. III describes the experimental setup, while
IV shows the results on all electrical feedback cooling and the prospects for
3D ground state cooling. We conclude in Sec. V with a brief discussion.

##  II Theory

The CoM motion along the $x$, $y$ and $z$-axes of an optically levitated
nanoparticle trapped by a strongly focused Gaussian beam can be effectively
modeled through a set of second-order Langevin equations,

$\displaystyle\ddot{x}(t)+\gamma_{m}\dot{x}(t)+\Omega_{x}^{2}x(t)=\frac{1}{m}F_{\text{th},x}(t)+b_{x}u_{x},$
(1a)
$\displaystyle\ddot{y}(t)+\gamma_{m}\dot{y}(t)+\Omega_{y}^{2}y(t)=\frac{1}{m}F_{\text{th},y}(t)+b_{y}u_{y},$
(1b)
$\displaystyle\ddot{z}(t)+\gamma_{m}\dot{z}(t)+\Omega_{z}^{2}z(t)=\frac{1}{m}F_{\text{th},z}(t)+b_{z}u_{z},$
(1c)

where $m$ is the particle’s mass, $\gamma_{m}$ the drag coefficient,
$\Omega_{i}$ the angular frequency along the $i$-axis and $F_{\text{th},i}$
represents the (white-noise) stochastic force on each axis due to residual gas
pressure in the vacuum chamber, satisfying

$\displaystyle\langle F_{\text{th},i}(t)\rangle$ $\displaystyle=0,$ (2a)
$\displaystyle\langle F_{\text{th},i}(t)F_{\text{th},j}(t+\tau)\rangle$
$\displaystyle=2m\gamma_{m}k_{B}T\delta_{ij}\delta(\tau),$ (2b)

where $k_{B}$ is the Boltzmann constant, $T$ the residual gas temperature,
$\delta_{ij}$ is the Kronecker delta and $i,j\in\\{x,y,z\\}$. The $b_{i}u_{i}$
terms in Eqs. (1) account for external forces that may influence the
particle’s motion, with $u_{i}$ representing the control signals defining
feedback forces acting on the trapped particle.

By defining the state vector

$\mathbf{x}(t)\equiv\begin{bmatrix}x(t)&y(t)&z(t)&\dot{x}(t)&\dot{y}(t)&\dot{z}(t)\end{bmatrix}^{T},$
(3)

one can then write Eqs. (1) in the state-variable representation [57],
resulting in the Multiple-Input-Multiple-Output (MIMO) system

$\mathbf{\dot{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t)+\mathbf{w}(t),$
(4)

where

$\mathbf{A}=\begin{bmatrix}\mathbf{0}_{3\times 3}&\mathbf{I}_{3}\\\
-\textrm{diag}(\mathbf{\Omega}^{2})&-\gamma_{m}\mathbf{I}_{3}\\\
\end{bmatrix},\,\mathbf{w}(t)=\frac{1}{m}\begin{bmatrix}\mathbf{0}_{3\times
1}\\\ \mathbf{F}_{\rm{th}}(t)\\\ \end{bmatrix},$ (5)

and

$\mathbf{B}=\begin{bmatrix}\mathbf{0}_{3\times 3}\\\
\mathrm{diag}(b_{x},b_{y},b_{z})\end{bmatrix},\,\mathbf{u}=\begin{bmatrix}u_{x}\\\
u_{y}\\\ u_{z}\end{bmatrix},$ (6)

with
$\mathbf{\Omega}^{2}=\begin{bmatrix}\Omega_{x}^{2}&\Omega_{y}^{2}&\Omega_{z}^{2}\end{bmatrix}^{T}$
and
$\mathbf{F}_{\rm{th}}(t)=\begin{bmatrix}F_{\rm{th},x}(t)&F_{\rm{th},y}(t)&F_{\rm{th},z}(t)\end{bmatrix}^{T}$.
Note that due to the geometry of the feedback actuators in our experiment, the
submatrix in $\mathbf{B}$ is not block diagonal, but assumes a more
complicated form; see Sec. III for more details.

Figure 1: Experimental setup. a) Simplified scheme of the setup. An optical
tweezer is assembled within a vacuum chamber, and a CCD is used for imaging of
the tweezed particle upon illumination with a
$532\text{\,}\mathrm{n}\mathrm{m}$ laser beam. The trapping lens is grounded,
and detection of forward-scattered light is used to generate the electrical
feedback signal sent to the electrodes. The collection lens works as the
$z$-electrode, whilst the board shown in b) is placed close to the trap’s
focus and contains the $x$\- and $y$-electrodes. The axes at the top left
indicate the orientation between the electrodes’ axes ($x^{\prime}$,
$y^{\prime}$) and the coordinate system of the detection.

Optimal control theory provides tools to find a control policy $\mathbf{u}(t)$
capable of minimizing the energy of a physical system. For linear systems,
such as the one described by Eq. (4), this is achieved by the LQR, a
controller where the optimization task targets the minimization of a quadratic
cost criterion $J$ of the form

$J=\frac{1}{2}\int_{0}^{\infty}[\mathbf{x}^{T}(t)\mathbf{Q}\mathbf{x}(t)+\mathbf{u}^{T}(t)\mathbf{R}\mathbf{u}(t)]\,dt,$
(7)

where $\mathbf{Q}$ is the weighting matrix and $\mathbf{R}$ is the control
effort matrix. The optimal control policy which minimizes Eq.(7) is [58]

$\mathbf{u}=-\mathbf{K}\mathbf{x},$ (8)

where $\mathbf{K}=\mathbf{R}^{-1}\mathbf{B}\mathbf{S}$ is the controller’s
gain matrix and $\mathbf{S}$ is the solution of the algebraic Riccati equation

$\mathbf{S}\mathbf{A}+\mathbf{A}^{T}\mathbf{S}+\mathbf{Q}-\mathbf{S}\mathbf{B}\mathbf{R}^{-1}\mathbf{B}^{T}\mathbf{S}=\mathbf{0}.$
(9)

Practical application of the LQR poses the significant challenge of obtaining
the complete state vector $\mathbf{x}$. Experimentally, access is not granted
to $\mathbf{x}$ but rather to a measurement vector $\mathbf{y}$, which is
related to the states according to

$\mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{m}(t),$ (10)

where $\mathbf{C}$ is known as the output matrix. The term $\mathbf{m}$ is the
measurement noise vector and can be expressed as
$\mathbf{m}=\begin{bmatrix}\zeta_{x}(t)&\zeta_{y}(t)&\zeta_{z}(t)\end{bmatrix}^{T}$.
Here $\zeta_{i}(t)$ are zero-mean white-noise processes with variance
$\sigma_{i}^{2}$, satisfying

$\displaystyle\langle\zeta_{i}(t)\rangle$ $\displaystyle=0,$ (11a)
$\displaystyle\langle\zeta_{i}(t)\zeta_{j}(t+\tau)\rangle$
$\displaystyle=\sigma_{i}^{2}\delta_{ij}\delta(\tau).$ (11b)

On the one hand, measurements of $x(t)$, $y(t)$ and $z(t)$ can be implemented
by collecting forward or backward-scattered light from the nanoparticle [50].
On the other hand, the velocities are not accessible experimentally. An
optimal estimation $\mathbf{\hat{x}}$ can be computed by applying real-time
filtering techniques to estimate $\mathbf{x}$. For linear dynamics where the
disturbances and measurement noises adhere to Eq. (2) and (11), $\mathbf{x}$
is best estimated using the Kalman filter [59, 60].

Implementing the Kalman filter significantly increases the complexity of the
feedback loop. As a simplification, it is possible to estimate the velocity as
being proportional to a delayed position measurement. This approach has proven
successful for cooling one of the spatial degrees of freedom of the levitated
nanoparticle [46], albeit increasing the minimal effective temperature
achievable. The effective temperature for each axis can be computed by using
the integral [18]

$T_{\rm{eff}}^{i}=\frac{m\Omega_{i}^{2}}{k_{B}}\int_{0}^{\infty}\Bigg{(}1+\frac{\Omega^{2}}{\Omega_{i}^{2}}\Bigg{)}S_{ii}(\Omega)\,d\Omega-\frac{1}{2},$
(12)

where $S_{ii}$ is the double-sided Power Spectral Density (PSD) for the
particle’s motion along the $i$-axis, expressed as

$S_{ii}=\frac{2\gamma_{m}k_{B}T}{m[(\Omega^{2}-\Omega_{i}^{2})^{2}+\gamma_{m}^{2}\Omega_{i}^{2}]}.$
(13)

##  III Experiment

The experimental setup is schematically illustrated in Fig. 1a. A CW laser at
$1550\text{\,}\mathrm{n}\mathrm{m}$ (RIO Orion) amplified by an Erbium-doped
fiber amplifier (Keopsys CEFA-C-BO-HP-SM) is used to produce a high-quality
Gaussian beam linearly polarized along the $x$ direction with a power of
$P_{t}\approx$2\text{\,}\mathrm{W}$$, at the output of a single-mode fiber.
The beam is focused by an aspheric lens (Thorlabs C330TM-C, NA = 0.68)
assembled inside a vacuum chamber, allowing for stable optical trapping. The
light scattered by the particle along the forward direction is collimated by a
collecting lens (Thorlabs C110TM-C, NA = 0.40). Silica nanoparticles (diameter
$143\text{\,}\mathrm{n}\mathrm{m}$, MicroParticles GmbH) are loaded into the
vacuum chamber by a nebulizer and trapped at atmospheric pressure. The trapped
particle oscillates with resonance frequencies along the three axes given by
$\Omega_{x}/2\pi=$96.24\text{\,}\mathrm{k}\mathrm{Hz}$$,
$\Omega_{y}/2\pi=$101.49\text{\,}\mathrm{k}\mathrm{Hz}$$ and
$\Omega_{z}/2\pi=$31.52\text{\,}\mathrm{k}\mathrm{Hz}$$.

Figure 2: Effect of delayed feedback forces. Comparison between experimental
results and theory (solid lines) is presented. Measurements were conducted at
room temperature ($293\text{\,}\mathrm{K}$) and a pressure of
$1.2\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$. Each data point
corresponds to 10,000 $50\text{\,}\mathrm{m}\mathrm{s}$-traces. The used gains
were $G_{x}=$(9.17\pm 0.98)\times
10^{-9}\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$$ and $G_{y}=$(8.97\pm
0.97)\times 10^{-9}\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$$. The gray shaded
area marks the region that could not be measured due to the minimal delay
imposed by the electronics. The horizontal axis, $\phi$, represents the phase
$\Omega_{i}\tau_{i}$ introduced by the delay. In the inset, the interval where
the delay induces cooling is presented with more detail.

Detection of transversal motion, $x(t)$ and $y(t)$, is carried out using
balanced photodiodes (Newport 2117-FC), while information about the
longitudinal $z(t)$ direction is obtained by direct intensity photodetection.
The optical trap is characterized through measurements of the particle’s
position PSDs for each direction. Information on the occupation numbers and
effective temperatures of each direction can also be obtained from the PSDs by
using Eq. (12).

A PCB containing two orthogonal pairs of electrodes, illustrated in Fig. 1b),
is placed in the vicinity of the optical trap’s focus, allowing for two-
dimensional electrical feedback control of the nanoparticle’s CoM motion. The
PCB is designed to be compatible with cage plate optical systems (Thorlabs
SP02). Note also that only coarse alignment of the PCB with respect to the
levitated nanoparticle is required, and this can be achieved by placing the
PCB near the optical focus. Due to the employed control method, coupling
between degrees of freedom in the transverse plane is compensated by the
calibration process.

Figure 3: All electrical cooling. a) Dependence between pressure and $x$, $y$
and $z$ effective temperatures. The grey shaded region shows a region of
instability, as discussed in the main text. b) PSD of the y motion.
Measurements were made at
$1.0\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$( ), $5.4\times
10^{-2}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$ ( ) and $1.2\times
10^{-4}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$ ( ).

A third pair of electrodes is implemented by applying an electric signal to
the mount of the collection lens, producing a voltage difference with respect
to the grounded trapping lens. Simulations conducted using the finite-elements
method have numerically demonstrated that this voltage difference establishes
a uniform electric field near the particle’s position [31]. The signal from
the detection is digitally processed by two FPGAs (STEMlab 125-14, Red Pitaya)
and analogically amplified before being fed back to the electrodes. We remove
any cross-talk between $z$ and $xy$ electrodes by digital filtering, which is
facilitated by the difference in characteristic frequencies between the
longitudinal and transversal degrees of freedom. Taking this and the geometry
of the actuators into consideration, the gain matrix assumes a block diagonal
form,

$\mathbf{K}=\begin{bmatrix}\mathbf{K}_{p,xy}&0&\mathbf{K}_{d,xy}&0\\\
\mathbf{0}_{1\times 2}&k_{p,z}&\mathbf{0}_{1\times 2}&k_{d,z}\end{bmatrix}.$
(14)

A detailed description of the analogical amplification and the digital
processing of the detection signal can be found in Appendix B. Digital
processing includes frequency filtering, delaying and application of
derivative/proportional gains to the signal. The choice of optimal gains was
based on control theory, as presented in Appendix A. Since the theory predicts
only a weak dependence of optimal gain on pressure, we consider a single gain
to be optimal throughout the experiment.

Appropriate calibration of the electrodes accounts for misalignment between
the electrodes’ axes and the mechanical modes, allowing for a partial
reconstruction of the $\mathbf{B}$ matrix, which assumes a $45^{\circ}$
rotated form with respect to the diagonal matrix given by Eqs. (1). During
calibration, the effect of the $z$-electrode was observed to be too weak, such
that only the $x$\- and $y$-electrodes could be calibrated. This has led to
applying the control LQR only to the $x$ and $y$ motion and a cold damping
protocol [47, 18] along the $z$ direction ($k_{p,z}=0$). We refer to Appendix
C for more information on the calibration procedure.

##  IV Results

Proper implementation of the control method as previously described requires
precisely delaying each detection signal. The delay characterization process
involves applying a force proportional to the delayed position independently
in the $x$ and $y$ directions. For instance, referring to Eq. (1), this
translates to $u_{x}=G_{x}x(t-\tau_{x})$ for the $x$ coordinate (and similarly
for $y$ and $z$). Each delay $\tau_{i}$ consists of two components, the
intrinsic electronic delay $\tau_{e,i}$, and an adjustable delay $\tau_{c,i}$.
Fig. 2 shows measures of $T_{\rm{eff}}^{x}$ and $T_{\rm{eff}}^{y}$ while
subjecting the particle to the delayed force. The controllable delay
$\tau_{c,i}$ was varied to span the range of $\tau_{i}$ from $\tau_{e,i}$ to
one period of oscillation ($\phi=2\pi$). The experimental results show
excellent agreement with the theoretical predictions from [49]. Furthermore,
this measurement allowed for the characterization of the electronic delays,
$\tau_{e,x}$ and $\tau_{e,y}$, both of which were determined to be
$0.639\text{\,}\mu\mathrm{s}$. We assume $\tau_{e,z}$ has the same value.

Figure 3.a) shows the results of 3D feedback cooling. The minimal effective
temperatures achieved in the experiment are $T_{\rm{eff}}^{x}=$(0.58\pm
0.12)\text{\,}\mathrm{K}$$, $T_{\rm{eff}}^{y}=$(0.55\pm
0.11)\text{\,}\mathrm{K}$$ and $T_{\rm{eff}}^{z}=$(3.63\pm
0.77)\text{\,}\mathrm{K}$$, for each of the three axes. The gray shaded area
in Fig. 3a) depicts an instability region observed near
$10^{-2}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, characterized by a
sudden increase in $T_{\rm{eff}}^{i}$. We attribute this phenomenon to
variations on the net charge of the nanoparticle [61]. The net charge acts as
a linear parameter affecting the input matrix, thus linearly impacting the
control gain. As electrode calibration was performed at high pressure
($>$1\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$$), for pressures
smaller than $0.01\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, it
cannot be assumed that the applied gain was optimal. Nonetheless, stable
cooling has been implemented by using only electrical actuators and the
application of LQR returned a gain matrix capable of handling any coupling
between degrees of freedom in the dynamics. The PSD of the CoM motion for the
$y$ direction under three distinct pressures is shown in Fig. 3.b). Feedback
cooling not only reduces the area of the PSD, from which the effective
temperatures are estimated, but also introduces a term which increases its
linewidth, as expected due to the presence of derivative terms in the
nanoparticle’s motion.

For pressures smaller than
$0.01\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, no instability has
been encountered, agreeing with results previously shown [62]. Therefore, the
control protocol employed should be capable of successfully controlling the
nanoparticle until the stochastic thermal force becomes negligible and the
dynamics starts to be dominated by measurement back-action and photon recoil
heating. When compared to parametric cooling, an all electrical approach is
advantageous since it avoids contamination of the signal by spurious
modulation signals, which are rendered unnecessary. Additionally, in contrast
to parametric cooling, the LQR employs a linear control law, thus not
affecting the overall linearity of the system.

Figure 4: Simulation of optimal all electrical 3D cooling with improved
trapping lens and detection scheme: expected thermal occupation numbers,
$\bar{n}$, as a function of pressure for the $x,y$ and $z$ directions. Dashed
line marks a single phonon. Error bars correspond to one standard deviation
over 30 simulation runs.

Since the LQR has been successfully employed in combination with Kalman filter
for ground-state cooling along the longitudinal axis [17], extending its
application as a 3D quantum control policy should be experimentally
achievable. By considering the electrode parameters presented in Appendix C
and the trapping and detection efficiency parameters reported in [17], we
numerically simulated 3D all electrical cooling of a trapped nanoparticle.
Figure 4 presents the expected final mean occupation numbers with our all
electrical controller. To account for quantum effects, the same parameters of
measurement uncertainty, detection efficiency $\eta_{z}$ and backaction
provided in [17] were taken into account in the simulation. Note that the
simulation considers a backward detection scheme, resulting in a higher
detection efficiency for the longitudinal axis compared to the transversal
axes, thereby leading to a smaller thermal occupancy for $z$. In contrast, the
experiment employed a forward detection scheme, therefore yielding the
opposite effect due to limited detection efficiency [50]. The detection
efficiency along the transverse axes, $\eta_{x}$ and $\eta_{y}$, were computed
by considering the expected proportion between the efficiency along $x$ and
$y$ and the longitudinal directions for the corresponding trap’s NA [50]. For
pressures on the order of
$1e{-10}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, the simulation
results agree with the experimental findings in [17]. It must be noted that,
for higher pressures, we expect that experimental imperfections increase the
minimum number of phonons . Moreover, while it is evident that in simulation
the thermal occupancy for $y$ exceeds that of $x$, the experimental results in
Fig. 3.a) shows the opposite. This most likely arises from experimental
imperfections due to detection efficiency in the $x$-axis.

##  V Conclusions

In conclusion, we have demonstrated an all electrical feedback cooling scheme
for reducing the CoM temperature of a levitated nanoparticle in high vacuum.
Through a simple custom-designed electrical actuator, we have shown sub-Kelvin
temperatures for the transverse directions of motion, avoiding the use of
nonlinear feedback cooling schemes such as parametric feedback cooling. This
greatly simplifies levitated optomechanics experiments by avoiding the need
for modulation of the trapping power. Numerical simulations point that future
improvements over our setup, in particular implementation of a higher NA
trapping lens and of the optimal backward detection scheme reported in [17],
should enable all electrical 3D cooling near the ground state, reaching
thermal occupation numbers below unity.

## Acknowledgments

We acknowledge Bruno Melo and Felipe Almeida for useful discussions. T.G.
acknowledges the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -
Brasil (CAPES) - Finance Code 001, Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq), Fundação de Amparo à Pesquisa do Estado do
Rio de Janeiro (FAPERJ Scholarship No. E-26/200.252/2023) and Fundação de
Amparo à Pesquisa do Estado de São Paulo (FAPESP process No. 2021/06736-5).
This work was supported by the Serrapilheira Institute (grant No. Serra –
2211-42299).

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##  Appendix A Digital control theory

In the study and analysis of physical systems, time is conventionally treated
as a continuous variable. However, when employing signal processing and
control methods, a transition to a discrete representation becomes necessary.
This is specially crucial when implementing systems on microprocessors or
FPGAs, where a set of instructions is executed based on a sampling frequency
$f_{s}$ [63]. The discrete-time formulation of a state-space model can be
obtained through the integral approximation, which is based on the assumption
that $\mathbf{x}$ and $\mathbf{u}$ remain constant during a sampling period
$T_{s}=1/f_{s}$. The system evolution is then considered to unfold at fixed
time-steps $t_{n}=nT_{s}$, leading to the following recursive equations

$\displaystyle\mathbf{x}_{n+1}$
$\displaystyle=\mathbf{A}_{d}\mathbf{x}_{n}+\mathbf{B}_{d}\mathbf{u}_{n}+\bar{\mathbf{w}}_{n},$
(15a) $\displaystyle\mathbf{y}_{n+1}$
$\displaystyle=\mathbf{C}_{d}\mathbf{x}_{n+1}+\bar{\mathbf{m}}_{n+1},$ (15b)

where $\mathbf{A}_{d}$, $\mathbf{B}_{d}$ and $\mathbf{C}_{d}$ can be expressed
in terms of their continuous analogues,

$\displaystyle\mathbf{A}_{d}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{T_{s}^{k}}{k!}\mathbf{A}^{k},$ (16a)
$\displaystyle\mathbf{B}_{d}$
$\displaystyle=(\mathbf{A}_{d}-\mathbf{I})\mathbf{A}^{-1}\mathbf{B},$ (16b)
$\displaystyle\mathbf{C}_{d}$ $\displaystyle=\mathbf{C}.$ (16c)

Also, $\mathbf{x}_{n}=\mathbf{x}(nT_{s})$ and
$\mathbf{u}_{n}=\mathbf{u}(nT_{s})$. The discrete disturbance and noise terms,
$\bar{\mathbf{w}}_{n}$ and $\bar{\mathbf{m}}_{n}$, represent discrete-time
white-noise processes adhering to conditions akin to those established in Eqs.
(2) and (11) in the main text. Considering
$\bar{\mathbf{w}}_{n}=\frac{1}{m}\begin{bmatrix}\mathbf{0}_{3\times
1}&\bar{\mathbf{F}}_{\rm{th},n}\end{bmatrix}^{T}$, with
$\bar{\mathbf{F}}_{\rm{th},n}=\begin{bmatrix}\bar{F}_{\rm{th},x,n}&\bar{F}_{\rm{th},y,n}&\bar{F}_{\rm{th},z,n}\end{bmatrix}^{T}$
and
$\bar{\mathbf{m}}_{n}=\begin{bmatrix}\bar{\zeta}_{x,n}&\bar{\zeta}_{y,n}&\bar{\zeta}_{z,n}\end{bmatrix}^{T}$,
the conditions are

$\displaystyle\langle\bar{F}_{\rm{th},i,k}\rangle$ $\displaystyle=0,$ (17a)
$\displaystyle\langle\bar{F}_{\rm{th},i,k}F_{\rm{th},j,k^{\prime}}\rangle$
$\displaystyle=2m\gamma_{m}k_{B}TT_{s}\delta_{ij}\delta_{kk^{\prime}},$ (17b)

and

$\displaystyle\langle\bar{\zeta}_{i,k}(t)\rangle$ $\displaystyle=0,$ (18a)
$\displaystyle\langle\bar{\zeta}_{i,k}\bar{\zeta}_{j,k^{\prime}}\rangle$
$\displaystyle=\frac{\sigma_{i}}{T_{s}}\delta_{ij}\delta_{kk^{\prime}}.$ (18b)

Similar to its continuous version, the LQR for discrete-time systems returns
an optimal control law, expressed as a linear combination on the states
$\mathbf{x}_{n}$,

$\mathbf{u}_{n}=-\mathbf{K}_{d}\mathbf{x}_{n},$ (19)

however, the expression for the controller’s gain changes to

$\mathbf{K}_{d}=(\mathbf{R}_{d}+\mathbf{B}_{d}^{T}\mathbf{S}_{d}\mathbf{B}_{d})^{-1}\mathbf{B}_{d}^{T}\mathbf{S}_{d}\mathbf{A}_{d},$
(20)

where $\mathbf{S}_{d}$ is the solution of the discrete algebraic Ricatti
equation

$\begin{split}\mathbf{S}_{d}&=\mathbf{A}_{d}^{T}\mathbf{S}_{d}\mathbf{A}_{d}+\mathbf{Q}_{d}\\\
&-\mathbf{A}_{d}^{T}\mathbf{S}_{d}\mathbf{B}_{d}(\mathbf{R}_{d}+\mathbf{B}_{d}^{T}\mathbf{S}_{d}\mathbf{B}_{d})^{-1}\mathbf{B}_{d}^{T}\mathbf{S}_{d}\mathbf{A}_{d},\end{split}$
(21)

and $\mathbf{Q}_{d}$ and $\mathbf{R}_{d}$ are the matrices defining the cost
function $J_{d}$ for the digital control law, which reads

$J_{d}=\frac{1}{2}\sum_{n=0}^{\infty}[\mathbf{x}_{n}^{T}\mathbf{Q}_{d}\mathbf{x}_{n}+\mathbf{u}_{n}^{T}\mathbf{R}_{d}\mathbf{u}_{n}].$
(22)

##  Appendix B Electronic setup

The control law defined in Eq. (19) was implemented using two Red Pitayas,
each equipped with a Xilinx Zynq 7010 FPGA and a 2 channel 14-bits ADC,
allowing for a maximum sampling frequency of
$125\text{\,}\mathrm{M}\mathrm{Hz}$ for two distinct inputs, $x_{a}$ and
$x_{b}$. The feedback loop incorporated a decimation block, increasing the
sampling time $T_{s}$ from $8.00\text{\,}\mathrm{n}\mathrm{s}$ to
$64.00\text{\,}\mathrm{n}\mathrm{s}$, enabling synchronous execution of more
complex tasks.

Figure 5: Digital electronic implementation. a) Block diagram illustrating the
FPGA implementation for stable control of the particle CoM motion. The digital
filters are responsible for signal conditioning. A Block Random Access Memory
allows the implementation of delay blocks, delaying the signal in multiples
($N_{a}$, $N_{b}$) of the sampling time. The delayed and non-delayed filtered
signals are then transmitted to the controllers to compute the output signals.
b) Bode plots for each notch filter $H_{z}$ and $H_{xy}$, depicting their
magnitude and phase behavior for the frequency range of interest.

In Fig. 5 a simplified block diagram of the main components implemented within
each FPGA is shown. The controller block is responsible for computing the
output signal $u_{a,n}$ and $u_{b,n}$, being equivalent to the following
expression

$\begin{bmatrix}u_{a,n}\\\
u_{b,n}\end{bmatrix}=\begin{bmatrix}k_{p,aa}^{d}&k_{p,ab}^{d}&k_{d,aa}^{d}&k_{d,ab}^{d}\\\
k_{p,ba}^{d}&k_{p,bb}^{d}&k_{d,ba}^{d}&k_{d,bb}^{d}\end{bmatrix}\begin{bmatrix}\tilde{x}_{a,n}\\\
\tilde{x}_{b,n}\\\ \tilde{x}_{a,n-N_{a}}\\\
\tilde{x}_{b,n-N_{b}}\end{bmatrix}.$ (23)

The signals $\tilde{x}_{a,n}$, $\tilde{x}_{b,n}$ result from passing the
inputs through a D.C block and a notch filter, both implemented by using
digital biquadratic filters. The constant $k_{p,ij}^{d}$ and $k_{d,ij}^{d}$
refer to the digital proportional and derivative gains. The signals
$\tilde{x}_{a,n-N_{a}}$, $\tilde{x}_{b,n-N_{b}}$ are the delayed positions,
serving as estimates of the particle’s velocity.

The notch filter transfer function is shown in Fig. 5.b). For the FPGA
processing the $x$ and $y$ signals, the transfer function used was $H_{xy}$ to
remove harmonic components near $\Omega_{z}$. In the other FPGA, a filter
$H_{z}$ was applied to remove any components sufficiently close to
$\Omega_{x}$ and to $\Omega_{y}$. The filter’s impact on the phase of each
signal is approximately constant near each resonance frequency, being included
in the overall intrinsic delay of the electronic setup, already described in
Section IV. The computed control signals were sent to non-inverting analog
amplifiers, providing a constant gain
$A=$5.00\text{\,}\mathrm{V}\mathrm{/}\mathrm{V}$$ with minimal phase impact
for signals with harmonic components from D.C up to
$150.00\text{\,}\mathrm{k}\mathrm{Hz}$.

##  Appendix C Model parameters

Implementation of LQR relies on the accurate extraction of the $\mathbf{A}$
and $\mathbf{B}$ matrices, essential for the correct computation of
$\mathbf{A}_{d}$ and $\mathbf{B}_{d}$. This appendix clarifies how the
parameters that allow the reconstruction of these matrices were extracted for
the experiment.

###  C.1 Detector calibration

Assuming the trapped nanoparticle reaches thermal equilibrium with the
residual gas in the vacuum chamber, its initial effective temperature along
the three axes is approximately $293\text{\,}\mathrm{K}$. Calibration of the
detection system involves establishing the linear relationship between the PSD
of the detector output for motion along the $i$-axis, denoted as
$S_{V_{i}V_{i}}(\Omega)$, and the displacement PSD for the same axis, denoted
as $S_{ii}(\Omega)$ [64],

$S_{V_{i}V_{i}}(\Omega)=(C_{Vm}^{i})^{2}S_{ii}(\Omega),$ (24)

with $C_{Vm}^{i}$ representing the calibration factor and $S_{V_{i}V_{i}}$
being defined by the Lorentzian function,

$S_{V_{i}V_{i}}(\Omega)=(C_{Vm}^{i})^{2}\frac{2\gamma_{m}k_{B}T}{m[(\Omega^{2}-\Omega_{i}^{2})^{2}+\gamma_{m}^{2}\Omega_{i}^{2}]}$
(25)

Calibration was done by collecting 10,000 traces, each with a duration of
$50\text{\,}\mathrm{m}\mathrm{s}$. The average PSDs were then fitted to
Equation (25), enabling the extraction of $C_{Vm}^{i}$, $\Omega_{i}$ and
$\gamma_{m}$. The coefficients were found to be

$C_{Vm}^{x}=$(6.87\pm 0.72)\times
10^{5}\text{\,}\mathrm{V}\mathrm{/}\mathrm{m}$$ $C_{Vm}^{y}=$(7.08\pm
0.75)\times 10^{5}\text{\,}\mathrm{V}\mathrm{/}\mathrm{m}$$
$C_{Vm}^{z}=$(1.07\pm 0.11)\times
10^{6}\text{\,}\mathrm{V}\mathrm{/}\mathrm{m}$$

###  C.2 Electrodes calibration

To compute the controller’s gain matrix $\mathbf{K}_{d}$ described in Appendix
A, it is necessary to measure the transduction coefficient $C_{NV}^{ij}$ that
provides the linear relation between the applied voltage across the electrodes
$j$ and the resulting force along the $i$-axis. From these, it is possible to
reconstruct the terms of the $\mathbf{B}$ matrix, which due to the geometry of
the actuators couples the $x$ and $y$-axes.

Figure 6: Electrode calibration. a) Calibration curves are presented for each
coefficient of the $xy$ plane. Each point corresponds to the analysis of 7000
traces with an individual duration of $50\text{\,}\mathrm{m}\mathrm{s}$. The
particle was driven with a sinusoidal signal at
$\Omega_{dr}/2\pi=$97.50\text{\,}\mathrm{k}\mathrm{Hz}$$. b) PSD of the
particle’s CoM motion under the action of a sinusoidal force. The dashed line
delineates the peak region from which the amplitude of the force $F_{0}$ can
be extracted.

Force calibration of the electrodes can be carried out by measuring the
particle’s response to sinusoidal voltage drives applied to an individual pair
of electrodes at known frequencies near each resonance [12]. The driving
voltage in the electrode $j$ introduces a sinusoidal force
$F_{i}^{j}\cos(\Omega_{dr}t)$ which can be observed within the PSD of the
driven CoM motion of the $i$ direction $S_{ii}^{j}$,

$S_{ii}^{j}=S_{ii}(\Omega)+S_{ii}^{j,el}(\Omega),$ (26)

where $S_{ii}(\Omega)$ follows Eq. (13) and $S_{ii}^{j,el}(\Omega)$ is

$S_{ii}^{j,el}(\Omega)=\frac{F_{i}^{j2}\tau_{el}\,\textrm{sinc}^{2}[(\Omega-\Omega_{dr})\tau_{el}]}{m^{2}[(\Omega^{2}-\Omega_{i}^{2})^{2}+\gamma_{m}^{2}\Omega^{2}]},$
(27)

with $\tau_{el}$ being the duration of the measure.

In Figure 6a), the calibration curves for each coefficient is shown, yielding

$C_{NV}^{xx}=$(2.83\pm 0.14)\times
10^{-16}\text{\,}\mathrm{N}\mathrm{/}\mathrm{V}$$ $C_{NV}^{xy}=$(2.18\pm
0.13)\times 10^{-16}\text{\,}\mathrm{N}\mathrm{/}\mathrm{V}$$
$C_{NV}^{yx}=$(2.21\pm 0.13)\times
10^{-16}\text{\,}\mathrm{N}\mathrm{/}\mathrm{V}$$ $C_{NV}^{yy}=$(2.36\pm
0.12)\times 10^{-16}\text{\,}\mathrm{N}\mathrm{/}\mathrm{V}$$

An example of one of the PSDs used for calibration is presented in Fig. 6b).

###  C.3 Gain matrix

After ensuring proper calibration of the detectors and actuators, computation
of the LQR gains becomes feasible. Analysis of the PSDs of the $x$, $y$ and
$z$ confirms the trapped nanoparticle’s oscillation frequencies
$\Omega_{x}/2\pi=$96.24\text{\,}\mathrm{k}\mathrm{Hz}$$,
$\Omega_{y}/2\pi=$101.49\text{\,}\mathrm{k}\mathrm{Hz}$$ and
$\Omega_{z}/2\pi=$31.52\text{\,}\mathrm{k}\mathrm{Hz}$$. Given the average
diameter of the nanoparticle as provided by the manufacturer, the its mass is
calculated to be $m\approx$3.37\text{\,}\mathrm{f}\mathrm{g}$$. The weighting
and cost-effort matrices used were

$\mathbf{R}_{d}=m\begin{bmatrix}\textrm{diag}(\mathbf{\Omega}^{2})&\mathbf{0}_{3\times
3}\\\ \mathbf{0}_{3\times 3}&I_{3}\\\ \end{bmatrix},$ (28)

and

$\mathbf{Q}_{d}=\frac{100}{m}\begin{bmatrix}\Omega_{x}^{-2}&0&0\\\
0&\Omega_{y}^{-2}&0\\\ 0&0&\Omega_{z}^{-2}\\\ \end{bmatrix}.$ (29)

These matrices were selected to ensure that the cost function $J_{d}$
possesses appropriate energy units, considering the states measured in S.I
units and $\mathbf{u}$ accounting for feedback forces. Such dimensional
considerations are crucial for converting the controller’s gain from the LQR
theory to the digital gains configured in the FPGA. The $\mathbf{B}$ matrix is
expressed as

$\mathbf{B}=\begin{bmatrix}\mathbf{0}_{3\times 3}\\\ \mathbf{B}_{xyz}\\\
\end{bmatrix}\textrm{, where
}\mathbf{B}_{xyz}=\begin{bmatrix}\mathbf{B}_{xy}&0\\\ \mathbf{0}_{1\times
2}&b_{z}\end{bmatrix}.$ (30)

Figure 7: Optimal gains dependence with pressure. The constant behavior for
values bellow $1\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$ allows one
to employ the same matrix $\mathbf{K}_{d}$ for the underdamped and undamped
regimes.

The submatrix $\mathbf{B}_{xy}$, expressed in
$\text{\,}\mathrm{k}\mathrm{g}^{-1}$, is determined by $m$ and the proportion
of the electrodes coefficients $C_{NV}^{ij}$,

$\mathbf{B}_{xy}=\frac{1}{m}\begin{bmatrix}-1&C_{NV}^{xy}/C_{NV}^{xx}\\\
C_{NV}^{yx}/C_{NV}^{xx}&C_{NV}^{yy}/C_{NV}^{xx}\end{bmatrix}.$ (31)

Without loss of generality, its terms were normalized by the biggest
transduction coefficient, $C_{NV}^{xx}$. The negative sign accounts for the
orientation of the electrodes axes, $x^{\prime}$ and $y^{\prime}$, as
illustrated in Fig. 1.

The final parameter required to fully describe the dynamics given by Eq. (4),
is $\gamma_{m}$. To assess the impact of varying it, we substitute the values
for the resonance frequencies, $\mathbf{B}$, $T_{s}$, $\mathbf{Q}_{d}$, and
$\mathbf{R}_{d}$ and compute $\mathbf{K}_{d}$ for different drag coefficients.
The results of this evaluation are depicted in Fig. 7. Notably, for pressures
below $1\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, the influence of
$\gamma_{m}$ on the controller’s gains is negligible. Therefore, under the
premise that pressure solely affects the drag coefficient, $\mathbf{K}_{d}$
can be computed only once, even as pressure reduces.

After completing the system characterization, with $\gamma_{m}$ considered as
zero, $\mathbf{K}_{d}$ can be properly computed. The next step involves
converting the theoretical gains into digital values configured within the
FPGA. The following expressions govern this conversion

$\displaystyle k_{p,ij}^{d}$
$\displaystyle=\frac{k_{p,ij}}{AC_{NV}^{xx}C_{Vm}^{j}},$ (32a) $\displaystyle
k_{d,ij}^{d}$
$\displaystyle=-\frac{\Omega_{j}k_{d,ij}}{AC_{NV}^{xx}C_{Vm}^{j}}.$ (32b)

Here, $\Omega_{j}$ emerges from estimating the velocity as proportional to the
delayed position, leading, for example, to $\dot{x}=-\Omega_{x}x(t-\tau_{x})$,
for a delay $\tau_{x}$. The factor $C_{NV}^{xx}$ arises from the
$\mathbf{B}_{xy}$ matrix normalization, while $C_{Vm}^{j}$ is used to convert
displacement in the $j$-axis to output voltages from its detector. In Table 1,
both theoretical and digital gains are presented.

Gain | LQR (Eq. 19) | Digital Gains (Eq. 32)
---|---|---
$k_{p,xx}$ | $-3.40\times 10^{-10}\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$ | $-0.35$
$k_{p,xy}$ | $\phantom{+}7.99\times 10^{-10}\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$ | $\phantom{+}0.80$
$k_{p,yx}$ | $\phantom{+}1.46\times 10^{-9\phantom{0}}\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$ | $\phantom{+}1.50$
$k_{p,yy}$ | $-1.15\times 10^{-9\phantom{0}}\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$ | $-1.15$
$k_{d,xx}$ | $-2.19\times 10^{-13}\text{\,}\mathrm{N}\cdot\mathrm{s}\mathrm{/}\mathrm{m}$ | $\phantom{+}136.45$
$k_{d,xy}$ | $\phantom{+}1.86\times 10^{-13}\text{\,}\mathrm{N}\cdot\mathrm{s}\mathrm{/}\mathrm{m}$ | $-119.14$
$k_{d,yx}$ | $\phantom{+}1.96\times 10^{-13}\text{\,}\mathrm{N}\cdot\mathrm{s}\mathrm{/}\mathrm{m}$ | $-122.22$
$k_{d,yy}$ | $\phantom{+}2.32\times 10^{-13}\text{\,}\mathrm{N}\cdot\mathrm{s}\mathrm{/}\mathrm{m}$ | $-148.23$
Table 1: Controller’s gains. Values returned by optimal control theory and
implemented values within the FPGA are shown according to the system
characterization and Eq. (32). The digital gains had to pass through a
conversion to a fixed-point representation during the VHDL implementation,
allowing arithmetical operations with minimal loss of numerical resolution
[65].


#  The graded Betti numbers of truncation of ideals in polynomial rings

Chwas Ahmed, Ralf Fröberg and Mohammed Rafiq Namiq  Chwas Ahmed, Department of
Mathematics, College of Science, University of Sulaimani, Kurdistan Region,
Iraq<EMAIL_ADDRESS>Ralf Fröberg, Department of mathematics,
Stockholm university, Sweden<EMAIL_ADDRESS>Mohammed Rafiq Namiq,
Department of Mathematics, College of Science, University of Sulaimani,
Kurdistan Region, Iraq<EMAIL_ADDRESS>
###### Abstract.

Let $R=\mathbb{K}[x_{1},\dots,x_{n}]$, a graded algebra $S=R/I$ satisfies
$N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution
is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$
such that $S$ satisfies $N_{k,p}$. Eisenbud and Goto have shown that for any
graded ring $R/I$, then $R/I_{\geq k}$, where $I_{\geq k}=I\cap M^{k}$ and
$M=(x_{1},\dots,x_{n})$, has a $k$-linear resolution (satisfies $N_{k,p}$ for
all $p$) if $k\gg 0$. For a squarefree monomial ideal $I$, we are here
interested in the ideal $I_{k}$ which is the squarefree part of $I_{\geq k}$.
The ideal $I$ is, via Stanley-Reisner correspondence, associated to a
simplicial complex $\Delta_{I}$. In this case, all Betti numbers of $R/I_{k}$
for $k>\min\\{\deg(u)\mid u\in I\\}$, which of course is a much finer
invariant than the index, can be determined from the Betti diagram of $R/I$
and the $f$-vector of $\Delta_{I}$. We compare our results with the
corresponding statements for $I_{\geq k}$. (Here $I$ is an arbitrary graded
ideal.) In this case we show that the Betti numbers of $R/I_{\geq k}$ can be
determined from the Betti numbers of $R/I$ and the Hilbert series of
$R/I_{\geq k}$.

###### Key words and phrases:

Truncation, Betti numbers, index, linear resolution, polarization, Hilbert
series

###### 2020 Mathematics Subject Classification:

Primary 13C40, 13D02; Secondary 13A02, 13C70.

##  1\. Introduction

Let $R=\mathbb{K}[x_{1},\dots,x_{n}]$ and $I$ be an ideal of $R$. Denote the
unique graded maximal ideal of $R$ by $M$. Green-Lazarsfeld introduced the
condition $N_{p}$ which means that the resolution of $R/I$ is 2-linear the $p$
first times. Thus $N_{1}$ means that the ideal is generated in degree 2,
$N_{2}$ means that furthermore all syzygies between the generators are linear.
The largest $p$ for which the graded algebra $R/I$ satisfies $N_{p}$ is called
the _index_ of $R/I$ (see [5, 6]). Later the condition $N_{k,p}$ was
introduced. This means that the resolution is $k$-linear the $p$ first times.
Thus $R/I$ satisfies $N_{k,1}$ if $I$ is generated in degree $k$, it satisfies
$N_{k,2}$ if furthermore the syzygies of the generators of $I$ are linear and
so on. We call the largest $p$ for which $R/I$ satisfies $N_{k,p}$ the
$k$-_index_ of $R/I$. That the $k$-index of $R$ is $\infty$ means that $R/I$
has a $k$-linear resolution. A quotient ring $S=R/I$ has a linear resolution
if all syzygies are linear. That is, the resolution of $S$ is $k$-linear if
$\operatorname{Tor}^{R}_{i,j}(S,\mathbb{K})=0$ if $j\neq k+i-1$ if $i>0$. For
a positive integer $k$, Eisenbud and Goto in [2] define $I_{\geq k}=I\cap
M^{k}$. They show that for any graded ring $R/I$, then $R/I_{\geq k}$ has a
$k$-linear resolution for $k\geq\operatorname{reg}(I)$ (see [2, Theorem 1.2]).
Moreover, a formula for the $k$-index is given by Eisenbud, Huneke, and Ulrich
in [1, Proposition 1.6]. For a monomial ideal $I$, let $I_{k}$ be the
squarefree part of $I_{\geq k}$. That is $I_{k}$ is the squarefree truncation
of $I$ past $k$. If $I\subseteq R$ is a squarefree monomial ideal and
$M_{k}=\left(u\mid u\text{ is a squarefree monomial in }M^{k}\right)$, then
$I_{k}=I\cap M_{k}$. In Theorem 2.5, we compute the graded Betti numbers of
$R/I_{k}$ for $k>\min\\{\deg(u)\mid u\in I\\}$. The Betti numbers provide a
deeper insight about the quotient ring than $k$-index invariant. For a
monomial ideal (not squarefree) $I$, in Corollary 3.4 we show that the
truncation of $I$ past $k$ has a linear resolution if and only if the
squarefree truncation of the polarization of $I$ past $k$,
$\mathcal{P}(I)_{k}$, has a linear resolution. Finally, for any graded ideal
$I$, in Theorem 4.1 we compute the graded Betti numbers of $R/I_{\geq k}$ for
$k>\min\\{\deg(u)\mid u\in I\\}$.

##  2\. The graded Betti numbers of $I_{k}$

In this section, let $I$ be a squarefree monomial ideal of $R$. Define
$J_{k}=\\{u\in M_{k}\mid u\notin I_{k}\\}$. Let $e_{u}=\operatorname{supp}(u)$
where $\operatorname{supp}(u)=\\{x_{i}\mid
u=x_{1}^{a_{1}}\dots,x_{n}^{a_{n}},a_{i}\neq 0\\}$ and $G(I)$ be the set of
minimal generators of $I$. The Stanley-Reisner complex of $I$ is the
simplicial complex consisting of faces which correspond to the squarefree
monomials not in $I$, that is

$\Delta_{I}=\\{e_{u}\subset\\{x_{1},\dots,x_{n}\\}\mid u\text{ is a squarefree
monomial not in }I\\}.$

An element $F$ of $\Delta_{I}$ is a face of $\Delta_{I}$ of dimension $|F|-1$.
A maximal face (with respect to inclusion) of $\Delta_{I}$ is a facet. Denote
the set of the facets of $\Delta_{I}$ by $\mathcal{F}(\Delta_{I})$. The
dimension of $\Delta_{I}$ is $\dim\Delta_{I}=\max\\{|F|\mid
F\in\mathcal{F}(\Delta_{I})\\}-1$. The pure $k$-skeleton of $\Delta_{I}$ is a
simplicial complex $\Delta_{I}^{[k]}$ whose facets are in
$\mathcal{F}(\Delta_{I})$ with $|F|=k+1$.

Note that $I_{k+1}=\left(ux_{i}\mid\deg(u)=k,u\in I_{k},x_{i}\notin
e_{u}\right)+\left(v\mid\deg(v)>k,v\in I_{k}\right).$

######  Lemma 2.1.

Let $I$ be a squarefree monomial ideal of $R$. Then for
$k\geq\min\\{\deg(u)\mid u\in I\\}$ we have $u\in J_{k+1}$ if and only if
$e_{u}\in\Delta_{I_{k}}^{[k]}$.

###### Proof.

($\Rightarrow$) Let $u\in J_{k+1}$. Assume that
$e_{u}\notin\Delta_{I_{k}}^{[k]}$. Then $u\in I_{k}$ and $u$ is a squarefree
monomial of degree $k+1$. Thus $u\in I_{k+1}$ which is a contradiction.

($\Leftarrow$) We have $u\in G(M_{k+1})$ since $e_{u}\in\Delta^{[k]}_{I_{k}}$.
We claim $u\notin I_{k+1}$. Suppose to the contrary that $u\in I_{k+1}$. Then
there exists $x_{t}\in e_{u}$ such that $\hat{u}_{t}\in I_{k}$. Hence
$\hat{u}_{t}\notin J_{k}$ for some $x_{t}\in e_{u}$, a contradiction. The
claim follows. ∎

In the following Proposition we find the facets, i.e. maximal faces, of
$\Delta_{I}$, then later use these facets to compute the Betti numbers of
$I_{k}$.

######  Proposition 2.2.

Let $I$ be a squarefree monomial ideal of $R$. Then for
$k\geq\min\\{\deg(u)\mid u\in I\\}$ we have

$\mathcal{F}(\Delta_{I_{k+1}})=\\{e_{u}\mid u\in
G(I_{k}),\deg(u)=k\\}\cup\\{e_{v}\mid
e_{v}\in\mathcal{F}(\Delta_{I_{k}}),\deg(v)\geq k\\}.$

###### Proof.

Let $P_{k}:=\\{e_{u}\mid u\in G(I_{k}),\deg(u)=k\\}$ and $Q_{k}:=\\{e_{v}\mid
e_{v}\in\mathcal{F}(\Delta_{I_{k}}),\deg(v)\geq k\\}$. Let
$F\in\mathcal{F}(\Delta_{I_{k+1}})$. Then $F=e_{u}$ for some $u\notin
I_{k+1}$. We need to consider the following three cases.

  * •

The case when $|F|>k$. If $e_{u}$ is not a face of $\Delta_{I_{k}}$, then
$u\in I_{k}$. But $\deg(u)\geq k+1$ so $u\in I_{k+1}$, and this contradicts
$e_{u}\in\Delta_{I_{k+1}}$. It remains to show that $F$ is maximal in
$\Delta_{I_{k}}$. Suppose $e_{ux}\in\Delta_{I_{k}}$ for some $x\notin e_{u}$.
Note that $ux\not\in I_{k+1}$, otherwise having $\deg(ux)>k+1$ implies $ux\in
I_{k}$. Hence $e_{ux}\in\Delta_{I_{k+1}}$, a contradiction. Therefore $F\in
Q_{k}$.

  * •

The case when $|F|=k$. If $u\notin I_{k}$, then $u\in J_{k}$. To obtain a
contradiction assume that $e_{ux}\in\Delta_{I_{k}}$ for some $x\notin e_{u}$.
Then $\deg({ux})=k+1$, and Lemma 2.1 implies $ux\in J_{k+1}$. Hence
$e_{ux}\in\Delta_{I_{k+1}}$, again this contradicts the assumption
$e_{u}\in\mathcal{F}(\Delta_{I_{k+1}})$. Therefore $e_{u}\in P_{k}$ or
$e_{u}\in Q_{k}$.

  * •

The case when $|F|<k$. We have $e_{ux}\notin\Delta_{I_{k+1}}$. Hence $ux\in
I_{k+1}$ with $\deg({ux})\leq k$ which is a contradiction.

Conversely, let $e_{u}\in P_{k}\cup Q_{k}$, we show that
$e_{u}\in\mathcal{F}(\Delta_{I_{k+1}})$ by considering the following three
cases:

  * •

When $|e_{u}|=k$. If $e_{u}\in P_{k}$, then $ux\in I_{k+1}$ for all $x\notin
e_{u}$. It follows that $e_{u}\in\mathcal{F}(\Delta_{I_{k+1}})$. Now if
$e_{u}\in Q_{k}$, then $u\not\in I_{k}$. Suppose that $ux\not\in I_{k+1}$ for
some $x\notin e_{u}$. Then $ux\notin I_{k}$ otherwise $ux\in I_{k+1}$. Hence
$e_{ux}\in\Delta_{I_{k}}$ which is a contradiction. Therefore $ux\in I_{k+1}$
for all $x\notin e_{u}$. Hence $e_{u}\in\mathcal{F}(\Delta_{I_{k+1}})$.

  * •

If $|e_{u}|=k+1$, then $e_{u}\in\Delta_{I_{k}}$. By Lemma 2.1 we have $u\in
J_{k+1}$, hence $e_{u}\in\Delta_{I_{k+1}}$. If $e_{u}$ is not a facet of
$\Delta_{I_{k+1}}$, then there an $x\notin e_{u}$ such that
$e_{ux}\in\Delta_{I_{k+1}}$. Then $ux\notin I_{k+1}$, and so $ux\notin I_{k}$.
Hence $e_{ux}\in\Delta_{I_{k}}$ some $x\notin e_{u}$ which is a contradiction.

  * •

When $|e_{u}|\geq k+2$. If $u\in I_{k+1}$ then $u\in I_{k}$, and so
$e_{u}\notin\Delta_{I_{k}}$, which is also a contradiction. The face $e_{u}$
is maximal in $\Delta_{I_{k+1}}$ by the same argument used in the case
$|e_{u}|=k+1$.

∎

For a fixed $t$, we shall write $(f_{-1}^{t},f_{0}^{t},\dots,f_{d-1}^{t})$ for
the $f$-vector of $\Delta_{I_{t}}$.

######  Corollary 2.3.

Let $I$ be a squarefree monomial ideal of $R$ and
$f(\Delta_{I})=(f_{-1},f_{0},f_{1},\dots,f_{d-1})$. Then for
$k\geq\min\\{|e_{u}|\mid u\in I\\}$ we have

$f(\Delta_{I_{k+1}})=\begin{cases}\left(\binom{n}{0},\binom{n}{1},\binom{n}{2},\dots,\binom{n}{k},f_{k},\dots,f_{d-1}\right)\quad&\text{
if }k\leq d,\\\
\left(\binom{n}{0},\binom{n}{1},\binom{n}{2},\dots,\binom{n}{k}\right)\quad&\text{
if }d<k<n.\end{cases}$

###### Proof.

By Proposition 2.2, we have $f^{k+1}_{k-1}=f^{k}_{k-1}+|\\{e_{u}\mid u\in
G(I_{k}),\deg(u)=k\\}|=|\\{e_{u}\mid u\in J_{k}\\}|+|\\{e_{u}\mid u\in
G(I_{k}),\deg(u)=k\\}|=\binom{n}{k}$. Then the result follows by induction. ∎

The following Lemma of Hochster provides a very useful explanation of the
graded Betti numbers of a Stanley-Reisner ring (see [8, Theorem 5.1] or [4,
Lemma 9]). Let $I$ be a squarefree monomial ideal of
$R=\mathbb{K}[x_{1},\dots,x_{n}]$. We write $\Delta_{W}$ for the simplicial
complex on the vertex set $W\subseteq\\{x_{1},\dots,x_{n}\\}$, whose faces are
$F\in\Delta_{I}$ with $F\subseteq W$.

######  Lemma 2.4 (Hochster).

Let $W\subseteq\\{x_{1},\dots,x_{n}\\}$ and $K_{R(W)}$ be the part of the
Koszul complex $K_{R}$ which is of degree $\delta(W)=(d_{1},\dots,d_{n})$,
where $d_{i}=1$ if $x_{i}\in W$ and $d_{i}=0$ otherwise. Then
$H_{i,\delta(W)}=H_{i}(K_{R(W)})\cong\tilde{H}_{|W|-i-1}\left(\Delta_{W};\mathbb{K}\right)$.

Now we come to our main result. We have a precise formula for the graded Betti
numbers of the graded algebra $R/I_{k}$ for $k>\min\\{\deg(u)\mid u\in I\\}$.

######  Theorem 2.5.

Let $I$ be a squarefree monomial ideal of $R$. Then for $k>\min\\{|e_{u}|\mid
u\in I\\}$ we have

$\beta_{i,i+j}(R/I_{k})=\begin{cases}1\quad&\mbox{ if }i=0,j=0,\\\
0\quad&\mbox{ if }i\neq 0,0\leq j\leq k-2,\\\ \alpha_{i,i+k-1}\quad&\mbox{ if
}i\neq 0,j=k-1,\\\ \beta_{i,i+j}(R/I)\quad&\mbox{ if }i\neq 0,j\geq
k.\end{cases}$

where

(2.1)
$\displaystyle\alpha_{i,i+j}=\sum_{r=0}^{i+j}(-1)^{j-r}\binom{n-r}{i+j-r}f_{r-1}^{k}+\sum_{\scriptstyle\ell+m=i+j\atop\scriptstyle\ell<i}(-1)^{\ell-i-1}\beta_{\ell,\ell+m}(R/I).$

###### Proof.

Let the $f$-vector of $\Delta_{I}$ be $(f_{-1},f_{0},f_{1}\dots,f_{d-1})$.
Then Corollary 2.3 implies that for $0\leq i\leq k-1$,
$f^{k}_{i-1}=\binom{n}{i}$. Then $\beta_{i,i+j}=0$ for all $0\leq j\leq k-2$
except $\beta_{0,0}=1$. Hence by Hochster’s formula, Lemma 2.4, the only
graded Betti numbers that change are

$\beta_{i,i+k-1}(R/I_{k})=\sum_{\scriptstyle|W|=k-1,\atop
W\subseteq\\{x_{1},\dots,x_{n}\\}}\dim_{\mathbb{K}}\tilde{H}_{k-i-2}\left(\Delta_{W};\mathbb{K}\right)=\alpha_{i,i+k-1}.$

where $\Delta_{W}$ is the simplicial complex on the vertex set $W$, whose
faces are $F\in\Delta_{I_{k}}$ with $F\subseteq W$. Thus
$\beta_{i,i+j}(R/I_{k})=\beta_{i,i+j}(R/I)$ for $j\geq k$. The Hilbert series
of $\mathbb{K}[\Delta_{I}]$ is

$H_{\mathbb{K}[\Delta_{I}]}(t)=\sum_{r=0}^{d}\frac{f_{r-1}t^{r}}{(1-t)^{r}}=\frac{\sum_{i}(-1)^{i}\sum_{j}\beta_{i,i+j}t^{i+j}}{(1-t)^{n}}.$

We have that

$\displaystyle\sum_{r=0}^{d}\frac{f_{r-1}t^{r}}{(1-t)^{r}}\times(1-t)^{n}=\sum_{s=0}^{n}\sum_{r=0}^{s}(-1)^{s-r}\binom{n-r}{s-r}f_{r-1}t^{s}.$

Then from the Hilbert series of $\mathbb{K}[\Delta_{I}]$ we have

$\sum_{s=0}^{n}\sum_{r=0}^{s}(-1)^{s-r}\binom{n-r}{s-r}f_{r-1}t^{s}=\sum_{i}(-1)^{i}\sum_{j}\beta_{i,i+j}t^{i+j}$

If $s=i+j$, then

$\sum_{r=0}^{s}(-1)^{s-r}\binom{n-r}{s-r}f_{r-1}=\sum_{i}(-1)^{i}\sum_{i+j=s}\beta_{i,s},\quad
s=0,1,\dots,n.$

Let $j=k-1$. Since $\beta_{i,i+j}(R/I_{k})\neq 0$ whenever $j\geq k-1$, we
have

$\beta_{i,i+j}(R/I_{k})=\sum_{r=0}^{i+j}(-1)^{j-r}\binom{n-r}{i+j-r}f_{r-1}^{k}+\sum_{\scriptstyle\ell+m=s\atop\scriptstyle\ell<i}(-1)^{\ell-i+1}\beta_{\ell,\ell+m}(R/I).$

∎

######  Corollary 2.6.

Let $I$ be a squarefree monomial ideal of $R$. If $I$ has a linear resolution,
then $I_{k}$ has a linear resolution for any positive integer $k$.

###### Proof.

Let $I$ be generated in degree $d$. If $k\leq d$, then $I_{k}=I_{d}=I$. For
the case $k>d$, one can apply Theorem 2.5. ∎

######  Corollary 2.7.

Let $I$ be a squarefree monomial ideal of $R$ and $\operatorname{reg}(I)=d$.
Then for any positive integer $k$ we have

$\operatorname{reg}(I_{k})=\begin{cases}d\quad\mbox{ if }d\geq k,\\\
k\quad\mbox{ otherwise}.\end{cases}$

###### Proof.

We have $\beta_{i,i+d}(I)\neq 0$ for some $i$. Theorem 2.5 implies that
$\operatorname{reg}(I_{k})=d$ whenever $d\geq k$ and
$\operatorname{reg}(I_{k})=k$ otherwise. ∎

######  Corollary 2.8.

Let $I$ be a squarefree monomial ideal of $R$. Then $\operatorname{reg}(I)=d$
if and only if $d$ is the smallest integer such that $d\geq\min\\{\deg(u)\mid
u\in I\\}$ and $I_{d}$ has a linear resolution. In particular, $I_{k}$ has a
linear resolution for all $k\geq\operatorname{reg}(I)$.

###### Proof.

We have $\beta_{i,i+d}(I)\neq 0$ for some $i$. Hence Theorem 2.5 implies that
$I_{d}$ has a $d$-linear resolution but $I_{d-1}$ does not have a linear
resolution.

Conversely, there is a non-negative integer $m$ such that $I=I_{d-m}$. By
Theorem 2.5 we have
$\beta_{i,i+d}(I)=\beta_{i,i+d}(I_{d-m})=\beta_{i,i+d}(I_{d})$. Thus
$\operatorname{reg}I=d$. ∎

######  Example 2.9.

Let
$I=\left(x_{1}x_{2}x_{3},x_{4}x_{5}x_{6}x_{7},x_{1}x_{2}x_{4}x_{5}x_{8}x_{9}\right)$
be an ideal of $R=\mathbb{K}[x_{1},\dots,x_{9}]$. Then the $f$-vector of
$\Delta_{I}$ is $(1,9,36,83,119,106,53,10)$ and the graded Betti numbers of
$R/I$ are $\beta_{0,0}=1$, $\beta_{1,3}=1$, $\beta_{1,4}=1$, $\beta_{1,6}=1$,
$\beta_{2,7}=2$, $\beta_{2,8}=1$ and $\beta_{3,9}=1$. Corollary 2.3 implies
that the $f$-vector of $\Delta_{I_{5}}$ is $(1,9,36,84,126,106,53,10)$. By
Theorem 2.5, the graded Betti numbers of $T=R/I_{5}$ for $j=4$ are determined
by

$\beta_{i,i+4}(T)=\sum_{r=0}^{4+i}(-1)^{4-r}\binom{9-4}{4+i-r}f_{r-1}^{5}+\sum_{\scriptstyle\ell+m=i+4\atop\scriptstyle\ell<i}(-1)^{\ell-i+1}\beta_{\ell,\ell+m}(R/I),$

and $\beta_{1,6}(T)=1$, $\beta_{2,7}(T)=2$, $\beta_{2,8}(T)=1$ and
$\beta_{3,9}(T)=1$. Then $\beta_{1,5}(T)=20$,
$\beta_{2,6}(T)=49+\beta_{1,6}=50$, $\beta_{3,7}(T)=53+\beta_{2,7}=55$,
$\beta_{4,8}(T)=30-\beta_{2,8}=29$ and $\beta_{5,9}(T)=7-\beta_{3,9}=6$. By
Corollary 2.8, $I_{7}$ has a linear resolution but $I_{6}$ does not have since
$\operatorname{reg}(I)=7$.

##  3\. Polarizations

Let $I=(u_{1},\dots,u_{t})$ be a monomial ideal of
$R=\mathbb{K}[x_{1},\dots,x_{n}]$ with $u_{j}=\prod_{i=1}^{n}x_{i}^{a_{i,j}}$
for $1\leq j\leq t$. For ${1\leq i\leq n}$, let $a_{i}=\max\\{a_{i,j}\mid
1\leq j\leq t\\}$. The polarization of $I$, denoted by $\mathcal{P}(I)$, is a
squarefree monomial ideal of a polynomial ring

$S=\mathbb{K}[x_{1,1},x_{1,2},\dots,x_{1,a_{1}},x_{2,1},x_{2,2},\dots,x_{2,a_{2}},\dots,x_{n,1},x_{n,2},\dots,x_{n,a_{n}}]$

with minimal generators $\mathcal{P}(u_{1}),\dots,\mathcal{P}(u_{t})$ where

$\mathcal{P}(u_{j})=\prod_{i=1}^{n}\prod_{l=1}^{a_{i,j}}x_{i,l},1\leq j\leq
t.$

In 1982, Fröberg proved the following Lemma.

######  Lemma 3.1 ([3]).

Let $I=(u_{1},\dots,u_{t})$ be a monomial ideal of $R$ such that for each $i$,
the variable $x_{i,a_{i}}$ appears in at least on of the monomials
$\mathcal{P}(u_{1}),\dots,\mathcal{P}(u_{t})$. Then for $1\leq i\leq n$ and
$1<j\leq a_{i}$, $f_{i,j}=x_{i,1}-x_{i,j}$ forms a regular sequence of degree
one in $S/\mathcal{P}(I)$. Moreover, $R/I\cong S/(\mathcal{P}(I)+J)$ where
$J=(f_{i,j}\mid 1\leq i\leq n,1<j\leq a_{i})$.

For any quotient ring, factoring out with a regular sequence of degree one
does not change the Betti numbers. That is, the graded Betti numbers of $R/I$
and $S/\mathcal{P}(I)$ are the same (see [3]). Hence $R/I$ has a linear
resolution if and only if $S/\mathcal{P}(I)$ has a linear resolution. We show
that it is also valid for componentwise linear ideals.

Herzog and Hibi in [7] defined the concept of (squarefree) componentwise
linear. A graded ideal $I\subseteq R$ is componentwise linear if $I_{<j>}$ has
a linear resolution for all $j$ where $I_{<j>}$ is the ideal generated by all
homogeneous polynomials of degree $j$ belonging to $I$. An ideal $I\subseteq
R$ is squarefree componentwise linear if $I_{[j]}$ has a linear resolution for
all $j$ where $I_{[j]}$ is the ideal generated by all squarefree homogeneous
polynomials of degree $j$ belonging to $I$. The following Theorem is standard
but we include for the sake of self-containment.

######  Theorem 3.2.

Let $R/I$ be a graded algebra with all minimal generators of $I$ of degree
$\geq k$. Then $R/I$ has a $k$-linear resolution if and only if
$\operatorname{reg}(I)=k$.

###### Proof.

Suppose $R/I$ has a $k$-linear resolution

$0\longleftarrow R/I\longleftarrow R\longleftarrow
R^{\beta_{1}}[-k]\longleftarrow
R^{\beta_{2}}[-k-1]\longleftarrow\cdots\longleftarrow
R^{\beta_{m}}[-k-m+1]\longleftarrow 0$

Thus reg$(R/I)=\max\\{k+i-i-1\\}=k-1$, so reg$(I)=k$ . If all generators of
$I$ are of degree $\geq k$ a resolution of $R/I$ looks like this:

$0\longleftarrow R/I\longleftarrow
R\longleftarrow\oplus_{j=1}^{j_{1}}R^{\beta_{1,k-1+j}}[-k+1-j]\longleftarrow\oplus_{j=1}^{j_{2}}R^{\beta_{2,k+j}}[-k-j]\longleftarrow\cdots$
$\cdots\longleftarrow\oplus_{j=1}^{j_{m}}R^{\beta_{m,k+m-2+j}}[-k-m+2-j]\longleftarrow
0$

If reg$(I)=k$ (so reg$(R/I)=k-1$) then $\\{k+i-2+j_{i}-i\\}\leq k-1$ for all
$i$, so $j_{i}=1$ for all $i$, so the resolution is linear. ∎

######  Proposition 3.3.

Let $I$ be monomial ideal $R$. Then the following conditions are equivalent:

  1. i)

$I$ is componentwise linear;

  2. ii)

$\mathcal{P}(I)$ is componentwise linear;

  3. iii)

$\mathcal{P}(I)$ is squarefree componentwise linear.

###### Proof.

$(i)\Leftrightarrow(ii)$ Let $k$ be a positive integer. By Lemma 3.1,
$\operatorname{reg}(I)=\operatorname{reg}(\mathcal{P}(I))$ and $I$ has a
linear resolution if and only if $\mathcal{P}(I)$ has a linear resolution.
Then it follows from Theorem 3.2 that $\mathcal{P}(I_{<k>})$ has a linear
resolution if and only if $\operatorname{reg}(I_{<k>})=k$. But Theorem 3.2
also implies that $\mathcal{P}(I)_{<k>}$ has a linear resolution if and only
if $\operatorname{reg}(\mathcal{P}(I)_{<k>})=k$. Hence $\mathcal{P}(I_{<k>})$
has a linear resolution if and only if $\mathcal{P}(I)_{<k>}$ has a linear
resolution. Then the proof follows from the definition of componentwise
linear.

$(ii)\Leftrightarrow(iii)$ See [7, Proposition 1.5]. ∎

######  Corollary 3.4.

Let $I$ be a monomial ideal of $R$. Then the following conditions are
equivalent:

  1. i)

$I_{\geq k}$ has a linear resolution;

  2. ii)

$\mathcal{P}(I)_{\geq k}$ has a linear resolution;

  3. iii)

$\mathcal{P}(I)_{k}$ has a linear resolution.

###### Proof.

Note that by the argument in the proof of Proposition 3.3 we have
$\mathcal{P}(I_{\geq k})$ has a linear resolution if and only if
$\mathcal{P}(I)_{\geq k}$ has a linear resolution. Then the proof is an
immediate consequence of the statement of Proposition 3.3. ∎

######  Example 3.5.

Let $J=\left(x_{1}^{3},x_{2}^{4},x_{1}^{2}x_{2}^{2}x_{3}^{2}\right)$ be an
ideal of $\mathbb{K}[x_{1},x_{2},x_{3}]$. The polarization of $J$ can be
thought as the ideal $I$ in Example 2.9. It follows from Corollaries 2.8 and
3.4 that $I_{\geq k}$ has a linear resolution for all $k\geq 7$ but it does
not have for $k\leq 6$.

##  4\. The graded Betti numbers of $I_{\geq k}$

In this section we consider quotients of $I_{\geq k}=I\cap M^{k}$, where $M$
is the graded maximal ideal, for any graded, not necessarily monomial, ideal.
We show that we can determine all graded Betti numbers of $R/I_{\geq k}$ if we
know the Betti numbers of $R/I$ and the Hilbert series of $R/I_{\geq k}$. The
following theorem is probably well known, but we haven’t found any reference.

######  Theorem 4.1.

Let $I$ be a graded ideal in $R=k[x_{1},\ldots,x_{n}]$. Let
$k=\min\\{\deg(u)\mid u\in I\\}$. If $j<i+k-1$, then $\beta_{i,j}(R/I_{\geq
k})=0$. If $j>i+k-1$, then $\beta_{i,j}(R/I_{\geq k})=\beta_{i,j}(R/I)$. If we
know the graded Betti numbers of $R/I$ and the Hilbert series of $R/I_{\geq
k}$, we can determine the graded Betti numbers of $R/I_{\geq k}$.

###### Proof.

We think about the Betti numbers as the dimension of the homology of the
Koszul complex of $R/I$. A “monomial” $mT_{l_{1}}\wedge\cdots\wedge T_{l_{i}}$
is of degree $(i,i+j)$ if $m\in R/I$ is of degree $j$, and elements of degree
$(i,i+j)$ are linear combinations of such monomials. Now
$\beta_{1,j}(R/I_{\geq k})=0$ if $j<k$ since the generators of $I_{\geq k}$
have degree $\geq k$, and then $\beta_{i,j}(R/I_{\geq k})=0$ if $j<i+k-1$,
since if $j_{i}$ is the smallest $j$ for which $\beta_{i,j}(R/I_{\geq k})\neq
0$, then $j_{i}<j_{i+1}$ because we have a minimal resolution. In degrees
$j\geq i-k-1$ $R/I_{\geq k}$ is identical with $R/I$, so if $j>i+k-1$
$\beta_{i,j}(R/I_{\geq k})=\beta_{i,j}(R/I)$. Since the Hilbert series of
$R/I_{\geq k}$ equals $\sum_{i=0}^{n}(-1)^{i}\beta_{i,i+j}(R/I_{\geq
k})t^{i+j}/(1-t)^{n}$, we can determine the remaining Betti numbers
$\beta_{i,i+k-1}(R/I_{\geq k})$ from the Hilbert series. ∎

We first give some corollaries.

######  Corollary 4.2.

We have that $k$-index of $R/I_{\geq k}$ is smaller or equal to the
$(k+1)$-index. In particular, if $R/I_{\geq k}$ has a linear resolution, then
$R/I_{\geq k+1}$ has a linear resolution.

######  Corollary 4.3.

Let $I$ be a graded ideal of $R$ and $\operatorname{reg}(I)=d$. Then for any
positive integer $k$ we have

$\operatorname{reg}(I_{\geq k})=\begin{cases}d\quad\mbox{ if }d\geq k,\\\
k\quad\mbox{ otherwise}.\end{cases}$

###### Proof.

We have $\beta_{i,i+d}(I)\neq 0$ for some $i$. Theorem 4.1 implies that
$\operatorname{reg}(I_{\geq k})=d$ whenever $d\geq k$ and
$\operatorname{reg}(I_{\geq k})=k$ otherwise. ∎

For a complete intersection generated in degree 2, the result is easy to
describe.

######  Corollary 4.4.

If $I$ is an artinian complete intersection generated in degree 2, then the
$k$-index of $R/I_{\geq k}$ is $k-1$ if $2\leq k\leq n$ and the $k$-index is
$\infty$ if $k\geq n+1$.

######  Corollary 4.5.

If $S=k[x_{1},\ldots,x_{n}]/((f_{1},\ldots,f_{n})\cap M^{k})$,
$f_{1},\ldots,f_{n}$ a complete intersection in degree $d$, then $S$ has a
linear resolution exactly for $k\geq nd-n+1$.

######  Corollary 4.6.

If $S=k[x_{1},\ldots,x_{n}]/(f_{1},\ldots,f_{n})$, where $f_{1},\ldots,f_{n}$
is a complete intersection in degree $d$, and
$T=k[x_{1},\ldots,x_{n}]/((f_{1},\ldots,f_{n})\cap M^{k})$. Suppose that
$k\leq n(d-1)$. Then $\beta_{i,j}(T)\neq 0$ only if $(i,j)=(0,0)$,
$(i,j)=(i,k+i-1)$ for $1\leq i\leq n$, and for $(i,j)=(r,dr)$ for
$\frac{k-1}{d-1}<r\leq n$. Furthermore, $\beta_{r,dr}={n\choose r}$ for those
$r$.

######  Example 4.7.

Let $S=k[x_{1},\ldots,x_{8}]/\left((x_{1}^{2},\ldots,x_{8}^{2})\cap
M^{5}\right)$. In $S$ we have all monomials of degree $<5$, but only the
squarefree in degrees 6,7, and 8, and nothing in degrees $>8$. Thus the
Hilbert series is

$1+{8\choose 1}t+{9\choose 2}t^{2}+{10\choose 3}t^{3}+{11\choose
4}t^{4}+{8\choose 3}t^{5}+{8\choose 2}t^{6}+{8\choose 1}t^{7}+{8\choose
0}t^{8}=$ $\frac{(1-t)^{8}(1+{8\choose 1}t+{9\choose 2}t^{2}+{10\choose
3}t^{3}+{11\choose 4}t^{4}+{8\choose 3}t^{5}+{8\choose 2}t^{6}+{8\choose
1}t^{7}+{8\choose 0}t^{8})}{(1-t)^{8}}=$
$\frac{1-736t^{5}+4200t^{6}-10528t^{7}+14910t^{8}-12832t^{9}+6720t^{10}-2016t^{11}+288t^{12}-8t^{14}+t^{16}}{(1-t)^{8}}.$

We have $\beta_{5,10}=56,\beta_{6,12}=28,\beta_{7,14}=8,\beta_{8,16}=1$. The
Hilbert series gives
$\beta_{0,0}=1,\beta_{1,5}=736,\beta_{2,6}=4200,\beta_{3,7}=10528,\beta_{4,8}=14910,\beta_{5,9}=12832,\beta_{6,10}=6720+56=6776,\beta_{7,11}=2016,\beta_{8,12}=288-28=260$.

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