# Unidirectionality of spin waves in Synthetic Antiferromagnets

F. Millo<EMAIL_ADDRESS>J.-P. Adam  C. Chappert
J.-V. Kim  A. Mouhoub  Université Paris-Saclay, CNRS, C2N, 91120 Palaiseau,
France  A. Solignac  SPEC, CEA, CNRS, Université Paris-Saclay, 91191 Gif-sur-
Yvette, France  T. Devolder  Université Paris-Saclay, CNRS, C2N, 91120
Palaiseau, France

###### Abstract

We study the frequency non-reciprocity of the spin waves in symmetric
CoFeB/Ru/CoFeB synthetic antiferromagnets stacks set in the scissors state by
in-plane applied fields. Using a combination of Brillouin Light Scattering and
propagating spin wave spectroscopy experiments, we show that the acoustical
spin waves in synthetic antiferromagnets possess a unique feature if their
wavevector is parallel to the applied field: the frequency non-reciprocity due
to layer-to-layer dipolar interactions can be so large that the acoustical
spin waves transfer energy in a unidirectional manner for a wide and bipolar
interval of wavevectors. Analytical modeling and full micromagnetic
calculations are conducted to account for the dispersion relations of the
optical and acoustical spin waves for arbitrary field orientations. Our
formalism provides a simple and direct method to understand and design devices
harnessing unidirectional propagation of spin waves in synthetic
antiferromagnets.

Spin waves (SW), introduced by F. Bloch[1], are the elementary excitations of
the magnetic order parameter. They have interesting properties such as non-
linear effects, anisotropic propagation and frequency non-reciprocity (NR) [2,
3, 4]. NR is the situation where the frequency $\omega$ of a SW of wavevector
$\vec{k}$ changes upon reversing the direction of phase propagation[5]. The
asymmetry of the dispersion relation $\omega(\vec{k})\neq\omega(-\vec{k})$ can
be harnessed to design devices with very peculiar behaviors such as magnonic
diodes [6, 7], directional spin wave emitters [8, 9], curved spin wave
antennas[10], chiral magnonic logic devices[11] and passive non-reciprocal
filters [12].

Layer-to-layer dipolar interactions in a multilayer can lead to large NR [13].
This requires some contrast of magnetization – orientation or magnitude –
within the thickness of the multilayer [14]. The contrast is maximal for
synthetic antiferromagnets (SAFs), i.e. two ferromagnetic layers separated by
a spacer layer that mediates an effective interlayer interaction of coupling
parameter $J<0$ favoring an antiparallel state. In SAFs, SWs have two
precession modes[13] namely acoustical (in-phase) and optical (out-of-phase).
Owing to the strong interest on non-reciprocal SWs modes, the analysis of the
dispersion relations in SAFs has already been undertaken by several authors
[4, 14, 15, 16, 17, 18, 19]. Analytical expressions were first proposed for a
symmetric SAF in refs. [14, 15] at zero applied field. Expressions of
$\omega(\vec{k})$ valid at finite fields were then provided by Ishibashi et
al. [16]. They were successfully used for a qualitative discussion of
experimental results. One special character of non-reciprocity is
unidirectionality, achieved when reversing the sign of the wavevector does not
reverse the direction of the group velocity $\vec{\nabla}_{\vec{k}}(\omega)$,
such that the energy carried by a wavepacket is truly "unidirectional": SW can
carry energy in one direction but not its opposite. The concept of
unidirectional SWs and its potential applications has been reviewed in ref.
[20], which mentions examples of successful implementations on various
configurations.
In this paper, we show that when a SAF is set in a scissors state, its
acoustical spin wave mode can have a very unique dispersion relation: there
exists field conditions and wavevector orientations for which
unidirectionality is achieved, and in addition is reconfigurable. We
demonstrate switchable unidirectionality of acoustical spin waves, in line
with previous observations of reconfigurable frequency non-reciprocity in SAF
(refs. [15, 16]). Furthermore, we develop an easy-to-use formalism validated
by simulations and measurements. Our results and the associated understanding
could then be used to design optimally non-reciprocal devices.
The paper is organized as follows. In section I, we evidence this unique NR
feature by Brillouin Light Scattering (BLS); we then design a device that
benefits from this feature. In this device, inductive propagating spin wave
spectroscopy (PSWS) experiments demonstrate that the energy flow associated
with SWs can be fully unidirectional. In section II we derive an approximate
description of the group velocities, enabling a physical understanding of the
peculiarities of the SW propagation within SAFs. These expressions are
compared with ref. [16] and with our full micromagnetic calculations in
section III. Finally, their accuracy is discussed.

##  I Evidence of unidirectionality of spin waves in symmetric SAF

###  I.1 Materials: synthetic antiferromagnets

We use symmetric SAFs of composition: substrate / Ta(6 nm) /
Co40Fe40B20($t_{\textrm{mag}}$) / Ru(0.7 nm) / Co40Fe40B20($t_{\textrm{mag}}$)
/ Ru(0.4 nm) / Ta(3 nm) where $t_{\textrm{mag}}=17$ nm. The properties of the
SAFs were characterized in refs. [21, 22], notably to derive the interlayer
exchange field $H_{J}=-\frac{2J}{\mu_{0}M_{s}t_{\textrm{mag}}}$, where $M_{s}$
is the saturation magnetization.
Two different substrates were needed: Y-cut LiNbO3 substrates, which are
optimal for microfabrication[23], are used for electrical measurements, while
naturally oxidized silicon substrates are optimal for Brillouin Light
Scattering experiments. The corresponding samples have slightly different
properties. Both samples have in-plane uniaxial anisotropies much smaller than
their respective spin-flop fields. They share the same saturation
magnetization $\mu_{0}M_{s}$ = 1.71 T and the same damping $\alpha\approx
0.0045$, although different interlayer exchange fields: $\mu_{0}H_{J}=148$ mT
for Y-cut LiNbO3 and $\mu_{0}H_{J}=78$ mT for oxidized silicon substrates
(more details are given in appendix). Since we will focus on SWs of long
wavelengths where the shapes of the dispersion relations are mainly determined
by layer-to-layer dipolar effects[5, 4, 15, 16, 24], we will consider that the
two samples are essentially comparable.

###  I.2 Spin wave dispersion relations

We measured the dispersion relations of the SWs within the SAF by performing
wavevector-resolved BLS spectroscopy. The experimental configuration is
sketched in Fig. 1(b). The applied field $\mu_{0}|\vec{H}_{0}|=$ 25 mT, higher
than the spin-flop field ($\mu_{0}|\vec{H}_{\textrm{SF}}|=$ 8 mT), was chosen
to set the SAF in one of the two degenerate scissors states[16] sketched in
Fig. 2(c). The BLS laser beam impinges the sample surface with an incidence
$\theta\in[5^{o}-30^{o}]$ with respect to the normal and we analyse the back-
scattered photons. The scattering conserves the total momentum, so the
wavenumber of the magnons annihilated (anti-Stokes process) or created (Stokes
process) in the scattering process obeys
${k}_{x}=\pm\frac{4\pi}{\lambda}\sin\theta\in[\pm 2,\pm
12]\;\textrm{rad/}\mu\textrm{m}$ with a frequency shift allowing to construct
$\omega(k)$. Fig. 1(c) reports the measured dispersion relations
$\omega(\vec{k})/2\pi$ for $\vec{k}_{x}\;||\;\vec{H}_{0}$. Two points are
worth emphasizing:

Figure 1: a) Multilayer stack of the synthetic antiferromagnet used for the
BLS experiments. b) BLS geometry: applied field $\vec{H}_{0}$ parallel to the
$\hat{\vec{x}}$ axis, plane of incidence (xz), wavelength of laser beam
$\lambda=532$ nm. c) Dispersion relations are measured by BLS for
$\vec{k}_{x}\parallel\vec{H}_{0}$. The uniform resonances (circles at $k$=0)
were measured by VNA-FMR. The error bars are the linewidths (full width at
half maximum). The black arrow emphasizes that the group velocity of the
acoustical SWs always points toward the positive side, irrespective of the
sign of the wavevector.

  * •

The optical spin wave mode has a tilted (i.e. non-symmetric) $\vee$-shaped
dispersion relation [see Fig. 1(c)]: it is non-reciprocal. The NR is classical
in the sense that the group velocity has always the same sign than the
wavevector, i.e. it is a "forward" wave, in agreement with the forthcoming
modeling (Table. 1 and section III).

  * •

In contrast, the dispersion relation of the acoustical spin wave branch is
quasi-linear in the range explored here [-12,12] rad/$\mu$m, a fact that is
very unusual for a wave. The group velocity of these SWs is always positive,
whatever the sign of the wavevector. This is a unique feature of the
acoustical SWs of a SAF set in the scissors state. As a result, the energy
carried by a wave-packet of acoustical SWs is expected to propagate in a one-
way manner.

It is also interesting to note that the frequency NR $\delta
f=\frac{1}{2\pi}(\omega(|\vec{k}|)-\omega(-|\vec{k}|))$ of the two modes are
of comparable magnitudes although with opposite signs. For $k=2$ rad/$\mu$m,
the NRs amount already to $\delta f_{ac}=1.35$ GHz and $\delta f_{op}=-1.81$
GHz. For $k=12$ rad/$\mu$m, the NRs are as large as $\delta f_{ac}=3.39$ GHz
and $\delta f_{op}=-4.29$ GHz. These very large values demonstrate that the
SAF has an exceptional NR.
Gallardo et al.[15] predicted by theoretical and micromagnetic simulations and
measured by BLS a high NR in frequency in a bilayer systems due to the layer-
to-layer dipolar interactions. They found that this high NR of acoustical SWs
can be controlled by equilibrium configuration and by the geometry of the
system. Our results fully agree with their study. For the acoustical mode, the
NR even achieves unidirectionality of the energy flow [see Fig. 1(c)]. Let us
illustrate this point using electrical measurements.

###  I.3 Energy flow carried by propagating spin waves

In this section, we perform propagating spin wave spectroscopy measurements to
identify in which direction the spin waves effectively transfer energy. The
measurement relies on the device sketched in Fig. 2(a) and (b). The device is
a SW conduit made of a $20\leavevmode\nobreak\ \mu$m-wide SAF stripe covered
by a thick insulating layer and two inductive antennas. These rf antennas have
a width of $w=1.8\leavevmode\nobreak\ \mu$m and a center-to-center distance of
6 $\mu$m in the $\hat{\vec{x}}$ direction. The antennas can be considered as
infinite in the $\hat{\vec{y}}$ direction and therefore they only couple to
SWs with wavevectors directed in the $\hat{\vec{x}}$ direction, as seen in
Fig. 2(b). When connected to a vector network analyzer, the antennas emit a
circulating rf field $\vec{H}^{rf}(x,z)$ with components along both
$\hat{\vec{x}}$ and $\hat{\vec{z}}$ axes. Thanks to these two components the
rf field can couple with the acoustical spin waves even when
$\vec{H}_{0}\parallel\hat{\vec{x}}$. The applied field
$\mu_{0}|\vec{H}_{0}|=41$ mT was chosen to lead to the same uniform resonance
acoustical frequency $f^{k=0}_{ac}=5.4$ GHz as in Fig. 1(c).

The 4 measurements of Fig. 2(d, e, f, g) are done for exactly the same applied
field $\vec{H}_{0}\parallel\hat{\vec{x}}$. This field sets the SAF in one of
two degenerate scissors state [see Fig. 2(c)] where the equilibrium
magnetizations of each layer [see Fig. 2(c)], $\vec{m}_{1}$ and $\vec{m}_{2}$,
scissors toward the direction of the applied field $\vec{H_{0}}$. The scissors
state that is effectively obtained is unknown:
$(\vec{m}_{1}\times\vec{m}_{2}).\hat{\vec{z}}$ can be either positive or
negative. However, a toggle switching experiment[25] can switch the scissors
from one state to the other.

The experiment is done in four steps. First we apply $\vec{H}_{0}$ in the
+$\hat{\vec{x}}$ direction and characterize the sample [see Fig. 2(d, e)]. The
SAF is in one of the two possible scissors states [see Fig. 2(c)]. Then, we
rotate $\vec{H}_{0}$ by 180o degrees. The scissors state follows the rotating
field. We then take back $\vec{H}_{0}$ to 0 mT, then apply again $\vec{H}_{0}$
in the +$\hat{\vec{x}}$ direction: the newly obtained "toggled" scissors state
[see Fig. 2(f,g)] is the inverse of the initial one. Finally, we perform the
characterization exactly as for the initial state.

Figure 2:  a) Stack of synthetic antiferromagnetic film patterned into devices
for propagating spin wave spectroscopy experiments. b) Geometry of the device.
Gray: single-wire antennas with
$r=6\;\mu\textrm{m}\;\textrm{and}\;w=1.8\;\mu\textrm{m}$. Antenna 1 (A1) and
Antenna 2 (A2) are connected to a VNA to collect transmission
[$S_{12},S_{21}$] parameters. c) Two possible scissors state of the system. d)
and e) Forward and backward transmission parameter measured for wave
propagating from A1-to-A2, A2-to-A1 respectively and at one of two possible
degenerate scissors state. f) and g) Forward and backward transmission
parameter measured for propagating SWs after toggling the scissors state.

Fig. 2(d) shows that in the first scissors state, the forward transmission
parameter is oscillatory and strong, while in the toggled scissors state [Fig.
2(f)] it is weak and non-oscillatory, with a quasi-Lorentzian shape. The
oscillatory character is indicative[26] that the forward transmission arises
from propagating waves. The quasi-Lorentzian shape is indicative [27] that it
arises from quasi-uniform SWs excited under the receiving antenna by the rf
field of the exciting antenna, cf. Fig. 2(e) and Fig. 2(f), without
significant propagation of the involved SWs.

This interpretation – propagating SWs in the sole forward direction – is
corroborated by the frequency contents of the transmission signals. Indeed the
backward signal is finite only near the uniform resonance $f_{ac}^{k=0}=5.4$
GHz within its linewidth. In contrast, the forward transmission parameter
spans over frequencies both much below and much above $f_{ac}^{k=0}$. This
stems from the linear character of the acoustical dispersion relation: the SWs
with $k<0$ have frequencies below $f_{ac}^{k=0}$ and the SWs with $k>0$ are
above $f_{ac}^{k=0}$. They have both a group velocity pointing towards the
receiving antenna, such that this leads to a one-way transfer of energy
between the antennas, in the same direction for frequencies above and below
$f_{ac}^{k=0}$.

Interestingly also, when toggling the SAF from one of degenerate scissors
state to the other, we observe that the direction in which the acoustical spin
waves can transfer energy is reversed [compare the panels of Fig. 2(d,f) and
Fig. 2(e,g)]. As we shall see later, this arises from the fact that the
dispersion relation of acoustical spin waves undergoes the transformation
$\vec{k}_{x}\rightarrow-\vec{k}_{x}$ when the two magnetizations of the SAF
are toggled while keeping $\vec{H}_{0}$ applied along $+\hat{\vec{x}}$
direction. This experiment can be interpreted as switching the
unidirectionality of the acoustical SWs, to be compared to the experiment of
ref. [16] that switches the non-reciprocity of optical SWs.

##  II Approximate group velocities

Let us model the dispersion relations of SW in SAF and their dependence over
the field orientation. We shall make approximations to get insightful
expressions.

Table 1: Group velocities of SWs in a SAF for a 2-macrospin ground state in the long wavelength limit. The SAF is isotropic and fully symmetrical. The field $\vec{H}_{0}$ is applied along the $\hat{\vec{x}}$ direction. The group velocities are meant as $\frac{d\omega}{dk}$, irrespective of the sign of $k$. They are given in units of $\frac{1}{2}\gamma_{0}M_{s}t_{\textrm{mag}}$. The colored cells highlight the situations when the SWs show frequency non-reciprocity. The $\dagger$ symbol recalls that the Taylor expansion to first order in $kt_{\textrm{mag}}$ is not legitimate when the acoustical eigenfrequency vanishes. Note that the sign of non-reciprocal terms within the group velocities are reversed when toggling the magnetizations of the two layers of the SAF. Field | $H_{0}=0$ | $0<H_{0}<H_{j}$ | $H_{0}\geq H_{j}$
---|---|---|---
State | Antiparallel | Scissors | Parallel
Acoustical branch |  |  |
$v_{g}^{\textrm{ac}}$, $\vec{H}_{0}\parallel\vec{k}_{x}$, $k_{x}>0$ | $-1$ $\dagger$ | $-\left(\sqrt{1-\left(\frac{H_{0}}{H_{j}}\right)^{2}}+\frac{H_{0}}{\sqrt{H_{j}(M_{s}+H_{j})}}\right)$ |  $-\sqrt{\frac{H_{0}}{H_{0}+M_{s}}}$ ($\Lambda-$shaped)
$v_{g}^{\textrm{ac}}$, $\vec{H}_{0}\parallel\vec{k}_{x}$, $k_{x}<0$ | $-1$ $\dagger$ | $-\left(\sqrt{1-\left(\frac{H_{0}}{H_{j}}\right)^{2}}-\frac{H_{0}}{\sqrt{H_{j}(M_{s}+H_{j})}}\right)$ |  $+\sqrt{\frac{H_{0}}{H_{0}+M_{s}}}$ ($\Lambda-$shaped)
$v_{g}^{\textrm{ac}}$, $\vec{H}_{0}\perp\vec{k}_{x}$, $\forall k_{x}$ | 0 | $\textrm{sgn($k_{x}$)}\frac{H_{0}}{H_{j}}\frac{M_{s}}{\sqrt{H_{j}(H_{j}+M_{s})}}$ | $\textrm{sgn($k_{x}$)}{\frac{M_{s}}{\sqrt{H_{0}(H_{0}+M_{s})}}}$
|  | ($\vee-$ shaped dispersion) | ($\vee-$ shaped dispersion)
$v_{g}^{\textrm{ac}}$, $\\{\vec{k}_{x},\vec{H}_{0}\\}=\varphi$, $k_{x}>0$ | $-\cos\varphi$ $\dagger$ | $v_{g}^{ac}\bigr{|}_{H_{0}=0}-\frac{H_{0}}{H_{j}}\left(\frac{H_{j}-(M_{s}+H_{j})\sin^{2}\varphi}{\sqrt{H_{j}(M_{s}+H_{j})}}\right)+O(H_{x}^{2})$ | $\left(\frac{\sqrt{M_{s}+H_{0}}}{\sqrt{H_{0}}}\sin^{2}\varphi-\frac{\sqrt{H_{0}}}{\sqrt{M_{s}+H_{0}}}\right)$
Optical branch |  |  |
$v_{g}^{\textrm{op}}$, $\vec{H}_{0}\parallel\vec{k}_{x}$, $k_{x}>0$ | $+\left(\sqrt{\frac{M_{s}}{H_{j}}}+1\right)$ | $+\left(\sqrt{\frac{{M_{s}}}{{H_{j}}}}+1\right)\sqrt{1-\left(\frac{H_{0}}{H_{j}}\right)^{2}}$ | 0
$v_{g}^{\textrm{op}}$, $\vec{H}_{0}\parallel\vec{k}_{x}$, $k_{x}<0$ | $-\left(\sqrt{\frac{M_{s}}{H_{j}}}-1\right)$ | $-\left(\sqrt{\frac{{M_{s}}}{{H_{j}}}}-1\right)\sqrt{1-\left(\frac{H_{0}}{H_{j}}\right)^{2}}$ | 0
$v_{g}^{\textrm{op}}$, $\vec{H}_{0}\perp\vec{k}_{x}$, $\forall k_{x}$ | 0 | 0 | 0
(flat dispersion) |  |  |
$v_{g}^{\textrm{op}}$, $\\{\vec{k}_{x},\vec{H}_{0}\\}=\varphi$, $k_{x}>0$ | $+\left(\sqrt{\frac{M_{s}}{H_{j}}}\cos\varphi+1\right)\cos\varphi$ | $v_{g}^{op}\bigr{|}_{H_{0}=0}+O(H_{0}^{2})$ | 0

###  II.1 Simplifying assumptions and methods

We describe the ground state in the 2-macrospin approximation, i.e. we assume
that the SAF comprises two films (labelled 1 and 2 [see Fig. 2(a)]) that are
uniformly magnetized across their thickness. For an applied field
$|H_{0}|<H_{J}$, the SAF is in one of the two degenerate scissors states [13,
28]. Note that to consider the other scissors state [obtained by swapping of
the magnetizations of the top (t) and bottom layers (b)], one just needs to
rotate the sample by 180o around the $\hat{\vec{y}}$ axis, which changes
$+\hat{\vec{x}}$ into $-\hat{\vec{x}}$ and $+\vec{k}_{x}$ into $-\vec{k}_{x}$.
The measurement geometry does not change in this process, but the top layer
become the bottom one and the antennas 1 and 2 get exchanged. The measured
dispersion relations have thus to obey the bottom-top swapping property:

$\omega_{tb}(\vec{k})=\omega_{bt}(-\vec{k}),$ (1)

as confirmed from the PSWS characterizations [see Fig. 2(d,e,f,g)]. The
dynamical matrix of the system can be calculated following the standard
methodology [5, 29, 30, 31] and must be augmented with the contribution of
finite wavevectors to the self- and mutual demagnetizing effects of the two
magnetic layers (see Eq. 25 and 26 of ref. [24] for a general description, or
ref. [16] for the specific case of a SAF set in scissors state). The two
eigenvalues of the dynamical matrix are the frequencies of the SWs: the
acoustical [13] SW mode for which the scissoring state undergoes essentially a
rigid rocking, and the optical SW mode for which the scissoring angle
breathes. At $k=0$ the eigenfrequencies reduce to:

$\frac{\omega_{\textrm{ac}}^{k=0}}{\gamma_{0}}=H_{0}\sqrt{\frac{M_{s}+H_{J}}{H_{J}}}\textrm{\leavevmode\nobreak\
and\leavevmode\nobreak\
}\frac{\omega_{\textrm{op}}^{k=0}}{\gamma_{0}}=\sqrt{\frac{M_{s}}{H_{J}}}\sqrt{H_{J}^{2}-H_{0}^{2}}$
(2)

where $\gamma_{0}$ is the gyromagnetic ratio.

Being solutions of a biquadratic equation, the expressions of the frequencies
of the optical and acoustical spin waves are heavy (see for instance Eq. 1 and
2 ref. [16]). To get linear in $kt_{\textrm{mag}}$ (simple to apply) formulas,
these frequencies were Taylor-expanded to first order in $kt_{\textrm{mag}}$
near $kt_{\textrm{mag}}=0$. The noticeable expressions of the group velocities
are gathered in Table 1, given in $\frac{1}{2}\gamma_{0}M_{s}t_{\textrm{mag}}$
units. We must again emphasize that we are in a dipolar dominated system where
at low $\vec{k}$ , $\lambda/t_{\textnormal{mag}}\gg 1$. For
$\vec{H}_{0}\perp\vec{k}_{x}$ and $\vec{H}_{0}\parallel\vec{k}_{x}$, the group
velocities have simple expressions that are consistent with previous reports
[23, 16]. For wavevectors directed in other directions, getting simple
formulas requires a further Taylor expansion assuming $H_{0}\ll H_{j}$ to
provide an analytical angular dependence of the $v_{g}$’s valid at low fields.

###  II.2 Peculiarities of the group velocities of a SAF in the scissors state

Several points are worth noticing in Table 1.

(i) At low applied fields, the group velocities of the acoustical SWs follow a
natural scale which is $\frac{1}{2}\gamma_{0}M_{s}t_{\textrm{mag}}$. It
amounts to 2.5 km/s for our material parameters. This velocity scale is
independent from the exchange coupling $J$.
In contrast, the group velocities of the optical SWs comprise an additional
dimensionless accelerating factor that depends on $J$. For
$\vec{k}\parallel\vec{H}_{0}$ this accelerating factor is either
$+\left(\sqrt{\frac{M_{s}}{H_{j}}}+1\right)$ or
$-\left(\sqrt{\frac{M_{s}}{H_{j}}}-1\right)$. These are respectively 5.6 and
-3.6 for our samples. This is in agreement with BLS study that concluded that
optical SWs are faster than acoustical ones when $\vec{k}\parallel\vec{H}_{0}$
[see Fig. 1].
(ii) In the scissors state the SWs in a SAF are almost always non-reciprocal.
The only reciprocal case is when $\vec{H}_{0}\perp\vec{k}_{x}$. The dispersion
is then $\vee$-shaped for the acoustical mode and flat for the optical mode.
In the other cases, despite its non-reciprocity, the dispersion relation of
the optical SWs is still conventional in the sense that the sign of group
velocity and the sign of $k$ are always correlated. For most111The optical
wave has a backward character only in the narrow range of orientations when
$\frac{\pi}{2}<\varphi<\cos^{-1}(-\sqrt{\frac{H_{j}}{M_{s}}})$. In this case
the group velocity is very close to vanish and propagation is not foreseen.
wavevector orientations, the dispersion relation of the optical SWs has the
shape of a tilted $\vee$: the group velocity and the wavevector are of same
sign and a SW transmits energy in the direction of its wavevector.
(iii) As anticipated from the experimental results, the most counter-intuitive
situation is that of acoustical SWs of wavevectors
$\vec{H}_{0}\parallel\vec{k}_{x}$ in the scissors state. In this case, the
dispersion relation is quasi-linear across $k=0$, such that the group velocity
points towards the same direction regardless of the sign of $k$. Acoustical
spin waves with $\vec{H}_{0}\parallel\vec{k}_{x}$ can transfer energy only in
one direction. Since toggling the SAF to the other scissor state is equivalent
to changing the sign of $k$, this also explains why the direction in which SWs
can transfer energy is reversed when toggling the SAF.
(iv) The non-reciprocities of the acoustical and optical modes defined as
$\delta f_{ac,op}=f_{ac,op}(k)-f_{ac,op}(-k)$ are predicted to be equal in
magnitude but opposite.
From Taylor-expanded expressions of the SW frequencies versus $\vec{k}_{x}$ we
find:

$\delta f_{op},\leavevmode\nobreak\ \leavevmode\nobreak\ -\delta
f_{ac}=\gamma_{0}M_{s}k\,t_{\textrm{mag}}\;\sqrt{1-\left(\frac{H_{0}}{H_{j}}\right)^{2}}\;\cos\varphi$
(3)

This expression is exact for $\phi=0$ and $\pi/2$ but defined at
$O(H^{2}_{0})$ otherwise.

(v) Finally we emphasize that the group velocity
$\vec{\nabla}_{\vec{k}}(\omega)$ is in general not colinear with the
wavevector, such that wavepackets with a 2D content of wavevectors will
propagate with potentially non trivial deformations like self-focusing and
caustics, as already mentioned in ref. [4].
The expressions gathered in Table 1 are convenient for the physical
understanding. They must however be considered with care: indeed the linear
Taylor expansion over $kt_{\textrm{mag}}$ leads to negative (unphysical)
frequencies when used at low fields. Besides, the expressions in Table 1
suffer from a fundamental limitation: the 2-macrospin ground state is
inaccurate for SAFs thicker than the exchange lengths [21]. In this case, a
gradient of the magnetization orientation can develop within each layer[21],
which renders incorrect the eigenfrequencies at $k=0$. The micromagnetic
configuration and its SWs must then be calculated numerically using a full
micromagnetic framework.

##  III Micromagnetic Simulations of the dispersion relations

In this section we will develop micromagnetic simulations based on the Mumax3
software [33] to provide more reliable dispersion relations of Si / SiOx / SAF
system. After describing the geometry, we detail how to selectively excite the
acoustic or optic modes. Then we describe a method that allows to
differentiate $+k$ (forward phase velocity) and $-k$ (backward) in any non-
reciprocal situation.

###  III.1 Simulation geometry

To mimic experiments, we first define a SAF slab of in-plane dimensions
{$l_{x}\times l_{y}$} where $l_{x}=l_{y}=4\pi\mu$m $\approx
12.6\leavevmode\nobreak\ \mu$m and total thickness of 34 nm. The slab is
replicated 10 times in $\hat{\vec{x}}$ and $\hat{\vec{y}}$ directions to
approximate an infinite thin film. To optimize the calculation time, each slab
is meshed into $512\times 64\times 16$ cells. The slices along the thickness
are numbered $i=1,...,16$ upward. An interlayer exchange coupling is applied
between slices $i=8$ and $i=9$ to account for the Ru layer whose thickness is
neglected. We first apply an in-plane static field $H_{0}$ and let the system
relax to its ground state. As in the experiments, the angle $\varphi$ is
defined between the SW wavevector ($k_{x}$ conventionnaly along $x$), and the
field $H_{0}$ [see Fig. 3(c)].

###  III.2 Methods

To identify the dispersion relations, we excite the sample with stimuli that
bring it out of equilibrium and we analyze the spectral content of the
magnetization response.

####  III.2.1 Mode-resolved dispersion relations

A frequently implemented stimulus is a short field pulse that imprints a
spatial periodicity and contains a broadband frequency content. We choose to
apply a pulsed RF field of the form:

$\vec{H}^{rf}(x,t,z)=h_{0}\,f(x)\,g(t)\,q(z)\,\hat{\vec{e}}_{z}$ (4)

where:

– the peak of the excitation field is $\mu_{0}h_{0}=1$ mT,
– the temporal shape of the pulse is $g(t)=\textrm{sinc}(\pi f_{c}t)$, with
its duration set by $f_{c}=$ 100 GHz [see Fig. 3(b)].
– The thickness profile of the stimulus is $q(z)$ [Fig. 3(e)].
– Its spatial profile is
$f(x)=\textrm{sin}(\vec{k}_{x}.\hat{\vec{e}}_{x}\leavevmode\nobreak\ x)$ [Fig.
3(a)].

The spatial periodicity of the stimulus along $\hat{\vec{e}}_{x}$ is chosen so
as to fit an integer number of wavelengths within the slab in the $x$
direction, by setting $\vec{k}_{x}=(2\pi n/l_{x})\hat{\vec{e}}_{x}$ with
$n\in\mathbb{N}$. This leads to a wavevector resolution of
$\frac{1}{2}\textrm{rad}/\mu\textrm{m}$.

The selective excitation of the acoustical mode is depicted in Fig. 3(f). To
excite the sole acoustical modes, the torque from the stimulus must be made
uniform across the thickness of the SAF. This is ensured by setting
$q(z)=q_{ac}(z)=1$ and an orientation of the rf field along
$\hat{\vec{e}}_{z}$ [see Fig. 3(f)]. Conversely, to excite the sole optical
mode one applies torques that are opposite in the two layers. For an rf field
along $\hat{\vec{e}}_{z}$, this condition is ensured by choosing
$q(z)=q_{op}(z)=\textrm{sgn}(i-8.5)$ where sgn is the signum function which is
$-1$ for the bottom layers ($i\leq 8$) and $+1$ for the top layers ($i>8$).
Finally, if one aims to excite both modes, one can just add the two stimuli
and use a thickness profile $q(z)=q_{op}(z)+q_{ac}(z)=\Theta(i-8.5)$, where
$\Theta$ is the Heaviside distribution.

Figure 3: Method used to construct the mode-resolved dispersion relations as
described in section III.2.1. (a) Spatial dependence and (b) time dependence
of the pulsed RF field [Eq. 4]. (c) Top view of the simulated system with
lateral dimensions of $(4\pi\times 4\pi)\leavevmode\nobreak\
\mu\textrm{m}^{2}$. The SAF is in the scissors state. The angle between SW
wavevector $\vec{k}$ and the DC field $\vec{H}_{0}$ is $\varphi$. (d) Sketched
time-resolved response of the magnetization to the stimulus of (a) for
$q(z)=\Theta(i-8.5)$. (e) Power spectral density (PSD) of this response after
apodization (Eq. 6). The maxima correspond to the acoustical and optical modes
sharing the common spatial periodicity defined in panel (a). (f): PSDs
obtained after the selective excitation of acoustical mode using a uniform
$q(z)$. Insets in (e) and (f): sketches of the thickness profiles of the
stimuli.

The magnetization response to the stimulus is simulated for $t_{s}=5$ ns. This
duration appears sufficient to reach the ground state [Fig. 3(d)]. We perform
a Fourier analysis of the magnetization response averaged over one row of
cells $\vec{m}_{i}(t)$, aligned in the direction perpendicular to the
wavevector; in practice[21] we choose the row $\\{i=1,\leavevmode\nobreak\
x=\frac{l_{x}}{2}\\}$. To minimize the spectrum leakage we apodize this
averaged moment with a Hann-window function:

$A_{\textrm{Hann}}(t)=\left\\{\begin{array}[]{ll}\cos^{2}\frac{\pi(t-t_{s}/2)}{t_{s}}&\textrm{for}\;0\leq
t\leq t_{s}\\\ 0&\textrm{otherwise}\end{array}\right.$ (5)

We then make a Fourier Transform of the apodized form of $\vec{m}_{i=1}(t)$:

$\tilde{\vec{m}}_{i=1}(f)=\mathcal{F}\left(A_{\textrm{Hann}}(t)\vec{m}_{i=1}(t)\right)$
(6)

where " $\,\tilde{}\,$ " recalls the complex-valued nature of frequency-domain
magnetization. The SWs frequencies are finally identified by the maxima of the
power spectra of the $x$ component of magnetization
$\,\tilde{\vec{m}}_{i=1}(f)\,$:

$\max\left(||\tilde{\vec{m}}^{x}_{i=1}(f)||^{2}\right).$ (7)

One representative result of this procedure (Eq. 4 to 7) is illustrated in
Fig. 3(e). When the thickness-resolved stimulus $q(z)$ is meant to excite both
optical and acoustical SWs, two eigenfrequencies are expected if the situation
is reciprocal. However in a non-reciprocal situation like ours, 4
eigenfrequencies are generally obtained. This arises because the spatial
stimulus $f(x)=\textrm{sin}(\vec{k}_{x}.\hat{\vec{e}}_{x}\leavevmode\nobreak\
x)$ is a standing wave composed of the sum of waves of wavevectors
$\vec{k}_{x}$ and $-\vec{k}_{x}$ [ notice that $f(x)\propto
e^{(i\vec{k}_{x}.\hat{\vec{e}}_{x}\leavevmode\nobreak\
x)}-e^{-(i\vec{k}_{x}.\hat{\vec{e}}_{x}\leavevmode\nobreak\ x)}$ ]. As a
result, even if exciting only the acoustical branch [Fig. 3(f)] or only the
optical branch (not shown), one is still left with a pair of eigenfrequencies
$\\{f_{0},f_{0}^{\prime}\\}$ that must be attributed to either
$\\{+\vec{k}_{x},-\vec{k}_{x}\\}$ or $\\{-\vec{k}_{x},+\vec{k}_{x}\\}$.

####  III.2.2 Sign-of-k-resolved dispersion relations

Getting sign-of-k-resolved dispersion relations requires to apply a stimulus
that exclusively excites either the $+\vec{k}_{x}$ or the $-\vec{k}_{x}$ spin
wave [Fig. 4(a)]. We thus replace the previously used broadband stimuli of Eq.
4 by traveling wave stimuli at the previously identified $f_{0}$. The new
stimuli have the form [see Fig. 4(a)]:

$\vec{H}^{rf}(x,t,z)=h_{0}\,q(z)\,\sin(\vec{k}_{x}.\hat{\vec{e}}_{x}\leavevmode\nobreak\
x-2\pi f_{0}t)\,\hat{\vec{e}}_{z}$ (8)

To ensure that at least one of these stimuli is resonant with $f_{0}$, we
investigate now signed wavevectors, i.e. spanning over the bipolar interval:

$\vec{k}_{x}=(2\pi n/l_{x})\hat{\vec{e}}_{x}\textrm{\leavevmode\nobreak\
\leavevmode\nobreak\ with\leavevmode\nobreak\ \leavevmode\nobreak\
}n\in\mathbb{Z}.$ (9)

A much smaller value of $\mu_{0}h_{0}=1\leavevmode\nobreak\ \mu$T is chosen
for the (continuously applied) stimulating field to remain in the linear
domain.
To determine whether a given $\\{\vec{k}_{x},f_{0}\\}$ combination is
resonant, the relevant information is the susceptibility i.e. the amplitude of
the magnetization response once the steady state regime is reached. As before,
we focus on one particular row of cells [ e.g. $\\{i=1,\leavevmode\nobreak\
x=\frac{l_{x}}{2}\\}$ ] on which we average the magnetizations. We apodize the
time response using Eq. 5 to minimize the spectral leakage and in addition, to
cancel the spectral contribution of the transient response of the
magnetization. This requires to calculate the magnetization response for a
longer duration, e.g. $t_{s}=10$ ns.

Figure 4: Method used to construct the sign-of-k-resolved dispersion relation,
as described in section III.2.2. a) Travelling wave like stimulus used to
excite. b) Real part of the dynamical magnetic response of the system at
$f_{0}=5$ GHz versus wavevector according to Eq. 10.

Since the power of the stimulus is fixed, the resonant or non-resonant
character of the dynamical magnetic response versus $\\{\vec{k}_{x},f_{0}\\}$
can be evaluated by simply plotting

$\mathcal{R}e\Big{(}\mathcal{F}\big{(}A_{\textrm{Hann}}(t).\vec{m}^{x}_{i=1}(t)\big{)}\Big{)}(f=f_{0})$
(10)

at a given frequency $f_{0}$ versus $\vec{k}_{x}.\hat{\vec{e}}_{x}$. The
system’s response for a frequency $f_{0}=5$ GHz versus the wavevector is shown
in Fig. 4(b). From this plot, we can conclude that the specific frequency
$f_{0}$ corresponds to $k_{x}=+3$ rad/$\mu$m and not to $k_{x}=-3$ rad/$\mu$m
[see Fig. 4(b)]. By performing this analysis for each SW frequency that was
identified from the broadband stimulus, we can assign each eigenfrequency to a
wavevector of the correct sign, and thus we can construct the dispersion
relation even in non-reciprocal cases.

###  III.3 Dispersion Relations

Fig. 5 shows the micromagnetic dispersion relations for
$\vec{k}_{x}\parallel\vec{H}_{0}$ and $\vec{k}_{x}\perp\vec{H}_{0}$ at
$\mu_{0}|\vec{H}_{0}|=25$ mT.
When $\vec{k}_{x}\parallel\vec{H}_{0}$, a strongly non-reciprocal behaviour
due to layer-to-layer dipolar interactions[15] is observed for both modes,
[see Fig. 5(a)]. The dispersion relation of the optical mode has the shape of
a tilted $\vee$. The dispersion relation of the acoustical mode still exhibits
its quite unique feature: it is essentially linear at low wavevectors.
Micromagnetic calculations confirm that the amplitude of the non-reciprocity
is very strong:
For instance for $|\vec{k}|=2$ and 12 rad/$\mu$m, it predicts respectively
$\delta f_{ac}=1.6$ and 7.2 GHz, and exactly opposite values $\delta
f_{op}=-1.6$ and $-7.2$ GHz. These equalities confirm the analytical
prediction of Eq. 3.
Conversely, when $\vec{k}_{x}\perp\vec{H}_{0}$, a perfectly reciprocal
behavior is observed [Fig. 5(b)]. An additional striking feature that is
specific to SAFs is that the optical mode has a very flat dispersion relation
(i.e. $V_{g}^{\perp}\approx 0$ km/s, $\forall k$).
To discuss these micromagnetic dispersion relations, we have superimposed the
predictions of two analytical 2-macrospin models based on dynamic matrix
theory[5, 29, 30, 31]: our formalism (Eq. 2 and Table 1) and that of Ishibashi
et al. [16]. These analytical models do not take into account the limited
intralayer exchange stiffness.

Figure 5: Comparison between our model (Table. 1, dashed lines), Ishibashi’s
analytical model[16] (continuous lines) and mumax3 simulation (squared dots)
adapted with our parameters of Si / SiOx / SAF system for an applied DC field
25 mT. a) Spin waves with $\vec{k}_{x}\parallel\vec{H}_{0}$ (non-reciprocal
situation). b) Spin waves with $\vec{k}_{x}\perp\vec{H}_{0}$ (reciprocal
situation). For the optical mode, the dashed and continuous lines are
superimposed.

##  IV Discussion

In this section, we compare the approximate models to the micromagnetic
calculations. The results for $\vec{k}_{x}\parallel\vec{H}_{0}$ and
$\vec{k}_{x}\perp\vec{H}_{0}$ shown in Fig. 5(a,b) are completed by additional
field magnitudes given in Supplemental Material [Fig. S1]. We also provide
angle-resolved dispersion relations for $\mu_{0}|\vec{H}_{0}|=25$ mT in
Supplemental Material [Fig. S2]. The fields span from 10 mT, for which the
scissors state is close to antiparallel with magnetizations that are quite
uniform across the two CoFeB layers, up to 75 mT, i.e. close to
$\mu_{0}H_{J}=78\leavevmode\nobreak\ \textrm{mT}$ where the SAF is close to
the parallel state but with a very significant gradient of the magnetization
orientation within each CoFeB layer[21]. Several points arise from these
comparisons:
(i) Uniform resonance ($k$ = 0). At low fields Eq. 2 correctly accounts for
the precession frequencies. As the applied field increases, the magnetizations
of each layer develop an increasing gradient within thickness, and Eq. 2
progressively fails to account for the two precession frequencies [see Fig.
S1]. This failure of the 2-macrospin model for nearly saturated thick SAF was
already noticed and discussed in ref. [21]. An unavoidable consequence of this
failure of the 2-macrospin model at large fields is that the group velocities
of Table 1 and of Ishibashi et al.[16], which are both calculated assuming a
2-macrospin ground state, are increasingly failing when approaching
saturation.
(ii) Concerning the optical mode, a thorough examination of Fig. 5 and Fig. S1
indicates that the three models agree for low and moderate applied fields,
i.e. $\mu_{0}H_{0}\leq 50\leavevmode\nobreak\ \textrm{mT}$ [see Fig. S1]. The
linear expansions of Table 1 are usable for wavevectors $k\in[-12,12]$
rad/$\mu$m when $\vec{k}_{x}\perp\vec{H}_{0}$: the dispersion relation
$f_{op}(k)$ is then very flat and one can hardly perceive some curvature of at
the largest wavevectors [see Fig. 5(b)]. For
$\vec{k}_{x}\parallel\vec{H}_{0}$, the linear expansions of Table 1 are usable
only up to $k\in[-3,3]$ rad/$\mu$m. Above 3 rad/$\mu$m, Ishibashi’s model
performs better and almost agrees with the result of micromagnetic
calculations [see Fig. 5(a)]. This agreement indicates that the intralayer
exchange stiffness –neglected in the approximate models– has little influence
on the optical dispersion relation at least up to 12 rad/$\mu$m.
(iii) Concerning the acoustical mode, for $\vec{k}_{x}\perp\vec{H}_{0}$, the
approximate models perform well at low and moderate applied fields. At fields
$H_{0}\leq 15$ mT [see Fig. S1(b),(f)], the predictions of Table I are closer
to full micromagnetic calculations than those of ref. [16]. At these fields
the branches of the $\vee$-shaped dispersion relation are remarkably close to
linear. The branches acquire progressively a curvature for $\mu_{0}H_{0}\geq
25$ mT [see Fig. 5(a)], and then Ishibashi’s model start performing better
than the expressions of Table I.
The case of $\vec{k}_{x}\parallel\vec{H}_{0}$ is far less satisfactory [Fig.
5(a)]. The approximate models perform reasonably well but only at very low
$k\in[-3,3]$ rad/$\mu$m, and provided that the applied field is low. For
instance at a field of 25 mT, a linear fit through positive wavevector yields
$V_{g}=2.95$ km/s for micromagnetic calculations [Fig. 5(a)] to be compared
with $V_{g}=2.57$ km/s from Table I and $V_{g}=2.4$ km/s from Ishibashi’s
analytical model. Besides, these three velocities are higher than the
experimental results which were $V_{g}(k_{x}>0)=1.25$ km/s for Si / SiOx / SAF
system studied by BLS [see Fig. 1(c)], and $V_{g}(k_{x}>0)=1.9$ km/s for
LiNbO3 / SAF system by propagating spin wave spectroscopy [see Fig.
2(d,e,f,g)].
There is thus a discrepancy for $\vec{k}_{x}\parallel\vec{H}_{0}$ both among
the models and with the experimental results. We believe that this is an
indication that the texture of the ground state –it being different from a
2-macrospin scissors– has a major impact on the dispersion relation of the
acoustical mode for this field orientation. As mentioned in section II.2, the
slope of the dispersion relation of the acoustical mode is set by the scaling
factor $\frac{1}{2}\gamma_{0}M_{s}t_{\textrm{mag}}$: it is proportional to the
thickness $t_{\textrm{mag}}$ of each layer; if a gradient of the magnetization
develops within each layer, the relevant thickness to be considered becomes
ill-defined. This is also probably the case in experimental samples where the
exchange stiffness is potentially non-uniform near the Ru barrier because of
composition gradients[22]. This has a substantial impact on the acoustical
dispersion relation for $\vec{k}_{x}\parallel\vec{H}_{0}$, which remains to be
fully expressed analytically.

##  V Conclusions

In summary, we have combined experiments, analytical modeling and numerical
simulations to study the spin waves in symmetric synthetic antiferromagnets in
the scissors state. The non-reciprocity comes from the layer-to-layer dipolar
interactions, which is the only contribution . Whatever the wavevector
orientation, the frequency non-reciprocities of the acoustical and optical
spin waves coincide in magnitude but have opposite signs. Our modeling
explains why the frequency non-reciprocity can reach unusually large
magnitudes, especially when the static field is oriented close to the spinwave
wavevector [Fig. S2]. In this case, the acoustical spin waves exhibit a unique
dispersion relation: it is close to linear around $k=0$. As a result,
reversing the sign of the SW wavevector does not reverse of the direction of
the group velocity, such that the energy flow associated to a wavepacket of
acoustical spin waves is truly unidirectional: these wavepackets can carry
energy in one direction but not the opposite. We have developed a simple
analytical model to approximate group velocities and evidenced switchable
unidirectionality of spin waves in synthetic antiferromagnets, a feature that
opens potential applications.

## Acknowledments

The authors acknowledge the French National Research Agency (ANR) under
contract No ANR-20-CE24-0025 (MAXSAW). This work was supported by a public
grant overseen by the French National Research Agency (ANR) as part of the
“Investissements d’Avenir” program (Labex NanoSaclay, reference:
ANR-10-LABX-0035, project SPICY). This work was done within the C2N micro
nanotechnologies platforms and partly supported by the RENATECH network and
the General Council of Essonne.

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### Appendix: Sample details

Piezoelectric Y-cut LiNbO3 substrates were used for the fabrication of devices
meant for electrical measurements. A strong interlayer exchange coupling
$J=-1.7$ mJ/m2 is obtained when using these substrates [21, 22], leading to an
interlayer exchange field $\mu_{0}H_{J}=-\frac{2J}{M_{s}t_{\textrm{mag}}}$
which amounts to 148 mT. The relatively low exchange stiffness
$A_{\textnormal{ex}}=16\;\textnormal{pJ/m}$ within this SAF allows for
gradients of magnetization orientation within the thicknesses of the CoFeB
layers [23, 21], which increases the saturation field up to
$\mu_{0}H_{\textrm{sat}}=251$ mT. There is a small (unwelcome) uniaxial
anisotropy field, with anisotropy field being $\mu_{0}H_{k}=3$ mT and spin-
flop field $\mu_{0}H_{\textrm{SF}}=19$ mT.

Si / SiOx substrates were used instead of the piezoelectric LiNbO3 substrates
when preparing samples meant for Brillouin Light scattering. Using Si / SiOx
substrates ensures that the BLS signals can be unambiguously attributed to
spin waves and not to the surface or bulk acoustical waves potentially present
in the LiNbO3 substrates. On these Si / SiOx substrates, a thermal treatment
can be applied to suppress any anisotropy[22]. It was effectively reduced to
only $\mu_{0}H_{k}=0.8$ mT. The thermal treatment also reduces the interlayer
exchange energy down to $J=-0.9$ mJ/m2 (i.e. $\mu_{0}H_{J}=78$ mT) while it
increases the exchange stiffness
$A_{\textnormal{ex}}=28.3\;\textnormal{pJ/m}$. Spin-flop field amounts to
$\mu_{0}H_{\textrm{SF}}=8$ mT. As a result of the strong intralayer exchange,
this SAF behaves more like a 2-macrospin system and its saturation field
$\mu_{0}H_{\textrm{sat}}=100$ mT is thus close to its $\mu_{0}H_{J}$.

Supplemental Material: Unidirectionality of spin waves in Synthetic
Antiferromagnets

F. Millo,${}^{1\textrm{, a)}}$ J.-P. Adam,1 C. Chappert,1 J.-V. Kim,1 A.
Mouhoub,1 A. Solignac,2 and T. Devolder1

1) Université Paris-Saclay, CNRS, C2N, 91120 Palaiseau, France

2)SPEC, CEA, CNRS, Université Paris-Saclay, 91191 Gif-sur-Yvette, France

FIG. S1: Comparison between our model (Table. I, dashed lines), Ishibashi’s
analytical model${}^{\textrm{16}}$ (continuous lines) and mumax3 simulation
(squared dots) adapted with parameters of Si / SiOx / SAF system for different
applied DC fields (column representation). a-d) $\vec{k}_{x}||\vec{H}_{0}$ is
the non-reciprocal situation. e-h) $\vec{k}_{x}\perp\vec{H}_{0}$ is the
reciprocal situation. For optical mode, the dashed and continuous lines are
superimposed. FIG. S2:  Dispersion relations for different angles $\varphi$
(defined between $\vec{k}_{x}$ and $\vec{H}_{0}$) at DC applied field
$\mu_{0}|\vec{H}_{0}|=25$ mT.


#  A uniform General Neron Desingularization in dimension one

Asma Khalid, Gerhard Pfister, and Dorin Popescu  Asma Khalid, Abdus Salam
School of Mathematical Sciences,GC University, Lahore, Pakistan
<EMAIL_ADDRESS>Gerhard Pfister, Department of Mathematics, University
of Kaiserslautern, Erwin-Schrödinger-Str., 67663 Kaiserslautern, Germany
<EMAIL_ADDRESS>Dorin Popescu, Simion Stoilow Institute of
Mathematics of the Romanian Academy, Research unit 5, University of Bucharest,
P.O.Box 1-764, Bucharest 014700, Romania<EMAIL_ADDRESS>
###### Abstract.

We give a uniform General Neron Desingularization for one dimensional local
rings with respect to morphisms which coincide modulo a high power of the
maximal ideal. The result has interesting applications in the case of Cohen-
Macaulay rings.

Key words  : Smooth morphisms, regular morphisms
2010 Mathematics Subject Classification: Primary 13B40, Secondary
14B25,13H05,13J15.

## Introduction

Solving equations is an old problem in mathematics. When we deal with
polynomial equations over Noetherian complete local rings then a special case
is the smooth case, that is a system where we may apply the implicit function
theorem. If possible one tries to replace an arbitrary polynomial system of
equations by a smooth one. This is done in the theorem below, which was used
for example in Artin’s Approximation Theory [2] (see also [12], [14]).

Let us recall some definitions. A ring morphism $u:A\rightarrow A^{\prime}$ of
Noetherian rings has regular fibers if for all prime ideals
${\mathfrak{p}}\in\operatorname{Spec}A$ the ring
$A^{\prime}/{\mathfrak{p}}A^{\prime}$ is a regular ring, i.e. its
localizations are regular local rings. It has geometrically regular fibers if
for all prime ideals ${\mathfrak{p}}\in\operatorname{Spec}A$ and all finite
field extensions $K$ of the fraction field of $A/{\mathfrak{p}}$ the ring
$K\otimes_{A/{\mathfrak{p}}}A^{\prime}/{\mathfrak{p}}A^{\prime}$ is regular.

A flat morphism of Noetherian rings $u$ is regular if its fibers are
geometrically regular. If $u$ is regular of finite type then $u$ is called
smooth. A localization of a smooth algebra is called essentially smooth. A
Henselian Noetherian local ring $A$ is excellent if the completion map
$A\rightarrow\hat{A}$ is regular.

######  Theorem 1.

(General Neron Desingularization, André [1], Popescu [12], [13], Swan [18],
Spivakovski [17]) Let $u:A\rightarrow A^{\prime}$ be a regular morphism of
Noetherian rings and $B$ an $A$-algebra of finite type. Then any $A$-morphism
$v:B\rightarrow A^{\prime}$ factors through a smooth $A$-algebra $C$, that is
$v$ is a composite $A$-morphism $B\rightarrow C\rightarrow A^{\prime}$.

The $A$-algebra $C$ is called a General Neron Desingularization (shortly GND)
and in the case when $(A^{\prime},{\mathfrak{m}}^{\prime})$ is local of
dimension one it is constructive (see [10], [9]). The construction depends on
formal power series defining $v$. We have to replace a given system of
polynomials by a smooth one using only a polynomial approximation of $v$.

The purpose of this paper is to give such construction which depends only on a
solution $\bar{y}$ in $A^{\prime}$ of $f$ modulo ${\mathfrak{m}}^{\prime m}$
for some $m>>0$ (see Theorem 2). Moreover, the constructed GND works for each
solutions $y$ of $f$ in $A^{\prime}$ lifting $\bar{y}$. We call such GND
uniform.

As a consequence (see Remark 4) we get [9, Theorem 14] (see also [10, Theorem
20]), that is there exists a linear map similar as in Greenberg’s case (see
[5]).

An extension of [14, Theorem 10] is Theorem 10, which shows that if
$A,A^{\prime}$ are Cohen-Macaulay rings and $v:B\rightarrow
A^{\prime}/{\mathfrak{m}}^{\prime 3k}$ an $A$-morphism, then the set of all
$A-$ morphisms $B\rightarrow A^{\prime}$ equivalent with $v$ modulo
${\mathfrak{m}}^{\prime 3k}$, is in bijection with
${\mathfrak{m}}^{3k}A^{\prime s}$ for some $s\in{\bf N}$. Theorem 10 has also
a variant in the case when $(A,{\mathfrak{m}})$ is Artinian and
$(A^{\prime},{\mathfrak{m}}^{\prime})$ is a Noetherian local ring of dimension
one, which says that for an $A$-algebra $B$ of finite type and $v:B\rightarrow
A^{\prime}/{\mathfrak{m}}^{\prime 3k}$ an $A$-morphism, the set of all
$A$-morphisms $B\rightarrow A^{\prime}$ equivalent with $v$ modulo
${\mathfrak{m}}^{\prime 3k}$, is in bijection with ${\mathfrak{m}}^{\prime
3k}A^{\prime s}$ for some $s\in{\bf N}$ (see Theorem 11).

We owe thanks to the Referees who showed us several misprints and had some
useful comments on our presentation.

##  1\. Uniform General Neron Desingularization

We begin recalling some definitions. Let $A$ be a Noetherian ring, $B=A[Y]/I$,
$Y=(Y_{1},\ldots,Y_{n})$. If $f=(f_{1},\ldots,f_{r})$, $r\leq n$ is a system
of polynomials in $I$ then we can define the ideal $\Delta_{f}$ generated by
all $r\times r$-minors of the Jacobian matrix $(\partial f_{i}/\partial
Y_{j})$. After Elkik [4] let $H_{B/A}$ be the radical of the ideal
$\sum_{f}((f):I)\Delta_{f}B$, where the sum is taken over all systems of
polynomials $f$ from $I$ with $r\leq n$. Then $B_{{\mathfrak{p}}}$,
${\mathfrak{p}}\in\operatorname{Spec}B$, is essentially smooth over $A$ if and
only if ${\mathfrak{p}}\not\supset H_{B/A}$ by the Jacobian criterion for
smoothness. Thus $H_{B/A}$ measures the non smooth locus of $B$ over $A$. $B$
is standard smooth over $A$ if there exists $f$ in $I$ as above such that
$B=((f):I)\Delta_{f}B$.

Now let us fix some notations and assumptions for the next theorem. Let
$u:A\rightarrow A^{\prime}$ be a flat morphism of Noetherian local rings of
dimension $1$ and $B,I$ be as above. Suppose that the maximal ideal
${\mathfrak{m}}$ of $A$ generates the maximal ideal of $A^{\prime}$ and the
completions of $A,A^{\prime}$ are isomorphic. Moreover suppose that
$A^{\prime}$ is Henselian and $u$ is a regular morphism. Let
$(0)=\cap_{{\mathfrak{p}}\in\operatorname{Ass}A}Q_{{\mathfrak{p}}}$ be a
reduced primary decomposition of $(0)$ in $A$, where $Q_{{\mathfrak{p}}}$ is a
primary ideal with $\sqrt{Q_{{\mathfrak{p}}}}={\mathfrak{p}}$. Let $e\in{\bf
N}$ be such that ${{\mathfrak{p}}}^{e}\subset Q_{{\mathfrak{p}}}$ for all
${\mathfrak{p}}\in\operatorname{Ass}A$.

######  Theorem 2.

With the notations $B,I,e$ fixed above, let $f=(f_{1},\ldots,f_{r})$, $r\leq
n$ be a system of polynomials in $I$, $M$ an $r\times r$-minor of the Jacobian
matrix $(\partial f_{i}/\partial Y_{j})$ and $k,c\in\bf N$. Suppose that there
exist $N\in((f):I)$ and an $A$-morphism $v:B\rightarrow
A^{\prime}/{\mathfrak{m}}^{(2e+1)k+c}A^{\prime}\cong
A/{\mathfrak{m}}^{(2e+1)k+c}$, given by $Y\rightarrow y^{\prime}\in A^{n}$,
such that the ideal generated by $d=(MN)(y^{\prime})$ in
$A/{\mathfrak{m}}^{(2e+1)k}$ contains
${\mathfrak{m}}^{k}/{\mathfrak{m}}^{(2e+1)k}$. Then there exists a $B$-algebra
$C$ which is standard smooth over $A$ with the following properties:

  1. (1)

Every $A$-morphism $v^{\prime}:B\rightarrow A^{\prime}$ with $v^{\prime}\equiv
v\ \mbox{modulo}\ d^{2e+1}A^{\prime}$ (that is $v^{\prime}(Y)\equiv v(Y)\
\mbox{modulo}\ d^{2e+1}A^{\prime}$) factors through $C$.

  2. (2)

Every $A$-morphism $v^{\prime}:B\rightarrow A^{\prime}$ with $v^{\prime}\equiv
v\ \mbox{modulo}\ {\mathfrak{m}}^{(2e+1)k}A^{\prime}$ factors through $C$.

  3. (3)

There exist an $A$-morphism $w:C\rightarrow A^{\prime}$ which makes the
following diagram commutative

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A/{\mathfrak{m}}^{(2e+1)k+c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A/{\mathfrak{m}}^{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}/{\mathfrak{m}}^{c}A^{\prime}}$

###### Proof.

We follow the proof of [14, Theorem 10] and [9, Theorem 2]. Set $P=NM$. As $A$
is not Artinian we see that $d$ is not nilpotent. We have
$(0:_{A}d^{e})=(0:_{A}d^{e+1})$ because of the choice of $e$. Note that
${\mathfrak{m}}^{k}\subset dA+{\mathfrak{m}}^{(2e+1)k+c}\subset
dA+{\mathfrak{m}}^{(2e+1)((2e+1)k+c)+c}\subset\ldots$. Thus
${\mathfrak{m}}^{k}\subset dA$, in particular
${\mathfrak{m}}^{(2e+1)k+c}\subset dA$.

Note that $dA/{\mathfrak{m}}^{(2e+1)k+c}$ is the ideal corresponding to
$v(NM)A^{\prime}/{\mathfrak{m}}^{(2e+1)k+c}A^{\prime}$ by the isomorphism
$A/{\mathfrak{m}}^{(2e+1)k+c}\cong
A^{\prime}/{\mathfrak{m}}^{(2e+1)k+c}A^{\prime}$.

We may assume that $M=\det((\partial f_{i}/\partial Y_{j})_{i,j\in[r]})$,
where $[r]=\\{1,\ldots,r\\}$. Let $H$ be the $n\times n$-matrix obtained by
adding down to $(\partial f/\partial Y)$ as a border the block
$(0|\mbox{Id}_{n-r})$ (we assume as above that $M$ is given by the first $r$
columns of the Jacobian matrix). Let $G^{\prime}$ be the adjoint matrix of $H$
and $G=NG^{\prime}$. We have

$GH=HG=NM\mbox{Id}_{n}=P\mbox{Id}_{n}$

and so

$d\mbox{Id}_{n}=P(y^{\prime})\mbox{Id}_{n}=G(y^{\prime})H(y^{\prime}).$

Let ${\bar{A}}=A/(d^{2e+1})$, ${\bar{B}}={\bar{A}}\otimes_{A}B$,
${\bar{A}}^{\prime}=A^{\prime}/(d^{2e+1}A^{\prime})$. Since
${\mathfrak{m}}^{k}\subset dA$ we see that $v$ defines modulo $d^{2e+1}$ a map
${\bar{v}}:{\bar{B}}\rightarrow{\bar{A}^{\prime}}$.

Let

(1) $h=Y-y^{\prime}-d^{e}G(y^{\prime})T$

where $T=(T_{1},\ldots,T_{n})$ are new variables. Thus

(2) $Y_{l}-y^{\prime}_{l}\equiv d^{e}G_{l}(y^{\prime})T\ \mbox{modulo}\ h_{l}$

for $l\in[n]$, where $G_{l}(y^{\prime})$ means the row $l$ of the matrix
$G(y^{\prime})$. Since

$f(Y)-f(y^{\prime})\equiv\sum_{j}\partial f/\partial
Y_{j}(y^{\prime})(Y_{j}-y^{\prime}_{j})$

modulo higher order terms in $Y_{j}-y^{\prime}_{j}$, by Taylor’s formula we
see that we have

$f(Y)-f(y^{\prime})\equiv\sum_{j}d^{e}\partial f/\partial
Y_{j}(y^{\prime})G_{j}(y^{\prime})T+d^{2e}Q=$
$d^{e}P(y^{\prime})T+d^{2e}Q=d^{e+1}(T+d^{e-1}Q)\ \mbox{modulo}\ h,$

where $Q\in T^{2}A[T]^{r}$ and modulo $h$ means modulo $(h)A[Y,T]^{r}$. This
is because we replace everywhere $Y_{j}-y^{\prime}_{j}$ by
$d^{e}G_{j}(y^{\prime})T$ modulo $h_{j}$ and use that $(\partial f/\partial
Y)G=(P\mbox{Id}_{r}|0)$. Since
$f(y^{\prime})\in{\mathfrak{m}}^{(2e+1)k+c}\subset d^{2e+1}{\mathfrak{m}}^{c}$
we have $f(y^{\prime})=d^{e+1}a$ for some $a\in d^{e}{\mathfrak{m}}^{c}A^{r}$.
Set $g_{i}=a_{i}+T_{i}+d^{e-1}Q_{i}$, $i\in[r]$ and $E=A[Y,T]/(I,g,h)$.
Clearly, it holds that $d^{e+1}g\subset(f,h)$ and $(f)\subset(h,g)$ .

Let $s$ be the first $r\times r$-minor of the Jacobian matrix of $g$. Thus
$s=\det(\mbox{Id}_{r}+d^{e-1}(\partial Q_{i}/\partial T_{j})_{i,j\in[r]})\in
1+(T)$ and $U=(A[T]/(g))_{s}$ is standard smooth. In particular, $U$ is flat
over $A$ and so $(0:_{U}d^{e})=(0:_{U}d^{e+1})$. We claim that
$E_{ss^{\prime}}\cong U_{s^{\prime}}$ for some $s^{\prime}\in 1+(d,T)$. It
will be enough to show that $I\subset(h,g)A[Y,T]_{ss^{\prime}}$ because then
we get $E_{ss^{\prime}}\cong A[Y,T]_{ss^{\prime}}/(g,h)\cong U_{s^{\prime}}$.
We have $PI\subset(f)\subset(h,g)$ and so $P(y^{\prime}+d^{e}L)I\subset(h,g)$
where $L=G(y^{\prime})T$. But $P(y^{\prime}+d^{e}L)$ has the form
$ds^{\prime}$ for some $s^{\prime}\in 1+(d^{e-1}T)\subset 1+(d^{e},T)$. It
follows that $s^{\prime}I\subset((h,g):d)A[Y,T]_{s}$, that is
$s^{\prime}IA[Y,T]_{s}/(h,g)\subset(0:_{A[Y,T]_{s}/(h,g)}\ d)\cong(0:_{U}d)$.
On the other hand, $I\equiv I(y^{\prime})\equiv 0$ modulo $(d^{e},h)$ because
$I(y^{\prime})\subset{\mathfrak{m}}^{(2e+1)k}\subset(d^{2e+1})$ since
${\mathfrak{m}}^{k}\subset(d)$ and so $s^{\prime}IA[Y,T]_{s}/(h,g)=0$ since
$(0:_{U}d^{e})=(0:_{U}d^{e+1})$ implies $(0:_{U}d)\cap d^{e}U=0$. Thus
$I\subset(h,g)A[Y,T]_{ss^{\prime}}$ and it follows that $C=E_{ss^{\prime}}$ is
a $B$-algebra smooth over $A$. Note that by construction $C$ is a smooth
$A$-algebra of standard form.

It remains to see that an arbitrary $A$-morphism $v^{\prime}:B\rightarrow
A^{\prime}$ with $v^{\prime}\equiv v$ modulo ${\mathfrak{m}}^{(2e+1)k}$
factors through $C$. We have $v^{\prime}(Y)\equiv y^{\prime}\ \mbox{modulo}\
{\mathfrak{m}}^{(2e+1)k}$ and so $v^{\prime}(Y)\equiv y^{\prime}\
\mbox{modulo}\ d^{2e+1}$. Thus there exists $\varepsilon\in d^{e}A^{\prime n}$
such that $v^{\prime}(Y)-y^{\prime}=d^{e+1}\varepsilon$. Then
$t:=H(y^{\prime})\varepsilon\in d^{e}A^{\prime n}$ satisfies

$G(y^{\prime})t=P(y^{\prime})\varepsilon=d\varepsilon$

and so

$v^{\prime}(Y)-y^{\prime}=d^{e}G(y^{\prime})t,$

that is $h(v^{\prime}(Y),t)=0$. Note that
$d^{e+1}g(t)\in(h(v^{\prime}(Y),t),f(v^{\prime}(Y)))=(0)$ and $g(t)\in
d^{e}A^{\prime r}$. It follows that $g(t)\in(0:_{A^{\prime r}}d^{e+1})\cap
d^{e}A^{\prime r}=0$ because
$(0:_{A^{\prime}}d^{e})=(0:_{A^{\prime}}d^{e+1})$. Thus $v^{\prime}$ factors
through $E$, that is $v^{\prime}$ is a composite map $B\rightarrow
E\xrightarrow{\alpha}A^{\prime}$, where $\alpha$ is a $B$-morphism given by
$T\rightarrow t$. As $\alpha(s)\equiv 1$ modulo $d$ and
$\alpha(s^{\prime})\equiv 1$ modulo $(d,t)$, $t\in dA^{\prime}$ we see that
$\alpha(s),\alpha(s^{\prime})$ are invertible because $A^{\prime}$ is local
and so $\alpha$ (thus $v^{\prime}$) factors through the standard smooth
$A$-algebra $C$. Thus (1) holds. Then (2) holds from (1), because in this case
we get $v^{\prime}\equiv v$ modulo $d^{2e+1}$ since ${\mathfrak{m}}^{k}\subset
dA$.

Now for (3) take the map ${\hat{w}}:C\cong U_{s^{\prime}}\rightarrow
A^{\prime}/{\mathfrak{m}}^{c}A^{\prime}$ given by
$(Y,T)\rightarrow(y^{\prime},0)$. Then the composite map $B\rightarrow
C\xrightarrow{\hat{w}}A^{\prime}/{\mathfrak{m}}^{c}A^{\prime}$ is given by
$Y\rightarrow v(Y)\ \mbox{modulo}\ {\mathfrak{m}}^{c}$. Since $C$ is standard
smooth, we may lift $\hat{w}$ to an $A$-morphism $w:C\rightarrow A^{\prime}$
by the Implicit Function Theorem. Clearly, $w$ makes the above diagram
commutative because $v(Y)\ \mbox{modulo}\ {\mathfrak{m}}^{c}$ corresponds to
$w(Y)\ \mbox{modulo}\ {\mathfrak{m}}^{c}A^{\prime}$ by the isomorphism
$A/{\mathfrak{m}}^{c}\cong A^{\prime}/{\mathfrak{m}}^{c}A^{\prime}$.

######  Remark 3.

The number $c$ in the above theorem is necessary only in (3) and for (1) (and
so (2)) we could take $c=0$ from the beginning. Since (1) and (3) have mainly
the same proof, we did not want to repeat the arguments and so we unify (1),
(3) in the above theorem, the disadvantage being to have $c$ not useful in
(1).

######  Remark 4.

The third statement of the above theorem gives a variant of [9, Theorem 14] in
the idea of Greenberg [5]. More precisely, let $(A,{\mathfrak{m}})$ be an
excellent Henselian local ring, $A^{\prime}={\hat{A}}$ and $B,\ I$ be as
above. Suppose that $\hat{y}$ is a solution of $I$ in
$A/{\mathfrak{m}}^{(2e+1)k}\cong{\hat{A}}/{\mathfrak{m}}^{(2e+1)k}{\hat{A}}$
such that there exist a system of polynomials $f=(f_{1},\ldots,f_{r})$ in $I$,
an $r\times r$ minor $M$ of the Jacobian matrix $(\partial f/\partial Y)$ and
$N\in((f):I)$ such that
$(MN)({\hat{y}}){\hat{A}}/{\mathfrak{m}}^{(2e+1)k}{\hat{A}}\supset{\mathfrak{m}}^{k}{\hat{A}}/{\mathfrak{m}}^{(2e+1)k}{\hat{A}}$.
As a consequence of Theorem 2 (3), every solution $\tilde{y}$ of $I$ in
$A/{\mathfrak{m}}^{(2e+1)k+c}$, $c\in{\bf N}$ lifting $\hat{y}$ can be lifted
to a solution $y$ of $I$ in $A$. When $A$ is a DVR then by Greenberg [5] there
exists a linear map $\nu:{\bf N}\rightarrow{\bf N}$ such that every solution
$\tilde{y}$ of $I$ in $A/{\mathfrak{m}}^{\nu(c)}$, $c\in{\bf N}$ can be lifted
to a solution $y$ of $I$ in $A$. By [8] and [12] there exist such maps $\nu$
(not necessarily linear) for every excellent Henselian local ring $A$. Our
above consequence gives similarly a linear map $\nu$ of a special form
$c\rightarrow(2e+1)k+c$ but this is not the map given by Greenberg because
here $k$ (and so $\nu$) depends on $\tilde{y}$.

######  Remark 5.

If $A$ is reduced then $e=1$ and $v:B\rightarrow
A^{\prime}/{\mathfrak{m}}^{3k+c}A^{\prime}$. In fact, the proof goes similarly
with $v:B\rightarrow A^{\prime}/{\mathfrak{m}}^{3k+c}A^{\prime}$ (in (1) with
$v:B\rightarrow A^{\prime}/d^{3}A^{\prime}$) if $A$ is Cohen-Macaulay, because
for the proof in this case it is enough to see that $d$ is regular which
follows from ${\mathfrak{m}}^{k}\subset(d)$.

The following proposition follows in particular from [4, Theorem 2] but it is
also a consequence of Theorem 2.

######  Proposition 6.

(Elkik) Let $(A,{\mathfrak{m}})$ be a Noetherian Henselian local ring of
dimension one and $B=A[Y]/I$, $Y=(Y_{1},\ldots,Y_{n})$ an $A$-algbra of finite
type. Then for every $k\in\bf N$ there exist two integers $m_{0},p\in\bf N$
such that if $y^{\prime}\in A^{n}$ satisfies ${\mathfrak{m}}^{k}\subset
H_{B/A}(y^{\prime})$ and $I(y^{\prime})\equiv 0$ modulo ${\mathfrak{m}}^{m}$
for some $m>m_{0}$ then there exists $y\in A^{n}$ such that $I(y)=0$ and
$y\equiv y^{\prime}$ modulo ${\mathfrak{m}}^{m-p}$.

###### Proof.

Suppose that $A^{\prime}=A$. In the notations of Theorem 2 given $k$ set
$m_{0}=p=(2e+1)k$ and suppose that $y^{\prime}\in A^{n}$ satisfies
${\mathfrak{m}}^{k}\subset H_{B/A}(y^{\prime})$ and $I(y^{\prime})\equiv 0$
modulo ${\mathfrak{m}}^{m}$ for some $m>m_{0}$. Let $v:B\rightarrow
A/{\mathfrak{m}}^{m}$ be given by $Y\rightarrow y^{\prime}$. Set $c=m-p$. By
Theorem 2 there exists a standard smooth $A$-algebra $C$ and a map
$w:C\rightarrow A$ which makes the diagram commutative from (3) of Theorem 2.
Let $y$ be the image of $Y$ by the composite map $B\rightarrow
C\xrightarrow{w}A$. Then we have $I(y)=0$ and $y\equiv y^{\prime}$ modulo
${\mathfrak{m}}^{c}={\mathfrak{m}}^{m-p}$.

######  Remark 7.

Proposition 6 still holds when $A$ is a Noetherian Henselian local ring of
arbitrary dimension (see [4, Theorem 2]). Thus for fixed $k\in\bf N$ and
$B=A[Y]/I$ as above there exist $m_{0},p\in\bf N$ such that the linear map
$c\rightarrow p+c$ behaves almost like the Artin function of $B$ when
$y^{\prime}\in A^{n}$ satisfies $H_{B/A}(y^{\prime})\supset{\mathfrak{m}}^{k}$
and $c\geq m_{0}-p$. As in Remark 4, $k$ depends on $y^{\prime}$ and we cannot
say that the Artin function of $B$ is linear.

Let $f$ be a system of $r$ polynomials from $I$ as above, $M$ an $r\times
r$-minor of $(\partial f/\partial Y)$ and $N\in((f):I)$. The above sentences
give the idea that there exists a $\rho\in\bf N$ such that if there exists an
$A$-morphism $v:B\rightarrow A/{\mathfrak{m}}^{\rho}$ with
$v(MN)\supset{\mathfrak{m}}^{k}/{\mathfrak{m}}^{\rho}$ for some $k\in\bf N$
then there exists a $B$-algebra $C$, which is smooth over $A$, such that any
$A$-morphism $v^{\prime}:B\rightarrow\hat{A}$ with $v^{\prime}\equiv v$ modulo
${\mathfrak{m}}^{\rho}\hat{A}$ factors through $C$, $\hat{A}$ being the
completion of $A$. In other words, we believe that Theorem 2 holds in
arbitrary dimension if the smooth locus of $B$ is big with respect to $v$,
that is if $v(H_{B/A}){\hat{A}}$ is ${\mathfrak{m}}{\hat{A}}$ primary. Note
that [16, Remark 4.7] (see also [9, Remark 16]) almost disagrees with our
belief.

##  2\. The Cohen-Macaulay case.

Here we present some consequences of Theorem 2 when $A$ is Cohen-Macaulay and
so $e=1$ by Remark 5.

######  Corollary 8.

With the assumptions and notations of Theorem 2, suppose that $A$ is Cohen-
Macaulay, and let $\rho:B\rightarrow C$ be the structural algebra map. Then
$\rho$ induces bijections $\rho^{*}$ given by $\rho^{*}(w)=w\circ\rho$,
between

  1. (1)

$\\{w\in\operatorname{Hom}_{A}(C,A^{\prime}):w\circ\rho\equiv v\
\mbox{modulo}\ d^{3}A^{\prime}\\}$ and
$\\{v^{\prime}\in\operatorname{Hom}_{A}(B,A^{\prime}):v^{\prime}\equiv v\
\mbox{modulo}\ d^{3}A^{\prime}\\},$

  2. (2)

$\\{w\in\operatorname{Hom}_{A}(C,A^{\prime}):w\circ\rho\equiv v\
\mbox{modulo}\ {\mathfrak{m}}^{3k}A^{\prime}\\}$ and
$\\{v^{\prime}\in\operatorname{Hom}_{A}(B,A^{\prime}):v^{\prime}\equiv v\
\mbox{modulo}\ {\mathfrak{m}}^{3k}A^{\prime}\\}.$

###### Proof.

  1. (1)

By Theorem 2, (1), also by Remark 5, $\rho^{*}$ is surjective. Let
$w,w^{\prime}\in\operatorname{Hom}_{A}(C,A^{\prime})$ be such that
$w\circ\rho=w^{\prime}\circ\rho\equiv v\ \mbox{modulo}\ d^{3}A^{\prime}$. In
particular $w|_{B}=w^{\prime}|_{B}$. In the notations which are introduced in
the proof of Theorem 2, and from equation 2 we have
$H(y^{\prime})(Y-y^{\prime})\equiv d^{2}T\operatorname{mod}h$ by using
$G(y^{\prime})H(y^{\prime})=d\mbox{Id}_{n}.$ Applying $w$ and $w^{\prime}$ on
the above congruence and subtracting we get
$H(y^{\prime})(w(Y)-w^{\prime}(Y))=d^{2}(w(T)-w^{\prime}(T))$, that is
$d^{2}(w(T)-w^{\prime}(T))=0$ because $w(Y)=w^{\prime}(Y)$ and so
$w(T)=w^{\prime}(T)$ because $d$ is regular on $A^{\prime}$ since $d$ is
regular in $A$ (see Remark 5) and $u$ is flat. It follows that $w=w^{\prime}$.

  2. (2)

Apply Theorem 2, (2) and Remark 5 for surjectivity, and injectivity goes as
above.

By construction, $C$ has the form $(A[T]/(g))_{ss^{\prime}}$ in the notations
of Theorem 2, where $s=\det(\partial g_{i}/\partial T_{j})_{i,j\in[r]}$. Let
${\tilde{w}}:C\rightarrow A^{\prime}$ be the unique $A$-morphism such that
${\tilde{w}}\circ\rho\equiv v\ \mbox{modulo}\ d^{3}A^{\prime}$ in the
notations of Corollary 8. We have
${\tilde{w}}(ss^{\prime})\not\in{\mathfrak{m}}A^{\prime}$.

######  Lemma 9.

There exist canonical bijections

  1. (1)

$A^{\prime
n-r}\rightarrow\\{w^{\prime}\in\operatorname{Hom}_{A}(C,A^{\prime}):w^{\prime}\equiv{\tilde{w}}\
\mbox{modulo}\ d^{3}A^{\prime}\\}.$

  2. (2)

${\mathfrak{m}}^{3k}A^{\prime
n-r}\rightarrow\\{w^{\prime}\in\operatorname{Hom}_{A}(C,A^{\prime}):w^{\prime}\equiv{\tilde{w}}\
\mbox{modulo}\ {\mathfrak{m}}^{3k}A^{\prime}\\}.$

###### Proof.

Let $w^{\prime}\in\operatorname{Hom}_{A}(C,A^{\prime})$ be with
$w^{\prime}\equiv{\tilde{w}}\ \mbox{modulo}\ d^{3}A^{\prime}$. We have

$g(w^{\prime}(T))\equiv g({\tilde{w}}(T))=0\ \mbox{modulo}\ d^{3}A^{\prime},$
$(ss^{\prime})(w^{\prime}(T))\equiv(ss^{\prime})({\tilde{w}}(T))\not\equiv 0\
\mbox{modulo}\ {\mathfrak{m}}.$

Set $V=(T_{1},\ldots,T_{r})$, $Z=(T_{r+1},\ldots,T_{n})$. Thus
$g(V,w^{\prime}(Z))=0$ has a unique solution (namely $w^{\prime}(V)$) in
${\tilde{w}}(Z)+d^{3}A^{\prime}$ by the Implicit Function Theorem. It follows
that the restriction

$\\{w^{\prime}\in\operatorname{Hom}_{A}(C,A^{\prime}):w^{\prime}\equiv{\tilde{w}}\
\mbox{modulo}\ d^{3}A^{\prime}\\}\rightarrow$
$\\{w^{\prime\prime}\in\operatorname{Hom}_{A}(A[Z],A^{\prime}):w^{\prime\prime}\equiv{\tilde{w}}|_{A[Z]}\
\mbox{modulo}\ d^{3}A^{\prime}\\}$

is bijective.

On the other hand, the map

$\\{w^{\prime\prime}\in\operatorname{Hom}_{A}(A[Z],A^{\prime}):w^{\prime\prime}\equiv{\tilde{w}}|_{A[Z]}\
\mbox{modulo}\ d^{3}A^{\prime}\\}\rightarrow{\tilde{w}}(Z)+d^{3}A^{\prime
n-r}$

given by $w^{\prime\prime}\rightarrow w^{\prime\prime}(Z)$ is a bijection too.
It follows that there exists a canonical bijection between
$\\{w^{\prime}\in\operatorname{Hom}_{A}(C,A^{\prime}):w^{\prime}\equiv{\tilde{w}}\
\mbox{modulo}\ d^{3}A^{\prime}\\}$ and ${\tilde{w}}(Z)+d^{3}A^{\prime n-r}$,
the last one being in bijection with $d^{3}A^{\prime n-r}$, that is with
$A^{\prime n-r}$, $d$ being regular in $A^{\prime}$. The proof of (2) goes
similarly.

The following theorem extends [14, Theorem 15].

######  Theorem 10.

With the assumptions and notations of Corollary 8 there exist canonical
bijections

  1. (1)

$A^{\prime
n-r}\rightarrow\\{v^{\prime}\in\operatorname{Hom}_{A}(B,A^{\prime}):v^{\prime}\equiv
v\ \mbox{modulo}\ d^{3}A^{\prime}\\}.$

  2. (2)

${\mathfrak{m}}^{3k}A^{\prime
n-r}\rightarrow\\{v^{\prime}\in\operatorname{Hom}_{A}(B,A^{\prime}):v^{\prime}\equiv
v\ \mbox{modulo}\ {\mathfrak{m}}^{3k}A^{\prime}\\}.$

For the proof apply Corollary 8 and the above lemma.

Let $(\Lambda,{\mathfrak{p}})$ be a local Artinian ring,
$(A^{\prime},{\mathfrak{m}}^{\prime})$ be a local Noetherian ring of dimension
$1$ and $u:\Lambda\rightarrow A^{\prime}$ a regular morphism. Suppose that
$A^{\prime}$ is Henselian and has the same residue field with $\Lambda$. Let
$F$ be an $\Lambda$-algebra of finite type, let us say $F=\Lambda[U]/J$,
$U=(U_{1},\ldots,U_{n})$. A constructive GND of $F$ with respect to an
$A$-morphism $F\rightarrow A^{\prime}$ follows easily from the classical Néron
Desingularization [7] as it is shown in [6]. Note that
$A^{\prime}/{{\mathfrak{p}}}A^{\prime}$ is a DVR because it is a one
dimensional regular local ring, $u$ being regular. Let $x\in A$ be a lifting
of a local parameter of $A/{\mathfrak{p}}$. Clearly, $x$ is a regular element
in $A$ and $\Lambda,A^{\prime}$ are Cohen-Macaulay rings. Set
$A=\Lambda[X]_{({\mathfrak{p}},X)}$, $B=A\otimes_{\Lambda}F$, and let
$u^{\prime}:A\rightarrow A^{\prime}$ be the regular morphism extending $u$ by
$X\rightarrow x$. Let $f=(f_{1},\ldots,f_{r})$, $r\leq n$ be a system of
polynomials from $J$ and $M$ an $r\times r$-minor of the Jacobian matrix
$(\partial f_{i}/\partial U_{j})$, $g:F\rightarrow
A^{\prime}/{\mathfrak{m}}^{\prime 3k}$ a $\Lambda$-morphism, and
$N\in((f):J)$. Suppose that $g(NM)A^{\prime}/{\mathfrak{m}}^{\prime
3k}\supset{\mathfrak{m}}^{\prime k}/{\mathfrak{m}}^{\prime 3k}$.

The following theorem extends [14, Corollary 16].

######  Theorem 11.

In the assumptions and notations from above, the set
${\mathcal{G}}=\\{g^{\prime}\in\operatorname{Hom}_{\Lambda}(F,A^{\prime}):g^{\prime}\equiv
g\ \mbox{modulo}\ {\mathfrak{m}}^{\prime 3k}\\}$ is in bijection with
${\mathfrak{m}}^{\prime 3k}A^{\prime n-r}$.

For the proof apply Theorem 10 to the morphism $A\otimes_{\Lambda}F\rightarrow
A^{\prime}$ induced by $g$ and $u^{\prime}$ and note that $A\otimes_{\Lambda}$
induces a bijection
$\operatorname{Hom}_{\Lambda}(F,A^{\prime})\rightarrow\operatorname{Hom}_{A}(B,A^{\prime})$
by adjunction.

##  3\. Algorithm Néron Desingularization

Input: $A=k[x]_{(x)}/J,$ $J=(h_{1},\ldots,h_{q}),\ h_{i}\in
k[x],x=(x_{1},\ldots,x_{t}),\ \dim(A)=1,$ k is a field.
$B=A[Y]/I,I=(f_{1},\ldots,f_{l}),f_{i}\in k[x,Y],Y=(Y_{1},\ldots,Y_{n})$,
integers $k$,$r$, $y^{\prime}\in{k[x]}^{n}$, $N\in(f_{1},\ldots,f_{r}):I$.

Output: A Neron Desingularization induced by $y^{\prime}$ or the message
“$y^{\prime}$, $N$, $(f_{1},\ldots,f_{r})$ are not well chosen”.

  1. (1)

Compute $M=\det((\partial f_{i}/\partial Y_{j})_{i,j\in[r]})$, $P:=NM$ and
$d:=P(y^{\prime})$

  2. (2)

$f:=(f_{1},\ldots,f_{r})$

  3. (3)

Compute $e$ such that $(0:_{A}d^{e})=(0:_{A}d^{e+1})$

  4. (4)

If $I(y^{\prime})\nsubseteq(x)^{(2e+1)k}+J$ or $(x)^{k}\nsubseteq(d)+J$,
return “$y^{\prime}$, $N$, $(f_{1},\ldots,f_{r})$ are not well chosen”

  5. (5)

Complete $(\partial f_{i}/\partial Y_{j})_{i\leq r}$ by
$(0|(\mbox{Id}_{n-r}))$ to obtain a square matrix $H$

  6. (6)

Compute $G^{\prime}$ the adjoint matrix of $H$ and $G:=NG^{\prime}$

  7. (7)

$h=Y-y^{\prime}-d^{e}G(y^{\prime})T,\ T=(T_{1},\ldots,T_{n})$

  8. (8)

Write $f(Y)-f(y^{\prime})=\sum_{j}d^{e}\partial f/\partial
Y_{j}(y^{\prime})G_{j}(y^{\prime})T+d^{2e}Q$

  9. (9)

Write $f(y^{\prime})=d^{e+1}a$

  10. (10)

For $i=1$ to $r$, $g_{i}=a_{i}+T_{i}+d^{e-1}Q_{i}$

  11. (11)

$E:=A[Y,T]/(I,g,h)$

  12. (12)

Compute $s$ the $r\times r$ minor defined by the first $r$ columns of
$(\partial g/\partial T)$

  13. (13)

Write $P(y^{\prime}+d^{e}G(y^{\prime})T)=ds^{\prime}$

  14. (14)

return $E_{ss^{\prime}}$.

Acknowledgements The first author gratefully acknowledges the support from the
ASSMS GC. University Lahore, for arranging her visit to Bucharest, Romania and
she is also grateful to the Simion Stoilow Institute of the Mathematics of the
Romanian Academy for inviting her.

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