# Real-time estimation of phase and amplitude with application to neural data

Michael Rosenblum  Department of Physics and Astronomy, University of Potsdam,
Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany  corresponding
author<EMAIL_ADDRESS>Arkady Pikovsky  Department of Physics and
Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-
Golm, Germany  Andrea A. Kühn  Movement Disorders and Neuromodulation Unit,
Department of Neurology, Charité – Universitätsmedizin Berlin, Charitéplatz 1,
D-10117 Berlin, Germany  Johannes L. Busch  Movement Disorders and
Neuromodulation Unit, Department of Neurology, Charité – Universitätsmedizin
Berlin, Charitéplatz 1, D-10117 Berlin, Germany

###### Abstract

Computation of the instantaneous phase and amplitude via the Hilbert Transform
is a powerful tool of data analysis. This approach finds many applications in
various science and engineering branches but is not proper for causal
estimation because it requires knowledge of the signal’s past and future.
However, several problems require real-time estimation of phase and amplitude;
an illustrative example is phase-locked or amplitude-dependent stimulation in
neuroscience. In this paper, we discuss and compare three causal algorithms
that do not rely on the Hilbert Transform but exploit well-known physical
phenomena, the synchronization and the resonance. After testing the algorithms
on a synthetic data set, we illustrate their performance computing phase and
amplitude for the accelerometer tremor measurements and a Parkinsonian
patient’s beta-band brain activity.

## Introduction

Estimation of the instantaneous phase and amplitude is essential for
electrical and mechanical engineering, synchronization studies of oscillatory
systems of different nature, time series analysis of physiological data, and,
in particular, for neuroscience [1, 2, 3, 4]. A special application is
developing efficient sensing algorithms for adaptive deep brain stimulation, a
recent advancement of a widely used treatment option for Parkinson’s disease
and other neurological disorders [5]. One of the directions in this
development is to adjust stimulation parameters according to a peripheral or
neurophysiological signal’s phase and amplitude computed on the fly [6, 7, 8,
9, 10, 11, 12]. The popular approach to phase and amplitude estimation is to
exploit the analytic signal approach based on the Hilbert Transform (HT) or,
equivalently, the wavelet transform with a complex wavelet [13, 14, 15, 1].
However, this widely-used tool is non-causal and therefore not appropriate for
real-time analysis, whereas causal estimation of phase and amplitude is often
crucial for closed-loop control of complex systems. Despite several attempts
to adapt the HT for a causal measurement [16, 17], the reliable and fast
estimation of phase and amplitude of real-world signals remains challenging.

In this study, we do not rely on the Hilbert Transform. Instead, we follow
another development line and extend our previous approach [18, 19] for real-
time estimation based on the oscillation theory and nonlinear dynamics. The
main idea is as follows. Suppose we have an oscillator, e.g., an electronic
circuit, which amplitude and phase we can monitor. Next, suppose we feed our
measurement to this oscillator. We choose the oscillator so that there is a
one-to-one correspondence between the oscillator’s phase and amplitude (what
we can monitor) and the measured signal (what we want to determine). If this
correspondence is achieved, we recompute the monitored state of the oscillator
into desired quantities. Thus, the oscillator acts as a measuring device.
Indeed, we will not exploit a physical oscillator but use a simple computer
program that simulates the oscillator’s dynamics. In our approach, we rely on
two well-known physical effects: linear resonance and synchronization. We
present and compare three techniques that provide a fast estimation of phases
and amplitudes, using only the past of the time series. We test the algorithms
on model data and apply them to neural time series. Namely, demonstrating our
approach’s efficiency, we causally estimate the phase and amplitude of the
accelerometer tremor measurements and Parkinsonian patients’ beta-band brain
activity and compare the results with non-causal HT-based analysis.

## Results

### Hilbert Transform vs causal estimation

A common practice in extracting the amplitude and the phase of an input signal
$s(t)$ is based on application of the Hilbert transform. The HT analysis
provides proper estimates for the instantaneous phase of $s(t)$ for narrow-
band one-component signals with slowly varying amplitude and frequency. We
emphasize that, though formally one can compute the HT for an arbitrary signal
$s(t)$, not in all cases the extraction of the amplitude and the phase will
lead to reasonable results. We also stress that HT is a non-causal operation:
to extract features at time instant $t^{\prime}$, one has to know both the
signals’ past $t<t^{\prime}$ and future $t>t^{\prime}$. For a further detailed
discussion of the HT’s properties and practical implementation, see Methods.

The main goal of this paper is not to extend HT approach to complex signals
(see [20, 21] for examples of such an extension), but to provide and explore
causal alternatives to the HT approach. In what follows, we use the HT-based
amplitude $a_{H}$ and the HT-based phase $\varphi_{H}$ as a “gold standard”
for testing our algorithms. Strictly speaking, this is reasonable for narrow-
band signals only. Definition and determination of the proper phase and
amplitude for a complex signal represent a challenging problem that we do not
address here. Instead, we use an operational approach: a proper amplitude
should correctly represent an envelope of the signal; a proper phase should
gain $2\pi$ at each oscillatory cycle. Below, we present three techniques for
causal computations and compare the computed amplitudes and phases with non-
causally obtained $a_{H},\varphi_{H}$.

### Causal estimation techniques

#### Phase locking approach

The first technique exploits the ideas from the synchronization theory. It is
well-known that a force $s(t)$ acting on a limit-cycle oscillator can entrain
(lock) it. It means that the oscillator’s frequency becomes equal to that of
the force, and their phases differ by a constant. Thus, the phase of the
locked limit-cycle oscillator will correspond to the phase of the signal. For
our purposes, it is helpful to use the simplest oscillator model, the so-
called phase oscillator. To ensure the phase-locking to the force, we have to
adjust the oscillator’s frequency to the signal’s frequency. We assume that we
do not know the latter a priori, but can only roughly estimate it. We propose
a simple approach that starts with this estimate and automatically tunes the
“device’s” frequency to ensure the locking and thus provides the instantaneous
phase $\varphi_{L}(t)$. The amplitude is not determined with this approach.
One can treat the suggested scheme as a software implementation of a phase-
locked loop [22]. Technically, the algorithm reduces to solving differential
equation incorporating measured data given at discrete time points; for
details of the technique and its implementation, see Methods.

#### Nonresonant linear filter

The second technique relies on the resonance effect. Our measuring “device”
consists now of two linear damped oscillators. The oscillators’ frequency is
much larger than the frequency of the signal, i.e., the system is far from
resonance. We choose the damping parameters to ensure that (i) phase of the
first linear oscillator equals that of the input and that (ii) amplitude of
the second one and the input relate by a known constant multiplicator. The
technique yields both phase $\varphi_{N}(t)$ and amplitude $a_{N}(t)$, where
the index $N$ stands for “non-resonant”.

#### Resonant linear filter

Our third approach adopts the technique used for model studies in our previous
publications [18, 19]. The corresponding “device” consists of a linear
oscillator in resonance with the measured signal and of an integrating unit.
It also provides both the phase and the amplitude that we denote as
$\varphi_{R}$ and $a_{R}$, respectively. The method exploits the known
relation between the resonant oscillator’s phase and amplitude and those of
the input. Additionally, the resonant oscillator acts as a bandpass filter for
experimental data.

Technically, both latest techniques require the numerical solution of linear
differential equations with the input signal $s(t)$. We present a detailed
description of the techniques and the developed numerical schemes in the
Methods section. Like the phase-locking algorithm, both techniques include an
automated frequency-tuning algorithm to adjust the systems to the a priori
unknown signal’s frequency.

### Testing the algorithms

#### Artificial data

First, we test our approach on artificial data. All algorithms work well with
simple narrow-band signals like slowly modulated sine waves. So, we do not
show these results and proceed with a more complicated case. The signal is

$s(t)=\left[1+0.95\cos(\Omega_{1}t)\right]\cdot\left[\cos(\psi(t))+0.2\cos(2\psi(t)+\pi/6)+0.1\cos(3\psi(t)+\pi/3)\right]\;,$
(1)

where

$\psi(t)=t+5\sin(\Omega_{2}t)\;,$

its waveform has three harmonics. The signal is amplitude- and phase-
modulated, see Fig. 1a; it is sampled with $\Delta=0.01$ to yield $s(\Delta
k)=s_{k}$. The frequencies are $\Omega_{1}=\sqrt{2}/30$,
$\Omega_{2}=\sqrt{5}/60$. Another test signal is $\bar{s}_{k}=s_{k}+\xi_{k}$,
where $\xi_{k}$ are Gaussian random numbers with zero mean and standard
deviation $0.05$, see Fig. 2a.

Figure 1: (a) The test signal $s$ according to Eq. (1) and its Hilbert
amplitude $a_{H}$ (red); one can see that $a_{H}$ does not represent a good
envelope for $s$. Panels (b, c, d) show the difference between the Hilbert
phase $\varphi_{H}$ and causally estimated phases ($\varphi_{L}$,
$\varphi_{N}$, and $\varphi_{R}$ are obtained by means of the locking-based
technique, non-resonant and resonant oscillator, respectively). Notice that in
(c) we show $(\varphi_{H}-\varphi_{N})\mod 2\pi$: within the first 20 time
units the phase difference decreases to $-14\pi$ until it saturates and
oscillates around $(\varphi_{H}-\varphi_{N})\mod 2\pi=0$.

Figure 2: (a) Short epoch of noisy data $\bar{s}$ (blue) and causally obtained
amplitudes $a_{N}$ (red) and $a_{R}$ (magenta); the latter provides the most
smooth envelope (cf. $a_{H}$ in Fig. 1a). When the signal’s amplitude is very
small, the noise dominates and phase determination becomes complicated. The HT
approach fails here (blue line in (b,c)), while both locked and resonant
oscillator “devices” provide reasonable results. The performance of the non-
resonant technique is poor: when the amplitude nearly vanishes, the phase
estimated by this technique (not shown) is not better than the Hilbert phase.

For testing all three algorithms, we set the initial frequency of the device
to be $10\%$ higher than the actual value; in this way, we imitate the
imprecision on initial frequency estimation. The other parameters are given in
the Methods.

##### Phase-locked oscillator.

Figure 1b shows that after a short transient, the phase determined by the
“device” becomes close to the Hilbert phase. The difference is due to the
presence of the harmonics (for a mono-component amplitude-modulated signal,
the phase difference is very small, its standard deviation is $0.03$). The
locking-based measurement’s advantage becomes evident in the noisy data, as
expected for a phase-locked loop approach: here, the real-time estimation
provides a reasonable phase even at low amplitudes, where the noise dominates,
see Fig. 2a,b.

##### Non-resonant linear oscillator (filter).

This approach works well in noise-free case (Fig. 1c). Initially, the
difference with $\varphi_{H}$ grows rapidly due to decaying high-frequency
oscillation with the natural frequency $\omega$ of the system. (We remind that
frequency $\omega$ of the non-resonant oscillator is much larger than the
input’s frequency.) Nevertheless, after the transient, the performance is
comparable or even better than that of two other techniques. In the presence
of noise the performance is poor because noisy perturbations excite high-
frequency oscillation with the frequency $\omega$. The amplitude estimation is
slightly worse than that via HT, see Fig. 2a. Thus, for processing without a
filter, this technique is not optimal.

##### Resonant linear oscillator (filter).

This technique also demonstrates efficient phase estimation, see Fig. 1d for
the noise-free data and Fig. 2c for the noisy case. Actually, the results for
the latter are hardly distinguishable from those in Fig. 2b, though the
frequency adaptation of the resonant oscillator is faster. The technique also
provides the instantaneous amplitude, see Fig. 2a. Notice that the Hilbert
amplitude $a_{H}$ for the multi-component signal $\bar{s}$ is not the perfect
envelope. The real-time amplitude $a_{R}$ is not perfect either, but is much
more smooth than $a_{H}$.

#### Estimation without an additional filter: tremor data

Tremor data was acquired from a patient with essential tremor using a 3D
accelerometer (TMSi, Oldenzaal, The Netherlands) attached to the right index
finger. The signal was sampled at 2048 Hz sampling rate using a Porti
amplifier (TMSi). The patient stretched their arms out in front of their chest
holding a bottle in order to provoke postural tremor. Only data from one axis
was analyzed. The tremor time series (Fig. 3a) is relatively simple: it
resembles the test signal from the previous section. The tremor data is
amplitude modulated, and its power spectrum also contains several harmonics
(Fig. 3b). However, the real-world signals are naturally more complicated than
the simple test data series Eq. (1). Analyzing them, one frequently faces
additional difficulties. For the tremor signal under consideration these
features are: a drift of the baseline; presence of epochs when the amplitude
vanishes; and outliers. To cope with the baseline fluctuation, we exploit the
following causal detrending algorithm. For each new measurement point, we
remove the mean value computed over several previous cycles. For the sake of
computational speed, we update this value several times per period. We denote
the detrended signal by $\tilde{s}_{k}$ (Fig. 3c).

Figure 3: The raw tremor data (a) and its power spectral density $S(f)$,
computed by the Burg algorithm (b). Panels (c,d) show some epochs of
$\tilde{s}_{k}$ (the original signal with the constant component and baseline
fluctuation removed on the fly), the Hilbert amplitude $a_{H}$ (red) and the
causal amplitude $a_{R}$ (magenta). The latter provides a smooth envelope and
does not reflect the outliers (d). Panels (e,f) demonstrate the Hilbert phase
$\varphi_{H}$ (blue) and causal phases $\varphi_{L}$ (red) and $\varphi_{R}$
(magenta). For epochs with vanishing amplitude, causal phases do not exhibit
noisy jumps as the Hilbert phase. The causal phases are stable with respect to
outliers (f).

Figure 3c compares the Hilbert amplitude $a_{H}$ with the causally computed
$a_{R}$. Like in the case of test data, the causally computed amplitude
$a_{R}$ is a better estimation of the envelope than $a_{H}$ is. Moreover,
computation of $a_{R}$ is stable with respect to outliers (Fig. 3d). Next, in
Fig. 3e,f we show the Hilbert phase $\varphi_{H}$ along with two causally-
obtained phases $\varphi_{L}$ and $\varphi_{R}$. We see that for large-
amplitude oscillation, all the phases practically coincide. When the amplitude
almost vanishes, the “device” of the locking-based method falls out of
synchrony and makes one cycle less. On the contrary, the resonant-oscillator
phase $\varphi_{R}$ and the Hilbert phase reveal the same number of cycles. We
emphasize, that both causal techniques are not sensitive to the outliers in
the original data, while the Hilbert approach is (see Fig. 3f). Finally, we
mention that the non-resonant oscillator technique works poorly for the
unfiltered tremor data. Only if we use the bandpass filter $4.5\pm 2$ Hz, then
this technique works perfectly.

#### Wide-band signal: beta-band brain activity

Elevated beta-band activity has been established as a marker for rigidity and
bradykinesia in patients with Parkinson’s Disease [23]. As such, it has been
employed in several studies of adaptive deep brain stimulation aiming for
automatic adjustment of stimulation parameters in response to beta-band
amplitude [7, 9]. Here we recorded LFP data from a patient with Parkinson’s
Disease on medication, 2 days after surgery for deep brain stimulation with
electrode cables being externalized. LFPs were acquired from the left
subthalamic nucleus via a 4-contact electrode (Model 3389, Medtronic,
Minneapolis, USA) using a D360 amplifier (Digitimer Ltd., Welwyn Garden City,
UK) and were digitized at 1 kHz sampling rate with a 1401 analog-to-digital
converter (CED Ltd., Cambridge, UK). A bipolar referencing scheme was adopted,
with the signal shown here originating from contacts one to two. During the
recording session, the patient was comfortably seated and was asked to rest
quietly.

The analysis of the beta-band brain activity requires a bandpass filter
preprocessing. We use a simple FIR filter with bandwidth $17\pm 4$ Hz
($281$-point filter was generated by the Matlab fir1 function). A causal
filtration introduces a delay, but this is a necessary price to be paid. Our
tests with this signal demonstrate that the non-resonant-oscillator technique
outperforms the other two that fail because of the substantial variation of
the signal’s amplitude. Noteworthy, due to the bandpass, the approach works
without the baseline correction and the frequency adaptation. Hence, the
algorithm implementation requires only a few lines of code. The results are
illustrated by Fig. 4.

Figure 4: Beta-band brain activity data. (a) Power spectral density $S(f)$ of
the raw (blue) and filtered (red) series. (b) Bandpass filtered series
(arbitrary units). (c) A short epoch of the filtered series (yellow) and two
their envelopes computed via the Hilbert Transform (blue) and by means of the
non-resonant oscillator (red). The envelopes practically overlap, as is also
illustrated by panel (d). Panel (e) demonstrates the difference between the
Hilbert phase $\varphi_{H}$ and the real-time phase $\varphi_{N}$ along with
the amplitude: one can see that the phase difference is negligible when the
amplitude does not vanish and is not small only when amplitude is on the level
of noise so that the phase is not well-defined; for these amplitude values,
reliable phase determination is not possible. (Here $\tilde{a}_{N}$ is the
amplitude rescaled for better visibility.) The probability distribution of the
phase difference in panel (f) confirms practical coincidence of $\varphi_{H}$
and $\varphi_{N}$.

## Discussion

To summarize, in this paper, we considered the problem of causal instantaneous
amplitude and frequency estimation. Contrary to attempts to adjust the
inherently non-causal analytic signal approach, we used oscillation theory
ideas. We presented and tested three algorithms. The first one exploits the
phase-locking property of nonlinear limit-cycle oscillators, while the other
two rely on linear resonance. Technically, we encountered solving differential
equations incorporating experimental data; we suggested efficient numerical
schemes to solve them. Below, we discuss the advantages and disadvantages of
all three techniques. Since these techniques aim at real-time applications, we
pay special attention to their computational efficiency.

The locking-based technique is similar to the phase-locked loop approach and
provides the phase only. It works well for relatively narrow-band signals like
the tremor data and does not require an additional bandpass filter. It can
easily incorporate the frequency adaptation that makes the algorithm able to
cope with the signal’s frequency’s slow variation. The technique does not work
well with the signals like beta-band activity, where both the frequency and
the amplitude vary relatively fast; this results in frequent loss of
synchrony. The algorithm is computationally efficient: the only demanding
operations for a new phase value are several computations of the sine
function.

As an advantage of the resonant oscillator technique, we mention that it
provides both the phase and the amplitude. It is stable to high-frequency
noise because the resonant oscillator acts as a weak bandpass filter. However,
it requires baseline correction because the integrator unit enhances the low-
frequency perturbations. The implementation described in the Methods section
is very efficient, provided the oscillator’s frequency does not change.
Adaptation of the frequency to that of the signal requires recomputation of
algorithm’s coefficients what slightly reduces the efficiency.

If extraction of the rhythm of our interest requires a bandpass filter, the
non-resonant oscillator approach is the best choice. Using the beta-band brain
activity as an illustrative example, we demonstrated that this causal
algorithm provides phase and amplitude very close to non-causally computed
Hilbert Transform-based values. An essential advantage of the algorithm is its
efficiency: computation of the new phase and amplitude values requires only
several arithmetic operations and two function calculations. This property
makes the algorithm especially appropriate for the real-time processing of
high-frequency signals.

We emphasize that all three techniques are much faster than approaches based
on the HT. For example, the technique introduced in [17] requires the
computation of the direct and inverse Fourier transform in a running window
plus additional filtration to compensate for the boundary effects and non-
causal nature of the HT; altogether, this computation is about thirty times
more demanding than an oscillator-based approach.

Taken together, the techniques suggested here may provide useful means for
real-time detection of phase and amplitude in the context of adaptive deep
brain stimulation. For example, estimation of instantaneous amplitude has been
implemented by interpolating beta-filtered and squared LFP data [24]. However,
this approach is not able to recover the signal’s phase simultaneously.
Additionally tracking phase may augment therapeutic benefit, though, by
delivering stimulation pulses at specific points within the oscillatory cycle.
This has been shown to effectively disrupt pathological oscillations [12, 17,
10, 11]. So far, to our knowledge, adaptive stimulation algorithms integrating
both the phase and amplitude of a signal have not been established yet.

Finally, we mention that our techniques do not cause any additional delay. The
processing delay is only due to a causal filter if the latter is required,
e.g., for brain activity data. To minimize the delay, one has to exploit
specially designed filters [25].

## Methods

In all cases the input is an oscillatory signal $s(t)$ is sampled with
interval $\Delta$. Thus, available is a sequence $s(t_{k})=s(k\Delta)=s_{k}$.

##### The non-causal Hilbert-Transform based approach.

In this approach one constructs the corresponding complex-valued analytic
signal $Z(t)=s(t)+\mathrm{i}\hat{s}(t)$, where
$\hat{s}(t)=\pi^{-1}\text{P.V.}\int_{-\infty}^{\infty}\frac{s(\tau)}{t-\tau}\mathrm{d}\tau$
is the HT of $s(t)$. Obviously, the HT is a non-causal operation; HT of a
finite-length time series yields spurious values for the boundaries. The
absolute value and argument of $Z(t)$ provide the instantaneous Hilbert
amplitude $a_{H}(t)$ and phase $\varphi_{H}(t)$, respectively. For a narrow-
band one-component signals like $s(t)=A(t)\cos[\omega(t)t]$, where $A(t)$,
$\omega(t)$ are slow functions of time, the analytic signal approach provides
$a_{H}\approx A$ and $\varphi_{H}\approx\int\omega(t)\mathrm{d}t$. For a
discussion of the HT’s practical implementation and technical hints we refer,
e.g., to [26, 1, 4]. In a practical implementation either a discrete
evaluation of the integral is performed, or a discrete Fourier transform is
used.

### Causal estimation of phase and amplitude

In these methods, we obtain the value of the instantaneous phase
$\varphi(t_{k})=\varphi_{k}$ and amplitude $a(t_{k})=a_{k}$ by using only the
current and the previous values of the signal, i.e.,
$s_{k},\;s_{k-1},\;\ldots$.

#### Measuring “device”: phase-locked oscillator

The synchronization theory says that an oscillatory force $s(t)$ acting on a
limit-cycle oscillator can entrain it, if the frequency of the force is close
to the natural frequency of the limit-cycle oscillator. It means that the
oscillator’s frequency becomes equal to that of the force, and their phases
fulfill the locking condition $\varphi-\theta\approx\mbox{const}$, where
$\theta$ and $\varphi$ are the oscillator’s and the signal’s phases. For our
purposes, it is appropriate to use the so-called phase oscillator. Its forced
dynamics is described by

$\dot{\theta}=\omega-\varepsilon\sin\theta\cdot s(t)\;,$ (2)

where $\varepsilon$ is a parameter that determines the coupling strength.
Consider first a harmonic force, $s=a\cos(\nu t)$. If $\omega=\nu$, then in
the locked state $\varphi\approx\theta$. However, $\nu$ is not known a priori,
but we assume that we can roughly estimate it. Let this initial guess be
$\nu_{0}$. Thus, we set initially $\omega=\nu_{0}$ and start our computation
with this value. To adapt the phase oscillator to the signal, we estimate the
frequency of the forced oscillator $\nu_{e}$ on the fly. (The index $e$ stands
for “estimated”). To this end, for the time instant $t$, we take previously
computed unwrapped phases $\theta$ for the time interval $[t-T_{e},t]$, where
$T_{e}\sim 2\pi/\nu_{0}$ (approximately one or two cycles). Assuming that
within this time interval
$\theta(t^{\prime})=\theta(t-T_{e})+\nu_{e}(t^{\prime}-t+T_{e})$, we compute
the frequency $\nu_{e}$ via the linear fit. Next, we update the oscillator’s
autonomous frequency as

$\omega\quad\longrightarrow\quad\omega+K(\nu_{e}-\omega)\;,$ (3)

where $K$ is a constant update factor. Adapting the frequency in this way, we
ensure a transition from the unlocked to locked dynamics. Performing frequency
estimation several times per cycle, we successfully estimate the phase of
signals with slowly drifting frequency. We expect that this algorithm also
works if the force’s amplitude $a$ slightly varies with time (so that the
oscillator still remains locked).

One can treat the suggested scheme as a software implementation of a phase-
locked loop [22], cf. also [27]. Practically, we have to solve the
differential Eq. (2) numerically, whereas its right-hand side is known only in
discrete time points $t_{k}$. The easiest way is to exploit Euler’s technique
with the integration step $\Delta$ to advance from the known value
$\theta_{k}$ at $t_{k}$ to the new phase $\theta_{k+1}$ at
$t_{k+1}=t_{k}+\Delta$, given a new measurement $s_{k+1}$. This technique may
work properly if $\Delta$ is sufficiently small; otherwise, the numerical
solution becomes unstable. (We recall that $\Delta$ is the fixed sampling
interval that cannot be made arbitrarily small). Therefore, to advance the
solution of Eq. (2) from $\varphi_{k}$ to $\varphi_{k+1}$ given new
measurement $s_{k+1}$, we use the parabolic approximation of $s(t)$ on the
interval $[t_{k},t_{k+1}]$ and then exploit the standard Runge-Kutta
algorithm. (Notice that the Runge-Kutta step can be much smaller than
$\Delta$.) The coefficients of the parabolic fit are computed from
$s_{k-1},s_{k},s_{k+1}$. Starting with some initial condition, e.g.,
$\theta_{0}=0$, we achieve, after a short transient (see examples in Fig. 1),
the synchronous state where $\varphi_{k}\approx\theta_{k}$. Notice that the
phase oscillator can be complimented by a low-pass filter:

$\dot{\varphi}=\omega+\varepsilon
w\;,\qquad\tau\dot{w}+w=-s(t)\sin\theta\;,\qquad$

then the scheme becomes the simplest traditional phase-locked loop [22, 4];
for $\tau\to 0$ it reduces to Eq. (2). This extension may be useful for the
data with strong high-frequency noise but it does not show any advantages for
the tremor data we use.

#### Measuring “device”: non-resonant linear oscillator

Our second measuring “device” is linear damped oscillator:

$\ddot{x}+\alpha\dot{x}+\omega^{2}x=s(t)\;,$ (4)

where $\omega$ and $\alpha$ are the frequency and the damping parameter of the
oscillator, respectively. Suppose first that $s(t)$ is harmonic,
$s(t)=a\cos(\nu t)=a\cos(\varphi(t))$. The well-known stationary solution of
linear Eq. (4) is $x=b\cos(\nu t+\beta)$, where
$b=\frac{a}{\sqrt{(\omega^{2}-\nu^{2})^{2}+(\alpha\nu)^{2}}}$ and
$\beta=\arctan\left[\frac{-\alpha\nu}{\omega^{2}-\nu^{2}}\right]$. Thus, the
forced system (4) oscillates with the force’s frequency $\nu$ and the
amplitude $b$. The dependencies of the amplitude ratio $b/a$ and of the phase
shift $\beta$ on the forcing frequency $\nu$ reflect the well-known resonance
effect. Knowing the oscillator’s state $x(t)=b\cos(\nu t+\beta)$,
$\dot{x}(t)=-b\nu\sin(\nu t+\beta)$ we find the amplitude and phase of the
external force:

$a(t)=\sqrt{x(t)^{2}+[\dot{x}(t)/\nu]^{2}}\sqrt{(\omega^{2}-\nu^{2})^{2}+(\alpha\nu)^{2}}\;,\qquad\varphi(t)=\arctan\left[\frac{-\dot{x}(t)}{\nu
x(t)}\right]-\beta\;.$ (5)

Thus, if our “device” yields $x(t)$ and $\dot{x}(t)$, then we easily compute
$a(t)$ and $\varphi(t)$, provided the frequency $\nu$ of the force is known.
However, it is not known but can only be roughly estimated. Moreover,
generally, it varies with time. Before we discuss how to cope with this fact,
we mention how we practically obtain $x(t)$ and $\dot{x}(t)$. Here, we make
use of the linearity of Eq. (4) and develop and exploit an efficient numerical
scheme, presented below. With this scheme, starting with some initial
conditions, e.g., $x_{0}=\dot{x}_{0}=0$, we obtain, after a short transient,
the forced solution of Eq. (4) and compute $a(t_{k})$ and $\varphi(t_{k})$
from Eqs. (5).

Now, we discuss how to choose the parameters of the measuring oscillator.
Recall that to compute $a,\varphi$, we need the value of the signal frequency
$\nu$. First, let us consider the amplitude measurement. Inspecting Fig. 5, we
see that if we take a large value of $\alpha$ (strongly damped oscillator) and
$\nu\ll\omega$ then the ratio $b/a$ is practically independent on $\nu$. Thus,
to compute the second square root in Eq. (5) we need only a very rough
estimate of $\nu$. For the phase estimation, the damping parameter shall be
different. Indeed, as follows from Fig. 5b, now we have to choose $\alpha$ to
be small, then the phase shift $\beta\approx 0$ in a wide range of $\nu$ and
the phase of the external force equals the phase of the oscillator. A
reasonable choice is to take the oscillator’s frequency $\omega$ about five
times larger than $\nu$.

Figure 5: Resonance curves for amplitude (left) and phase shift (right) for
the linear oscillator with frequency $\omega=5$. Blue and red curves
correspond to the weakly ($\alpha=0.2$) and strongly ($\alpha=6$) damped
oscillator, used for the phase and amplitude measurement, respectively (these
parameters were used to process the artificial data). The domains where
$b/a\approx\text{const}$ (for the red curve) and $\beta\approx 0$ (for the
blue curve) are marked by red and blue arrows, respectively. These are
intervals of signal frequency $\nu$ where the algorithm’s performance is good.
We see that $\omega\approx 5\nu$ is a reasonable choice.

Thus, we neglect $\beta$ in the expression for $\varphi$ and compute
$\sqrt{(\omega^{2}-\nu^{2})^{2}+(\alpha\nu)^{2}}$ in the expression for $a$
only once, using an initial guess $\nu_{0}$ for frequency. However, the terms
$\sqrt{x(t)^{2}+[\dot{x}(t)/\nu]^{2}}$ and
$\arctan\left(\frac{-\dot{x}(t)}{\nu x(t)}\right)$ essentially depend on
$\nu$. Imprecise estimation of $\nu$ and/or its variation with time results in
spurious oscillations of the estimated amplitude and phase. To remove these
oscillations and to make the results less dependent on the initial frequency
estimation, we improve this estimation by computing the signal frequency in
real-time as discussed in the precious section. We start with an initial guess
$\nu_{0}$ and begin the computation of $a(t)$, $\varphi(t)$. After a transient
time of the order of several cycles, we begin with the frequency estimation to
obtain $\nu_{e}$. (The estimation is performed in exactly same way as for the
phase-locked oscillator.) We use $\nu_{e}$ instead of $\nu_{0}$ and, in this
way, significantly improve the real-time computation of the amplitude and
frequency. Performing frequency estimation several times per cycle, we track
the slowly drifting signal frequency. (Notice that for the bandpass filtered
signal used in the last example, the algorithms works well without any
frequency correction. Here we set $nu$ equal to the center of the bandpass.)

In summary, for the amplitude measurement, we need a strongly damped
oscillator, while for the phase estimation, we need an oscillator with small
damping. If we need both measurements simultaneously, then the solution is to
exploit two “devices” concurrently. Though Eqs. (5) are derived for the
harmonic force, we expect that they approximately hold for signals with a slow
variation of the amplitude and frequency. The presented numerical tests
confirm this expectation.

#### Measuring “device”: resonant linear oscillator

Here we adopt the technique used for model studies in our previous
publications [18, 19]. The device consists of a linear oscillator in resonance
with the measured signal and an integrating unit:

$\displaystyle\ddot{x}+\alpha\dot{x}+\omega^{2}x=s(t)\;,$ (6)
$\displaystyle\mu\dot{z}+z=\dot{x}\;.$ (7)

The role of the harmonic oscillator Eq. (6) is twofold. First, it acts as a
bandpass filter and the damping factor $\alpha$ determines the width $\delta
f\approx\alpha/2\pi$ of the bandpass. Second, the harmonic oscillator yields
signal $\dot{x}$ which phase is close to that of the input $x(t)$, provided
the frequency $\omega$ is close to the mean frequency of $s(t)$. (This
condition also ensures that the band-pass is centered at the frequency of the
signal.) Next, for $\mu\gg 1$ Eq. (7) acts as the integrating unit. Its output
is shifted by $\pi/2$ with respect to $\dot{x}$. It is useful to rescale
$\dot{x}$, $z$ so that their amplitudes are close to that of $s(t)$. For this,
we compute $u=\alpha\dot{x}$ and $w=\alpha\omega_{0}\mu z$ and obtain the
instantaneous phase and amplitude of $s(t)$ as $\varphi(t)=\arctan(w/u)$,
$a(t)=\sqrt{u^{2}+w^{2}}$.

### Solving the linear oscillator equation for discrete input signal

Here we present the numerical scheme for solving Eq. (4), adopted to the
situation where the input signal is available at a finite sampling rate. Using
the substitution $y=xe^{\gamma t}$, where $\gamma=\alpha/2$, one obtains

$\ddot{y}+\eta^{2}y=s(t)e^{\gamma t}=S(t)\;,\quad\text{with
}\eta^{2}=\omega^{2}-\gamma^{2}\;.$

Another standard substitution

$y=0.5(Ae^{\mathrm{i}\eta t}+A^{*}e^{-\mathrm{i}\eta
t})\;,\qquad\dot{y}=0.5\mathrm{i}\omega(Ae^{\mathrm{i}\eta
t}-A^{*}e^{-\mathrm{i}\eta t})\;,$ (8)

where $A^{*}$ denotes complex conjugate of $A$, yields
$\dot{A}=-\mathrm{i}\eta^{-1}S(t)e^{-\mathrm{i}\eta t}$. Integrating we obtain

$A_{k+1}=A_{k}-\mathrm{i}\eta^{-1}e^{-\mathrm{i}\eta
t_{k}}\int_{0}^{\Delta}S(t_{k}+\tau)e^{-\mathrm{i}\eta\tau}\mathrm{d}\tau=A_{k}-\mathrm{i}\eta^{-1}e^{-\mathrm{i}\eta
t_{k}}I\;.$

Next, locally interpolating the measured signal $s(t)$ by a parabola going
through the points $s_{k-1},s_{k},s_{k+1}$ we compute the integral
$I=\int_{0}^{\Delta}S(t_{k}+\tau)e^{-\mathrm{i}\eta\tau}\mathrm{d}\tau$. As a
result, we obtain the following practical scheme for integration of Eq. (4).
First, we pre-compute the coefficients $C_{1,2,3}$:

$C_{1}=\mathrm{i}\eta^{-1}e^{-\gamma\Delta}(I_{2}\Delta-
I_{3})/2\Delta^{2}\;,\quad
C_{2}=\mathrm{i}\eta^{-1}(I_{3}/\Delta^{2}-I_{1})\;,\quad
C_{3}=-\mathrm{i}\eta^{-1}e^{\gamma\Delta}(I_{2}\Delta+I_{3})/2\Delta^{2}\;,$

where

$I_{1}=i\eta^{-1}(e^{-\mathrm{i}\eta\Delta}-1)\;,\quad
I_{2}=\eta^{-2}[e^{-\mathrm{i}\eta\Delta}(1+\mathrm{i}\Delta\eta)-1]\;,\quad
I_{3}=\eta^{-3}[e^{-\mathrm{i}\eta\Delta}(\Delta\eta(2+i\Delta\eta)-2\mathrm{i})+2\mathrm{i}]\;.$

Next, for given $x_{k},\dot{x}_{k}$ and the new measurement $s_{k+1}$ we
compute $x_{k+1},\dot{x}_{k+1}$ in three steps:

  1. 1.

Compute $A_{k}=x_{k}-\mathrm{i}(\dot{x}_{k}+\gamma x_{k})/\eta$.

  2. 2.

Compute $A_{k+1}=A_{k}+C_{1}s_{k-1}+C_{2}s_{k}+C_{3}s_{k+1}$.

  3. 3.

Compute
$x_{k+1}=\mbox{Re}(A_{k+1}e^{\mathrm{i}\eta\Delta})e^{-\gamma\Delta}\;,\qquad\dot{x}_{k+1}=\left[0.5\mathrm{i}\eta(A_{k+1}e^{\mathrm{i}\eta\Delta}-A^{*}_{k+1}e^{-\mathrm{i}\eta\Delta})-\gamma\mbox{Re}(A_{k+1}e^{\mathrm{i}\eta\Delta})\right]e^{-\gamma\Delta}\;.$
We emphasize that all coefficients can be pre-computed, so that the
integration step requires only a few summations and multiplications.

The extension of the approach to solving Eqs. (6,7) of the resonant oscillator
is straightforward. The equation (6) of the linear oscillator is solved as
just described. Then the known points
$\dot{x}_{k-1},\dot{x}_{k},\dot{x}_{k+1}$ provide the local parabolic
approximation of the function $\dot{x}$ in Eq. (7). The solution of this
linear equation is readily obtained by variation of the constant and reads:

$z_{k+1}=(z_{k}-a+b\mu-2c\mu^{2})e^{-\Delta/\mu}+a-b\mu+2c\mu^{2}+b\Delta-2c\mu\Delta+c\Delta^{2}\;,$

with $a=\dot{x}_{k}$, $b=(\dot{x}_{k+1}-\dot{x}_{k-1})/2\Delta$,
$c=(\dot{x}_{k-1}-2\dot{x}_{k}+\dot{x}_{k+1})/2\Delta^{2}$.

To summarize the presentation of numerical techniques, we emphasize that if
the sampling rate is very high, the differential equations in all presented
algorithms can be fast and easily solved by the midpoint or predictor-
corrector technique [28]. However, the stability of these simple methods is
not guaranteed.

### Choosing parameters

##### Locking-based oscillator.

The main parameter is the forcing coefficient $\varepsilon$ (Eq. 2). The
larger $\varepsilon$ the faster the system synchronizes. On the other hand,
the forcing term $\varepsilon\sin\theta\cdot s(t)$ shall be small enough to
ensure the monotonicity of the phase $\theta$. So, if $s(t)=a\cos\varphi$ then
(Eq. 2) reads $\dot{\theta}=\omega+\frac{\varepsilon
a}{2}\sin(\varphi-\theta)-\frac{\varepsilon a}{2}\sin(\varphi+\theta)$; in the
locked state $\varphi\approx\theta$ the first sine term vanishes and
$\dot{\theta}>0$ if $\varepsilon a<2\omega$. In the artificial data example we
checked that he values $0.4\leq\varepsilon\leq 0.8$ provide good result. In
this example, the amplitude goes up to $\approx 2$ and the frequency is about
1, so that the above condition is satisfied. The parameter of the frequency
adaptation shall be $K\leq 1$; the values between 0.5 and 1 work well. The
data in Fig. 5 correspond to $\varepsilon=0.8$, $K=1$. For the tremor data
$\omega\approx 2\pi\cdot 4.5$, $\varepsilon=30$, and $K=0.5$. In both cases we
updated the frequency $\omega$ 20 times per period using the preceding phases
$\theta$ in the interval that is about one cycle long.

##### Non-resonant oscillator.

To process the artificial signal we exploited $\alpha_{a}=6$ (amplitude
measurement), $\alpha_{p}=0.2$ (phase measurement). For the LFP example, the
parameters were $\alpha_{a}=80$, $\alpha_{p}=10$. In both cases we took
$\omega=5\nu$, where $\nu=1$ and $\nu=2\pi\cdot 17$, respectively.
Practically, we have to choose $\alpha_{p}$ so that
$|\tan\beta|\approx|\beta|\ll 1$. For the common choice $\omega=5\nu$ it
reduces to the condition $|\beta|=\alpha_{p}/24\nu\ll 1$. (Notice that the
smaller $\alpha$ the longer the transient.) Similarly, we have to choose
$\alpha_{a}$ so that the derivative of the resonance curve
$\left.\frac{d}{d\nu}(a/b)\right|_{\nu=\omega/5}=\left.\frac{2(\omega^{2}-\nu^{2})\nu-\alpha_{a}^{2}\nu}{[(\omega^{2}-\nu^{2})^{2}+\alpha_{a}^{2}\nu^{2}]^{3/2}}\right|_{\nu=\omega/5}\approx
0$ what yields $\alpha_{a}\approx 7\nu$. For a bandpass filtered signal, the
algorithm is not very sensitive to $\alpha_{a}$. So, for $\alpha_{a}\approx
0.75\nu$ in the LFP example, variation of the resonance curve within the
bandpass is less than 4%.

##### Resonant oscillator.

Here we always take $\alpha=0.3\omega$ what corresponds to bandpass width
about $30\%$ of the central frequency $\omega$. The requirement for the
parameter $\mu$ is $\mu\gg 1$; in all computations we choose $\mu=500$.

## Acknowledgements

We thank Michael Feldman for useful discussions. The work was supported by
Deutsche Forschungsgemeinschaft (Project-ID 424778381 – TRR 295 Retune).

## Author contributions statement

M.R. and A.P. designed the algorithms. M.R. performed the computations,
prepared the final figures, and wrote the manuscript with the contribution and
corrections from all other authors. J.L.B. contributed LFP and accelerometer
data. All authors participated in the evaluation and discussion of the
results.

## Additional information

The authors declare no competing interests. The Matlab scripts for all
algorithms can be obtained from the authors upon request.

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# On the depth of tensor products of modules

Arash Sadeghi  School of Mathematics, Institute for Research in Fundamental
Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran<EMAIL_ADDRESS>
###### Abstract.

The depth of tensor product of modules over a Gorenstein local ring is
studied. For finitely generated nonzero modules $M$ and $N$ over a Gorenstein
local ring $R$, under some assumptions on the vanishing of finite number of
Tate and relative homology modules, the
$\operatorname{depth}_{R}(M\otimes_{R}N)$ is determined in terms of the
$\operatorname{depth}_{R}(M)$ and $\operatorname{depth}_{R}(N)$.

###### Key words and phrases:

depth formula, $\operatorname{\mathcal{G}}$-relative homology, Tate homology.
This research was supported by a grant from IPM

###### 2010 Mathematics Subject Classification:

13D07; 13D05

##  1\. Introduction

For finitely generated modules $M$ and $N$ over a commutative noetherian local
ring $R$, the pair $(M,N)$ is said to satisfy the _depth formula_ provided

$\operatorname{depth}R+\operatorname{depth}_{R}(M\otimes_{R}N)=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N).$

The depth formula was first studied by Auslander. In [3], he proved that the
pair $(M,N)$ satisfies the depth formula provided that the projective
dimension of $M$ is finite and $M$ and $N$ are Tor-independent, i.e.,
$\operatorname{Tor}^{R}_{i}(M,N)=0$ for all $i>0$. Three decades later Huneke
and Wiegand proved that the depth formula holds for Tor-independent modules
over complete intersection local rings [11]. Araya and Yoshino [1], and
Iyengar [14] independently generalized Auslander’s result for modules of
finite complete intersection dimension. The finite complete intersection
dimension condition was then relaxed by Christensen and Jorgensen [10]. More
precisely, they proved that the pair $(M,N)$ satisfies the depth formula
provided that the Gorenstein dimension of $M$ is finite,
$\operatorname{Tor}^{R}_{i}(M,N)=0$ for all $i>0$ and
$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for all $i\in\mathbb{Z}$.
Recently, Celikbas, Liang and Sadeghi established that the depth formula holds
under weaker assumptions [7]:

######  Theorem 1.1.

Let $M$ and $N$ be $R$–modules such that $M$ has finite Gorenstein dimension.
Then the depth formula holds provided the following conditions hold.

  1. (i)

$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for all $i>0$.

  2. (ii)

$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for all $i\leq 0$.

In this paper, our aim is to improve Theorem 1.1 over Gorenstein local rings;
rings over which each module has finite Gorenstein dimension. Let us remark
that the condition (i) in Theorem 1.1 is equivalent to the vanishing of
$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)$ for all $i=1,\ldots,d$, where
$d=\dim(R)$. However the condition (ii) cannot be obtained by the vanishing of
finitely many Tate homology modules
$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)$ in general. The main feature of
our result is that we require the vanishing of only finitely many Tate
homology modules for the depth formula to hold. More precisely, we prove the
following result; (see also Corollary 4.3 for a more general case).

######  Theorem 1.2.

Let $R$ be a Gorenstein local ring of dimension $d$ and let $M$ and $N$ be
$R$–modules. Assume that the following conditions hold:

  1. (i)

$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for all $i=1,\ldots,d$.

  2. (ii)

$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for all $i=-d,\ldots,0$.

Then
$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N)$,
i.e., the pair $(M,N)$ satisfies the depth formula.

The organization of the paper is as follows. In Section 2, we collect
preliminary notions, definitions and some known results which will be used in
this paper. In Section 3, we study the depth of tensor products of modules
over Cohen-Macaulay local rings. As a consequence, in Corollary 3.5, we obtain
a bound for the depth of tensor products of modules under some conditions
weaker than those in Theorem 1.2. In Section 4, we prove our main result.

##  2\. Notations and preliminary results

Throughout the paper, $R$ denotes a commutative noetherian local ring and all
$R$–modules are tacitly assumed to be finitely generated.

The notion of the Gorenstein dimension was introduced by Auslander [2], and
developed by Auslander and Bridger in [4].

######  Definition 2.1.

An $R$–module $M$ is said to be of Gorenstein dimension zero (or totally
reflexive) whenever the canonical map $M\rightarrow M^{**}$ is an isomorphism
and $\operatorname{Ext}^{i}_{R}(M,R)=0=\operatorname{Ext}^{i}_{R}(M^{*},R)=0$
for all $i>0$.

The Gorenstein dimension of $M$, denoted
$\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)$, is defined to be the
infimum of all nonnegative integers $n$ such that there exists an exact
sequence

$0\rightarrow G_{n}\rightarrow\cdots\rightarrow G_{0}\rightarrow M\rightarrow
0,$

where each $G_{i}$ are totally reflexive.

For a finite presentation $P_{1}\overset{f}{\rightarrow}P_{0}\rightarrow
M\rightarrow 0$ of an $R$–module $M$, its transpose, $\operatorname{Tr}M$, is
defined as $\operatorname{coker}f^{*}$, where
$(-)^{*}:=\operatorname{Hom}_{R}(-,R)$. Therefore we have the following exact
sequence:

(2.1) $0\rightarrow M^{*}\rightarrow P_{0}^{*}\rightarrow
P_{1}^{*}\rightarrow\operatorname{Tr}M\rightarrow 0.$

Note that $\operatorname{Tr}M$ is unique up to projective equivalence.

In the following, we summarize some basic facts about Gorenstein dimension;
see, for example, [4] for more details.

######  Theorem 2.2.

For an $R$–module $M$, the following statements hold:

  1. (i)

$\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)=0$ if and only if
$\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(\operatorname{Tr}M)=0$;

  2. (ii)

(Auslander-Bridger formula) If $M$ has finite Gorenstein dimension, then

$\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)=\operatorname{depth}R-\operatorname{depth}_{R}(M).$

  3. (iii)

$R$ is Gorenstein if and only if
$\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)<\infty$ for all finitely
generated $R$–module $M$.

Tate (co)homology for modules of finite Gorenstein dimension was studied by
Avramov and Martsinkovsky in [6]. A complex $\mathbf{T}$ of free $R$–modules
is called _totally acyclic_ if
$\operatorname{H}_{n}(\mathbf{T})=0=\operatorname{H}_{n}(\operatorname{Hom}_{R}(\mathbf{T},R))$
for all $n\in\mathbb{Z}$. A _complete resolution_ of an $R$-module $M$ is a
diagram

$\mathbf{T}\overset{\vartheta}{\longrightarrow}\mathbf{P}\overset{\pi}{\longrightarrow}M,$

where $\pi$ is a projective resolution, $T$ is a totally acyclic complex,
$\vartheta$ is a morphism, and $\vartheta_{i}$ is an isomorphism for $i\gg 0$.
An $R$-module has finite Gorenstein dimension if and only if it has a complete
resolution.

Let $M$ be an $R$-module with a complete resolution
$\mathbf{T}\rightarrow\mathbf{P}\rightarrow M$. For an $R$-module $N$, _Tate
homology_ of $M$ and $N$ is defined as

$\widehat{\operatorname{Tor}}_{i}^{R}(M,N)=\operatorname{H}_{i}(\mathbf{T}\otimes_{R}N)\text{
for }i\in\mathbb{Z}.$

Also, for an $R$-module $N$, _Tate cohomology_ of $M$ and $N$ is defined as

$\widehat{\operatorname{Ext}}^{i}_{R}(M,N)=\operatorname{H}^{i}(\operatorname{Hom}_{R}(\mathbf{T},N))\text{
for }i\in\mathbb{Z}.$

By construction, there are isomorphisms

(2.2)
$\widehat{\operatorname{Tor}}_{i}^{R}(M,N)\cong\operatorname{Tor}_{i}^{R}(M,N)\text{
and
}\widehat{\operatorname{Ext}}^{i}_{R}(M,N)\cong\operatorname{Ext}^{i}_{R}(M,N),$

for all $i>\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)$. In the
following we summarize some basic properties about Tate homology which will be
used throughout the paper; see [6] and [9] for more details.

######  Theorem 2.3.

Let $M$ and $N$ be $R$–modules such that $M$ has finite Gorenstein dimension.
Then the following statements hold:

  1. (i)

If either $\operatorname{pd}_{R}(M)<\infty$ or
$\operatorname{pd}_{R}(N)<\infty$, then
$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for all $i\in\mathbb{Z}$.

  2. (ii)

If $\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(N)<\infty$, then
$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)\cong\operatorname{\widehat{Tor}}_{i}^{R}(N,M)$
for all $i\in\mathbb{Z}$.

  3. (iii)

If $\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)=0$, then
$\widehat{\operatorname{Tor}}_{i}^{R}(M,N)\cong\widehat{\operatorname{Ext}}^{-i-1}_{R}(M^{*},N)$
for all $i\in\mathbb{Z}$.

  4. (iv)

If $N$ has finite injective dimension, then
$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for all $i\in\mathbb{Z}$.

A sequence $\eta$ of $R$-modules is called _$\operatorname{\mathcal{G}}$
-proper_ if the induced sequence $\operatorname{Hom}_{R}(M,\eta)$ is exact for
every totally reflexive $R$-module $M$.

A _$\operatorname{\mathcal{G}}$ -proper resolution_ of an $R$-module $M$ is a
resolution $\mathbf{L}\rightarrow M\rightarrow 0$ by totally reflexive
$R$-modules such that the augmented resolution $\mathbf{L}^{+}$ is a
$\operatorname{\mathcal{G}}$-proper sequence. Every module of finite
Gorenstein dimension has a $\operatorname{\mathcal{G}}$-proper resolution.

Let $M$ be an $R$-module with a $\operatorname{\mathcal{G}}$-proper resolution
$\mathbf{L}\rightarrow M\rightarrow 0$. The _$\operatorname{\mathcal{G}}$
-relative_ homology of $M$ and $N$ is defined as

$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N):=\operatorname{H}_{i}(\mathbf{L}\otimes_{R}N)\
\text{for }i\in\mathbb{Z}.$

It follows from the definition that
$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for all
$i>\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)$.

Next we summarize some basic properties about relative homology which will be
used throughout the paper; see [6] and [13] for more details.

######  Theorem 2.4.

Let $M$ and $N$ be $R$–modules such that $M$ has finite Gorenstein dimension.
The following statements hold:

  1. (i)

If either $\operatorname{pd}_{R}(M)$ or $\operatorname{pd}_{R}(N)$ is finite,
then

$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=\operatorname{Tor}_{i}^{R}(M,N)\text{
for all }i\geq 0.$

  2. (ii)

There is an exact sequence:

$\cdots\rightarrow\operatorname{\mathcal{G}Tor}_{2}^{R}(M,N)\rightarrow\operatorname{\widehat{Tor}}_{1}^{R}(M,N)\rightarrow\operatorname{Tor}_{1}^{R}(M,N)\rightarrow\operatorname{\mathcal{G}Tor}_{1}^{R}(M,N)\rightarrow
0.$

  3. (iii)

If $\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(N)<\infty$, then
$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)\cong\operatorname{\mathcal{G}Tor}_{i}^{R}(N,M)$
for all $i\geq 0$.

  4. (iv)

If $N$ has finite injective dimension, then

$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=\operatorname{Tor}_{i}^{R}(M,N)\text{
for all }i\geq 0.$

The following results will be used throughout the paper.

######  Lemma 2.5.

Let $M$ and $N$ be $R$-modules such that $M$ has finite Gorenstein dimension.
Assume that either $\operatorname{pd}_{R}(M)$ or $\operatorname{id}_{R}(N)$ is
finite. If $\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i>0$, then
$\operatorname{depth}R+\operatorname{depth}_{R}(M\otimes_{R}N)=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N).$

###### Proof.

See [3, Theorem 1.2] and [15, Theorem 2.13]. ∎

######  Lemma 2.6.

[7, Corollary 3.18] Let $M$ and $N$ be $R$–modules such that $M$ has finite
Gorenstein dimension. Then

$\sup\\{i\mid\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)\neq
0\\}\leq\sup\\{\operatorname{depth}R_{\mathfrak{p}}-\operatorname{depth}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})-\operatorname{depth}_{R_{\mathfrak{p}}}(N_{\mathfrak{p}})\mid\mathfrak{p}\in\operatorname{Spec}R\\}.$

##  3\. Depth formula and vanishing of Tate homology

The following lemma plays a crucial role in this paper. Before giving a proof
we recall that, for $R$–modules $M$ and $N$, the following sequence is exact:

(3.1) $0\rightarrow\operatorname{Ext}^{1}_{R}(\operatorname{Tr}M,N)\rightarrow
M\otimes_{R}N\overset{\eta}{\rightarrow}\operatorname{Hom}_{R}(M^{*},N)\rightarrow\operatorname{Ext}^{2}_{R}(\operatorname{Tr}M,N)\rightarrow
0,$

where $\eta$ is the evaluation map [4, Proposition 2.6].

######  Lemma 3.1.

Let $M$ be a totally reflexive $R$–module and let $N$ be an $R$–module. Assume
$n$ is a nonnegative integer such that
$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for $-n\leq i\leq 0$. Then
$\operatorname{depth}_{R}(M\otimes_{R}N)\geq\min\\{n+1,\operatorname{depth}_{R}(N)\\}$.

###### Proof.

As $M^{*}=\Omega^{2}\operatorname{Tr}M$, by Theorem 2.3(iii), Theorem 2.2(i)
and (2.2), we have the following isomorphisms

$\begin{array}[]{rl}\operatorname{\widehat{Tor}}_{i}^{R}(M,N)&\cong\widehat{\operatorname{Ext}}^{-i-1}_{R}(M^{*},N)\\\
&\cong\widehat{\operatorname{Ext}}^{-i+1}_{R}(\operatorname{Tr}M,N)\\\
&\cong\operatorname{Ext}^{-i+1}_{R}(\operatorname{Tr}M,N)=0,\end{array}$

for all $-n\leq i\leq 0$. If $n=0$ then we get the following exact sequence:
$M\otimes_{R}N\hookrightarrow\operatorname{Hom}_{R}(M^{*},N)$ by (3.1.1) and
the exact sequence (3.1). Therefore,
$\operatorname{Ass}_{R}(M\otimes_{R}N)\subseteq\operatorname{Ass}_{R}(N)$ and
so the assertion is clear. If $n\geq 1$ then by (3.1.1) and the exact sequence
(3.1) we get the following isomorphism,

(3.1.2) $M\otimes_{R}N\cong\operatorname{Hom}_{R}(M^{*},N).$

Hence,
$\operatorname{depth}_{R}(M\otimes_{R}N)=\operatorname{depth}_{R}(\operatorname{Hom}_{R}(M^{*},N))\geq\min\\{2,\operatorname{depth}_{R}(N)\\}$
and so the assertion is clear for $n=1$. Now let $n>1$ and let
$\mathbf{P}\rightarrow M^{*}\rightarrow 0$ be a free resolution of $M^{*}$.
Note that $\operatorname{Ext}^{i}_{R}(M^{*},N)=0$ for $1\leq i\leq n-1$ by
(3.1.1). Hence, by applying the functor $\operatorname{Hom}_{R}(-,N)$ to the
free resolution of $M^{*}$, we get the following exact sequence:

(3.1.3)
$0\rightarrow\operatorname{Hom}_{R}(M^{*},N)\rightarrow\operatorname{Hom}_{R}(P_{0},N)\rightarrow\cdots\rightarrow\operatorname{Hom}_{R}(P_{n},N).$

Note that
$\operatorname{depth}_{R}(\operatorname{Hom}_{R}(P_{i},N))=\operatorname{depth}_{R}(N)$
for all $i\geq 0$. It follows easily from the exact sequence (3.1.3) that

(3.1.4)
$\operatorname{depth}_{R}(\operatorname{Hom}_{R}(M^{*},N))\geq\min\\{n+1,\operatorname{depth}_{R}(N)\\}.$

Now the assertion follows from (3.1.2) and (3.1.4). ∎

######  Theorem 3.2.

Let $R$ be a Cohen-Macaulay local ring of dimension $d$ and let $M$ be an
$R$–module of finite Gorenstein dimension and of depth $n$. Assume that $N$ is
an $R$–module and that the following conditions hold:

  1. (i)

$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for $-n<i\leq 0$.

  2. (ii)

$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for $1\leq i\leq d-n$
(equivalently, for all $i>0$).

Then either
$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N)$
or
$\operatorname{depth}_{R}(M\otimes_{R}N)\geq\operatorname{depth}_{R}(M)$ .

###### Proof.

Assume that
$\operatorname{depth}_{R}(M\otimes_{R}N)<\operatorname{depth}_{R}(M)$. We must
prove that the pair $(M,N)$ satisfies the depth formula. Without loss of
generality, we may assume that $R$ is Cohen-Macaulay ring with canonical
module. Therefore, by [5, Theorem A], $N$ has a maximal Cohen-Macaulay
approximation:

(3.2.1) $0\longrightarrow Q\longrightarrow Y\longrightarrow N\longrightarrow
0,$

Where $Y$ is maximal Cohen-Macaulay and $Q$ has finite injective dimension.
The exact sequence (3.2.1) induces the following long exact sequence

(3.2.2)
$\cdots\rightarrow\operatorname{\widehat{Tor}}_{i}^{R}(M,Q)\rightarrow\operatorname{\widehat{Tor}}_{i}^{R}(M,Y)\rightarrow\operatorname{\widehat{Tor}}_{i}^{R}(M,N)\rightarrow\cdots,$

of Tate homology modules by [9, Proposition 2.8]. As
$\operatorname{id}_{R}(Q)<\infty$,
$\operatorname{\widehat{Tor}}_{i}^{R}(M,Q)=0$ for all $i\in\mathbb{Z}$ by
Theorem 2.3(iv). Therefore,
$\operatorname{\widehat{Tor}}_{i}^{R}(M,Y)\cong\operatorname{\widehat{Tor}}_{i}^{R}(M,N)$
for all $i\in\mathbb{Z}$ by (3.2.2). In particular, by (i)

(3.2.3) $\operatorname{\widehat{Tor}}_{i}^{R}(M,Y)=0\text{ for }-n<i\leq 0.$

As $\operatorname{id}_{R}(Q)<\infty$, the exact sequence (3.2.1) is proper and
it induces the following long exact sequence

$\begin{array}[]{rl}\cdots\rightarrow\operatorname{\mathcal{G}Tor}_{1}^{R}(M,Q)\rightarrow\operatorname{\mathcal{G}Tor}_{1}^{R}(M,Y)\rightarrow\operatorname{\mathcal{G}Tor}_{1}^{R}(M,N)\rightarrow\\\
M\otimes_{R}Q\rightarrow M\otimes_{R}Y\rightarrow M\otimes_{R}N\rightarrow
0,\end{array}$

of relative homology modules by the homology version of [6, Proposition 4.4]
(see also [6, Remark 7.4]). By (ii),
$\operatorname{\mathcal{G}Tor}_{1}^{R}(M,N)=0$ and so we have the following
exact sequence

(3.2.5) $0\rightarrow M\otimes_{R}Q\rightarrow M\otimes_{R}Y\rightarrow
M\otimes_{R}N\rightarrow 0,$

by (3.2.4). As $Y$ is maximal Cohen-Macaulay,

(3.2.6) $\operatorname{\mathcal{G}Tor}_{i}^{R}(M,Y)=0\text{ for all }i>0,$

by Lemma 2.6. It follows from (3.2.4), (3.2.6) and (ii) that
$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,Q)=0$ for all $i>0$. Therefore, by
Lemma 2.5 and Theorem 2.4(iv)

(3.2.7)
$\operatorname{depth}R+\operatorname{depth}_{R}(M\otimes_{R}Q)=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(Q).$

Now consider the following exact sequence

(3.2.8) $0\longrightarrow X\longrightarrow G\longrightarrow M\longrightarrow
0,$

where $\operatorname{pd}_{R}(X)<\infty$ and
$\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(G)=0$. As $X$ has a finite
projective dimension, $\operatorname{\widehat{Tor}}_{i}^{R}(X,Y)=0$ for all
$i\in\mathbb{Z}$ by Theorem 2.3(i). Therefore, the exact sequence (3.2.8)
induces the following isomorphism
$\operatorname{\widehat{Tor}}_{i}^{R}(M,Y)\cong\operatorname{\widehat{Tor}}_{i}^{R}(G,Y)$
for all $i\in\mathbb{Z}$ by [9, Proposition 2.9]. In particular, by (3.2.3) we
have

(3.2.9) $\operatorname{\widehat{Tor}}_{i}^{R}(G,Y)=0\text{ for }-n<i\leq 0.$

As $Y$ is maximal Cohen-Macaulay, it follows from (3.2.9) and Lemma 3.1 that

(3.2.10) $\operatorname{depth}_{R}(G\otimes_{R}Y)\geq n.$

As $\operatorname{pd}_{R}(X)<\infty$ and $Y$ is maximal Cohen-Macaulay,
$\operatorname{Tor}_{i}^{R}(X,Y)=0$ for all $i>0$ by [16, Lemma 2.2] and so

(3.2.11)
$\operatorname{depth}_{R}(X\otimes_{R}Y)=\operatorname{depth}_{R}(X)=n+1.$

by Lemma 2.5. As $\operatorname{pd}_{R}(X)<\infty$, the exact sequence (3.2.8)
is proper and it induces the following long exact sequence

(3.2.12) $0\rightarrow X\otimes_{R}Y\rightarrow G\otimes_{R}Y\rightarrow
M\otimes_{R}Y\rightarrow 0.$

by the homology version of [6, Proposition 4.6] and (3.2.6). It follows easily
from (3.2.10), (3.2.11) and the exact sequence (3.2,12) that

(3.2.13) $\operatorname{depth}_{R}(M\otimes_{R}Y)\geq n.$

As $\operatorname{depth}_{R}(M\otimes_{R}N)<n$, by (3.2.13) and the exact
sequence (3.2.5)

(3.2.14)
$\operatorname{depth}_{R}(M\otimes_{R}N)=\operatorname{depth}_{R}(M\otimes_{R}Q)-1.$

Note that $\operatorname{depth}_{R}(N)=\operatorname{depth}_{R}(Q)-1$.
Therefore the assertion follows from (3.2.14) and (3.2.7). ∎

As a consequence we have the following result.

######  Corollary 3.3.

Let $R$ be a Cohen-Macaulay local ring of dimension $d$ and let $M$, $N$ be
$R$–modules. Assume that $M$ is totally reflexive and that
$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for $-d<i\leq 0$. Then either
$M\otimes_{R}N$ is maximal Cohen-Macaulay or
$\operatorname{depth}_{R}(M\otimes_{R}N)=\operatorname{depth}_{R}(N)$.

###### Proof.

First note that $\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for all $i>0$,
because $M$ is totally reflexive. Therefore, by Theorem 3.2, either the pair
$(M,N)$ satisfies depth formula or
$\operatorname{depth}_{R}(M\otimes_{R}N)\geq\operatorname{depth}_{R}(M)$. Now
the assertion follows from Auslander-Bridger formula. ∎

######  Theorem 3.4.

Let $R$ be a Cohen-Macaulay local ring of dimension $d$ and let $M$, $N$ be
$R$–modules. Assume that $M$ has finite Gorenstein dimension and that the
following conditions hold:

  1. (i)

$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for $-d<i\leq 0$.

  2. (ii)

$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for all $i>0$.

Then either
$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N)$
or
$\operatorname{depth}_{R}(M\otimes_{R}N)\geq\operatorname{depth}_{R}(N)$.

###### Proof.

Assume that
$\operatorname{depth}_{R}(M\otimes_{R}N)<\operatorname{depth}_{R}(N)$. We must
prove that the pair $(M,N)$ satisfies the depth formula. Without loss of
generality, we may assume that $R$ is Cohen-Macaulay ring with canonical
module. Consider the following exact sequence,

(3.4.1) $0\longrightarrow T\longrightarrow G\longrightarrow M\longrightarrow
0,$

where $G$ is totally reflexive and $T$ has finite projective dimension.
Therefore, $\operatorname{\widehat{Tor}}_{i}^{R}(T,N)=0$ for all
$i\in\mathbb{Z}$ by Theorem 2.3(i) and so the exact sequence (3.4.1) induces
the following isomorphism

(3.4.2)
$\operatorname{\widehat{Tor}}_{i}^{R}(G,N)\cong\operatorname{\widehat{Tor}}_{i}^{R}(M,N)\text{
for all }i\in\mathbb{Z},$

by [9, Proposition 2.9]. Now consider the following exact sequence

(3.4.3) $0\longrightarrow N\longrightarrow I\longrightarrow L\longrightarrow
0,$

where $L$ is maximal Cohen-Macaulay and $I$ has finite injective dimension
(see [5, Theorem A]). By Theorem 2.3(iv),
$\operatorname{\widehat{Tor}}_{i}^{R}(G,I)=0$ for all $i\in\mathbb{Z}$. Hence,
the exact sequence (3.4.3) induces the following isomorphism

(3.4.4)
$\operatorname{\widehat{Tor}}_{i}^{R}(G,N)\cong\operatorname{\widehat{Tor}}_{i+1}^{R}(G,L)\text{
for all }i\in\mathbb{Z},$

by [9, Proposition 2.8]. It follows from (i), (3.4.2) and (3.4.4) that

(3.4.5) $\operatorname{\widehat{Tor}}_{i}^{R}(G,L)=0\text{ for }-d+1<i\leq 1.$

As $G$ is totally reflexive, $\operatorname{Tor}_{1}^{R}(G,L)=0$ by (2.2) and
(3.4.5). Hence, the exact sequence (3.4.3) induces the following exact
sequence:

(3.4.6) $0\rightarrow N\otimes_{R}G\rightarrow I\otimes_{R}G\rightarrow
L\otimes_{R}G\rightarrow 0.$

It follows from Lemma 3.1 and (3.4.5) that
$\operatorname{depth}_{R}(L\otimes_{R}G)\geq d-1$. Therefore, by the exact
sequence (3.4.6),

(3.4.7)
$\operatorname{depth}_{R}(N\otimes_{R}G)\geq\min\\{\operatorname{depth}_{R}(I\otimes_{R}G),d\\}=\operatorname{depth}_{R}(I\otimes_{R}G).$

As $\operatorname{id}_{R}(I)<\infty$ and $G$ is totally reflexive,
$\operatorname{Tor}_{i}^{R}(G,I)\cong\operatorname{\mathcal{G}Tor}_{i}^{R}(G,I)=0$
for all $i>0$ by Theorem 2.4(iv) and Lemma 2.6. Hence, by Lemma 2.5 the pair
$(G,I)$ satisfies depth formula and so by Auslander-Bridger formula we have

(3.4.8)
$\operatorname{depth}_{R}(I\otimes_{R}G)=\operatorname{depth}_{R}(I)=\operatorname{depth}_{R}(N).$

As $\operatorname{pd}_{R}(T)<\infty$, the exact sequence (3.4.1) is proper and
it induces the following long exact sequence

$\begin{array}[]{rl}\cdots\rightarrow\operatorname{\mathcal{G}Tor}_{1}^{R}(T,N)\rightarrow\operatorname{\mathcal{G}Tor}_{1}^{R}(G,N)\rightarrow\operatorname{\mathcal{G}Tor}_{1}^{R}(M,N)\rightarrow\\\
T\otimes_{R}N\rightarrow G\otimes_{R}N\rightarrow M\otimes_{R}N\rightarrow
0,\end{array}$

of relative homology modules. By (ii) and the long exact sequence (3.4.9) we
obtain the following exact sequence

(3.4.10) $0\rightarrow T\otimes_{R}N\rightarrow G\otimes_{R}N\rightarrow
M\otimes_{R}N\rightarrow 0.$

As $G$ is totally reflexive, $\operatorname{\mathcal{G}Tor}_{i}^{R}(G,N)=0$
for all $i>0$. Hence, it follows from (ii), Theorem 2.4(i) and the exact
sequence (3.4.9) that
$\operatorname{Tor}_{i}^{R}(T,N)\cong\operatorname{\mathcal{G}Tor}_{i}^{R}(T,N)=0$
for all $i>0$. Therefore, by Lemma 2.5 and Auslander-Buchsbaum formula

$\begin{array}[]{rl}\operatorname{depth}_{R}(T\otimes_{R}N)=\operatorname{depth}_{R}(N)-\operatorname{pd}_{R}(T)\\\
=\operatorname{depth}_{R}(N)-\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)+1.\end{array}$

As $\operatorname{depth}_{R}(M\otimes_{R}N)<\operatorname{depth}_{R}(N)$, it
follows from (3.4.7), (3.4.8) and the exact sequence (3.4.10) that

(3.4.12)
$\operatorname{depth}_{R}(T\otimes_{R}N)=\operatorname{depth}_{R}(M\otimes_{R}N)+1.$

Now the assertion follows from (3.4.11), (3.4.12) and Auslander-Bridger
formula. ∎

The following is an immediate consequence of Theorem 3.2 and Theorem 3.4.

######  Corollary 3.5.

Let $R$ be a Cohen-Macaulay local ring of dimension $d$ and let $M$, $N$ be
$R$–modules. Assume that $M$ has finite Gorenstein dimension and that the
following conditions hold:

  1. (i)

$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for $-d<i\leq 0$.

  2. (ii)

$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for all $i>0$.

Then at least one of the following holds:

  1. (1)

$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N)$.

  2. (2)

$\operatorname{depth}_{R}(M\otimes_{R}N)\geq\max\\{\operatorname{depth}_{R}(M),\operatorname{depth}_{R}(N)\\}$.

In particular, if either $M$ or $N$ is maximal Cohen-Macaulay then either
$\operatorname{depth}_{R}(M\otimes_{R}N)=\operatorname{depth}_{R}(N)$ or
$\operatorname{depth}_{R}(M\otimes_{R}N)=\operatorname{depth}_{R}(M)$.

The following example shows that the depth formula may not hold under the
assumptions of Corollary 3.5. In the next Section, we prove that the depth
formula holds under some stronger conditions.

######  Example 3.6.

[12] Let $R=k[[x,y]]/(xy)$ and let $M=R/(x)$, $N=R/(x^{2})$. Then $R$ is a
complete intersection ring with an isolated singularity of dimension one. Note
that $M\otimes_{R}N\cong M$. Therefore, $M$ and $M\otimes_{R}N$ are maximal
Cohen-Macaulay but $N$ is not maximal Cohen-Macaulay. In other words, the pair
$(M,N)$ does not satisfy the depth formula. Now we prove that the assumptions
of Corollary 3.5 hold. First note that
$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for all $i>0$, because $M$ is
totally reflexive. Consider the following exact sequence

(3.6.1)
$0\rightarrow\operatorname{Ext}^{1}_{R}(\operatorname{Tr}M,N)\rightarrow
M\otimes_{R}N\rightarrow\operatorname{Hom}_{R}(M^{*},N)\rightarrow\operatorname{Ext}^{2}_{R}(\operatorname{Tr}M,N)\rightarrow
0.$

Let
$\mathfrak{p}\in\operatorname{Ass}_{R}(\operatorname{Ext}^{1}_{R}(\operatorname{Tr}M,N))$.
As $M\otimes_{R}N$ is maximal Cohen-Macaulay, it follows from the exact
sequence (3.6.1) that
$\mathfrak{p}\in\operatorname{Ass}_{R}(M\otimes_{R}N)\subseteq\operatorname{Ass}R$
which is a contradiction, because $R$ has an isolated singularity. Therefore,
by Theorem 2.3(iii) and (2.2),
$\operatorname{\widehat{Tor}}_{0}^{R}(M,N)\cong\widehat{\operatorname{Ext}}^{-1}_{R}(M^{*},N)\cong\operatorname{Ext}^{1}_{R}(\operatorname{Tr}M,N)=0$.

##  4\. Main result

In order to prove our main result, Theorem 1.2, we will prove two Theorems
which are of independent interest.

######  Theorem 4.1.

Let $R$ be a Cohen-Macaulay local ring of dimension $d$ and let $N$ be an
$R$–module of depth $n$. Assume that $M$ is an $R$–module of finite and
positive Gorenstein dimension and that the following conditions hold:

  1. (i)

$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for $-n\leq i\leq 0$.

  2. (ii)

$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for all $i>0$.

Then
$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N)$.

###### Proof.

Without loss of generality, we may assume that $R$ is Cohen-Macaulay ring with
canonical module. Consider the following exact sequence:

(4.1.1) $0\longrightarrow M\longrightarrow P\longrightarrow X\longrightarrow
0,$

where $\operatorname{pd}_{R}(P)<\infty$ and $X$ is totally reflexive (see [8,
3.3]). As $P$ has finite projective dimension,
$\operatorname{\widehat{Tor}}_{i}^{R}(P,N)=0$ for all $i\in\mathbb{Z}$ by
Theorem 2.3(i). Therefore, the exact sequence (4.1.1) induces the following
isomorphism
$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)\cong\operatorname{\widehat{Tor}}_{i+1}^{R}(X,N)$
for all $i\in\mathbb{Z}$. In particular,

(4.1.2) $\operatorname{\widehat{Tor}}_{i}^{R}(X,N)=0\text{ for }-n+1\leq i\leq
1,$

by (i). It follows from (4.1.2) and Lemma 3.1 that

(4.1.3)
$\operatorname{depth}_{R}(X\otimes_{R}N)\geq\operatorname{depth}_{R}(N)=n.$

As $X$ is totally reflexive, by (4.1.2) and (2.2),

(4.1.4)
$\operatorname{Tor}_{1}^{R}(X,N)\cong\operatorname{\widehat{Tor}}_{1}^{R}(X,N)=0.$

Therefore, the exact sequence (4.1.1) induces the following exact sequence:

(4.1.5) $0\longrightarrow M\otimes_{R}N\longrightarrow
P\otimes_{R}N\longrightarrow X\otimes_{R}N\longrightarrow 0.$

Consider the exact sequence $0\rightarrow\Omega P\rightarrow F\rightarrow
P\rightarrow 0$, where $F$ is free. Taking the pull-back of the two maps into
$P$, we get the following commutative diagram

$\setcounter{MaxMatrixCols}{12}\begin{CD}&&&&&&&&\\\ \ \ &&&&0&&0\\\
&&&&@V{}V{}V@V{}V{}V\\\ &&&&\Omega P@>{=}>{}>\Omega P\\\
&&&&@V{}V{}V@V{}V{}V\\\ \ \ &&0@>{}>{}>Z@>{}>{}>F@>{}>{}>X@>{}>{}>0&\\\
&&&&@V{}V{}V@V{}V{}V@V{\parallel}V{}V\\\ \ \
&&0@>{}>{}>M@>{}>{}>P@>{}>{}>X@>{}>{}>0&\\\ &&&&@V{}V{}V@V{}V{}V\\\ \ \
&&&&0&&0\\\ \end{CD}$

The usual properties of the pull-back and the Snake Lemma show that the rows
and columns of this diagram are exact. Note that $Z$ is totally reflexive and
so $\operatorname{\mathcal{G}Tor}_{i}^{R}(Z,N)=0$ for all $i>0$. As $\Omega P$
has finite projective dimension, the exact sequence $0\rightarrow\Omega
P\rightarrow Z\rightarrow M\rightarrow 0$ is proper. Therefore the above
diagram induces the following commutative diagram with exact rows

$\setcounter{MaxMatrixCols}{11}\begin{CD}&&&&&&&&\\\ \ \
&&0@>{}>{}>\operatorname{\mathcal{G}Tor}_{1}^{R}(M,N)@>{}>{}>\Omega
P\otimes_{R}N@>{}>{}>Z\otimes_{R}N&\\\
&&&&@V{}V{}V@V{\parallel}V{}V@V{f}V{}V\\\ \ \
&&0@>{}>{}>\operatorname{Tor}_{1}^{R}(P,N)@>{}>{}>\Omega
P\otimes_{R}N@>{}>{}>F\otimes_{R}N&\\\ \end{CD}$

It follows easily from (4.1.4) that $f$ is a monomorphism. By (ii),
$\operatorname{\mathcal{G}Tor}_{1}^{R}(M,N)=0$. It follows from the above
commutative diagram that

(4.1.6) $\operatorname{Tor}_{1}^{R}(P,N)=0.$

Now consider the maximal approximation of $N$,

(4.1.7) $0\longrightarrow Q\longrightarrow Y\longrightarrow N\longrightarrow
0,$

where $Y$ is maximal Cohen-Macaulay and $Q$ has finite injective dimension.
The exact sequence (4.1.1) induces the following long exact sequence

(4.1.8)
$\cdots\rightarrow\operatorname{Tor}_{i}^{R}(M,Q)\rightarrow\operatorname{Tor}_{i}^{R}(P,Q)\rightarrow\operatorname{Tor}_{i}^{R}(X,Q)\rightarrow\cdots.$

As we have seen in the proof of Theorem 3.2,

(4.1.9)
$\operatorname{Tor}_{i}^{R}(M,Q)\cong\operatorname{\mathcal{G}Tor}_{i}^{R}(M,Q)=0\text{
for all }i>0.$

Since $\operatorname{id}_{R}(Q)<\infty$ and $X$ is totally reflexive,

(4.1.10)
$\operatorname{Tor}_{i}^{R}(X,Q)\cong\operatorname{\mathcal{G}Tor}_{i}^{R}(X,Q)=0\text{
for all }i>0,$

by Theorem 2.4(iv). It follows from (4.1.8), (4.1.9) and (4.1.10) that

(4.1.11) $\operatorname{Tor}_{i}^{R}(P,Q)=0\text{ for all }i>0.$

As $Y$ is maximal Cohen-Macaulay and $P$ has finite projective dimension,

(4.1.12) $\operatorname{Tor}_{i}^{R}(P,Y)=0\text{ for all }i>0.$

by [16, Lemma 2.2]. The exact sequence (4.1.7) induces the following long
exact sequence

(4.1.13)
$\cdots\rightarrow\operatorname{Tor}_{i}^{R}(P,Y)\rightarrow\operatorname{Tor}_{i}^{R}(P,N)\rightarrow\operatorname{Tor}_{i-1}^{R}(P,Q)\rightarrow\cdots.$

It follows from (4.1.6), (4.1.11), (4.1.12) and (4.1.13) that
$\operatorname{Tor}_{i}^{R}(P,N)=0$ for all $i>0$. Hence, by Lemma 2.5, and
the Auslander-Buchsbaum formula,

$\begin{array}[]{rl}\operatorname{depth}_{R}(P\otimes_{R}N)=\operatorname{depth}_{R}(N)-\operatorname{pd}_{R}(P)\\\
=\operatorname{depth}_{R}(N)-\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)<\operatorname{depth}_{R}(N).\end{array}$

It follows from (4.1.3), (4.1,14) and the exact sequence (4.1,5) that

$\operatorname{depth}_{R}(M\otimes_{R}N)=\operatorname{depth}_{R}(P\otimes_{R}N)=\operatorname{depth}_{R}(N)-\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M).$

Therefore the assertion follows from the Auslander-Bridger formula. ∎

######  Theorem 4.2.

Let $R$ be a Cohen-Macaulay local ring of dimension $d$ and let $M$, $N$ be
$R$–modules. Assume that $M$ is totally reflexive. If
$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for $-d\leq i\leq 0$, then
$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N)$.

###### Proof.

Without loss of generality, we may assume that $R$ is Cohen-Macaulay ring with
canonical module. Consider the following exact sequence:

(4.2.1) $0\longrightarrow N\longrightarrow Q\longrightarrow X\longrightarrow
0,$

where $\operatorname{id}_{R}(Q)<\infty$ and $X$ is maximal Cohen-Macaulay (see
[5]). The exact sequence (4.2.1) induces the following long exact sequence

(4.2.2)
$\cdots\rightarrow\operatorname{\widehat{Tor}}_{i+1}^{R}(M,X)\rightarrow\operatorname{\widehat{Tor}}_{i}^{R}(M,N)\rightarrow\operatorname{\widehat{Tor}}_{i}^{R}(M,Q)\rightarrow\cdots,$

of Tate homology modules by [9, Proposition 2.8]. As
$\operatorname{id}_{R}(Q)<\infty$,
$\operatorname{\widehat{Tor}}_{i}^{R}(M,Q)=0$ for all $i\in\mathbb{Z}$ by
Theorem 2.3(iv). Therefore,
$\operatorname{\widehat{Tor}}_{i+1}^{R}(M,X)\cong\operatorname{\widehat{Tor}}_{i}^{R}(M,N)$
for all $i\in\mathbb{Z}$ by (4.2.2). In particular, by our assumption

(4.2.3) $\operatorname{\widehat{Tor}}_{i}^{R}(M,X)=0\text{ for }-d<i\leq 1.$

As $M$ is totally reflexive, $\operatorname{Tor}_{1}^{R}(M,X)=0$ by (2.2) and
(4.2.3). Therefore, the exact sequence (4.2.1) induces the following exact
sequence

(4.2.4) $0\longrightarrow M\otimes_{R}N\longrightarrow
M\otimes_{R}Q\longrightarrow M\otimes_{R}X\longrightarrow 0.$

As $Q$ has finite injective dimension and $M$ is totally reflexive,
$\operatorname{Tor}_{i}^{R}(M,Q)\cong\operatorname{\mathcal{G}Tor}_{i}^{R}(M,Q)=0$
for all $i>0$ by Theorem 2.4(iv) and Lemma 2.6. Therefore,

(4.2.5)
$\operatorname{depth}_{R}(M\otimes_{R}Q)+d=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(Q),$

by Lemma 2.5. Note that, by Auslander-Bridger formula, $M$ is maximal Cohen-
Macaulay. Hence, it follows from (4.2.5) and the exact sequence (4.2.1) that

(4.2.6)
$\operatorname{depth}_{R}(M\otimes_{R}Q)=\operatorname{depth}_{R}(Q)=\operatorname{depth}_{R}(N).$

On the other hand, by (4.2.3) and Lemma 3.1,

(4.2.7) $\operatorname{depth}_{R}(M\otimes_{R}X)=d.$

It follows from (4.2.6), (4.2.7) and the exact sequence (4.2.4) that
$\operatorname{depth}_{R}(M\otimes_{R}N)=\operatorname{depth}_{R}(N)$. In
other words, the pair $(M,N)$ satisfies the depth formula. ∎

Note that every finitely generated module over a Gorenstein local ring has
finite Gorenstein dimension. Therefore, Theorem 1.2 is a special case of the
following result.

######  Corollary 4.3.

Let $R$ be a Cohen-Macaulay local ring and let $M$ and $N$ be $R$–modules such
that $M$ has finite Gorenstein dimension. Set
$m=\max\\{\operatorname{depth}_{R}(M),\operatorname{depth}_{R}(N)\\}$ and
$n=\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)$. Assume that the
following conditions hold:

  1. (i)

$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for $-m\leq i\leq 0$.

  2. (ii)

If $n>0$, assume $\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for $1\leq
i\leq n$.

Then
$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N)$.

###### Proof.

If $n=0$ then $M$ is maximal Cohen-Macaulay by Auslander-Bridger formula and
so $m=\dim R$. Hence, the assertion follows from Theorem 4.2. If $n>0$ then
the assertion is clear by Theorem 4.1. ∎

As an application, we have the following result.

######  Corollary 4.4.

Let $R$ be a Cohen-Macaulay local ring and let $M$ and $N$ be $R$–modules such
that $M$ has finite Gorenstein dimension. Set
$m=\max\\{\operatorname{depth}_{R}(M),\operatorname{depth}_{R}(N)\\}$ and
$n=\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)$. Assume that the
following conditions hold:

  1. (i)

$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for $-m\leq i\leq n$.

  2. (ii)

If $n>0$, assume $\operatorname{Tor}_{i}^{R}(M,N)=0$ for $1\leq i\leq n$.

Then
$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N)$.

###### Proof.

If $n=0$, then $\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for all $i>0$
and $M$ is maximal Cohen-Macaulay by Auslander-Bridger formula. Hence, the
assertion is clear by Theorem 4.2. Assume that $n>0$. It follows from (i),
(ii) and Theorem 2.4(ii) that $\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$
for $1\leq i\leq n$. Now the assertion is clear by Corollary 4.3. ∎

We finish this section by proving a vanishing result for local rings that are
not necessarily Cohen-Macaulay.

######  Theorem 4.5.

Let $R$ be a local ring of depth $d$ and let $M$ and $N$ be $R$–modules of
finite Gorenstein dimension. Assume that the following conditions hold:

  1. (i)

$\operatorname{\widehat{Tor}}_{i}^{R}(M,N)=0$ for $-d\leq i\leq 0$.

  2. (ii)

$\operatorname{\mathcal{G}Tor}_{i}^{R}(M,N)=0$ for $1\leq i\leq d$.

Then either
$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N)$
or $\operatorname{depth}_{R}(M\otimes_{R}N)>d$.

###### Proof.

Assume that $\operatorname{depth}_{R}(M\otimes_{R}N)\leq d$. We must prove
that $(M,N)$ satisfies the depth formula. If $M$ and $N$ are totally
reflexive, then the assertion follows from Lemma 3.1 and Auslander-Bridger
formula. Assume that either $M$ or $N$ is not totally reflexive. Note that by
Theorem 2.3(ii) and Theorem 2.4(iii), Tate and relative homology are balanced
for modules of finite Gorenstein dimension. Therefore, without loss of
generality we may assume that
$\operatorname{\mathcal{G}\textnormal{-dim}}_{R}(M)>0$. Consider the following
exact sequence:

(4.5.1) $0\longrightarrow Q\longrightarrow Y\longrightarrow N\longrightarrow
0,$

where $\operatorname{pd}_{R}(Q)<\infty$ and $Y$ is totally reflexive. By
applying the exact sequence (4.5.1) instead of the exact sequence (4.1.7) in
the proof of Theorem 4.1, similarly one can deduce the assertion. ∎

## Acknowledgments

I am grateful to Olgur Celikbas for his comments on an earlier version of this
paper and for his help in finding Example 3.6.

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