text:
\begin{document}

\begin{frontmatter}

\title{Supplementary Variable Method for Developing Structure-Preserving Numerical Approximations to Thermodynamically Consistent Partial Differential Equations}

\author[mymainaddress,mymainaddress2]{Yuezheng Gong}
\author[mysecondaryaddress,mymainaddress2]{Qi Hong}
\author[mythirdaryaddress]{Qi Wang}
\cortext[mycorrespondingauthor]{Corresponding author}
\ead{qwang@math.sc.edu}
\address[mymainaddress]{College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China}
\address[mysecondaryaddress]{Beijing Computational Science Research Center, Beijing 100193, China}
\address[mymainaddress2]{{Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems}}
\address[mythirdaryaddress]{Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA}
\begin{abstract}

We present a new temporal discretization paradigm  for developing energy-production-rate preserving numerical approximations to thermodynamically consistent partial differential equation systems, called the supplementary variable method.   The central idea behind it is to introduce a supplementary variable to the thermodynamically consistent model to make the over-determined equation system, consisting of the thermodynamically consistent PDE system, the energy definition and the energy dissipation equation, structurally stable. The supplementary variable allows one to retain the consistency between the energy dissipation equation and the PDE system after the temporal discretization. We illustrate the method using a dissipative gradient flow model. Among virtually infinite many possibilities, we present two ways to add the supplementary variable in the gradient flow model to develop energy-dissipation-rate preserving algorithms. Spatial discretizations are carried out using the pseudo-spectral method. We then compare the two new schemes with the energy stable SAV scheme and the fully implicit Crank-Nicolson scheme. The results favor the new schemes in the overall performance. This new numerical paradigm can be applied to any  thermodynamically consistent models.
\end{abstract}

\begin{keyword}
Supplementary variable method, thermodynamically consistent models, gradient flows, energy-production-rate preserving schemes, finite difference methods, pseudo-spectral methods.
\end{keyword}

\end{frontmatter}

\section{Introduction}
Nonequilibrium phenomena require dynamical models derived from laws and principles of nonequilibrium thermodynamics to describe. The laws of thermodynamics, especially, the second law of thermodynamics or the equivalent generalized Onsager principle are fundamental laws/principles for developing such models for nonequilibrium phenomena not far from equilibria \cite{Onsager1931a,Onsager1931b,Yang&Li&Forest&Wang2016}. Assuming the thermodynamic variables describing nonequilibrium phenomena of a certain system are denoted as  $\Phi\in {\bf R}^n$, where $n$ is the number of the variables, and the free energy of the system is described by a functional $F[\Phi]$ in isothermal cases (for nonisothermal cases, we use an entropy functional instead). The Onsager linear response theory yields the dynamical equation
\ben
\bR \cdot \dot{\Phi}=-\bM \cdot \mu,\quad \mu = \frac{\delta F}{\delta \Phi},\label{gfeq}
\een
where $\bM$ and $\bR$ are operators, $\frac{\delta F}{\delta \Phi}$ is the variation of $F$, $\bM^{-1} \bR$ is known as the friction operator and  $\bR^{-1} \bM$ the mobility operator when they exist. This equation system provides relaxation dynamics for the nonequilibirum  state to return to equilibria in dissipative systems or to oscillate in nondissipative systems. For simplicity, we consider the case of  $\bR=\bI$ in this paper so that $\bM$ is the mobility operator.

In general, the mobility operator can be decomposed into two parts:
\ben
\bM=\bM_{a}+\bM_{s},
\een
where $\bM_a$ is the antisymmetric part and $\bM_s$ is the symmetric part. The time rate of change of the free energy  is given by
\ben
\dfrac{\d F}{\d t}= -\int_{\Omega} \frac{\delta F}{\delta \Phi} \bM \frac{\delta F}{\delta \Phi} \textrm{d}\bx=-\int_{\Omega} \frac{\delta F}{\delta \Phi} \bM_s \frac{\delta F}{\delta \Phi} \textrm{d}\bx ,\label{energy-dissipation-rate}
\een
under proper boundary conditions on $\Phi$.
If $\bM_a=0$ and $\bM_s\geq 0$,  the system is called a dissipative system since $\d F/ \d t \leq 0$. While only $\bM_s=0$,  the system is a conservative system because $\d F/ \d t=0$.
Apparently, \eqref{energy-dissipation-rate} is an equation  deduced from \eqref{gfeq}. It can therefore be viewed as a consistent constraint for \eqref{gfeq}.  When approximating equation \eqref{gfeq} numerically, one would like to preserve energy dissipation property at the discrete level, i.e., to arrive at an approximate equation to \eqref{energy-dissipation-rate} that is derivable from the approximate equation of \eqref{gfeq}. A numerical algorithm of such a property is called an energy stable algorithm or scheme.

In many thermodynamical models given in the form of \eqref{gfeq}, the total free energy is given by
\beq
F = \frac{1}{2}(\Phi, \cL \Phi ) + \big( f (\Phi, \nabla \Phi), 1\big),\label{free-energy}
\eeq
where $(\bullet,\bullet)$ is the  inner product in a $L^2$ space,  $\cL$ is a linear, self-adjoint, positive definite operator and $f$ or $(f,1)$ is bounded below for the physically accessible states of $\Phi$. This is also known as the gradient flow model in the literature. In this  model, if we view $F$ as one of the thermodynamical variables, \eqref{gfeq}, \eqref{energy-dissipation-rate}, and \eqref{free-energy} constitute an {\it over-determined system} of equations with $n+2$ equations and $n+1$ unknowns $(\Phi, F)$. This system of equations is consistent and solvable should gradient flow model \eqref{gfeq} is solvable with free energy \eqref{free-energy} and proper initial and boundary conditions. However, the structural consistency among the equations in the system can be easily broken under small perturbations to the system, leading to a system that is not solvable because of inconsistency among the equations. We label this scenario in the over-determined system of equations as structural unstable.

Traditionally, to retain consistency among the equations in the over-determined system in numerical approximations, one  discretizes \eqref{gfeq} and \eqref{free-energy}, and then try to put together a consistent discretized energy dissipation equation or inequality for \eqref{energy-dissipation-rate} to arrive at energy stable algorithms. However, there is no guarantee that this approach would be  successful. Then, there came about the convex splitting  method for the gradient flow with a free energy given as a difference of two convex functions, in which {\it  one implements a mismatched temporal discretization on the two convex parts of the free energy, respectively, to achieve energy stability} \cite{Eyre1998}; and the stabilizer approach \cite{Yang09}, in which {\it one modifies the gradient flow equation by adding higher order perturbations}. All these approaches try to put together a consistent, discrete  energy dissipation equation  following the traditional approach mentioned above within the over-determined partial differential equation system.

Recently, the energy quadratization (EQ) approach coined by Xiaofeng Yang, Jia Zhao and Qi Wang following the work of Ganzales and Tierra on liquid crystal models, provides a systematic approach to derive energy-dissipation-rate preserving schemes for thermodynamically consistent models \cite{Guill2013On,Yang2017Numerical,Zhao2017Numerical}. In this approach, {\it one introduces new unknowns (known as auxiliary variables) to reformulate the energy into a quadratic form and in the meantime supplement the system with the same number of dynamical equations for the new auxiliary variables}. This process is known as the EQ reformulation, which effectively embeds the gradient flow model into a higher dimensional phase space with a quadratic free energy while retaining the thermodynamically consistent structure and property (including the energy dissipation property.) The EQ reformulated gradient flow model together with the definition of the quadratic free energy and the energy dissipation equation constitute {\it a reformulated  over-determined, consistent and solvable gradient flow system.} One designs linear, energy stable schemes from the EQ reformulated gradient flow model. Analogously, the SAV method follows the same idea \cite{shen2018scalar}.

We illustrate the idea using an example here.  We assume $f$ only depends on $\Phi$, but not its spatial derivatives. In this case, one introduces a $q$ variable to transform the free energy into a quadratic form:
\ben
q = \sqrt{2\left(f +\frac{A}{|\Omega|}\right)}, \quad
F = \frac{1}{2}(\Phi, \cL \Phi) + \frac{1}{2}\|q\|^2 - A,\label{q-f-eq}
\een
where  $A$ is a constant large enough to make $q$ well-defined.
Then, one takes the time derivative of the algebraic equation of $q$ to arrive at an equivalent dynamical equation of $q$. The EQ-reformulated  system in a higher dimensional phase space recasts into
\beq \label{eq:gradient-flow-EQ-scalar}
\left\{
\bea{l}
\frac{\partial}{\partial t}\Phi = -\bM \Big(\cL \Phi + q \frac{\partial q}{\partial \Phi} \Big), \\
\frac{\partial}{\partial t} q = \frac{\partial q}{\partial \Phi} \cdot \frac{\partial \Phi}{\partial t}.
\eea
\right.
\eeq
Denoting $\Psi = \left( \Phi,\; q \right)^T$, the above system is rewritten into \cite{Gong&Zhao&WangSISC2020}
\beq \label{eq:gradient-flow-EQ-full}
\frac{\partial}{\partial t} \Psi = -\cN \cB \Psi,
\eeq
where $\cN = \cA^* \bM \cA,$ $\cA^*$ is the adjoint operator of $\cA,$ $$\cA = \left (\bI_n \quad \frac{\partial q}{\partial \Phi} \right)_{n,n+1},\quad \cB = \diag(\cL, 1)_{n+1,n+1},$$  $\cB$ is a linear, self-adjoint, positive definite operator.
The energy dissipation rate is given by
\beq
\frac{\d F}{\d t} = \Big( \frac{\delta F}{\delta \Psi} , \frac{\partial\Psi}{\partial t} \Big) = -\Big( \cB \Psi , \cN  \cB\Psi \Big) \leq 0.\label{eq:energy-diss}
\eeq
Recently, Gong, Zhao and Wang showed that one can devise arbitrarily high order energy stable schemes to solve the over-determined system consisting of \eqref{q-f-eq}, \eqref{eq:gradient-flow-EQ-full} and \eqref{eq:energy-diss} \cite{Gong&Zhao&WangSISC2020,Gong&ZhaoAML2019,Gong&Zhao&WangCPC2020,Gong&Zhao&WangArXiv2019}. In contrast to the traditional method, the EQ approach guarantees the consistency and solvability in the discretized system due to the reformulated free energy is quadratic  whereas the traditional approach may not.
Most recently, a class of new methods rooted in the scalar auxiliary variable approach have been devised to ensure linearity as well as energy stability in the numerical schemes developed for gradient flow models \cite{AkrivisLiLi:2019,Yang&g2020}.

Here, we take look at the issue of developing energy stable numerical approximations from a new perspective. We begin with the over-determined, consistent system of equations consisting of \eqref{gfeq}, \eqref{energy-dissipation-rate}, and \eqref{free-energy}. Now that this system is structurally unstable, it can easily lose the consistency among the equations subject to any small perturbations. How can we make it structurally stable. One way to make it structurally stable is to extend the system by adding supplementary variables to make it well-determined, i.e., the number of unknowns equals to the number of equations. There are virtually unlimited number of ways this can be done. To be consistent with the original problem, we accomplish this using perturbations and require that the well-determined, perturbed system include the original system as a special case. We would like to take a new point of view towards next present a  new paradigm to devise energy dissipation rate preserving schemes for the over-determined system consisting of \eqref{gfeq}, \eqref{energy-dissipation-rate} and \eqref{free-energy} to retain structural consistency and solvability in the over-determined system. We name it the supplementary variable method ({\bf SVM}). We will show that this includes the newly proposed Lagrange multiplier method  based on the SAV approach by Cheng, Liu and Shen as well as the generalized SAV method developed by Yang and Dong lately as special cases \cite{ChengA,Yang&g2020}. This method originates from the idea on how to enforce structural stability in the over-determined equation system by augmenting it with supplementary variables. Specifically, we perturb the system consisting of \eqref{gfeq}, \eqref{energy-dissipation-rate} and \eqref{free-energy} by adding a perturbation:
\ben
\left \{
\bea{l}
\dot{\Phi}=-\bM \cdot \mu,\quad
 \mu = \frac{\delta F}{\delta \Phi}+\alpha g[\Phi],\\
F = \frac{1}{2}(\Phi, \cL \Phi ) + \big( f (\Phi, \nabla \Phi), 1\big)+\alpha h(t),\\
\dfrac{\d F}{\d t}=-\int_{\Omega} \frac{\delta F}{\delta \Phi} \bM_s \frac{\delta F}{\delta \Phi}+\alpha j(t),
\eea\right.\label{extended}
\een
where $\alpha(t)(g[\Phi], h(t), j(t))^T$ is the perturbation, $\alpha(t)$ is a function of time,  $g[\Phi]$ is  a prescribed functional of $\Phi$, $h$ and $j$ are prescribed functions of t. $\alpha(t)$ is called the supplementary variable. If $\alpha(t)=0$, this perturbed  equation system reduces to the original, consistent and over-determined system. By adding $\alpha$, however,  we  lift the dynamical system into a higher dimensional phase space  to remove its over-determinedness and to effectively provide it with structural stability. The gained structural stability warrants us to obtain energy-dissipation-rate preserving numerical algorithms for \eqref{gfeq} under very general conditions so long as we keep the perturbation the same order as the truncation error when we discretize the system. We next detail the implementation of this method in one simple gradient flow case.

\section{Supplementary variable method--a new paradigm for developing energy-dissipation-rate preserving numerical algorithms}

 We explicitly rewrite system \eqref{gfeq}, \eqref{energy-dissipation-rate} as follows
\ben
&& \partial_t \Phi = -\bM \big( \cL\Phi + f'(\Phi) \big),\label{Phi_eq} \\
&& \mu = \cL\Phi + f'(\Phi),\label{mu_eq}\\
&& \frac{\textrm{d} F}{\textrm{d} t} = - ( \mu, \bM\mu).\label{F_eq}
\een
To derive energy-dissipation-rate preserving numerical approximations, we firstly  approximate the energy-dissipation-rate equation of the system given in  \eqref{Phi_eq}-\eqref{F_eq} as follows
\ben
&& \frac{\widetilde{\Phi}^{n+\frac{1}{2}}-\Phi^n}{\tau/2} = - \bM \left(\cL \widetilde{\Phi}^{n+\frac{1}{2}} + f'(\overline{\Phi}^{n+\frac{1}{2}})\right),  \label{Phi_tilde}\\
&& \mu^{*} = \cL \widetilde{\Phi}^{n+\frac{1}{2}} + f'(\widetilde{\Phi}^{n+\frac{1}{2}}),\label{mu_star}\\
&& \frac{\widetilde F^{n+1}-F[\Phi^n]}{\tau} = - ( \mu^*, \bM\mu^*),\label{energy-constraint}
\een
where $\tau$ is the time step,
\ben
\overline{\Phi}^{n+\frac{1}{2}} = \frac{3\Phi^n - \Phi^{n-1}}{2}, \quad F[\Phi^n] = \frac{1}{2}(\Phi^n, \cL \Phi^n ) + \big( f (\Phi^n), 1\big).
\een
It follows from \eqref{Phi_tilde} that
\ben
\label{Phi_tilde_ex} \widetilde{\Phi}^{n+\frac{1}{2}} = \left(1+\frac{\tau}{2}\bM\cL\right)^{-1} \Big( \Phi^n - \frac{\tau}{2}\bM f'(\overline{\Phi}^{n+\frac{1}{2}}) \Big), \quad
{\widetilde F^{n+1}}=F[\Phi^n]-{\tau} ( \mu^*, \bM\mu^*).
\een
This is a 2th order approximation to the energy-dissipation-rate equation with a truncation error in the order of $O(\tau^2)$.

Secondly,

we modify \eqref{Phi_eq} by a time-dependent supplementary variable $\alpha(t)$ together with a user supplied functional $g[\Phi]$:
\beq \label{LagrangePDEs}
\left\{
\bea{l}
\partial_t \Phi = -\bM \big( \cL\Phi + f'(\Phi) \big) + \alpha g[\Phi], ~ t\in (t^n, t^{n+1}], \\
F[\Phi(t^{n+1})] = \widetilde F^{n+1},
\eea
\right.
\eeq
where $g[\Phi]$ is a given functional that may depend on $\Phi$ and its derivatives. There is a grate deal of flexibility in determining how to modify the gradient flow model with a supplementary variable. It is clearly an open problem for this approach.   We simply present two special implementations in this study to illustrate the idea.

\begin{itemize}
\item One is
	\beq\label{gc1} g[\Phi] =  \bM f'(\Phi),\eeq which leads to
	\beq\label{modified_eq1} \partial_t \Phi = -\bM \Big( \cL\Phi + (1-\alpha)f'(\Phi) \Big). \eeq
 Here, the chemical potential is modified into
 \beq
 \label{shen_mu} \mu =  \cL\Phi + (1-\alpha)f'(\Phi).
\eeq
\item
	The other is
	\beq
\label{gc2} g[\Phi] =   -\bM \Big(  \cL\Phi + f'(\Phi)  \Big),
\eeq which implies
	 \beq
\label{modified_eq2} \partial_t \Phi = -( 1 +\alpha ) \bM \big( \cL\Phi + f'(\Phi) \big).
\eeq
Here, the mobility  is modified into $(1+\alpha)\bM$.
\end{itemize}

We note that a system consisting of  \eqref{F_eq}, \eqref{modified_eq1} and \eqref{shen_mu}  is equivalent to the reformulated model proposed in \cite{ChengA} using what they call the Lagrange multiplier method. The supplementary variable $\alpha$ can also be viewed as a perturbation variable since when $\alpha(t)=0$, the modified/perturbed system reduces to the original one. {\it By supplementing the over-determined system with a new variable without introducing an equation, we effectively remove the over-determinedness by making the modified system a well-determined in a higher dimensional phase space! In addition, the modified model reduces to the original one at $\alpha=0$. } We reiterate that there is  a great deal of flexibility to place the supplementary variable in the gradient flow model, which differentiates it from the Lagrange multiplier method and the generalized SAV method.

Thirdly, we apply the implicit-explicit Crank-Nicolson scheme in time to \eqref{LagrangePDEs} to arrive at the following semi-discrete system
\beq \label{scheme}
\begin{cases}
	\delta_t^+ \Phi^n = - \bM \left(\cL \Phi^{n+\frac{1}{2}} + f'(\widetilde{\Phi}^{n+\frac{1}{2}})\right) + \alpha^{n+\frac{1}{2}} g[\widetilde{\Phi}^{n+\frac{1}{2}}],\\
	F[\Phi^{n+1}] = \widetilde F^{n+1}, \quad \delta_t^+ \Phi^n = \frac{\Phi^{n+1}-\Phi^n}{\tau}, \quad \Phi^{n+\frac{1}{2}} = \frac{\Phi^{n+1}+\Phi^n}{2},
\end{cases}
\eeq
where $\widetilde{\Phi}^{n+\frac{1}{2}},~ \widetilde F^{n+1}$ have been obtained from \eqref{Phi_tilde_ex}.
\begin{remark}
	We apply a special implicit-explicit Crank-Nicolson scheme to the system consisting of \eqref{modified_eq1}, \eqref{shen_mu} and \eqref{F_eq} and obtain the following discrete method
	 \beq \label{shen_scheme}
	 \begin{cases}
	 	\delta_t^+ \Phi^n = - \bM \mu^{n+\frac{1}{2}},\\
	 	\mu^{n+\frac{1}{2}} = \cL \Phi^{n+\frac{1}{2}} + (1-\alpha^{n+\frac{1}{2}}) f'(\overline{\Phi}^{n+\frac{1}{2}}),\\
	 	\dfrac{F[\Phi^{n+1}]-F[\Phi^n]}{\tau} = - ( \mu^{n+\frac{1}{2}}, \bM\mu^{n+\frac{1}{2}}).
	 \end{cases}
	 \eeq
	 It is not difficult to prove that the scheme \eqref{shen_scheme} is equivalent to the Lagrange multiplier approach proposed in \cite{ChengA}.

\end{remark}

Next we discuss how to solve system \eqref{scheme} efficiently . Let
\ben
&& \widehat{\Phi}^{n+1} = (1+\frac{\tau}{2}\bM\cL)^{-1} \Big( (1-\frac{\tau}{2}\bM\cL)\Phi^n - \tau \bM f'(\widetilde{\Phi}^{n+\frac{1}{2}}) \Big),\label{Phi_hat}\\
&& w^n = (1+\frac{\tau}{2}\bM\cL)^{-1} g[\widetilde{\Phi}^{n+\frac{1}{2}}],\label{w_eq}\\
&& \beta = \tau \alpha^{n+\frac{1}{2}}.
\een
It follows from the first equation of \eqref{scheme} that
\beq
\label{get_Phi} \Phi^{n+1} = \widehat{\Phi}^{n+1} + \beta w^n.
\eeq
Then we plug $\Phi^{n+1}$ into the second equation in \eqref{scheme} to obtain
\beq\label{beta_eq} F[\widehat{\Phi}^{n+1} + \beta w^n] = \widetilde F^{n+1}, \eeq
which is an algebraic equation for $\beta.$ In general, it can have multiple solutions, but one of them must be close to 0 and it approaches to zero as $\tau \to 0$. So, we solve for this solution using an iterative method such as the Newton iteration with 0 as the initial condition, it generally converges to a solution close to 0 when $\tau$ is not too large. After obtaining $\beta$, we update $\Phi^{n+1}$ using \eqref{get_Phi}. Following the work of Refs. \cite{calvo2006on,calvo2010projection}, the existence of solution $\beta$ is guaranteed under the conditions of the following theorem.

\begin{thm}\label{thm:order}
	If $\left( \frac{\delta F}{\delta \Phi}[\Phi^n] , g[\Phi^n] \right) \neq 0,$ there exists a $\tau^*>0$ such that \eqref{beta_eq} defines a unique function $\beta = \beta (\tau)$ for all $\tau\in[0,\tau^*]$ and scheme \eqref{scheme} with \eqref{Phi_tilde}-\eqref{energy-constraint} is of order $O(\tau^2)$.
\end{thm}

\begin{proof}
	For $\tau,\beta$ in a neighborhood of $(0,0)$, we define the real function
\ben
u(\tau,\beta) = F[\widehat{\Phi}^{n+1} + \beta w^n] -\widetilde  F^{n+1} = F[\widehat{\Phi}^{n+1} + \beta w^n] - F[\Phi^n] + \tau(\mu^* , \bM \mu^*),
\een
where $\mu^*$ is calculated in the first step.
	It follows from  \eqref{mu_star}, \eqref{Phi_tilde_ex}, \eqref{Phi_hat} and \eqref{w_eq} that
\ben
u(0,0) = 0, \quad \frac{\partial u}{\partial \beta}(0,0) = \left( \frac{\delta F}{\delta \Phi}[\Phi^n] , g[\Phi^n] \right) \neq 0.
\een
	According to the implicit function theorem, there exists a $\tau^*>0$ such that  equation $u(\tau,\beta) = 0$ defines a unique smooth function $\beta = \beta (\tau)$ satisfying $\beta(0) = 0$ and $u\big(\tau,\beta(\tau)\big) = 0$ for all $\tau\in[0,\tau^*]$.

	Since $\widehat{\Phi}^{n+1}$ satisfies the following scheme
	\beq \frac{\widehat{\Phi}^{n+1}-\Phi^n}{\tau} = - \bM \left(\cL\frac{\widehat{\Phi}^{n+1}+\Phi^n}{2} + f'(\widetilde{\Phi}^{n+\frac{1}{2}})\right),\eeq
	through local truncation error analysis we have
	\ben
\widehat{\Phi}^{n+1} = \Phi(t^{n+1}) + O(\tau^3), \quad F[\widehat{\Phi}^{n+1} ] = F[\Phi(t^{n+1})] + O(\tau^3).
\een
In addition, we expand
\ben
u(\tau,\beta) = u(\tau,0) +  \beta \frac{\partial u}{\partial \beta}(\tau,0) + O(\beta^2),
\een
 with
	\ben
\bea{l}
u(\tau,0) = F[\widehat{\Phi}^{n+1}] - \widetilde F^{n+1} = O(\tau^3),\\[0.1cm]
\dfrac{\partial u}{\partial \beta}(\tau,0) = \dfrac{\partial u}{\partial \beta}(0,0) + O(\tau).
\eea
\een
	Then $\beta = \beta(\tau) = O(\tau^3)$ and the proposed scheme is of order $O(\tau^2)$.
\end{proof}

\begin{remark}
	If \eqref{gc2} is chosen, i.e. $g[\Phi] = -\bM\frac{\delta F}{\delta \Phi}[\Phi],$ then the sufficient condition in Theorem \ref{thm:order} reduces to $\left( \frac{\delta F}{\delta \Phi}[\Phi^n], \bM\frac{\delta F}{\delta \Phi}[\Phi^n] \right) \neq 0,$ which is usually satisfied when the steady state is not reached.
\end{remark}

\section{Numerical results}
We present a numerical example to demonstrate the practicability, accuracy, as well as energy stability of the proposed schemes. For simplicity, we use periodic boundary conditions for the numerical example below and name the resulting scheme {\bf SVM-I}  and {\bf SVM-II} when \eqref{gc1} and \eqref{gc2} are adopted, respectively.

\begin{example}[Cahn-Hilliard model]
	We consider the following Cahn-Hilliard phase field model with phase variable $\phi$
	\begin{align*}
	\partial_t \phi = \lambda \Delta \left(   - \varepsilon^2 \Delta \phi + \phi^3 - \phi  \right),
	\end{align*}
	where the free  energy is given by
	\begin{align*}
	F = \frac{ \varepsilon^2 }{2} \|  \nabla \phi \|^2 + \frac{1}{4}(  \phi^2 - 1 )^2.
	\end{align*}
	First of all, we present the  mesh refinement test in time to confirm the order of accuracy of the schemes. We set the computational domain as $\Omega = [0, 1]^2$ and the parameter values as $\varepsilon = 10^{-2}$ and $ \lambda = 10^{-3}$, respectively.
	This model is discretized spatially using a Fourier pseudo-spectral method with $256 \times 256$ spatial meshes.  We use initial condition $\phi = 0.25 \sin (2 \pi x) \cos(2 \pi y)$ and time steps $\tau  = 1.25e-2 \times \frac{1}{2^{k - 1}}$, $k = 1, 2, 3, \cdots$. The errors are calculated as the difference between the solution of the coarse time step and that of the adjacent finer time step.
	Both the discrete $L^2$ and $L^{\infty}$ errors of numerical solution $\phi$ at $t = 1$  are shown in Figure \ref{fig:test-order}, where  we observe that  the proposed schemes  yield second order convergence rates in time.

	\begin{figure}[H]
		\centering
		\subfigure[$L^2$ error.]{
			\includegraphics[width=0.35\textwidth,height=0.35\textwidth]{Figure/Mesh_refinement_test/L2_Order.png}
		}\quad
		\subfigure[$L^{\infty}$ error.]{
			\includegraphics[width=0.35\textwidth,height=0.35\textwidth]{Figure/Mesh_refinement_test/Lmax_Order.png}
		}
		\caption{Mesh refinement test in time for the two schemes. Here, we fix the number of spatial meshes at $N = 256$. Second order convergence rates are confirmed. } \label{fig:test-order}
	\end{figure}

	Next, we examine the computational efficiency of the two schemes: {\bf SVM-I} and {\bf SVM-II}.  We compare the two proposed schemes with the SAV scheme using the Crank-Nicolson method {\bf SAV-CN} \cite{shen2018scalar} and the fully implicit Crank-Nicolson scheme {\bf FICN} \cite{du1991numerical}. The result in the total CPU time to solve for  the solution using each scheme is summarized in Figure \ref{fig:test-CPU}. We observe that {\bf SVM-I} and {\bf SVM-II} for this model are less efficient than scheme {\bf SAV-CN}, but much more efficient than scheme {\bf FICN}. The price we pay using the proposed scheme in CPU time is that we have to solve a scalar nonlinear equation at each time step. In contrast, {\bf SAV} scheme solves a linear system while {\bf FICN} solves  a nonlinear system.

	\begin{figure}[H]
		\centering
		\subfigure{
			\includegraphics[width=0.60\textwidth,height=0.35\textwidth]{Figure/CPU/CPU.png}
		}
		\caption{Comparison of CPU time in logarithmic scale using the four different numerical methods with two sets of spatial meshes up to  $t = 1$, where the time step is chosen as $\tau = 1.0e-2$. The charts show that the two proposed schemes performs equally well, more slowly than {\bf SAV-CN}, but much faster than {\bf FICN}.} \label{fig:test-CPU}
	\end{figure}

Finally, we compare {\bf SVM-I}, {\bf SVM-II} and {\bf SAV-CN} in their accuracy in resolving the energy-dissipation-rate. We do it using a spatial discretization on $128^2$ meshes in $\Omega=[0,1]^2$. The parameter values are $\lambda = 1$ and $\varepsilon = 1.0e-2$.  We use the following initial condition \cite{Gong&Zhao&WangArXiv2019}
\begin{align*}
	\phi(x, y, 0) = 0.05 \left(  \cos(6\pi x) \cos(8 \pi y) + (\cos(8 \pi x) \cos(6 \pi y))^2 + \cos(2 \pi x -10 \pi y) \cos(4\pi x - 2\pi y)   \right).
\end{align*}
The initial profile undergoes a fast coarsening dynamics such that the algorithms must use small time steps in order to capture the correct coarsening dynamics.  The simulation results are depicted in Figure \ref{fig:test-fast-coarsening}, where the snapshots of $\phi(x,y ,t)$ at $t = 0.1$ are shown using different schemes and time steps.  We observe that scheme {\bf SAV-CN} predicts the correct solution at time step $\tau = 1.5625e-6$ but fails at time step $\tau = 3.125e-6$. For {\bf SVM-I}, it predicts correct numerical result  at time step  $\tau = 5.0e-5$, and {\bf SVM-II} performs even  better with time step $\tau = 2.0e-4$.
The errors in the total volume and the energy are plotted in Figure \ref{fig:test-energy-volume}. The numerical results show that both schemes conserve the total volume and capture the energy-dissipation-rate correctly using relatively larger time steps compared to the {\bf SAV} scheme. In addition, the supplementary variable $\alpha(t)$ remains close to zero except at a few initial time spots.

		\begin{figure}[H]
		\centering
				\subfigure[$\phi$ at $t = 0.1$ using {\bf SAV-CN} with time steps:  $\tau = 3.125e-6$, $\tau = 1.5625e-6$ and $\tau=1.0e-6$ (reference) (from left to right).]{
			\includegraphics[width=0.20\textwidth,height=0.20\textwidth]{Figure/fast_coarsening/phi_SAV_dt3_125e_06.png}
			\includegraphics[width=0.20\textwidth,height=0.20\textwidth]{Figure/fast_coarsening/phi_SAV_dt1_5625e_06.png}
			\includegraphics[width=0.20\textwidth,height=0.20\textwidth]{Figure/fast_coarsening/phi_reference.png}
		}
		\subfigure[ $\phi$ at $t = 0.1$ using {\bf SVM-I} with time steps:  $\tau = 5.0e-5$, $\tau = 4.0e-5$ and $\tau=1.0e-6$ (reference) (from left to right).]{
	       \includegraphics[width=0.20\textwidth,height=0.20\textwidth]{Figure/fast_coarsening/phi_G1_dt5e_05.png}
			\includegraphics[width=0.20\textwidth,height=0.20\textwidth]{Figure/fast_coarsening/phi_G1_dt4e_05.png}
			\includegraphics[width=0.20\textwidth,height=0.20\textwidth]{Figure/fast_coarsening/phi_reference.png}
		}
		\subfigure[$\phi$ at $t = 0.1$ using {\bf SVM-II} with time steps: $\tau = 2.0e-4$, $\tau = 1.5625e-4$ and $\tau=1.0e-6$ (reference) (from left to right).]{
	\includegraphics[width=0.20\textwidth,height=0.20\textwidth]{Figure/fast_coarsening/phi_G2_dt2e_04.png}
	\includegraphics[width=0.20\textwidth,height=0.20\textwidth]{Figure/fast_coarsening/phi_G2_dt1_5625e_04.png}
	\includegraphics[width=0.20\textwidth,height=0.20\textwidth]{Figure/fast_coarsening/phi_reference.png}
}
		\caption{Comparison of three schemes at different time steps. The first sub-figure in each row indicates the ``maximum possible" time step to predict correct dynamics. These snapshots show that both schemes  {\bf SVM-I} and {\bf SVM-II} perform better than {\bf SAV-CN} and {\bf SVM-II} performs the best.  \label{fig:test-fast-coarsening} }
	\end{figure}

	\begin{figure}[H]
	\centering
	\subfigure[]{
	\includegraphics[width=0.30\textwidth,height=0.30\textwidth]{Figure/fast_coarsening/energy.png}
}
	\subfigure[]{
		\includegraphics[width=0.30\textwidth,height=0.30\textwidth]{Figure/fast_coarsening/volume.png}
	}
	\subfigure[]{
\includegraphics[width=0.30\textwidth,height=0.30\textwidth]{Figure/fast_coarsening/Supplementary_Variable.png}
}
	\caption{(a)Time evolution of the free energy computed using the three schemes with various time steps. The subfigure shows that {\bf SVM-I} and {\bf SVM-II} can work well with much larger time steps than {\bf SAC-CN} does, while {\bf SVM-II} performs the best. (b)
		Time evolution of the error in the total volume using {\bf SVM-I} and {\bf SVM-II} with $\tau = 5.0e-5$ and $\tau = 2.0e-4$, respectively.  The results show that the proposed schemes preserve the total volume very well. (c) The evolution of supplementary variable $\alpha$. This subfigure indicates  $\alpha$  may fluctuate near zero initially, but eventually, settles down close to zero.
		\label{fig:test-energy-volume}}
\end{figure}
	\end{example}

\section{Conclusion}
 The numerical results demonstrate the proposed schemes based on the supplementary variable approach can  predict fast coarsening dynamics of the Cahn-Hilliard model accurately and outperform  {\bf SAV-CN}  in solution accuracy and {\bf FICN} in efficiency.	Here we simply present two convenient implementations of supplementary variables for developing energy-dissipation-rate preserving schemes. There can be many other ways guided by this paradigm to achieve energy stability, better computational efficiency and accuracy. We expect to see more property preserving numerical algorithms developed for thermodynamically consistent models guided by this paradigm in the future.

\section*{Acknowledgment}
Research is partially supported by the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems (202001, 202002), the Natural Science Foundation of Jiangsu Province (award BK20180413),  the National Natural Science Foundation of China (award 11801269 and  NSAF-U1930402) and National Science Foundation of US (award DMS-1815921 and  OIA-1655740) and a GEAR award from SC EPSCoR/IDeA Program.


TITLE: Proving that 17 does not divide $c^2$ for given equation
QUESTION [0 upvotes]: I'm trying to prove that $17$ does not divide $c^2$, given $c^2=a^2-7b^2$, where $a$, $b$, and $c$ are three relatively prime positive integers.
I initially tried to look at the four cases given by the two possible forms of $a^2$ and $b^2$, namely, $4k+1$ and $4k$ (all squares are of one of these two forms). So we have:
(1) $a^2\equiv 0$ and $b^2\equiv 0$ (mod 4);
(2) $a^2\equiv 1$ and $b^2\equiv 0$ (mod 4);
(3) $a^2\equiv 0$ and $b^2\equiv 1$ (mod 4); and
(4) $a^2\equiv 1$ and $b^2\equiv 1$ (mod 4).
For cases (1), (3), and (4), I was able to find a contradiction, if we assume 17 does indeed divide $c^2$. For example, for case (1), we know $a^2-7b^2 \equiv 0$, thus, if $c=17x$ for some nonzero integer $x$, then $289x^2 \equiv 0$ i.e. $x^2\equiv 0$. $x$ was defined not to be zero, so we have a contradiction.
But for case (2), this method doesn't work. We have $a^2-7b^2 \equiv 1$ and thus $x^2\equiv 1$, which doesn't seem to be a contradiction. Or is it? I'd love some direction here; or re-direction, if this is just totally the wrong way of trying to approach this problem.

REPLY [4 votes]: Suppose $a^2\equiv7b^2; a,b\not\equiv0\bmod17$.  Then $b$ has a $\bmod17$ multiplicative inverse $d$ so $a^2d^2\equiv(ad)^2\equiv7\bmod 17$, making $ad$ a nonzero $\bmod 17$ residue whose square is $\equiv7$.  Now try to find what this nonzero residue could be.  Uh-oh.


TITLE: What's the power series solution of $y' = x^2\,y$
QUESTION [1 upvotes]: My attempted solution

I'm not sure how to factor out $x$. In the other problems I've solved, both series terms have the same power of $x$. but in this problem one series contains $x^{n+2}$ and the other contains $x^n$.

REPLY [1 votes]: Make a change of variable in the summation index. For instance, letting $n+2=k$ we have
$$
\sum_{n=0}^\infty c_nx^{(n+2)}=\sum_{k=2}^\infty c_{k-2}x^k.
$$


text:
\begin{document}

\title[Frobenius map, D-modules and local cohomology]{A note on the use of Frobenius map and D-modules in local cohomology
}

\begin{abstract}
The Frobenius depth denoted by $\Fdepth$ defined by
Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a
different way, for rings of positive characteristic. The first aim
of the present paper is to compare the $\Fdepth$ with formal grade
and reprove some results of Lyubeznik using formal local cohomology.
Then the endomorphism rings of local cohomology modules will be
considered. As an application we reprove the results due to
Huneke-Koh in positive characteristic and Lyubeznik in
characteristic zero on the annihilators of local cohomology modules.
\end{abstract}
 \maketitle

\section{Introduction} \label{sect1}

Let $Y$ be a closed subscheme of $\mathbb{P}^n_k$, the projective
space over a field $k$ of characteristic $p>0$. Vanishing of
$H^i(\mathbb{P}^n-Y,\mathcal{F})$ for all coherent sheaves
$\mathcal{F}$ was asked by Grothendieck (\cite{Gr68}). Among the
attempts to answer the mentioned question Hartshorne and Speicer in
\cite{HS} used the notion of Frobenius depth of $Y$ to give an
essentially complete solution to this problem.

To be more precise, Let $Y$ be a Noetherian scheme of finite
dimension, whose local rings are all of characteristic $p>0$. Let $y
\in Y$ be a (not necessarily closed) point. Let $d(y)$ be the
dimension of the closure $\{y\}^-$ of the point $y$. Let
$\mathcal{O}_y$ be the local ring of $y$, let $k_0$ be its residue
field, let $k$ be a perfect closure of $k_0$, and let
$\widehat{\mathcal{O}}_y$, be the completion of $\mathcal{O}_y$.
Choose a field of representatives for $k_0$ in
$\widehat{\mathcal{O}}_y$. Then we can consider
$\widehat{\mathcal{O}}_y$ as a $k_0$-algebra, and we let $A_y$ be
the local ring $\widehat{\mathcal{O}}_y \otimes_{k_0} k$ obtained by
base extension to $k$. Let $Y_y=\Spec A_y$ and let $P$ denote its
closed point. So, the Frobenius depth of $Y$ denoted by $\Fdepth
Y$ is the largest integer $r$ (or $+\infty$) such that for all
points $y \in Y$, one has $H^i_P(Y_y,\mathcal{O}_{Y_y})_s=0$ (the
stable part of $H^i_P(Y_y,\mathcal{O}_{Y_y})$) for all $i < r-d(y)$.

 From the local
algebra point of view, Grothendieck's problem is stated to find
conditions under which  $H^i_I(M)=0$ for all $i>n\ (n \in
\mathbb{Z})$ and all $A$-modules $M$, where $A$ is a commutative
Noetherian local ring and $I \subset A$ is an ideal. For an
$A$-module $M$, we denote by $H^i_I(M)$ the $i$th local cohomology
module of $M$ with respect to $I$. For more details the reader may
consult \cite{Gr67} and \cite{Br-Sh}. From the celebrated result of
Hartshorne (cf. \cite[pp. 413]{Hart68}), it is enough to find
conditions for the vanishing of $H^i_I(A)$. In this direction, for a
local ring $(A,\fm)$, Lyubeznik in \cite{Lyu06} using the Frobenius
map from $ H^i_{\fm}(A)$ to itself defined the Frobenius depth of
$A$ denoted by $\Fdepth A$ as the smallest $i$ such that for every
iteration of Frobenius map, $H^i_{\fm}(A)$ does not go to zero. It
is noteworthy to say that the Lyubeznik's $\Fdepth$ coincides with
the notion of $\Fdepth$ defined by Hartshorne and Speiser, whenever
$A$ admits a surjection from a regular local ring and $Y=\Spec A$
(cf. \cite[Corollary 6.3]{Lyu06}).

Consider the family of local cohomology modules $\{H^i_{\fm}
(M/I^nM)\}_{n \in \mathbb{N}}$ where, $(A,\fm)$ is not necessarily
of characteristic $p>0$. For every $n \in \mathbb{N}$ there is a
natural homomorphism
$$H^i_{\fm} (M/I^{n+1}M) \rightarrow H^i_{\fm} (M/I^nM)$$
(induced from the natural projections $M/I^{n+1}M \rightarrow
M/I^nM$) such that the family forms a projective system. The
projective limit ${\vpl}_nH^i_{\fm}(M/I^n M)$ is called the $i$th
formal local cohomology of $M$ with respect to $I$ (cf. \cite{Sch}).
Formal local cohomology modules were used by Peskine and Szpiro in
\cite{P-S} when $A$ is a regular ring of prime characteristic. It is
noteworthy to mention that if $U = \Spec(A) \setminus \{\fm\}$ and
$( \widehat{U},\mathcal{O}_{\widehat{U}})$ denotes the formal
completion of $U$ along $V(I) \setminus \{\fm \}$ and For an
$A$-module $M$, $\mathcal{F}$ denotes the associated sheaf on $U$
and $\widehat{\mathcal{F}}$ denote the coherent
$\mathcal{O}_{\widehat{U}}$-sheaf associated to ${\vpl}_n M/I^n M$,
they have described the formal cohomology modules
$H^i(\widehat{U},\mathcal{O}_{\widehat{U}})$ via the isomorphisms
$H^i(\widehat{U},\mathcal{O}_{\widehat{U}})\cong
{\vpl}_nH^i_{\fm}(M/I^n M)$, $i \geq 1$. See also \cite[proposition
(2.2)]{Og} when $A$ is a Gorenstein ring.

The formal grade, $\fgrade(I,M)$, is defined as the index of the
minimal nonvanishing formal cohomology module, i.e., $\fgrade(I,M) =
\inf \{i \in \mathbb{Z}|\ \ {\vpl}_nH^i_{\fm}(M/I^n M) \neq 0\}$.
One way to check out vanishing of local cohomology modules is the
following duality
\begin{equation}\label{duality}
    {\vpl}_nH^i_{\fm}(A/I^n) \cong \Hom_A(H^{\dim A-i}_{I}(A),E(A/\fm)),
\end{equation}
where $(A,\fm)$ is a Gorenstein local ring and $E(A/\fm)$ denotes
the injective hull of the residue field (cf. \cite[Remark
3.6]{Sch}). To be more precise, in this case the last non vanishing
amount of $H^{i}_{I}(A)$ may be described with the $\fgrade(I,A)$.
Thus, it motivates us to consider the invariants $\Fdepth$ and
$\fgrade$ to shed more light on the notion of Frobenius depth from a
different point of view. For this reason, in Section $2$, we bring
some auxiliary results and among them we examine the structure of
${\vpl}_nH^i_{\fm}(A/I^n)$ as a unite $A[F^e]$-module (Theorem
\ref{unit}). In Section $3$, we show that the $F$-nilpotency of
$H^i_{\fm}(A/I),\ (i \in \mathbb{Z})$ is equivalent to the vanishing
of ${\vpl}_nH^i_{\fm}(A/I^n)$ for a regular local $F$-finite ring
$A$ (Proposition \ref{f-nil}). It helps us to reprove
\cite[Corollary 3.2, Lemma 2.4]{Lyu06}, see Corollary \ref{Lyu3.2}
and Proposition \ref{lyu4.2}. Finally, we compare the Frobenius
depth with the formal grade and depth, (cf. Theorem \ref{comparison}
and Corollary \ref{connected}).

Studying of the endomorphism rings of local cohomology modules are
of interest in recent years. See for instance Hochster-Huneke
\cite{Ho-Hu}, Hellus-St\"{u}ckrad \cite{He-S}, Schenzel
\cite{Sch09}, \cite{Sch10}, Eghbali-Schenzel \cite{E-Sch},
Mahmood-Schenzel \cite{Ma-Sch}. In Section $4$, we examine the
endomorphism rings of local cohomology modules,
$\Hom_R(H^i_I(R),H^i_I(R))$, where $R$ is a regular local ring
containing a field in both characteristic zero and $p>0$. Our way to
prove the results is to use the so-called $\fD$-modules and
Frobenius map. These methods have played decisive roles in many
subsequent studies in the rings of characteristic zero and positive
characteristic. As an application we reprove the results due to
Huneke-Koh \cite[Lemma 2.2]{Hu-K} in positive characteristic and
Lyubeznik \cite[Corollary 3.6]{Lyu93} in characteristic zero on the
annihilators of local cohomology modules (cf. Corollaries
\ref{annloc.0} and \ref{annloc.p}).

\section{Auxiliary Results in positive characteristic}

Throughout this section all rings are assumed to contain a field of
positive characteristic. The symbol $A$ will always denote a
commutative Noetherian ring of finite characteristic. We adapt the
notation from \cite{Bl} and except for notation we mostly follow
Lyubeznik \cite{Lyu97}. We let $F=F_A$ the Frobenius map on $A$,
that is $F:A \rightarrow A$, with $a \mapsto a^p,\ a \in A$. We
denote the $e$th iterate of the Frobenius map by $A^e$ which is the
$A-A$-bimodule. As a left $A$-module it is $A$ and as a right
$A$-module we have $m.a=a^{p^e}m$ for $m \in A^e$. We say $A$ is
F-finite, whenever $A^e$ is a finitely generated right $A$-module.

\begin{remark}\label{Hart-Speis}
Let us recall from \cite[Proposition 1.1(a)]{HS} that for a ring $A$
which is either a localization of an algebra of finite type over a
perfect field $k$, or a complete local ring containing a perfect
field $k$ as its residue field, then $A$ is $F$-finite.
\end{remark}

In the present section, among our results we recall various results
due to Hartshorne-Speiser \cite{HS}, Peskine-Szpiro \cite{P-S},
Lyubeznik \cite{Lyu97} and Blickle \cite{Bl}.

Peskine and Szpiro in \cite{P-S} defined the Frobenius functor as
follows:

\begin{definition} The Frobenius functor is the right exact functor
from $A$-modules to $A$-modules given by
$$F^{\ast}_A M:=A^1 \otimes_A M.$$
Its $e$th power is $F^{e\ast}_A M=A^e \otimes_A M$. For brevity we
often write $F^{e\ast}$ for $F^{e\ast}_A $ when there is no
ambiguity about the ring $A$.
\end{definition}

It follows from the definition that $F^{e\ast}$ commutes with direct
sum, direct limit and localization. By a theorem of Kunz \cite{Ku69}
the Frobenius functor is flat whenever $A$ is a regular ring, hence
in this case $F^{e\ast}$ will be exact and immediately one has
$F^{e\ast} A=A^e \otimes_A A \cong A$ and $F^{e\ast} I=I^{[p^e]}$ an
ideal of $A$ generated by $p^e$th powers of the elements of $I$.

\begin{definition} \label{def.unit} An $A[F^e]$-module is an $A$-module $M$ together with an $A$-linear map
$$\mathcal{V}^e_M:F^{e\ast}_AM \rightarrow M.$$
A morphism between two $A[F^e]$-modules $(M,\mathcal{V}^e_M)$ and
$(N,\mathcal{V}^e_N)$ is an $A$-linear map $\varphi : M \rightarrow
N$ such that the following diagram commutes:

$$\xymatrix{
F^{e\ast}M \ar[d]^{\mathcal{V}^e_M} \ar[r]^{F^{e\ast}(\varphi)} &F^{e\ast}N\ar[d]^{\mathcal{V}^e_N}\\
M \ar[r]^{\varphi} &N}.$$ In fact, we can consider $F^eM$ as a
$p^e$-linear map from $M \rightarrow M$; as such it is not
$A$-linear but we have $F^e(am) = a^{p^e}F^e(m),\ a \in A^e, m \in
M$. Furthermore, an $A[F^e]$-module $(M,\mathcal{V}^e_M)$ is called
a unite $A[F^e]$-module if $\mathcal{V}^e_M$ is an isomorphism (cf.
\cite[page 17, Definition 2.6]{Bl}).
\end{definition}

\begin{remark}
Since $F^{e\ast}A$ is canonically isomorphic to $A$, then $A$ is a
unite $A[F^e]$-module, but an ideal $I \subset A$ is not a unite
$A[F^e]$-module in general, as the inclusion $I^{[p^e]} \subseteq I$
can be strict. For a multiplicatively closed subset $S$ of $A$, the
structural map $\mathcal{V}^e_{S^{-1}A}:A^e \otimes S^{-1}A
\rightarrow S^{-1}A$ is an isomorphism (see \cite{Bl}).
\end{remark}

The following definition is introduced in \cite{HS}:

\begin{definition} \label{forgetful} Let $(M,\mathcal{V}^e)$ be an $A[F^e]$-module.
We define $G(M)$ as the inverse limit generated by the structural
map $\mathcal{V}^e$, i.e.
$$G(M):={\vpl}(\cdots \rightarrow F^{3e\ast}M \stackrel{F^{2e\ast}\mathcal{V}^e}{\longrightarrow}
 F^{2e\ast}M \stackrel{F^{e\ast}\mathcal{V}^e}{\longrightarrow} F^{e\ast}M \stackrel{\mathcal{V}^e}{\longrightarrow} M).$$
\end{definition}
Note that there are natural maps $\pi_e: G(M) \rightarrow
F^{e\ast}M$. Moreover, the maps $F^{e\ast}\pi_r: F^{e\ast}G(M)
\rightarrow F^{e(r+1)\ast}M$ are compatible with the maps defining
$G(M)$ and thus by the universal property of inverse limits, the map
$F^{e\ast}G(M) \rightarrow G(M)$ defines the natural $A[F^e]$-module
structure on $G(M)$.

\begin{prop}  \label{4.1}(\cite[Proposition 1.2]{HS} and \cite[Proposition 4.1]{Bl}) Let $A$ be regular and $F$-finite and $M$ an
$A[F^e]$-module. Then $G(M)$ is a unite $A[F^e]$-module.
\end{prop}

In order to extend the Matlis duality functor $D(-)=\Hom(-, E_A)$
where, $E_A$ is the injective hull of the residue field, the functor
$\mathcal{D}$ from $A[F^e]$-modules to $A[F^e]$-modules is defined
as follows ( \cite[Section 4]{Bl}):

Let $(M, \mathcal{V}^e)$ be an $A[F^e]$-module. We define
$$\mathcal{D}(M)={\vil}(D(M) \stackrel{D(\mathcal{V}^e)}{\longrightarrow}
 D(F^{e\ast}M) \stackrel{D(F^{e\ast}(\mathcal{V})}{\longrightarrow} D(F^{2e\ast}M) \longrightarrow \ldots).$$
An element $m \in M$ is called $F$-nilpotent if $F^{re}(m)=0$ for
some $r$. Then $M$ is called $F$-nilpotent if $F^{er}(M)=0$ for some
$r \geq 0$. It is possible that every element of $M$ is
$F$-nilpotent but $M$ itself is not.

Below, we recall some properties of the functor $\mathcal{D}$.

\begin{prop} \label{4.16-4.17}Let $A$ be a complete regular local ring.

(a) On the subcategory of $A[F^e]$-modules which are cofinite (i.e.
satisfy the descending chain condition for submodules) as
$A$-modules,  the functor $\mathcal{D}$ is exact and
$\mathcal{D}(N)$ is a finitely generated unite $A[F^e]$-module for
every cofinite module $N$ (cf. \cite[Theorem 4.2(i)]{Lyu97} and
\cite[Proposition 4.16]{Bl}).

(b) Let $M$ be an $A[F^e]$-module that is finitely generated or
cofinite as an $A$-module. Then $\mathcal{D}(\mathcal{D}(M))\cong
G(M)$ (cf. \cite[Proposition 4.17]{Bl}).

(c) Let $M$ be an $A[F^e]$-module which is a cofinite $A$-module.
Then $M$ is $F$-nilpotent if and only if $\mathcal{D}(M) = 0$ (cf.
\cite[Theorem 4.2(ii)]{Lyu97} and \cite[Proposition 4.20]{Bl}).
\end{prop}

Let $A$ be a regular local ring and let $I$ be an ideal of $A$. As
we have seen (following \cite[page 19]{Bl}) that $A$ is an
$A[F^e]$-module and this structure comes down to its localization.
The local cohomology modules $H^i_I(A)$ of $A$ with support in $I$
can be calculated as the cohomology modules of the \v{C}ech complex
$$\check{C}(A; x_1,\ldots, x_n) = A \rightarrow A_{x_i} \rightarrow A_{x_ix_j} \rightarrow A_{x_1x_2\cdots x_n}$$
where, $I$ is generated by $x_1,x_2,\ldots, x_n$. As the category of
unite $A[F^e]$-modules is an abelian subcategory of the category of
$A[F^e]$-modules, the modules $H^i_I(A)$ are unite $A[F^e]$-modules
for all $i \in \mathbb{Z}$. For formal local cohomology modules, the
situation is a bit more complicated, however, we show that these
kind of modules have unite $A[F^e]$-modules structure, where $A$ is
$F$-finite.

\begin{thrm} \label{unit} Let $(A,\fm)$ be a
regular $F$-finite local ring. Then
$$G(H^i_{\fm}(A/I)) \cong
{\vpl}_nH^i_{\fm}(A/I^{n}),\ i \in \mathbb{Z}$$ which is a unite
$A[F^e]$-module. In particular, $$ {\vpl}_nH^i_{\fm}(A/I^{n}) \cong
H^i_{\fm}(\hat{A^I}),\ i \in \mathbb{Z}$$ as $A[F^e]$-modules, where
$\hat{A^I}$ is the completion of $A$ along $I$.

\end{thrm}

\begin{proof}
By what we have seen above,  $H^i_{\fm}(A/I)$ is an $A[F^e]$-module
for all $i \in \mathbb{N}$. Now, we may apply functor $G(-)$ to
$H^i_{\fm}(A/I)$:
$$G(H^i_{\fm}(A/I))=
{\vpl}(\cdots \stackrel{F^{2e\ast}\mathcal{V}^e}{\longrightarrow}
 H^i_{\fm}(A/I^{[p^{2e}]}) \stackrel{F^{e\ast}\mathcal{V}^e}{\longrightarrow} H^i_{\fm}(A/I^{[p^{e}]}) \stackrel{\mathcal{V}^e}
 {\longrightarrow} H^i_{\fm}(A/I)).$$
By virtue of Proposition \ref{4.1}, $G(H^i_{\fm}(A/I))$ is a unite
$A[F^e]$-module. Notice that, the right hand side is nothing but
${\vpl}_eH^i_{\fm}(A/I^{[p^e]})$.

On the other hand, one has ${\vpl}_nH^i_{\fm}(A/I^{n}) \cong
\Hom_A(H^{\dim A-i}_{I}(A),E)$, where $E:=E(A/\fm)$ is the injective
hull of the residue field. The natural map
$$F^{e \ast} \Hom_A(H^{\dim A-i}_{I}(A),E) \rightarrow \Hom_A(H^{\dim A-i}_{I}(A),E)$$
by sending $r \otimes \varphi$ to $rF^{e \ast}(\varphi)$ is an
isomorphism of $A[F^e]$-modules ($r \in A$ and $\varphi \in
\Hom_A(H^{\dim A-i}_{I}(A),E)$). To this end note that $H^{\dim
A-i}_{I}(A)$ and $E \cong H^{\dim A}_{\fm}(A)$ carry natural unite
$A[F^e]$-structure. Thus, ${\vpl}_nH^i_{\fm}(A/I^{n})$ is a unite
$A[F^e]$-module for each $i \in \mathbb{Z}$.

In order to complete the proof, it is enough to show that
${\vpl}_eH^i_{\fm}(A/I^{[p^e]}) \cong {\vpl}_nH^i_{\fm}(A/I^{n})$.
For this reason, consider the decreasing family of ideals
$\{I^{[p^e]}\}_e$. Clearly, its topology is equivalent to the
$I$-adic topology on $A$. Thus, by \cite[Lemma 3.8]{Sch} there
exists a natural isomorphism
$${\vpl}_eH^i_{\fm}(A/I^{[p^e]}) \cong
{\vpl}_nH^i_{\fm}(A/I^{n})$$ for all $i \in \mathbb{Z}$.

The last part follows by \cite[Proposition 2.1]{HS}.
\end{proof}

\section{Frobenius depth} \label{sect2}

Let $A$ be a regular local $F$-finite ring of characteristic $p>0$
and let $I$ be an ideal of $A$. As we have seen in the previous
section the formal local cohomology modules, are unite
$A[F^e]$-modules. In the present section we use the unite $A[F^e]$
structure of ${\vpl}_nH^i_{\fm}(A/I^{n})$ in order to prove our
results.

\begin{prop} \label{f-nil} Let $(A,\fm)$ be a
regular local and $F$-finite ring. Then
${\vpl}_nH^i_{\fm}(A/I^{n})=0$ if and only if $H^i_{\fm}(A/I)$ is
$F$-nilpotent.
\end{prop}

\begin{proof}

By the assumptions $A$ is $F$-finite that is $A^e$ is a finitely
generated $A$-module. Then tensoring with $A^e$ commutes with the
inverse limit, as $A^e$ is a free right $A$-module (cf.
\cite[Proposition 1.1(b)]{HS}). Thus, we have
$$A^e \otimes_A \widehat{A}=A^e \otimes_A ({\vpl}_n A/I^n) \cong {\vpl}_n A^e/A^eI^n= {\vpl}_n A^e/I^{n[p^e]}A^e \cong \widehat{A^e}.$$
On the other hand, since the Frobenius action is the same in both
$H^i_{\fm}(A/I)$ and $H^i_{\widehat{\fm}}(\widehat{A}/I
\widehat{A})$, so we may assume that $A$ is a complete regular local
$F$-finite ring.

As $H^i_{\fm}(A/I)$ is an $A[F^e]$-module which is a cofinite
$A$-module, then $\mathcal{D}(H^i_{\fm}(A/I))$ is a finitely
generated unite $A[F^e]$-module (cf. \ref{4.16-4.17}(a)) and
therefore
$$\mathcal{D}(\mathcal{D}(H^i_{\fm}(A/I)))\cong
D(\mathcal{D}(H^i_{\fm}(A/I))).$$ As the functor $D(-)$ transforms
direct limits to inverse limits, then
\begin{eqnarray*}
  D(\mathcal{D}(H^i_{\fm}(A/I)))\!\!\!\! &=&\!\!\!\! D({\vil}(D(H^i_{\fm}(A/I))
  \rightarrow D(F^{e\ast} H^i_{\fm}(A/I)) \rightarrow D(F^{2e\ast} H^i_{\fm}(A/I)) \rightarrow \cdots)  \\
            \!\!\!\! & \cong &\!\!\!\! {\vpl}(\cdots \rightarrow D(D(F^{2e\ast} H^i_{\fm}(A/I)))
  \rightarrow D(D(F^{e\ast} H^i_{\fm}(A/I))) \rightarrow
  D(D(H^i_{\fm}(A/I)))) \\
  \!\!\!\! & \cong &\!\!\!\! {\vpl}_eH^i_{\fm}(A/I^{[p^e]}).
\end{eqnarray*}
As we have seen in the proof of Theorem \ref{unit},
${\vpl}_eH^i_{\fm}(A/I^{[p^e]}) \cong {\vpl}_nH^i_{\fm}(A/I^{n})$.
Therefore,
 $H^i_{\fm}(A/I)$ is $F$-nilpotent if and only if
$\mathcal{D}(H^i_{\fm}(A/I))=0$ (\ref{4.16-4.17}(c)) if and only if
${\vpl}_nH^i_{\fm}(A/I^{n})=0$.
\end{proof}

\begin{remark}
Notice that in the Proposition \ref{f-nil} the $F$-finiteness of $A$
is vital, because it guarantees the $A[F^e]$ structure of the
modules ${\vpl}_nH^i_{\fm}(A/I^{n})$. However, in the light of
\cite[Lemma 4.12]{Bl} if $H^{\dim A-i}_I(A)$ is cofinite, then the
module ${\vpl}_nH^i_{\fm}(A/I^{n})$ is $A[F^e]$-module.
\end{remark}

\begin{cor} \label{Lyu3.2}(\cite[Corollary 3.2]{Lyu06}) Let $(A,\fm)$ be a
 regular local ring and $I$ an ideal of $A$. Then $H^{\dim A-i}_{I}(A)=0$ if
and only if $F^{er}:H^i_{\fm}(A/I) \rightarrow H^i_{\fm}(A/I)$ is
the zero map for some $r>0$.
\end{cor}

\begin{proof} As $\widehat{A}$ is a faithful flat $A$-module then by passing to the completion
we may assume that $A$ is complete regular local ring. Let $H^{\dim
A-i}_{I}(A)=0$, so it is cofinite. Then by the duality
(\ref{duality}) in the introduction, one has
${\vpl}_nH^i_{\fm}(A/I^{n})=0$. Hence, Proposition \ref{f-nil}
implies that $H^i_{\fm}(A/I)$ is $F$-nilpotent.

Conversely, assume that $H^i_{\fm}(A/I)$ is $F$-nilpotent. Then by
Propostion \ref{4.16-4.17}(c) $\mathcal{D}(H^i_{\fm}(A/I))=0$ and
therefore one has $H^{\dim A-i}_{I}(A)=0$. To this end note that,
$\mathcal{D}(H^{\dim A-i}_{\fm}(A/I^{[p^e]})) \cong
\Ext^{i}_A(A/I^{[p^e]},A)$ for all $e \geq 0$ and the Frobenius
powers of $I$ are cofinal with its ordinary powers.
\end{proof}

In the light of Corollary \ref{Lyu3.2}, Lyubeznik \cite{Lyu06}
defined the $\Fdepth$ of a local ring in order to give a solution to
Grothendieck's Problem.

\begin{definition} Let $(A,\fm)$ be a local ring. The $\Fdepth$ of
$A$ is the smallest $i$ such that $F^{er}$ does not send
$H^i_{\fm}(A)$ to zero for any $r$.
\end{definition}

One of elementary properties of $\Fdepth$ shows that $\Fdepth A$ is
equal to the $\Fdepth$ of its $\fm$-adic completion, $\widehat{A}$
(cf. \cite[Proposition 4.4]{Lyu06}) because $H^i_{\fm}(A)\cong
H^i_{\widehat{\fm}}(\widehat{A})$. In the next result we give an
alternative proof of \cite[Lemma 4.2]{Lyu06} to emphasize that
$\Fdepth$ of $A$ is bounded above by its Krull dimension.

\begin{prop} \label{lyu4.2} Let $(A,\fm)$ be a
 local ring and $I$ an ideal of $A$. Then $F^{er}$ does not send $H^{\dim A}_{\fm}(A)$ to zero for any $r$.
 In particular, $0 \leq Fdepth A \leq \dim A$.
\end{prop}

\begin{proof} Since the Frobenius action is the same in both
$H^i_{\fm}(A)$ and $H^i_{\widehat{\fm}}(\widehat{A})$, so we may
assume that $A$ is a complete local ring. Thus, by the Cohen's
Structure Theorem $A \cong R/J$, where $R$ is a complete regular
local ring and $J \subset R$ and ideal. In the contrary, assume that
$H^{\dim A}_{\fm}(R/J)$ is $F$-nilpotent. Then, $\mathcal{D}(H^{\dim
A}_{\fm}(R/J))=0$ (cf. \ref{4.16-4.17}(c)) and with a similar
argument given in the proof of Corollary \ref{Lyu3.2}, one can
deduce the vanishing of $H^{0}_{J}(R)$. Hence, by virtue of
(\ref{duality}), in the introduction, one has ${\vpl}_nH^{\dim A}
_{\fm}(R/J^{n})=0$ which is a contradiction (cf. \cite[Theorem
4.5]{Sch}).
\end{proof}

To investigate the other properties of $\Fdepth$, in the next
Theorem we compare the Frobenius depth of $A$ and $A^{sh}$. For this
reason, let us recall some preliminaries. For a local ring $A$ we
denote by $A^{sh}$ the strict Henselization of $A$. A local ring
$(A,\fm,k)$ is said to be strictly Henselian if and only if every
monic polynomial $f(T)\in A[T]$ for which $\overline{f(T)} \in k[T]$
is separable splits into linear factors in $A[T]$. For more advanced
expositions on this topic we refer the interested reader to \cite
{Mi}.

\begin{prop} \label{Hensel}  Let $(A,\fm)$ be a complete local ring. Then
  $\Fdepth A=\Fdepth A^{sh}$.
\end{prop}

\begin{proof} First assume that $A$ is a regular local ring. We show that $\Fdepth A/I=\Fdepth
(A/I)^{sh}$. Put $\Fdepth A/I=t$. Then, by virtue of Corollary
\ref{Lyu3.2} one has $H^{i}_{I}(A)=0$ for all $i
> \dim A-t$. Due to the faithfully flatness of the inclusion $A
\rightarrow A^{sh}$ and the fact that $A^{sh}$ is a regular local
ring, it implies that $H^{i}_{I}(A^{sh})=0$ for all $i
> \dim A-t$. Again, using Corollary
\ref{Lyu3.2}, it follows that $\Fdepth (A/I)^{sh} \geq t$. To this
end note that $\dim A=\dim A^{sh}$ and $(A/I)^{sh}=A^{sh}/IA^{sh}$.
With the similar argument one has $\Fdepth A/I \geq \Fdepth
(A/I)^{sh}$. This completes the assertion.

Since $A$ is a complete local ring, then by virtue of Cohen's
Structure Theorem, $A$ is a homomorphic image of a regular local
ring $R$, i.e. $A=R/J$ for some ideal $J$ of $R$. Now, we are done
by the previous paragraph. To this end note that
$$\Fdepth A =\Fdepth R/J=\Fdepth (R/J)^{sh}=\Fdepth A^{sh}.$$
\end{proof}

\begin{remark} From Proposition \ref{Hensel} and \cite[Proposition 4.4]{Lyu06} one may
deduce that
$$\Fdepth A=\Fdepth \widehat{((\hat{A})^{sh})},$$
where, $A$ is a local ring.
\end{remark}

In the following, we compare the invariants $\depth$, $\Fdepth$ and
$\fgrade$. Let us recall that the formal grade, $\fgrade(I,R)$, is defined as the index of the minimal nonvanishing
formal cohomology module, i.e., $\fgrade(I,R) = \inf \{i \in \mathbb{Z}|\ \ {\vpl}_nH^i_{\fm}(R/I^n ) \neq 0\}$.

\begin{thrm} \label{comparison}  Let $(A,\fm)$ be a  local $F$-finite ring and let $I$ be
an ideal of $A$. Then
 $$ \fgrade(I,A) \leq \depth A \leq \Fdepth A.$$
\end{thrm}

\begin{proof} We have ${\vpl}_nH^i_{\fm \hat{A}}(\hat{A}/I^n \hat{A}) \cong {\vpl}_nH^i_{\fm}(A/I^n
A)$ (cf. \cite[Proposition 3.3]{Sch}) and $\Fdepth A \cong \Fdepth
\hat{A}$ (cf. \cite[Proposition 4.4]{Lyu06}). As $\fm$-adic
completion of an $F$-finite ring is again $F$-finite, Then, we may
assume that $A$ is a complete local $F$-finite ring. Then $K=A/\fm$
is $F$-finite, then so is every finitely generated algebra over $K$.

By the Cohen's Structure Theorem there exists an $F$-finite regular local ring $(R,\fn)$ with
$A \cong R/J$ where, $J$ is an ideal of $R$.

 Put $\fgrade(J,R)=t$. Then by
definition ${\vpl}_nH^i_{\fn}(R/J^n)=0$ for all $i <t$. It follows
from the Proposition \ref{f-nil} that $H^{i}_{\fn}(R/J)$ is
$F$-nilpotent for all $i<t$, i.e. $\Fdepth R/J \geq t$. With a
similar argument and again using Proposition \ref{f-nil} we have
$\fgrade(J,R)\geq \Fdepth R/J$. Thus, $\fgrade(J,R)= \Fdepth R/J$.

Now, we are done by \cite[Lemma 4.8(b)]{Sch}, \cite[Remark 3.1]{E13}
and the previous paragraph. To this end note that
\begin{eqnarray*}
  \fgrade(I,A) \leq \depth A  &\leq&  \fgrade(J,R)  \\
             &=& \Fdepth R/J \\
             &=& \Fdepth A.
\end{eqnarray*}
\end{proof}

Let $A$ be a complete local ring containing a perfect field $k$ as
its residue field, then $A$ satisfies the condition of Theorem
\ref{comparison}. To this end, note that by Remark \ref{Hart-Speis},
$A$ is F-finite. Furthermore, by virtue of Cohen's Structure Theorem
$A \cong R/J$ for some ideal $J \subset R$, where
$R=k[[x_1,\ldots,x_n]]$ is a regular $F$-finite ring.

\begin{cor} \label{connected}
 Let $(A,\fm)$ be a regular local and $F$-finite ring. Then we have
$$\depth A/I \leq \fgrade(I,A)=\Fdepth A/I \leq \dim A/I.$$

\end{cor}

\begin{proof}
The assertion follows from what we have seen in the proof of Theorem
\ref{comparison} and
 \cite[Remark 3.1]{E}.
\end{proof}

\begin{remark}
(a) The necessary and sufficient conditions for small values of the
$\Fdepth$ of $A$ is given in \cite[Corollary 4.6]{Lyu06}.
\begin{itemize}
  \item [(1)] $\Fdepth A>0$ if and only if $\dim A>0$.
  \item [(2)] $\Fdepth A>1$ if and only if $\dim A \geq 2$ and the punctured spectrum of $A$ is formally
geometrically connected.
\end{itemize}

Now, let $A$ be $F$-finite and $\depth A=0 < \dim A$.
 Then, one has $\fgrade(I,A)=0 < \Fdepth
 A$. It shows that the inequality in Theorem \ref{comparison}
can be strict.

(b) Keep the assumptions in Corollary \ref{connected}, if $\Fdepth
A/I>1$, then by \cite[Lemma 5.4]{Sch}, one has
$\Supp_{\hat{A}}(\hat{A}/I \hat{A})\setminus \{\hat{\fm}\}$ is
connected. To this end, note that $\hat{A}$ is a local ring so it is
indecomposable.
\end{remark}

\section{Endomorphism rings of local cohomology modules}

Let $R$ be a commutative algebra and $k \subset R$ be a field. We
denote by $\End_k (R)$ the $k$-linear endomorphism ring. The ring of
$k$-linear differential operators $\fD_{R|k} \subseteq \End_k (R)$
generated by the $k$-linear derivations $R \rightarrow R$ and the
multiplications by elements of $R$. By a $\fD_{R|k}$-module we
always mean a left $\fD_{R|k}$-module. The injective ring
homomorphism $R \rightarrow \fD_{R|k}$ that sends $r$ to the map $R
\rightarrow R$ which is the multiplication by $r$, gives $\fD_{R|k}$
a structure of $R$-algebra. Every $\fD_{R|k}$-module $M$ is
automatically an $R$-module via this map. The natural action of
$\fD_{R|k}$ on $R$ makes $R$ a $\fD_{R|k}$-module. If $R=k[[x_1,
\ldots,x_n]]$ is a formal power series ring of $n$ variables $x_1,
\ldots, x_n$ over $k$, then $\fD_{R|k}$ is left and right
Noetherian. Moreover, $\fD_{R|k}$ is a simple ring. Noteworthy, the
local cohomology module $H^{i}_{I}(R),\ i \in \mathbb{Z}$ is a
finitely generated $\fD_{R|k}$-module. For a more advanced
exposition based on differential operators and undefined concepts
the interested reader might consult \cite{Bj}. The similar results
are true whenever $R=k[x_1, \ldots,x_n]$ is a polynomial ring of $n$
variables $x_1, \ldots, x_n$ over $k$ (cf. \cite{Cou}). For a quick
introduction in this topic we refer the reader to \cite{Lyu00}. For
brevity we often write $\fD_R$ for $\fD_{R|k}$ when there is no
ambiguity about the field $k$.

Finally, we bring the following Lemma will be used later.

\begin{lem} \label{B} Let $R$ be a commutative ring containing a field $k$. Let $M$ be both an $R$-module and a
$\fD_R$-module. Then $\Ann_{\fD_R}M =0$ implies $\Ann_R M =0$.
\end{lem}

\begin{proof} Let $r \in \Ann_R M$ be an arbitrary element. As the endomorphism $\varphi_r :R \rightarrow
R$ with $\varphi_r(s) =rs$ for all $s \in R$ is an element of
$\fD_R$ so from $rsM =0$ (for all $s \in R$) we have $\varphi_r(s)M
=0$. That is $\varphi_r(s) \in \Ann_{\fD_R}M =0$, i.e. $rs =
\varphi_r(s) =0$, for all $s \in R$. Hence, we have $r = 0$, as
desired.
\end{proof}

\subsection{Characteristic $0$}

Throughout this subsection $k$ denotes a field of characteristic
zero. Let $R$ be either $k[|x_1, \ldots,x_n|]$, a formal power
series ring or $k[x_1, \ldots,x_n]$ a polynomial ring of $n$
variables $x_1, \ldots, x_n$ over $k$. It is known that the local
cohomology module $H^{i}_{I}(R),\ i \in \mathbb{Z}$ is a holonomic
$\fD_{R}$-module \cite{Lyu93}, i.e. it is a finitely generated
$\fD_{R}$-module with $d(H^{i}_{I}(R))=n$. To this end note that
$d(H^{i}_{I}(R))$ is the so-called Bernstein dimension of
$H^{i}_{I}(R)$ which is by the definition the Krull dimension of the
characteristic variety $\Char (H^{i}_{I}(R))$ (see \cite{Bj} and
\cite{Cou}). In the following result we show that this is not the
case for its endomorphism ring.

For the convenience of the reader let us recall two useful facts.

\begin{fact} \label{Bjork}(cf. \cite{Bj} and \cite{Cou})

\begin{enumerate}
  \item For any nonzero finitely generated $\fD_R$-module $M$, one has $n \leq d(M)
\leq 2n$.
  \item $d(\fD_R)=2n$.
\end{enumerate}
\end{fact}

\begin{prop} \label{A} Let $R$ be as above. Suppose that $H^{i}_{I}(R)\neq 0$, then
 $d(\Hom_{k}(H^{i}_{I}(R),H^{i}_{I}(R)))=2n$. In
  particular $\Hom_{k}(H^{i}_{I}(R),H^{i}_{I}(R))$ is not a
  holonomic $\fD_R$-module.
\end{prop}

\begin{proof} As $H^{i}_{I}(R)$ is a finitely generated $\fD_R$-module, then so is
$\Hom_{k}(H^{i}_{I}(R),H^{i}_{I}(R))$. As $H^{i}_{I}(R)$ is a
non-zero $k$-vector space, it has a non-trivial endomorphism ring.
Consider the homomorphism
\begin{equation}\label{hom}
   \fD_R \stackrel{f}{\longrightarrow} \Hom_{k}(H^{i}_{I}(R),H^{i}_{I}(R))
\end{equation}
of $\fD_R$-modules defined by $f(P)(m) = Pm,\ P \in \fD_R$ and $m
\in H^{i}_{I}(R)$ for all $i \in \mathbb{Z}$. The homomorphism $f$
is injective, as $\fD_R$ is a simple ring. In the light of
(\ref{hom}) and the fact that $d(\fD_R)=2n$, the dimension of
$\Hom_{k}(H^{i}_{I}(R),H^{i}_{I}(R))$ is at least $2n$. Hence, the
assertion follows from Fact (\ref{Bjork}).
\end{proof}

\begin{thrm} \label{annhom}
Let $R$ be either $k[|x_1, \ldots,x_n|]$, a formal power series ring
or $k[x_1, \ldots,x_n]$ a polynomial ring of $n$ variables $x_1,
\ldots, x_n$ over $k$. Suppose that $H^i_I(R)\neq 0$. Then
$$\Ann_{R}(\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R)))=0.$$
\end{thrm}

\begin{proof}
As $\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R))$ is a non zero $\fD_R$-module and
$\fD_R$ is a simple ring one has
$\Ann_{\fD_R}(\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R)))=0$. Now, we are
done by Lemma \ref{B}.
\end{proof}

Now, we may recover a result of Lyubeznik.

\begin{cor} \label{annloc.0} (cf. \cite[Corollary 3.6]{Lyu93})
Let $(R,\fm)$ be a regular local ring containing a field of
characteristic zero. Suppose that $H^i_I(R)\neq 0$. Then
$\Ann_{R}H^{i}_{I}(R)=0$.
\end{cor}

\begin{proof}
Let $\hat{R}$ be the $\fm$-adic completion of $R$. It is known that
$$\Ann_{R}(H^{i}_{I}(R))=
\Ann_{\hat{R}}(H^{i}_{I\hat{R}}(\hat{R}))$$ and $H^i_I (R)\otimes_R
\hat{R} \cong H^i_{I\hat{R}}(\hat{R}) \neq 0$, because of the
faithfully flatness of $\hat{R}$. Then, we may assume that $R$ is
complete. Now we are done by virtue of Cohen's Theorem and Theorem
\ref{annhom}. To this end, let $f \in
\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R)) $ be an arbitrary element and $r
\in \Ann_{R}(H^{i}_{I}(R))$ with $rz=0$, for every $z \in
H^{i}_{I}(R)$. Hence, for every $z \in H^{i}_{I}(R)$ one has
$rf(z)=f(rz)=0$ so $\Ann_{R}(H^{i}_{I}(R)) \subseteq
\Ann_{R}(\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R)))$.
\end{proof}

\subsection{characteristic $p$}

Throughout this subsection $R$ is a commutative Noetherian ring
containing a field of positive characteristic.

\begin{prop} \label{Blick.Ffinite}
Let $(R,\fm)$ be a $F$-finite regular local ring. Then
$\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R))$ is a unite $R[F^e]$-module.
\end{prop}

\begin{proof} As mentioned before (page 5), for a given $i$, $H^{i}_{I}(R)$ is a
unite $R[F^e]$-module. Now we are done by \cite[Corollary 4.10]{Bl}.
\end{proof}

\begin{lem} \label{endo} Let $M$ be an Artinian $R$-module ($R$ is an arbitrary ring with no restriction on its characteristic).
Then there exists an  isomorphism
$$\begin{array}{ll} \ \Phi: \Hom_R (M, M) \rightarrow \Hom_{\widehat{R}} (D(M), D(M)),
\end{array}$$
which sends every $\varphi \in \Hom_R (M, M)$ to $\Phi (\varphi):
\theta \mapsto \theta \circ \varphi$, where $\theta \in D(M)$.

Whenever $R$ is complete and $M$ is a finitely generated $R$-module,
we have again such isomorphism.
\end{lem}

\begin{proof} We prove for the case $M$ is an Artinian module.
For a finite module over a complete local ring the argument is the
same.
 It is clear that the homomorphism $\Phi$ is well-defined. It is enough to show that it is injective and onto.

 ($\Phi$ is injective) Let $0\neq \varphi$ so there exists a
nonzero elemnet $x \in M$ such that $\varphi(x) \neq 0$ in $M$. The
exact sequence $ \ 0 \rightarrow \left\langle  \varphi(x)
\right\rangle \rightarrow M \rightarrow
 (M/\left\langle  \varphi(x) \right\rangle):=L \rightarrow 0$
implies the following short exact sequence $ \ 0 \rightarrow  D(L)
\rightarrow D(M) \rightarrow  D(\left\langle
 \varphi(x) \right\rangle) \rightarrow 0$. It is clear that $D(\left\langle  \varphi(x) \right\rangle) \neq 0$, otherwise $D(L)
\cong D(M)$. Using Matlis duality it implies that $L \cong M$ so
$\varphi(x)=0$, that is contradiction.
   Thus, there exists a nonzero map $\theta: \left\langle
   \varphi(x) \right\rangle \rightarrow E(R/\fm)$. We may extend it to
    the nonzero map $\theta: M \rightarrow E(R/\fm)$ so $\theta \circ \varphi \neq 0$, as required.

 ($\Phi$ is onto) Let $ \psi \in \Hom_{\widehat{R}} (D(M), D(M))$.
We define $\psi$ to send $\theta \in D(M)$ to $\theta \circ
D(\psi)$, where $D(\psi) : D(D(M)) \rightarrow D(D(M))$. As $M$ is
Artinian, $D(D(M)) \cong M$ so $D(\psi): M \rightarrow M$. Then $
\Phi(D(\psi))=\theta \circ D(\psi)= \psi_{\theta}$. Thus, $
\Phi(D(\psi))=\psi$, i.e. $ \Phi$ is onto.
\end{proof}

In the following, the endomorphism ring
$\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R))$ is a unite $R[F^e]$-module
without any need for $R$ to be $F$-finite.

\begin{prop}
Let $(R,\fm)$ be a regular local ring. Suppose that $H^{i}_{I}(R)$
is an Artinian $R$-module. Then
$\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R))$ is a unite $R[F^e]$-module.
\end{prop}

\begin{proof}
By virtue of Lemma \ref{endo} and the fact that $D(H^{i}_{I}(R))$ is
finitely generated one has
$$F^e(\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R))) \cong
\Hom_{R}(F^e(D(H^{i}_{I}(R))),F^e(D(H^{i}_{I}(R))).$$ It follows
from \cite[Lemma 4.12]{Bl} that
$$\Hom_{R}(F^e(D(H^{i}_{I}(R))),F^e(D(H^{i}_{I}(R))) \cong
\Hom_{R}(D(F^e(H^{i}_{I}(R))),D(F^e(H^{i}_{I}(R))).$$ As
$H^{i}_{I}(R)$ is a unite $R[F^e]$-module, then
$$\Hom_{R}(D(F^e(H^{i}_{I}(R))),D(F^e(H^{i}_{I}(R))) \cong
\Hom_{R}(D(H^{i}_{I}(R)),D(H^{i}_{I}(R))).$$ Once again, using Lemma
\ref{endo} the proof is complete.
\end{proof}

Now we are ready to prove the main result of this subsection but
before it we need some preliminaries. It is noteworthy to say that
unite $R[F^e]$-modules have the structure of $\fD$-modules in the
case of characteristic $p>0$ (cf. \cite{Lyu97} and \cite{Bl}).

\begin{definition}
A reduced $F$-finite ring is said to be strongly $F$-regular, if for
all $c \in R$ not in any minimal prime, there exists $q=p^e$ such
that the map $R \rightarrow R^{1/q}$ sending $1 \mapsto c^{1/q}$
splits as an $R$-module homomorphism. Here, $R^{1/q}$ is the
over-ring of $q$th roots of elements in $R$.
\end{definition}

The strongly $F$-regular rings were introduced by Hochster and Huneke
in \cite{Ho-Hu2}. It is known that a regular, $F$-finite ring is
strongly $F$-regular.

\begin{definition}
An $F$-finite ring R has finite F-representation type if there
exists a finite set $\mathcal{S}$ of isomorphism classes of $R$-
modules such that any indecomposable $R$-module summand of
$R^{1/q}$, for any $q = p^e$, is isomorphic to some element of
$\mathcal{S}$.
\end{definition}

Note that an $F$-finite regular ring has finite $F$-representation
type, see \cite{Sm-Va} for more information.

\begin{thrm} \label{charp.endo}
Let $(R,\fm)$ be a complete ring. If $R$ is strongly $F$-regular and
has finite $F$-representation type. Suppose that
$\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R))$ is a unite $R[F^e]$-module.
Then $\Ann_R(\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R)))=0$.
\end{thrm}

\begin{proof}
As $\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R))$ is a unite $R[F^e]$-module,
it then follows from \cite[Proposition 3.6]{Bl} that
$\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R))$ has a $\fD_R$-module
structure. On the other hand by \cite[Theorem 4.2.1]{Sm-Va}, $\fD_R$
is a simple ring. Then one has
$$\Ann_{\fD_R}\Hom_{R}(H^{i}_{I}(R),H^{i}_{I}(R))=0.$$ Now we are done
by Lemma \ref{B}.
\end{proof}

As a consequence of Theorem \ref{charp.endo} we can recover a result
of Huneke-Koh.

\begin{cor} \label{annloc.p}(cf. \cite[Lemma 2.2]{Hu-K})
Let $(R,\fm)$ be a regular local ring containing a field $k$ of
characteristic $p>0$. If $H^{i}_{I}(R)\neq 0$,
then $\Ann_R H^{i}_{I}(R)=0$.
\end{cor}

\begin{proof}
Once again similar to what we have seen in the proof of Theorem
\ref{annloc.0} we may assume that $R$ is a complete local ring. Without loss of generality, we may
assume that $k=R/\fm$. Using a suitable  \textit{gonflement} of $R$
there exists a regular local ring $(S,\fn)$ such that $S/\fn$ is the
algebraic closure of $k$ and $(S,\fn)$ is faithful flat over
$(R,\fm)$, see \cite[Chapter IX, Appendice 2]{Bou}. By faithful flatness of $S$ one has $S/\fm S=S/\fn$.
 By passing over $S$ we may assume that $k$ is an algebraically closed field.
Therefore, $k$ is a perfect field. In the light of Remark
\ref{Hart-Speis} we observe that $R$ is $F$-finite. Now the claim
follows from the Proposition \ref{Blick.Ffinite} and Theorem
\ref{charp.endo}.
\end{proof}

\proof[Acknowledgements] I am deeply grateful to the referee for
his/her careful reading of the manuscript, appropriate and constructive suggestions to improve the paper.
 This project is based on a question asked by Professor
Josep \`{A}lvarez-Montaner on the comparison of $\Fdepth$ and
$\fgrade$ when the author was visiting the Universitat
Polit\`{e}cnica de Catalunya, in September 2013. Hereby, I would
like to express my thanks to the Universitat Polit\`{e}cnica and
specially, to Josep, for their support and warm hospitality. I am
also strongly indebted with Josep for several fruitful discussions.
I would like to mentioned that when the author was finishing this
project, he started a joint project with Alberto F. Boix, and gave a
characteristic free proof of the result of Huneke-Koh and Lyubeznik
on the annihilator of local cohomology modules.

config: arxiv


TITLE: complex torus has topological genus one
QUESTION [4 upvotes]: could any one give me a hint how to show a complex torus has topological genus one by constructing an explicit homeomorphism to $S^1\times S^1$?
Complex Torus: $\mathbb{C}/L$, where $L=\{\mathbb{Z}\omega_1+\mathbb{Z}\omega_2\}$.
Thank you.

REPLY [7 votes]: From a topological viewpoint $\Bbb C/L$ is indistinguishable from $\Bbb R^2/\Bbb Z^2$.
But then it should be clear that
$$
\frac{\Bbb R^2}{\Bbb Z^2}\simeq\frac{\Bbb R}{\Bbb Z}\times\frac{\Bbb R}{\Bbb Z}
\simeq S^1\times S^1.
$$


text:
\section{Preliminaries}
\label{sec:2}
There are many approaches to formulate ``Discrete Surface Theory'' on different motivations.
In \cite{Kotani-Naito-Omori}, a discrete surface is defined as a trivalent graph in $\R^3$
so that the tangent space is assigned at each vertex as the unique plane
determined by the three nearest neighbor vertices.
We briefly review their discussions and results.

\subsection{Discrete surface in $\R^3$ and their curvatures}
Let $X=(V,E)$ be a trivalent topological graph,
where $V$ denotes the set of vertices, $E$ denotes the set of edges.
The origin and the terminus of an edge $e$ are denoted by $o(e)$ and $t(e)$, respectively.
For any $v\in V$, $E_{v}$ refers to the set of edges that emerge from $v$.
\par
It is convenient to introduce a notion of a ``face'' although $X$ is a discrete object.
For a circuit, a closed simple curve without self-intersections,
we define a \emph{face} $f$ as an ordered set
$\{ v_0, \ldots, v_{n-1} \}$ of vertices in the circuit
and the set $F$ of \emph{faces}.
\par
Given a trivalent topological graph $X$,
we define a \emph{discrete surface} $M$ in $\R^3$
by a piecewise linear map
$\Phi\colon X\to\R^3$ with $M = \Phi(X)$.
Here we mean by ``piecewise linear'',
the image of each edge $e = (v_{0}, v_{1})$
is given by the line segment connecting two vertices
$\Phi (v_{0})$ and $\Phi(v_{1})$.
\begin{definition}[Discrete Surface]
  An injective piecewise linear realization
  $\Phi\colon X \to \R^3$
  of a trivalent graph $X = (V,E)$
  is said to be a \textnormal{discrete surface} in $\R^3$, if
  \begin{enumerate}
  \item
    for all $v\in V$ at least two elements of $\{\Phi(e) \mid e\in E_v\}$
    are linearly independent in $\R^3$,
  \item
    $\Phi(X)$ is locally oriented, that is, the order of the three edges
    is assumed to be assigned to each vertex of $X$.
  \end{enumerate}
\end{definition}

Let
$\Phi \colon X \to M=(\MV,\ME)\subseteq \R^3$
be a discrete surface, in which
$\MV = \Phi(V)$ denotes the set of vertices of $M$,
$\ME = \Phi(E)$ denotes the set of edges of $M$
and
$\MF$ denotes the set of polygonal faces of $M$.
In particular,
We do not assume the image of a face lie on a plane or a continuous surface.
\par
As you see, we consider $X$ and $M$ discrete sets
and often identify them with the sets of vertices $V$ and $\MV$, respectively.
\par
Let
$\v{v}=\Phi(v)$
and
$\v{e}=\Phi(e)$
as the corresponding vertex and edge in $M$ for $v\in V$ and $e \in E$. Let
$E_{v}=\{e_1,e_2,e_3\}$ be the oriented edges at $v$
and let
$v_i$ be the tail vertex of each $e_i$.
The \emph{tangent plane} $T_{\v{v}}M$
is defined as the plane with $n(\v{v})$ as its \emph{unit normal vector} $n(\v{v})$ at $\v{v}\in M$ is given by
\begin{equation*}
  n(\v{v})=
  \frac{\v{e}_{1} \times \v{e}_{2} + \v{e}_{2} \times \v{e}_{3} + \v{e}_{3} \times \v{e}_{1}}
  {|\v{e}_{1} \times \v{e}_{2} + \v{e}_{2} \times \v{e}_{3} + \v{e}_{3} \times \v{e}_{1}|},
  \quad (\v{e}_i = \Phi(e_i) ).
\end{equation*}
It is perpendicular to the triangle $\triangle(\v{v}_{1},\v{v}_{2},\v{v}_{3})$ with $\v{v}_i = \Phi(v_i)$.
\par

The \emph{first} and \emph{second fundamental form} of
$\v{v}\in M$ are given by,
respectively
\begin{equation*}
  \begin{aligned}
    \first(\v{v})=
    &\begin{pmatrix}
      \langle \v{e}_{2} - \v{e}_{1}, \v{e}_{2} - \v{e}_{1} \rangle,
      &\langle \v{e}_{2} - \v{e}_{1}, \v{e}_{3} - \v{e}_{1} \rangle\\
      \langle \v{e}_{3} - \v{e}_{1}, \v{e}_{2} - \v{e}_{1} \rangle,
      &\langle \v{e}_{3} - \v{e}_{1}, \v{e}_{3} - \v{e}_{1} \rangle
    \end{pmatrix},\\
    \second(\v{v})=
    &\begin{pmatrix}
      - \langle \v{e}_{2} - \v{e}_{1}, \v{n}_{2} - \v{n}_{1} \rangle,
      & - \langle \v{e}_{2} - \v{e}_{1}, \v{n}_{3} - \v{n}_{1} \rangle \\
      - \langle \v{e}_{3} - \v{e}_{1}, \v{n}_{2} - \v{n}_{1} \rangle,
      & - \langle \v{e}_{3} - \v{e}_{1}, \v{n}_{3} - \v{n}_{1} \rangle
    \end{pmatrix}.
  \end{aligned}
\end{equation*}
where $\v{n}_{i}=n(\v{v}_{i})$, $i=1,\,2,\,3$.
Note that $\second(\v{v})$ is not necessarily symmetric.
\par
\begin{definition}[Curvatures]
  Let
  $\Phi\colon X \to M$ be a discrete surface.
  Then for each vertex $\v{v}\in M$,
  the
  \textnormal{Gauss curvature $K(\v{v})$}
  and
  \textnormal{mean curvature $H(\v{v})$}
  are represented as follows, respectively
  \begin{equation}
    \begin{aligned}
      &K(\v{v}) = \det[\first(\v{v})^{-1}\second(\v{v})],\\
      &H(\v{v}) = \frac{1}{2} {\tr}[\first(\v{v})^{-1}\second(\v{v})].
    \end{aligned}
  \end{equation}
\end{definition}
\begin{definition}[Discrete Minimal Surface]
  A discrete surface $\Phi\colon X \to M$
  is called a \textnormal{discrete minimal surface}
  if its mean curvature vanishes at each vertex.
\end{definition}

\subsection{Discrete harmonic and minimal surfaces}
Given a trivalent graph $X$ with weight
$m\colon E \to \R_{+}$, satisfying
$m(e) = m(\bar{e})$,
where $\bar{e}$ is the reverse edge of $e$.
\par
Let $\Phi\colon X \to M$ be a discrete surface in $\E^3$.
For a finite subgraph $X' = (V', E') \subset X$,
we define
the \emph{Dirichlet energy}
$E_{D}(\Phi |_{X'})$ given as the sum of square norm of all edges,
i.e.,
\begin{equation*}
  E_{D}(\Phi |_{X'}) = \sum_{e\in E'} m(e)|\Phi(e)|^2.
\end{equation*}
A realization of a graph $X$
that minimizes the Dirichlet energy defined above for arbitrary finite subgraphs
is called as a \emph{harmonic realization} \cite{Kotani-Sunada}
or an \emph{equilibrium placement} \cite{Delgado-Friedrichs}.
\begin{prop}[Harmonic Discrete Surface (\!\!{\cite[Definition 3.15]{Kotani-Naito-Omori}})]
  A discrete surface $\Phi\colon X  \to \R^3$
  is harmonic with respect to the weight $m$, when it satisfies
  \begin{equation}\label{balancing}
    m(e_{v,1})\Phi(e_{v,1}) + m(e_{v,2})\Phi(e_{v,2}) + m(e_{v,3})\Phi(e_{v,3}) = 0,
  \end{equation}
  for any $v\in V$,  and $E_{v}=\{e_{v,1},e_{v,2},e_{v,3}\}$.
\end{prop}
The equation (\ref{balancing}) is called the \emph{balancing condition}, and plays an important role later on.
\begin{prop}[\!\!{\cite[Proposition 3.16]{Kotani-Naito-Omori}}]
  \label{prop-a}
  Let
  $F\colon X \to \R^3$ be a discrete harmonic surface with respect to the weight $m$,
  for $v\in V$ and $E_{v}=\{e_1,e_2,e_3\}$,
  the Gauss curvature $K(\v{v})$
  and
  the mean curvature $H(\v{v})$
  are respectively given by
  \begin{equation*}
    \begin{aligned}
      K(\v{v})
      & = -\frac{m_1 + m_2 + m_3}{2A(x)^2}
      \sum_{i,j,k} \frac{\langle \v{e}_{i},\v{n}_{j} \rangle \langle \v{e}_{j},\v{n}_{i}\rangle}{m_j},\\
      H(\v{v})
      & = \frac{m_1 + m_2 + m_3}{2A(x)^2}
      \sum_{i,j,k} \frac{\langle \v{e}_{i},\v{e}_{j}\rangle(\langle \v{e}_{i},\v{n}_{j}\rangle
        + \langle \v{e}_{j},\v{n}_{i} \rangle)}{m_j},
    \end{aligned}
  \end{equation*}
  where $m_i=m(e_i)$, $(i,\,j,\,k)$ is the alternate of $(1,\,2,\,3)$.
\end{prop}
By Proposition \ref{prop-a}, we notice that
a discrete harmonic surface may not be minimal.
The following theorem provides a sufficient condition
for a discrete harmonic surface
which has vanished mean curvature at each vertex.

\begin{thm}[\!\!{\cite[Theorem 3.17]{Kotani-Naito-Omori}}]
  A discrete harmonic surface
  $\Phi\colon X  \to \R^3$ is minimal
  if for any $v\in V$
  and
  $E_{v}=\{e_1,e_2,e_3\}$
  \begin{equation*}
    \langle \Phi(e_1), \Phi(e_2) \rangle
    + \langle \Phi(e_2), \Phi(e_3)\rangle
    + \langle \Phi(e_3), \Phi(e_1)\rangle = 0.
  \end{equation*}
  In particular,
  if $m\colon E \to \R_{+}$ is constant,
  the equation above is equivalent to
  \begin{equation*}
    |\Phi(e_1)| = |\Phi(e_2)| = |\Phi(e_3)|.
  \end{equation*}
\end{thm}
\par
In the present paper, we always use $m \equiv 1$ from now on.