11institutetext: Pembroke College, Cambridge, CB2 1RF 22institutetext: Apoha
Ltd, London, Acklam Rd, W10 5JJ 33institutetext: Department of Chemical
Engineering, University of Cambridge, CB 30 AS, UK

# A complexity perspective on fluid mechanics

Saksham Sharma 1133<EMAIL_ADDRESS><EMAIL_ADDRESS>Giulia Marcucci 22
Adnan Mahmud 33

###### Abstract

This article attempts to use the ideas from the field of complexity sciences
to revisit the classical field of fluid mechanics. For almost a century, the
mathematical self-consistency of Navier-Stokes equations has remained elusive
to the community of functional analysts, who posed the Navier-Stokes problem
as one of the seven millennium problems in the dawn of 21st century. This
article attempts to trace back the historical developments of fluid mechanics
as a discipline and explain the consequences of not rationalising one of the
commonly agreed upon tenets - continuum hypothesis \- in the community. The
article argues that ‘fluids’ can be treated as ‘emergent’ in nature, in that
the atoms and molecules in the nanometre length scale can likely be correlated
with the continuum physics at the microscale. If this is the case, then one
might start trying to find a theoretical framework that models the emergence
of fluids from atoms, effectively solving the multi-scale problem using a
single abstract framework. Cantor set with layers $N$ (N can have up to two
orders of magnitude) is presented as a potential contender (analytical
framework) for connecting the energy in molecular level $C_{1}$ at length
scale $l_{cut}$ to the energy at continuum level $C_{N}$ with length scale
$L$. Apart from fluid mechanics, Cantor set is shown to represent the
conceptual understanding of VLSI hardware design ($N=5$). Apart from Cantor
set, an experimental technique of architecting metafluids is also shown to
solve emergence experimentally (i.e. connect physics at specific lower scales
to higher scales).

###### keywords:

fluid mechanics; Navier-Stokes equations; complexity

##  1 Introduction

Matter, as we see it, with naked eyes or with external aids (to name a few,
optical lenses, image reconstructions from electromagnetic radiations) can be
modeled as existing in three-dimensional space ($\mathcal{R}^{3}$) and
amenable to differential calculus treatment. This article focuses on fluid
particularly as the matter and reflects on the analytical treatment of its
dynamics.

The primary question of interest is: if one is given a continuum 111 of fluid
in $\mathcal{R}^{3}$, which is external to the observer, does there exist an
abstract framework onto which the physics of fluids can be encoded? Is the
mathematical framework, built off orthogonal coordinate systems in
$\mathcal{R}^{3}$, self-consistent to model the dynamics of fluid? More
fundamental question could be: what is “fluid” fundamentally composed of?
As philosophical and fundamental these questions might sound, implicit answers
to them, commonly accepted in scientific community, have dictated the course
of scientific research for past almost two centuries. Starting with Euler, it
was conceptualised that the ‘fluid’ is composed of tiny parcels of continuum
(of length scale significantly higher - 2 or more orders of magnitude - than
molecular length scale) such that the Newton’s second laws of motion can be
applied in point-wise fashion inside the parcel [17, 51]. Such a treatment led
to the development of ‘hydraulic engineering’, ‘fluid mechanics’, ‘continuum
mechanics’ as separate disciplines (that we see now) and the use of calculus
in the continua of fluid became more and more accepted [27].

What is seen as modern ‘fluid mechanics’ discipline is equally the result of
the ideas which were not accepted by the academic community over generations.
A prime example is the ‘Laplace’s demon’ which was an optimistic hope of
Pierre-Simon Laplace to predict the future of a physical system given its past
and the classical laws that if obeys [36]. This hope died many deaths with the
discovery of laws of thermodynamics, quantum mechanics, and with the
construction of Turing machine. To give a sample argument against Laplace’s
demon, Josef Rukavicka [38] and David Wolpert [52] noted that the construction
of Turing machine and inference devices which perform observation and
prediction of the natural world would lead to a fundamental mathematical
structure in its logic, called “halting problem”. This problem prevents the
existence of a master rule or law that tells when the output of the device
would be fixed and determinate, and hence the Laplace’s demon is bound to
fail. Even after the failure of Laplace’s demon, the quest shifted towards
connecting the continuum (macroscopic) laws of physics to atomic/molecular
contributions. French physicists and mathematicians in the 18th century hugely
debated the molecular origins of continuum forces, as discussed by Darrigol
[14]. Navier incorporated the molecular forces that are relevant only during
the deformation of a fluid, Poisson, on the other hand, summed up the
molecular forces by the surrounding molecules acting on a given molecule.
Cauchy’s formulation was quite similar to Poisson, which is sometimes referred
to as the ‘Cauchy-Poisson’ theory. Later on, Poisson offered a fluid theory of
motion by treating a fluid like a solid, which experiences stresses during its
motion so that the fluid stress is related to the its rate of deformation,
which went on to become Navier-Stokes equation with additional pressure
gradient term.

The formulation of Navier-Stokes equations as the descriptor of motion of
fluids established a boundary between the disciplines: one disciplines is
focused on analysing fluids at a continuum level and the other at an
atomic/molecular level. However, it does not mean that the two fields are not
connected with one another. To give an example, for an ideal gas $G$ at a
thermodynamic equilibrium, if $G$ has temperature $T$ and the molecules that
combine to form $G$ have average kinetic energy $E$, then $E=k_{B}TN_{DOF}$.
Another example, for fluid $F$ with surface tension $\gamma$ composed of
molecules in two layers separated at a distance $d_{0}$, is that the surface
energy can be defined as: $\gamma=A/d_{0}^{6}$ where $d_{0}$ is the
intermolecular distance between two layers. This means that it is possible, in
some cases, to find laws that relation the dynamics at atomic level with the
dynamics at continuum level. However, there still is a strong boundary in the
level of treatment between continuum mechanics and atomic/molecular physics:
in modern era of science, this led to distinction between fluid mechanics and
statistical physics as separate disciplines [25, 26].

Though development of distinct disciplines has led to many development in both
fronts, it has also led to many unresolved problems in each discipline that
has captured attention of physicists for a century or more. For continuum
level fluid mechanics, the biggest unresolved problem is the Navier-Stokes
regularity problem [19]. In a loose sense, the problem is that of finding a
proof that the solutions to Navier-Stokes equations either blow up or remain
finite after a certain finite time given a set of initial datum. On the other
hand, for statistical physics, one of the unresolved problem is finding an
analytical treatment of the dynamics of a large N-particle system, also
described by BBGKY set of equations [21]. It is interesting to note that both
problems suffer from a solution only because the problem is restricted at the
first place to a certain scale of analysis: ‘continuum’ for Navier-Stokes
problem and ‘atomic/molecular’ for the BBGKY problem. It would be ideal to
have an abstract framework which connects any arbitrary two different physical
scales, and bringing more unified picture of the physical problem concerned.

Even without the absence of such a framework, much of the past many decades of
research in mathematical fluid dynamics has been influenced by the idea of
complexity/emergence. Towards the end of 20th century, Phillip W. Anderson in
his much celebrated essay “More is Different” [2] argued the twin difficulties
of scale and complexity, when attempting to explain a natural phenomenon in
terms of known fundamental laws. In his own words, “the ability to reduce
everything to simple fundamental laws does not imply the ability to start from
those laws and reconstruct the universe”, meaning that at every new scale a
new set of laws emerge. This scientific philosophy of “complexity science” has
pervaded numerous disciplines over the past 50 years, such as origin of life,
nonlinear synchronisation, self-assembly in active systems, ants and termites
self-organising colonies, and network science, as recently summarised in a
recent article [47] by eight eminent complexity scientists. On fluids,
Anderson wrote in his essay that, “We have already excluded the apparently
unsymmetric cases of liquids, gases, and glasses. (In any real sense they are
more symmetric.)”. The contention here is that in a practical sense, fluids
are quite predictable (upon the application of continuum hypothesis) hence,
they can be assumed to be symmetric. However, this symmetry holds true as long
as the laws of continuum mechanics are attempted to understand at a certain
length scale. Any fluid related phenomenon breaks its symmetry and becomes
difficult to be modelled when it is seen emergent in nature and one attempts
to find a model that encompasses multiple scales beginning from
atomic/molecular contributions.

Apart from emergence, there are few more perspectives posed in the past few
decades that challenges our classical treatment of ‘fluids’. Primary notion
that is being challenged is to see mathematical physics (framework to model
fluids) as fundamentally continuum in nature; it is not the case in
mathematics because of the invent of Turing machines [18] and not in the
physics of fluids because the basic units of fluids are discrete set of
interacting atoms. There is even more convincing argument that the physics is
fundamentally information-theoretic in origin. The primary argument for this
is the choice of doing an experiment that yields an observation which
eventually lets one record what was not measured by the equipment before. In
nutshell, the decision to perform an experiment yields an information about
the ‘fluid’ which then yields the governing equations or algorithm describing
the physics of the experiment. Such an observer-participatory process to yield
information and then the physics can offer another way of looking at ‘fluid
mechanics’, where instead of treating ‘fluid’ parcels as fundamentally
continuum of numbers, one treats them as fundamentally ‘bits’ and discrete
chunks of information in nature. If one accepts these arguments, then one is
liable to argue ‘fluid mechanics’ as somewhat digital in nature. For past few
decades, numerous works are pointing towards the creation of ‘Digital Fluid
Mechanics (DFM)’ as a potential discipline which has a potential to thrive
amongst existing scientific communities with variety of theoretical,
experimental, empirical, and numerical approaches. It should be noted that
while machine learning methods can be seen as contenders for empirical and
numerical approaches within DFM context, experimental approaches remain the
same as conceived before in traditional fluid mechanics research, and theory
of neural networks can be considered as the theoretical framework for DFM in
this regard [3].

Keeping aside the potential creation of DFM as a discipline and unsolved
problems that fluid mechanics and statistical physics are plagued with, there
are many experimental observations using fluids that can be discussed as
classroom examples of emergence. For example, Couette flow occuring when a
fluid is placed between two rotational cylinders at different velocities gives
rise to the roll of vortices when the velocity gradient is above a threshold.
Another example is the Benard cells formed when a layer of fluid between two
horizontal plates is heated from below to lead to the emergence of convection
rolls, in the form of multiple attractors - the end result is dependent upon
chance fluctuations and is often unpredictable [5]. Over the past one decade,
there has been an upsurge in the discovery of nonequilibrium phenomenon
centred around the idea of emergence [40]. To give an example, Palacci et al.
(2013) reported that synthetic photoactivated colloidal particles have a
dynamic assembly in the form of periodic crystals that results from the
competition between the self-propulsion of particles and attractive
interactions caused by the osmotic and phoretic effects [35].

This main result presented in this article is an abstract framework - a Cantor
set - that can be used to connect atomic/molecule scale to continuum scale and
present a more unified approach to understanding the dynamics of fluids. The
ideas from the field of complexity sciences are introduced in Section 2, along
side justifying the need for a framework that can be used to model emergent
phenomenon. By drawing focus on fluid mechanics, the unsolved Navier-Stokes
regularity problem (discussed in Section 3) is used as a case example which
has been attempted in the past by functional analysis communities to be solved
using traditional mathematical methods (assuming fluid is composed of
continuum at the core) discussed in Section 4. New ideas on potentially
solving the Navier-Stokes problem is discussed in Section 6. Section 7 is on
Cantor set which can be used to model ‘fluids’ as discrete, information-
theoretic and computational in nature composed of atoms/molecules. It is being
shown through an example calculation that it is possible to model the
emergence of ‘fluids’ from ‘atoms’ by modeling the interaction between atoms
composed in different discrete Cantor sets as ‘rotary logic gates’. Section 8
is on the potential usage of “Cantor sets” in VLSI hardware design problem (to
show its appeal outside the field of physics). Section 9 is on the existing
engineering techniques of solving emergent problems without using an abstract
framework using physical learning [45] and architecting metamaterials [6].

##  2 Revisiting complexity sciences

A system is a collection of interacting or interdependent components that
operate in accordance with a set of rules to produce a holistic entity.
Conversely to the conventional definition, a system with reference to its
individual components, complexity or complex system defies component-based
characterization since no component is independent of the behaviour of the
other components. These components establish networks of interactions (often
with only a few components involved in numerous interactions) and are capable
of generating novel information. Therefore, it is not feasible to strictly
formulate the collection’s properties based just on knowledge of its
components. Hence, the investigation of complex science requires, in general,
a new mathematical framework and, in specific, a new sets of measurement
metrics. This section of the paper analyses the types of measurement metrics
available in the literature for complex systems.
Complex science derives its multidisciplinary nature from the premise that
shared properties connect systems across fields and thus justifying the search
for universal sets of measuring metrics (modelling techniques in general)
applicable to complex systems stemming from all scientific and professional
domains covering physics, biology, medicine, engineering, ecology, social
sciences, finance, business, management, politics, psychology, anthropology,
and more. Due to the multidisciplinary nature of complex science and
ubiquitous characteristics such as emergence 222The characteristics of a
system that are not evident from its components in isolation, but arise from
their interactions, dependencies, or linkages when brought together in a
system. , nonlinearity 333A system in which a change in the input size does
not result in a proportional change in the output size., and adaptation 444The
ability to adapt and gain knowledge from experience. make defining measuring
metrics for complex science discipline-wise is not the optimised method.
Moreover, relatively recent work on the analytical framework of complex
networks has resulted in a considerable reaffirmation of similarities between
complex systems in several scientific fields [1].

If the measurement metrics are sectioned from the perspective of the
definition of a complex system, then the majority of existing complexity
metrics fall into two categories [43]:

1\. Complexity as Randomness: metrics in this category try to capture the
randomness, information content, or description length of a system or process,
with random processes being the most complicated since they defy compression
the most. This is because compression relies on exploiting a skewed
distribution in the data, and randomness and skewness are inversely
proportional to each other.

2\. Complexity as Structure and Information: metrics in this category
conceptualise complexity as distinct from randomness. In this context, complex
systems include a great deal of structure or information, frequently spanning
numerous temporal and spatial scales. Within this set of metrics, very complex
systems fall halfway between highly ordered (regular) and highly disordered
systems (random).

According to Seth Lloyd [30], contemporary researchers were asking the same
questions regarding the complexity of their respective research subjects;
nonetheless, the solutions they provided for how to assess complexity are
quite comparable. Therefore, if the measurement metric is sectioned from the
perspective of the questions researchers frequently ask to quantify the
complexity, then the metrics can be sectioned into four main categories:

  1. 1.

Metrics aiming to measure the difficulty in creating a complex system.

  2. 2.

Metrics aiming to measure the difficulty in creating a complex system.

  3. 3.

Metrics aiming to measure the degree of organisation of a complex system. This
criterion may further be divided up into two quantities:

    1. (a)

Effective Complexity: metrics aiming to measure the difficulty in describing
the organisational structure of a complex system.

    2. (b)

Mutual Information: metrics aiming to evaluate the information shared between
the parts of a complex system as the result of this organizational structure.

  4. 4.

Non- quantitative metrics.

A non-exhaustive list of metrics, sectioned with the latter method is
visualised in Figure 1.

Figure 1: A non-exhaustive list of measuring metrics for complex system [30].

##  3 Navier-Stokes regularity problem

Navier-Stokes equations (NSE) are used to model the dynamics of a given fluid
given by

$\partial_{t}\vec{u}(t,x)=\nu\Delta\vec{u}-\sum_{i=1}^{3}u_{i}\partial_{i}\vec{u}-\vec{\nabla}p+\vec{f}$
(1) $\nabla.\vec{u}=\sum^{3}_{i=1}\partial_{i}u_{i}=0$ (2)

where $\vec{u}(t,x)$ is the velocity vector field in time $t$ and space $x$,
$\nu$ is the kinematic viscosity, $\vec{f}$ is the external body force. The
initial conditions of the velocity vector field are given as

$\vec{u}(0,x)=\vec{u}_{0}(x)$ (3)

where Eq. 1 and Eq. 2 are defined for $t\geq 0$ and $x\in\mathcal{R}^{n}$.

The problem statement posed by the Clay Mathematical Institute [19] is to
prove either that the solutions to Eq. 1-2 remain smooth or they become
singular for a given set of initial datum $\vec{u}_{0}(x)$ and after a certain
finite-time $T^{*}$.

##  4 Efforts to solve the problem

Attempts to find solutions to the Navier-Stokes equations (NSE) can be traced
back to the first half of the 20th century. In 1911, Oseen [34] used an
explicit tensor $\mathcal{O}_{\nu}$, called the Oseen tensor to convert NSE
into an integro-differential equation and find the solution

$\begin{split}\vec{u}(t,x)=&\int_{\mathcal{R}^{3}}W_{\nu}(t,x-y)\vec{u}_{0}(y)dy+\int_{0}^{t}\int_{\mathcal{R}^{3}}\mathcal{O}_{nu}(t-s,x-y)\\\
&\left(\vec{f}(s,y))-\sum_{i=1}^{3}u_{i}(s,y)\partial_{i}\vec{u}(s,y)\right)dyds\end{split}$
(4)

where $W_{\nu}$ is the heat kernel associated to heat equation
$\partial_{t}T=\nu\Delta T$ where $T$ is temperature. While Oseen derived the
solution of (4) for the time interval $[0,T]$, the question of what can be
said about possible blowup at $z_{0}$ 555A function $f(z)$ is said to blowup
when $f(z)\to\infty$ at $z=z_{0}$.  of such solutions was left unanswered. In
1934, Leray [28] used an estimate of the energy
$\int_{\mathcal{R}^{3}}|\vec{u}(t,x)|^{2}dx$ to find the conditions that
prevent the blowup. After realising that the $L^{2}$ norm 666An $L^{2}$ norm
or the Euclidean norm of a function gives the length of a vector
$\mathbf{x}=(x_{1},x_{2},...,x_{n})$ as:
$||\mathbf{x}||_{2}=\sqrt{x_{1}^{2}+x_{2}^{2}+..+x_{n}^{2}}$, for a manifold
$\mathcal{R}^{n}$ is not controllable enough to prevent blowup, Leray
introduced a new kind (in fact, concept) of solutions, called weak solutions.
These solutions have generalised derivatives defined in a Sobolev space
$H^{1}$ 777Sobolev space is the vector space of functions defined by a norm
which is the combination of $L^{p}$ norms and the derivatives of $L^{p}$ norms
to a given order. Effectively, Leray proved that for an initial datum
$u_{0}\in L^{2}$, a global weak solution $\vec{u}(t,x)\in
L_{t}^{\infty}L_{x}^{2}$ exists with an additionally regularity
$L_{t}^{2}H_{x}^{1}$. Ever since such a formulation, the fundamental problem
of proving uniqueness of the weak solution and that of globalness of strong
solutions has remained unsolved.

Another parameter of paramount importance in NSE analysis is vorticity,
$\omega$, given by $\nabla\times u$. In terms of $\omega$, NSE can be written
as

$\partial_{t}\omega+u\cdot\nabla\omega-\nu\Delta\omega=\omega\cdot\nabla u$
(5)

where the RHS term is called the stretching term. For 2D flow, the RHS is zero
because of right angle between the vorticity vector and the flow field plane.
As a result, vorticity is controlled in magntiude: if $\nu=0$, vorticity
distribution is conserved and $L^{p}$ norms do not change in time; if $\nu\neq
0$, the norms are non-increasing functions of time. This leads to the regular
solutions of NSE and Euler equations, in 2D case, without any formation of
singularities. In 3D NSE, the direction of the vorticity vector has an
influence on the possibility of the formation of singularities. Constantin,
Fefferman in 1993 [12] proved that if the angle between unit vorticity vectors
at positions $x$ and $y$ is $\varphi(x,y,t)$, then for given constants
$\Omega$ and $\rho$, if magnitude of the vorticity at both locations is above
$\Omega$, then

$\left|\sin\varphi(x,y,t)\right|\leq\frac{|x-y|}{\rho}$ (6)

where the above equality holds true at high values of vorticity and the
direction of vorticity is coherent so that the singularities can not form. The
physical reason behind this result is the local alignment (vortex tubes being
parallel or antiparallel) which regularises the nonlinearity in NSE.

(1) can be collapsed into a seminlinear heat equation

$\partial_{t}u+B(u,u)=\nu\Delta u$ (7)

where $B(u,v)$ is the nonlinear quadratic function and a bilinear operator
with the general form

$\displaystyle B(u,v):=\frac{1}{2}\mathcal{P}((u.\nabla)v+(v.\nabla)u)$ (8a)
$\displaystyle\langle B(u,u),u\rangle=0$ (8b)

and $\mathcal{P}$ is the orthogonal projection ontp divergence-free vector
fields, also called Leray projection. Dimensional analysis heuristics can be
performed to suggest that NSE might not have much global regularity
properties, and are rather supercritical 888Supercritical differential
equations have a property that they blowup in finite-time. In hydrodynamics
literature, it is used in the sense of supercritical pitchfork or hopf
bifurcation. in nature. Supposing: $\nu=1$; $u(x,t)$ concentrated at a
frequency scale, $N(t)$, and amplitude, $A(t)$, gives

$\displaystyle\nu\Delta u\approx N(t)^{2}A(t)$ (9a) $\displaystyle
B(u,u)=\mathcal{P}(u.\nabla u)\approx N(t)A(t)^{2}$ (9b)

such that the viscosity dominates nonlinear effects when $A(t)\ll N(t)$ and
vice-versa. A power-law model of nonlinear dynamics: $A(t)\approx
N(t)^{\theta}$ for $\theta>1$ gives

$\displaystyle\partial_{t}A(t)\approx O(N(t)A(t)^{2})\approx
O(N(t)^{1+2\theta})$ (10a) $\displaystyle\partial_{t}N(t)\approx
O(N(t)^{1+\theta})$ (10b)

which alludes that if frequency $N(t)$ is of increasing nature, then the
blowup at a finite-time could happen. Now, if the solution $u$ is assumed to
exist in a spatial set of intermittency 999In the turbulent fluid flows,
intermittency is defined as the pause and start of velocity flow signals. The
recorded intermittent signals of the vorticity field $\omega$ is given in [31,
see fig. 2(a)]. dimension $0\leq\alpha\leq 3$, it should be supported in a
ball of volume $N(t)^{-3+\alpha}$. As a result, the energy
$\lim_{\mathcal{R}^{3}}|u(t,x)|^{2}$ is of the order $\approx
A(t)^{2}N(t)^{-3+\alpha}\approx N(t)^{-3+\alpha+2\theta}$. The energy identity
suggests that the exponent of $N(t)$ need to be negative as $N(t)\to\infty$
(blowup), giving the following constraint

$\theta\leq\frac{3}{2}-\frac{\alpha}{2}$ (11)

alongside the constraint $\theta>1$ for nonlinear effects to dominate. While
high intermittency $\alpha\geq 1$ is not possible for nonlinear effects to
dominate [8], if $\alpha<1$, then blowup is potentially possible. An extreme
blowup is possible at $\theta=3/2$ and $\alpha=0$, which means that the
solution $u$ is concentrated in a single ball of size $1/N(t)$. Interestingly,
there is no result available on the proof that finite time blowup for such a
case as $t\to T_{*}<\infty$ is not possible.
A dyadic shell model is often used to model the case: $\theta=3/2,\alpha=0$.
Introduced in Katz, Pavlovic [23], the model is given by

$\partial_{t}X_{n}=\lambda^{n-1}X_{n-1}^{2}-\lambda^{n}X_{n}X_{n+1}-\lambda^{4n/5}X_{n}$
(12)

for a constant $\lambda>0$, $X_{n}$ is the energy of fluid at scale $n-1$
which gets diffused to scale $n$, and the last term in the RHS is the
dissipation term. The total energy of the fluid is given by
$\sum_{n}X_{n}^{2}$. The analog of blowup in this model is when energy is
moved to higher values of $n$, ideally $n\to\infty$. At models corresponding
to five and higher dimensions, Cheskidov in 2018 [11] showed that when the 4/5
exponent is replaced with an exponent below 2/3, the equations can blowup.
However, in 2011, Barbata, Morandin, Romito [4] proved for a large class of
intial datum that the dyadic shell model admitted global regular solutions
when $X_{n}$ exhibited exponential decay as $n\to\infty$, or in other words,
are smooth. This was because the energy dispersed too quickly into a broad
spectrum of several higher frequency scales, where each frequency mode was
activated with amplitude scale such that the effects of viscosity remained
stronger in comparison to the nonlinear effects, to prevent blowup and not let
the solutions dissipate to infinitely larger frequency scales.

One type of scaling of NSE is given as

$v(x,t)\mapsto v_{\lambda}(x,t)=\lambda v(\lambda x,\lambda^{2}t)$ (13)

and there are various partial regularity results known for this. One of the
most prominent one is called Ladyzhenskaya-Prodi-Serrin conditions ([24],
[37], [39]), which states that if the Leray weak solution lies in
$L^{p}_{t}L^{q}_{x}$ with $2/p+3/q\leq 1$, then the solution is unique and
smooth in positive time. The endpoint in the above equality for $p=\infty$ and
$q=3$, given for $L^{\infty}_{t}L^{3}_{x}$, was proved by Escauriaza-
Seregin-S̆verák [16]. It should be noted that while the natural scaling of
weak solutions has a regularity for the condition, $2/p+3/q=3/2$, the energy
equality holds in NSE with an additional regularity for the condition,
$2/p+3/q=5/4$ [41]. This means that there is a gap open between the natural
scaling of the equations and the kinetic energy, which can lead to non-
uniqueness of weak solutions. In fact, this nonuniqueness was hinted by
Jia-S̆verák [22], where the authors proved that in the class
$L_{t}^{\infty}L_{x}^{3,\infty}$, Leray solutions are nonunique if a certain
spectral assumption holds for a linearised Navier-Stokes operator. In 2019,
Buckmaster and Vicol [7] proved a stronger result that the weak solutions to
the 3D NSE, $v\in C_{t}^{0}H_{x}^{\beta}$ (where $H^{\beta}$ is the $L^{2}$
space with the regularity index $\beta$), are nonunique. The nonuniqueness
refers to ill-posedness and existence of infinitely many solutions so that a
clear statement about the nature of the solution can not be made. The idea
used in the proof of this work is the convex integration method [15] and the
authors expect that these tools might in the future establish nonuniqueness of
Leray weak solutions.

##  5 Nature of the problem

Interestingly, it is not the first time that such a question - on whether an
equation yields physically reasonable answers for an arbitrary set of initial
values - has been asked. There have been many problems from other fields in
physics and mathematics that resemble to this problem. These include the ray
tracing problem in 3D optical systems; recurrent neural networks; nonabelian
topological quantum-field theory; stability of $n-$ body systems; and quantum
spectral gap problem, ask a similar, generalised, question. For example, the
quantum gap spectral problem states that given a quantum many-body
Hamiltonian, is the system that it describes gapped or gapless? In fact, as
recently as 2015, this question was proved to be undecidable (in other words,
not answerable) [13]. The notion of the undecidability of a system goes back
to the times of the Russell paradox - a paradox inherent in the statement
“This statement is false” formulated by Bertrand Russell. He showed that this
statement leads to a self-contradiction because the statement predicates
itself. Later on, an abstract computing machine, the Turing machine, devised
by Alan Turing, was marred by the same self-referential paradox, termed the
Halting problem. Turing showed that the question of whether for any arbitrary
pair of programs and inputs, the machine will run or halt is unanswerable;
meaning that no program exists to tell the consistency of all possible pairs
of programs and given inputs. Turing replaced the notion of a ‘question’ with
a ‘program’ to arrive at the result. It has been more than 80 years since this
abstract machine was designed, and even now, the lack of the self-consistency
of this machine has had enormous consequences on the self-consistency of
various other topics in physics. The quantum gap spectral problem was shown to
be unanswerable by formulating it in Turing machine terms, thus bringing in
the pathological nature of the machine. In a loose sense, one way to visualise
this pathology is by considering an ancient symbol ouroboros which is that of
a serpent eating its own tail. To ask a question on a problem by asking
whether the problem is solvable or not for arbitrary cases is akin to a snake
evading or attacking its own tail. This notion of accurate solvability for
Navier-Stokes equations (NSE) has been the subject of enquiry for almost a
century, long before the Navier-Stokes problem became a Clay problem.

##  6 A new scheme to look at the problem

In Section 4, numerous efforts to either prove uniqueness or nonuniqueness of
the solutions to NSE were outlined. This section, specifically, highlights a
new scheme to solve the problem by using ideas from theoretical computer
science.
In 2015, Terence Tao [48] introduced an improved variant to the dyadic shell
model by shunning off most of the nonlinear terms and incorporating only one
pair of adjacent modes $X_{n},X_{n+1}$ which experience a nonlinear and energy
interactions at an instant. An improved version of 12 is given as

$\begin{gathered}\partial_{t}X_{n}=-\lambda^{2n\alpha}X_{n}+1_{n-1=n(t)}\lambda^{n-1}X^{2}_{n-1}\\\
-1_{n=n(t)}\lambda^{n=n(t)}\lambda^{n}X_{n}X_{n+1}\end{gathered}$ (14)

where $n:[0,T_{*})\to\mathcal{Z}$ is a piecewise constant function that
describes the mode pair ($X_{n(t)},X_{n(t)+1}$) allowed to interact at a given
time $t$. A system of ODEs were constructed to simulate 14 by using a sequence
of “quadratic circuits” connected in series fashion. Each circuit is composed
of “quadratic logic gates” which simulate a certain form of basic quadratic
non-linear interaction. As shown in Fig. 3(c), the gates transfer energy from
one mode to another with “amplifier” and “rotor” gates. Combination of these
gates with certain coupling constants aid in energy transfer from frequency
scale $n$ to $n+1$. A bootstrap argument is used to construct a blowup
solution to the truncated system of ODEs, interpreted as “averaged Navier-
Stokes equations”, concisely given by

$\partial_{t}u+\tilde{B}(u,u)=\nu\Delta u$ (15)

and exhibit blowup for values close to $\theta_{3/2}$ and $\alpha=0$ scenarios
discussed previously. It is also important to note that the the dynamics of 15
is approximately and discretely self-similar in time. Also, noteworthy is that
the perfectly continuous self-similar solutions to NSE are not possible, as
proved in Necas-Ruzicka-Svreak [33] and Escuriaza-Seregin-Sverak [16]. When
the viscosity effects are of the lower order, such as in the case of
$\theta=3/2$ and $\alpha=0$, viscosity can been a perturbative term. As a
result, instead of the NSE, the spotlight then is on the Euler equations
($\nu=0$), specifically ones when the blowup solutions were found. When one
considers this equation, Kelvin’s circulation theorem is given by

$\Gamma(t)=\int_{C}\textbf{u}.dn=\int_{S}\omega$ (16)

which means that the line integral of $\vec{u}$ around a curve $C$ that loops
the fluid is equivalent to the surface integral of the curl of velocity field
(vorticity). Euler equations written in terms of vorticity is given as

$\displaystyle\partial_{t}\omega+\mathcal{L}_{u}\omega=0$ (17a) $\displaystyle
u=T\omega$ (17b)

where $\mathcal{L}$ is the Lie derivative 101010Lie derivative represents the
change in the tensor field, given by the directional derivatives of each
component, along the flow fields which are defined by another vector field.
w.r.t $u$ and $T\omega=-\nabla\times\Delta^{-1}(*\omega)$ is the Biot-Savart
operator 111111In fluid mechanics, for the equations: $\nabla\times u=\omega$,
$\nabla.u=0$, the Biot-Savart operator $T$ maps $\omega$ into $\vec{u}$. This
formulation is a concise encoding of the conservation laws that Euler
equations exhibit. Tao in 2016 [48] constructed a blowup for artificial
version of Euler equations where the Bio-Savart operator is replaced by a
truncated linear operator of the form $\tilde{T}$. The blowup scenario is
called a “vortex neck pinch” where the streamlines of the vortices are pinched
into a ring of radius $O((T_{*}-T)^{1/2})$ (Fig. 2) when the blowup is
approached at time $T_{*}$, with $u\sim(T_{*}-t)^{-1/2}$ and
$\omega\sim(T_{*}-t)^{-1}$. The blowup is similar to the discretely self-
similar NSE blowup scenario with $\theta=1,\alpha=0$. The pinchoff shown in
this artificial construction of Euler equations is similar to the those
observed in numerical studies [32].

Given that there is some possibility that the Euler equations can exhibit
blowup in finite blowup, a conjecture was postulated by Tao stating that

###### Conjecture 1

There exists some smooth manifold $M=(M,g)$ (of unspecified dimension) and a
smooth solution $\vec{u}(x,t),p$ to the Euler equations (with suitable decay
at infinity) that blows up in finite time.

The potential route to prove (1) is by carefully designing the metric $g$ and
the dimension $M$ so as to “program” various types of behavior that enables
blowup. It should be carefully noted that no corners or boundaries should
exist in the domain, that might yield a blowup, and ideally, fluid in an
infinite and smooth domain. The rational behind designing the manifold is akin
to constructing a computer program that operates on the fluid and cause it to
blowup.

There are two important features that are required in the solutions to the
NSE. The first one is universality and the second one is the scalar
invariance. In 2017, Tao proved the following theorem [49]

###### Theorem 1

Let $\partial_{t}u=B(u,u)$ be an ODE such that $B:V\times V\to V$ is a
bilinear form and $V$ is a finite-dimensional inner product space. The
conservation law is: $\langle B(u,u),u=0\rangle$. Then there exists a compact
Riemannian manifold $(M,g)$ and a linear isometry $T:V\to C^{\infty}(M)$ that
maps solutions to $\partial_{t}u=B(u,u)$ to solutions to the Euler equation on
M.

where a symmetry reduction from finite-dimensional ODE to Euler equations is
performed. For a specific $V$, there exists a manifold such that $V$ is
embeddable into smooth vector fields. Once this is possible, then any ODE of
the above type can be translated to a smooth vector field and then the
manifold can be created such that it exhibits finite time blowup, albeit for
fixed number of frequency scales. In 2019, Tao proved another theorem [50]
which establishes further than Euler equations (or flows) can be seen as
universal. The theorem states that

###### Theorem 2

If $d\geq 2$, then there is a somewhere dense set (in the smooth topology) of
incompressible flows $X:[0,T]\to\Gamma(T(R/Z)^{d})$ which can be lifted to an
Eulerisable flow on a warped product of $(R/Z)^{d}$ and another torus.

where vector field is Eulersiable if $X:[0,T]\to\Gamma(T(R/Z)^{d})$, meaning
that it solves the Euler equations on manifold $M$ with some metric $g$. The
theorem, in summary, proved that in a torus, all vector fields, though not
Eulerisable, some of them can be lifted up and mapped to a warped product
121212Warped product is a Riemannian manifold whose geometry can be decomposed
into a cartesian product of the $y$ and $x$ geometry where the term containing
$x$ is warped and is rescaled by a function of the other coordinate $y$. of
the torus such that the flows become Eulerisable. The outcome of these two
theorems is the first of many works by Cardona, Miranda, Peralta-Salas, Presas
[9] where the authors proved that

###### Theorem 3

Every geodesible vector field $X$ on a compact manifold $M$ can be obtained as
the pullback $X=\phi^{*}Y$ of a stationary Eulerisable flow $Y$ on another
manifold $N$ with respect to some embedding $\phi:M\to N$.

where the stationary vector fields (geodesic fields 131313Geodesic fields on a
Riemannian manifold are the vector fields having integral curves that are
geodesics, meaning, the curves that connect two points on a surface by the
shortest path.) can be mapped to stationary solutions to the Euler equations,
by embedding the manifold $M$ to bigger manifold $N$. Such flows are also
called Beltrami flows and they are related to Reeb vector fields using ideas
from contact geometry. In 2021, Torres de Lizaur [29] proved that

###### Theorem 4

Given any vector field $X$ on a compact manifold $N$, there exists a
Riemannian manifold (M,g) and an invariant manifold $\tilde{N}$ diffeomorphic
to $N$ in the phase space $\Gamma(TM)$ of the Euler equations on $(M,g)$, such
that the Euler flow on $\tilde{N}$ is arbitrarily close to $X$ in the smooth
topology after identifying $\tilde{N}$ with $N$.

such that the perturbed vector field in a manifold $\tilde{N}$ can be
embdedded in a higher dimensional manifold $N$ such that the solutions to
Euler equations on $M$ are quite close to those in $N$ which is stable upon
perturbation. One corollary of this work is that there are manifolds $(M,g)$
such that Euler dynamics in these manifolds are chaotic in that they exhibit
horseshoe map, homoclinic orbits, and similar features of a chaotic system
which are stable upon perturbations. In 2021, Cardona, Miranda, Peralta-Salas,
Presas [10] proved that

###### Theorem 5

There exists a Riemannian manifold $(M,g)$ whose Euler flow is Turing-
complete: the halting problem for any given Turing machine is equivalent to
the Euler flow for a certain initial condition associated to that machine
entering a certain region.

where the authors constructed solutions for the steady Euler flow (without
viscosity) on a Riemannian 3-sphere $\mathcal{S}^{3}$ that are Turing-
complete. Here, Turing completeness means that for given points on the sphere,
the problem of bounding the dynamical trajectory of those points, as the
equation evolves, is an undecidable problem. Following this work, the authors
extended the analysis to a standard Euclidean 3-dimensional space. The tools
used to prove such a result are contact topology, symplectic geometry,
h-principle, generalised shift maps, and Cantor sets.

Figure 2: Cantor set framework consisting of $N$ layers with length scale $L$
of the top-most one and $l_{cut}$ of the bottom-most one. There are $n$
discrete elements in the bottom-most layer, and the number of elements reduce
by a factor of 2 as one goes bottom-up layer by layer. Each layer is
represented by $C_{i}$, where $0\leq i\leq N$.

##  7 Cantor set and rotor gate analysis

We are interested to analyse two length scales: $l_{cut}$ and $L$, such that
$l_{cut}<<L$ (lower by few orders of magnitude). For example, for a liquid
drop of radius 1 mm sitting on a hydrophobic surface, $L=1$ mm and
$l_{cut}=10^{-9}$ nm; $l_{cut}$ is at a molecular scale, $L$ is at a continuum
scale. Consider that all the atoms at the length scale $l_{cut}$ can be
categorised and put into a collection of countably infinite sets - called
Cantor sets - denoted by $\mathcal{C}_{k}\,(1\leq k\leq N)$. $N$ is the number
of layers between $l_{cut}$ and $L$, so that at the length scale $L$, all the
atoms can be collected and put in single Cantor set denoted by
$\mathcal{C}_{N}$. From Fig. 1, it is apparent, geometrically, that the sets
at layer $k$ merge pairwise, to form the sets at layer $k+1$. Thus, if the
number of subsets in $C_{1}$ are $n$, the the number of subsets in $C_{2}$ are
$n/2$, and so on, until $C_{N}$. Since $C_{N}$ is a single set with just one
subset (itself), $n/2^{N}=1$. Rearranging it

$N=\log n_{2}$ (18)

At a general length scale, say $l$ ($l_{cut}\leq l\leq L$), with the atoms
categorised in the set $C_{k}$, let us call each subset in $C_{k}$ as a
“mode”. Consider any two adjacent modes in $C_{k}$, say $x$ and $y$ with
energies $e_{x}$ and $e_{y}$. At an initial time step $t_{0}$, the modes are
represented by $x(t_{0})$ and $y(t_{0})$. Consider them rotating around the
origin at a constant angular rate $\alpha z(t_{0})$ such that

$x(t)=x(t_{0})\cos(\alpha z(t_{0}))(t-t_{0})-y(t_{0})\sin(\alpha
z(t_{0})(t-t_{0}))$ (19) $y(t)=y(t_{0})\cos(\alpha
z(t_{0}))(t-t_{0})+x(t_{0})\sin(\alpha z(t_{0})(t-t_{0}))$ (20)
$z(t)=z(t_{0})$ (21)

where the mode $z\in C_{k+1}$ is fixed while $x,y\in C_{k}$ is not. In fact,
starting at $t_{0}$ until time $T_{k}$, the two modes $x$ and $y$ interchange
energy with each other and thus mix to $emerge$ and form the mode $z$ at the
layer $C_{k+1}$. This process can be viewed as $z$ mode driving the exchange
of energy between $x$ and $y$ modes. It is to be noted that though it is the
choice of the reader to see the Cantor set in either top-down or bottom-up
fashion, throughout this article, we choose to discuss emergence as a bottom-
up process, starting from $l_{cut}$ and reaching layer-by-layer until $L$.
Equipartition theorem in statistical mechanics gives us

$\alpha z(x^{2}-y^{2})=\partial_{t}(xy)$ (22)

which can be written as

$\alpha\int^{T_{k}}_{t_{0}}z(t)(x^{2}-y^{2})dt=x(T_{k})y(T_{k})-x(t_{0})y(t_{0})$
(23)

such that $x^{2}+y^{2}=E$ and constant $z$ gives

$\frac{1}{T-T_{0}}\int^{T_{k}}_{t_{0}}x^{2}(t)dt=\frac{E}{2}+O(\frac{E}{\alpha|z(t_{0}|)(T-T_{0})|})$
(24)

where $E=e_{x}+e_{y}$ and $e_{x}=e_{y}$. As a result,

$e_{x}=e_{y}=e[C_{k}]\approx\frac{e[C_{k+1}]}{2}$ (25)

Using Eq.25 and Eq.18, one gets the general formula

$e[C_{N}]\approx 2^{N}e[C_{1}]$ (26)

which relates the energy of a discrete block in $C_{1}$ (molecular scale) to
the energy in $C_{N}$ (continuum scale).

Figure 3: A rotor gate present between 2 layers $C_{k}$ and $C_{k+1}$ such
that $x$ and $y$ modes operate to start dynamics in the $z$ mode, which we
term as emergence of physics at layer $k+1$ from layer $k$.

##  8 Different abstraction levels in VLSI hardware design as cantor set

Cantor set as a framework can be used to model and understand phenomenon
exhibiting emergence in disciplines other than physics. This section is on the
VLSI hardware design problem and how it can be modelled using Cantor set
framework.

In 1964, an American semiconductor company, General Microelectronics (GMe)
manufactured first-ever MOS (metal-oxide-semi conductor) integrated chips
which composed of more than 10,000 transistors in a single chip. Later on,
VLSI (very large scale integrated) chips with millions and billions of MOS
transistors on them became the core element of semi-conductor industry.

One of the big challenges in the industry is something called, top-design
problem, which is to efficiently divide the task of making VLSI chips in
different categories. Gajski-Kuhn chart [20] is one visual description that
identifies three different domains of behaviour - behaviour, structure, layout
- that goes top-down to more refined abstraction levels. It is apparent from
the Fig. 4 that these domains, structural, can be expressed conceptually in a
Cantor set framework with N = 5. The interpretation from this representation
is that the knowledge and the workforce focused on the transistor level can
not individually contribute anything of value to the knowledge and workforce
at one level above (gates, flipflops). Same argument can be applied for the
layers above. It is the collective knowledge at one level that is of use to a
level above. This is what is emergence, the impossibility of using information
from single unit to say anything meaningful of the layer above. However, it is
to noted that Cantor set framework for VLSI hardware design is only a
conceptual representation, not a quantitative one.

Figure 4: VLSI hardware design top-down hierarchy represented in the form of a
Cantor set. There are 5 layers in the structural hierarchy and each of the
layer operates in a fashion such that the knowledge of individual element at a
certain layer can not help to deduce the knowledge of an arbitrary element at
the layer above.

##  9 If not cantor set framework, then what?

While the notion of correlating arbitrary lower scales to higher scale might
seem to be theoretical in nature, it can also be looked from an engineering
and design lens. Recently, Song et al. (2022) [42] showed that the linear
viscoelasticity of arrested soft materials (gels, glasses) at a macroscopic
level is correlated to the microscopic dynamics of the material. At the
macroscopic level, the stress relaxation of the system is studied using
rheometry techniques. On the other hand, at the microscopic level, the
superdiffusive dynamics of microscopic clusters are studied by perturbing the
material using X-ray photo correlation spectroscopy. The nonlinear phenomenon
at the smaller scale is physically correlated to the linear phenomenon at the
macroscopic scale. An experimental understanding of this correlation can
inform design experts to interfere and perturb the lower scale phenomenon in
such a way that the desired output at the macroscopic level is reached. The
techniques of physical learning (supervised or unsupervised) [44, 46] could be
useful here to train and adapt a physical system to reach a desired target
property by experimentally perturbing its internal nodes and adjusting the
weights of the network in accordance with the physical law governing the
emergent process.

## Dedication

This article is dedicated to Sir George Gabriel Stokes, a physicist who made
seminar contributions in the field of fluid mechanics. He served as the
Lucasian Professor of Mathematics from 1849 until 1903, and served as the
Master of Pembroke College from 1902-1903, at the University of Cambridge. The
article written above is the result of never-ending source of inspiration that
S.S. received by glancing fondly over the magnanimous portrait of Stokes in
the dining hall of Pembroke.

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By [AHR1, Proposition 9.1] (combined with Lemma 3.4), this problem is
equivalent to showing that for any orbit $C$ and any
$\lambda\in\mathbf{X}^{+}_{C}$, the scheme-theoretic support of the object
$\mathfrak{E}_{\lambda}=\pi_{*}\widetilde{\mathfrak{E}}_{w_{0}\lambda}$ is
equal to $\overline{C}$. We already know from [AHR1, Theorem 9.3(1)] that the
set-theoretic support at least contains $\overline{C}$, so it is enough to
show that the scheme-theoretic support is contained in $\overline{C}$. In
fact, we will prove a slightly stronger statement, replacing
$\mathbf{X}^{+}_{C}$ by $\mathbf{X}^{+}_{\overline{C}}$: we will show that

(7.2) if $\lambda\in\mathbf{X}^{+}_{\overline{C}}$, then the scheme-theoretic
support of $\mathfrak{E}_{\lambda}$ is contained in $\overline{C}$.

Suppose $C=C_{I}$ for some $I\subset S$. By Lemma 7.5,
$D^{\mathrm{b}}_{\overline{C_{I}}}\mathsf{Coh}^{G\times\mathbb{G}_{\mathrm{m}}}(\mathcal{N})$
is generated by the objects

$\pi_{J*}\widetilde{\mathfrak{E}}_{J,\mu}\\{n\\},$

for $n\in\mathbb{Z}$, $\mu\in\mathbf{X}^{+,\mathrm{reg}}_{J}$ where $J$
satisfies $C_{J}\subset\overline{C_{I}}$. These are silting objects by
Corollary 6.9. Let us set

$\mathbf{X}^{\prime}:=\left\\{\lambda\in\mathbf{X}^{+}\mid\text{$\mathfrak{E}_{\lambda}$
is a summand of some
$\pi_{J*}\widetilde{\mathfrak{E}}_{J,\mu}\\{n\\}$}\right\\}$

Since $\pi_{J}$ factors through the inclusion map
$\overline{C_{I}}\hookrightarrow\mathcal{N}$, all objects in the collection
$\\{\mathfrak{E}_{\lambda}\\}_{\lambda\in\mathbf{X}^{\prime}}$ have scheme-
theoretic support contained in $\overline{C_{I}}$.

It follows from [AHR1, Theorem 9.3(1)] that
$\mathbf{X}^{\prime}\subseteq\mathbf{X}^{+}_{\overline{C}}$. To prove (7.2),
we must show that $\mathbf{X}^{\prime}=\mathbf{X}^{+}_{\overline{C}}$. To
accomplish this, we will proceed with a $K$-theoretic argument. The set

(7.3) $\\{[\mathfrak{E}_{\lambda}]\\}_{\lambda\in\mathbf{X}^{+}}$

is a $\mathbb{Z}[t,t^{-1}]$-basis for the Grothendieck group
$K\left(D^{\mathrm{b}}\mathsf{Coh}^{G\times\mathbb{G}_{\mathrm{m}}}(\mathcal{N})\right)$
(where the action of $t$ is induced by twisting with $\\{1\\}$). Moreover,
Lemma 7.5, for any object $\mathcal{F}\in
D^{\mathrm{b}}_{\overline{C_{I}}}\mathsf{Coh}^{G\times\mathbb{G}_{\mathrm{m}}}(\mathcal{N})$,
the class $[\mathcal{F}]$ lies in the span of the subset

(7.4) $\\{[\mathfrak{E}_{\lambda}]\\}_{\lambda\in\mathbf{X}^{\prime}}.$

Two other bases for
$K\left(D^{\mathrm{b}}\mathsf{Coh}^{G\times\mathbb{G}_{\mathrm{m}}}(\mathcal{N})\right)$
are

(7.5)
$\\{[\mathfrak{L}_{\lambda}]\\}_{\lambda\in\mathbf{X}^{+}}\qquad\text{and}\qquad\\{[\overline{\nabla}_{\lambda}]\\}_{\lambda\in\mathbf{X}^{+}}.$

Because both the $\mathfrak{L}_{\lambda}$’s are obtained from the the pre-
exceptional set $\\{\overline{\nabla}_{\lambda}\\}_{\lambda\in\mathbf{X}^{+}}$
by a recollement construction, the transition matrix between these bases is
“upper-triangular”: we have

$[\mathfrak{L}_{\lambda}]=[\overline{\nabla}_{\lambda}]+\sum_{\begin{subarray}{c}\mu\in\mathbf{X}^{+}\\\
\mu<\lambda\end{subarray}}a_{\lambda,\mu}(t)[\overline{\nabla}_{\mu}].$

By the same reasoning, the transition matrix between (7.3) and the second
basis in (7.5) is also upper-triangular. It follows that the transition matrix
between (7.3) and the first basis in (7.5) is upper-triangular: we have

(7.6)
$[\mathfrak{L}_{\lambda}]=[\mathfrak{E}_{\lambda}]+\sum_{\begin{subarray}{c}\mu\in\mathbf{X}^{+}\\\
\mu<\lambda\end{subarray}}b_{\lambda,\mu}(t)[\mathfrak{E}_{\mu}].$

Now suppose that $\lambda\in\mathbf{X}^{+}_{\overline{C_{I}}}$. Then the
object $\mathfrak{L}_{\lambda}$ is supported on $\overline{C_{I}}$, so it lies
in the span of (7.4). Since $[\mathfrak{E}_{\lambda}]$ occurs with nonzero
coefficient in (7.6), we conclude that $\lambda\in\mathbf{X}^{\prime}$, and
hence that $\mathbf{X}^{\prime}=\mathbf{X}^{+}_{\overline{C}}$. ∎

######  Corollary 7.7.

Conjecture 7.1 holds for $\mathbf{G}=GL_{N}(\Bbbk)$ and $p>N$.

###### Proof.

Suppose $\mu\in\mathbf{X}^{+}$ belongs to the lower closure of
$w_{\lambda}\cdot_{p}\mathcal{C}_{p}$ for some $\lambda\in\mathbf{X}_{C}$. The
main result of [H] already tells us that the $\Bbbk[\mathcal{N}]$-module
$\operatorname{Ext}_{\mathbf{G}_{1}}^{*}(\mathsf{T}(\mu),\mathsf{T}(\mu))$ has
set-theoretic support equal to$\overline{C}$, so we need only prove that its
scheme-theoretic support is reduced. Write
$\mathsf{T}(\mu)^{*}\otimes\mathsf{T}(\mu)$ as a sum of indecomposable tilting
modules, say

$\mathsf{T}(\mu)^{*}\otimes\mathsf{T}(\mu)=\mathsf{T}(\nu_{1})\oplus\cdots\oplus\mathsf{T}(\nu_{k}).$

We have

$\operatorname{Ext}_{\mathbf{G}_{1}}^{*}(\mathsf{T}(\mu),\mathsf{T}(\mu))\cong\operatorname{Ext}_{\mathbf{G}_{1}}^{*}(\Bbbk,\mathsf{T}(\mu)^{*}\otimes\mathsf{T}(\mu))\cong\bigoplus_{i=1}^{k}\operatorname{Ext}_{\mathbf{G}_{1}}^{*}(\Bbbk,\mathsf{T}(\nu_{i})).$

By Theorem 7.6, every nonzero term in the last direct sum above has reduced
scheme-theoretic support. ∎

##  Appendix A Equivariant dg modules

Let $H$ be an algebraic group over $\Bbbk$, and let
$X=\operatorname{Spec}(\mathbf{A})$ be an affine
$H\times\mathbb{G}_{\mathrm{m}}$-variety over $\Bbbk$. In other words,
$\mathbf{A}$ is a graded commutative finitely generated reduced
$\Bbbk$-algebra equipped with a rational $H$-action. We assume throughout that
the grading on the ring $\mathbf{A}$ is concentrated in nonnegative degrees.

Let $\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$ be the
abelian category of $H\times\mathbb{G}_{\mathrm{m}}$-equivariant
$\mathbf{A}$-modules, or, equivalently, of graded $H$-equivariant
$\mathbf{A}$-modules. Each object
$M\in\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$ comes
equipped with an “internal grading” $M=\bigoplus_{j\in\mathbb{Z}}M_{j}$, where
each $M_{j}$ is a rational $H$-module. For
$M\in\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$, we set

$M\langle 1\rangle=M\otimes\Bbbk_{-1},$

so that the internal grading of $M\langle 1\rangle$ is given by $(M\langle
1\rangle)_{j}=M_{j+1}$.

Given a (possibly infinite) collection $(M^{i})_{i\in I}$ of rational
$H$-modules, the product vector space $\prod_{i\in I}M^{i}$ carries a (not
necessarily rational) action of the abstract group $H(\Bbbk)$. A vector
$v\in\prod_{i\in I}M^{i}$ is said to be _rational_ if it is contained in a
finite-dimensional $H(\Bbbk)$-stable subspace on which $H$ acts algebraically.
The subspace consisting of rational vectors is denoted by

$\sideset{}{{}^{\mathrm{rat}}}{\prod}_{i\in I}M^{i}\subset\prod_{i\in
I}M^{i}.$

This is a rational $H$-module. The same notion makes sense for
$H\times\mathbb{G}_{\mathrm{m}}$-modules. If the $M^{i}$ are objects of
$\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$, then
because the $H\times\mathbb{G}_{\mathrm{m}}$-action on $\mathbf{A}$ is
rational, it is easy to see that $\prod^{\mathrm{rat}}_{i\in I}M^{i}$ is an
$\mathbf{A}$-submodule of $\prod_{i\in I}M^{i}$. Thus, in this context,
$\prod^{\mathrm{rat}}_{i\in I}M^{i}$ is again an object of
$\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$.

Let
$\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}^{{\mathrm{fg}}}$
be the subcategory consisting of those modules that are finitely generated
over $\mathbf{A}$. Of course, we may identify

$\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}=\mathsf{QCoh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)\qquad\text{and}\qquad\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}^{{\mathrm{fg}}}=\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X).$

Next, let
$C^{++}\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$ be the
category of chain complexes $M=(\cdots\to M^{-1}\to M^{0}\to M^{1}\to\cdots)$
that are “cohomologically doubly bounded below”: that is, the cohomology
modules
$H^{i}(M)\in\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$
vanish for $i\ll 0$, and there is an integer $N$ such that $H^{i}(M)_{j}=0$
for all $j<N$ and all $i\in\mathbb{Z}$. We do not impose any boundedness
conditions on the underlying terms $M^{i}$. However, every chain complex in
$C^{++}\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$ is
quasi-isomorphic to one whose terms do satisfy such boundedness conditions.
Denote the derived category of
$C^{++}\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$ by

$D^{++}\mathsf{QCoh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)$

It is well known that the category
$\mathsf{QCoh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)=\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$
has enough injectives. (This follows from the fact that $\mathsf{QCoh}(X)$ has
enough injectives, and that the forgetful functor
$\mathsf{QCoh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)\to\mathsf{QCoh}(X)$ has an
exact right adjoint, namely, the “averaging” functor.) Therefore, any complex
in $C^{++}\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$ is
quasi-isomorphic to a bounded-below chain complex of injectives. (However, the
internal grading of injective modules is usually _not_ bounded below.)

Next, regard the graded ring $\mathbf{A}$ as a dg ring with zero differential.
Let $\mathbf{A}\text{-}\mathsf{dgmod}_{H}$ denote the category of
$H$-equivariant dg modules. Let
$\mathbf{A}\text{-}\mathsf{dgmod}_{H}^{+}\subset\mathbf{A}\text{-}\mathsf{dgmod}_{H}$
be the subcategory consisting of dg modules whose cohomology $H^{\bullet}(M)$
is bounded below, and then let
$\mathbf{A}\text{-}\mathsf{dgmod}_{H}^{+,{\mathrm{fg}}}\subset\mathbf{A}\text{-}\mathsf{dgmod}_{H}^{+}$
be the subcategory consisting of modules $M$ for which $H^{\bullet}(M)$ is
finitely generated over $\mathbf{A}$. We denote by

$D^{+,H}_{\mathsf{QCoh}}(X)\qquad\text{and}\qquad
D^{+,{\mathrm{fg}},H}_{\mathsf{Coh}}(X)$

the derived categories of $\mathbf{A}\text{-}\mathsf{dgmod}_{H}^{+}$ and
$\mathbf{A}\text{-}\mathsf{dgmod}_{H}^{+,{\mathrm{fg}}}$, respectively.

We define

$\xi:C^{++}\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}\to\mathbf{A}\text{-}\mathsf{dgmod}_{H}^{+}$

to be the functor that sends a chain complex $M=(\cdots\to M^{-1}\to M^{0}\to
M^{1}\to\cdots)$ in
$C^{++}\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$ to the
dg module given by

$\xi(M)^{n}=\sideset{}{{}^{\mathrm{rat}}}{\prod}_{i+j=n}M^{i}_{j}.$

It is easy to see that

(A.1)
$H^{n}(\xi(M))\cong\sideset{}{{}^{\mathrm{rat}}}{\prod}_{i+j=n}H^{i}(M)_{j}\cong\bigoplus_{i+j=n}H^{i}(M)_{j}.$

(The “doubly bounded below” condition ensures that this direct sum is finite.)
In particular, $\xi$ sends acyclic complexes to acyclic complexes, so it
induces functors

$\displaystyle\xi$
$\displaystyle:D^{++}\mathsf{QCoh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)\to
D^{+,H}_{\mathsf{QCoh}}(X),$ $\displaystyle\xi$
$\displaystyle:D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)\to
D^{+,{\mathrm{fg}},H}_{\mathsf{Coh}}(X).$

Lastly, we define

$D^{H}_{\mathsf{Coh}}(X)=\begin{array}[]{c}\text{the full triangulated
subcategory of $D^{+,{\mathrm{fg}},H}_{\mathsf{Coh}}(X)$ generated by}\\\
\text{the essential image of
$\xi:D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)\to
D^{+,{\mathrm{fg}},H}_{\mathsf{Coh}}(X)$.}\end{array}$

It seems likely that one has
$D^{H}_{\mathsf{Coh}}(X)=D^{+,{\mathrm{fg}},H}_{\mathsf{Coh}}(X)$ in many
cases of interest, but we will not try to prove this claim here.

######  Lemma A.1.

  1. (1)

If $I$ is a bounded-below chain complex of injective objects in
$C^{++}\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$, then
$\xi(I)$ is a $K$-injective object in
$\mathbf{A}\text{-}\mathsf{dgmod}_{H}^{+}$.

  2. (2)

Every object $M\in\mathbf{A}\text{-}\mathsf{dgmod}_{H}^{+}$ admits a quasi-
isomorphism $M\to I$ where $I\in\mathbf{A}\text{-}\mathsf{dgmod}_{H}^{+}$ is
$K$-injective.

###### Proof.

(1) Let $I=(\cdots\to I^{-1}\to I^{0}\to I^{1}\to\cdots)$ be a bounded-below
chain complex of injectives in
$C^{++}\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$.
Assume without loss of generality that $I^{i}=0$ for $i<0$. Let
$M\in\mathbf{A}\text{-}\mathsf{dgmod}_{H}$ be an acyclic dg modules, and let
$f:M\to\xi(I)$ be a morphism of $H$-equivariant dg $\mathbf{A}$-modules. We
must show that $f$ is null-homotopic.

For each $i$, $\xi(I^{i})$ is a dg module with zero differential. The
underlying graded $\mathbf{A}$-module of $\xi(I)$ (ignoring the differential)
is given by

$\xi(I)=\sideset{}{{}^{\mathrm{rat}}}{\prod}_{i\in\mathbb{Z}}\xi(I^{i})[-i],$

and its differential $d_{\xi(I)}$ is a product of maps of graded
$\mathbf{A}$-modules

$d_{\xi(I)}^{j}:\xi(I^{j})[-j]\to\xi(I^{j+1})[-j].$

Let $d_{M}:M\to M[1]$ be the differential of $M$. The map $f:M\to\xi(I)$ is a
product of maps

$f^{j}:M\to\xi(I^{j})[-j]\qquad\text{such that}\qquad
d_{\xi(I)}^{j}f^{j}=f^{j+1}d_{M}.$

Below, we will define a collection of maps

(A.2) $s^{j}:M\to\xi(I^{j})[-j-1]\qquad\text{such that}\qquad
d_{\xi(I)}^{j-1}s^{j-1}+s^{j}d_{M}=f^{j}$

for all $j\in\mathbb{Z}$. Let
$s=\prod_{i\in\mathbb{Z}}s^{i}:M\to\prod_{i\in\mathbb{Z}}\xi(I^{i})[-i-1]$.
Since $M$ is a rational $H\times\mathbb{G}_{\mathrm{m}}$-module, the image of
$s$ consists of rational vectors. That is, we have a map $s:M\to\xi(I)[-1]$
such that $d_{\xi(I)}s+sd_{M}=f$, as desired.

We define the $s^{j}$’s by induction on $j$. For $j<0$, we must have
$s^{j}=0$, and the equation in (A.2) holds trivially. Suppose now that $s^{j}$
is defined for all $j<N$ in such a way that (A.2) holds. Note that

$(f^{N}-d_{\xi(I)}^{N-1}s^{N-1})\circ
d_{M}=d_{\xi(I)}^{N-1}f^{N-1}-d_{\xi(I)}^{N-1}(f^{N-1}-d_{\xi(I)}^{N-2}s^{N-2})=0.$

It follows that $f^{N}-d_{\xi(I)}^{N-1}s^{N-1})$ induces a map
$g:M/\operatorname{im}d_{M}\to\xi(I^{N})[-N]$. Since $M$ is acyclic, we have
$\operatorname{im}d_{M}=\ker d_{M}$, and $M/\operatorname{im}d_{M}\cong M/\ker
d_{M}\cong\operatorname{im}d_{M}$. Regard $\operatorname{im}d_{M}$ as a
submodule of $M[1]$. Using the fact that $\xi(I^{N})$ is an injective module,
we can extend $g:\operatorname{im}d_{M}\to\xi(I^{N})[-N]$ to a map
$s^{N}:M[1]\to\xi(I^{N})[-N]$. This map satisfies the equation in (A.2) by
construction.

(2) If $M$ lies in the image of $\xi$, say, $M=\xi(\tilde{M})$, then the claim
follows from part (1), since $\tilde{M}$ admits an injective resolution in
$C^{++}\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$. For
general $M$ with differential $d_{M}:M\to M[1]$, consider the short exact
sequence of $\mathbf{A}$-modules $0\to\ker d_{M}\to
M\to\operatorname{im}d_{M}\to 0$. (Here, we are exploiting the fact that
$\mathbf{A}$ has zero differential.) This gives rise to a distinguished
triangle in $D^{+,H}_{\mathsf{QCoh}}(X)$. Now, $\ker d_{M}$ and
$\operatorname{im}d_{M}$ are both in the image of $\xi$ (since they have zero
differential), so they admit $K$-injective resolutions, and hence so does $M$.
∎

######  Proposition A.2.

The functor
$\xi:D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)\to
D^{H}_{\mathsf{QCoh}}(X)$ is a degrading functor with respect to
$\langle-1\rangle[1]$. In other words:

  1. (1)

There is a natural isomorphism $\xi(M)\cong\xi(M\langle-1\rangle[1])$ such
that the map

$\bigoplus_{n\in\mathbb{Z}}\operatorname{Hom}(M,N\langle-n\rangle[n])\xrightarrow{\sim}\operatorname{Hom}(\xi(M),\xi(N))$

is an isomorphism for all $M,N\in
D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)$.

  2. (2)

The image of $\xi$ generates $D^{H}_{\mathsf{Coh}}(X)$ as a triangulated
category.

###### Proof.

Part (2) holds by definition. For part (1), we rely on the well-known fact
that $D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)$ is
equivalent to the full subcategory of
$D^{++}\mathsf{QCoh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)$ consisting of
objects with bounded, coherent cohomology. To find
$\operatorname{Hom}(M,N\langle-n\rangle[n])$, we may replace $N$ by an
injective resolution in
$C^{++}\mathbf{A}\text{-}\mathsf{mod}_{H\times\mathbb{G}_{\mathrm{m}}}$. Then
$\xi(N)$ is $K$-injective. The claim follows from an analogue of (A.1) applied
to the chain complex $R\operatorname{Hom}(M,N)$. ∎

######  Lemma A.3.

Let $f:X\to Y$ be an $H$-equivariant closed immersion of affine $H$-varieties.

  1. (1)

There is a functor $\bar{f}_{*}:D^{H}_{\mathsf{Coh}}(X)\to
D^{H}_{\mathsf{Coh}}(Y)$ such that the following diagram commutes:

${D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)}$${D^{H}_{\mathsf{Coh}}(X)}$${D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(Y)}$${D^{H}_{\mathsf{Coh}}(Y)}$$\scriptstyle{\xi_{X}}$$\scriptstyle{f_{*}}$$\scriptstyle{\bar{f}_{*}}$$\scriptstyle{\xi_{Y}}$

  2. (2)

Assume that $\Bbbk[X]$ admits a bounded resolution by
$H\times\mathbb{G}_{\mathrm{m}}$-equivariant free $\Bbbk[Y]$-modules. Then
there is a functor $\bar{f}^{*}:D^{H}_{\mathsf{Coh}}(Y)\to
D^{H}_{\mathsf{Coh}}(X)$ such that the following diagram commutes:

${D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(Y)}$${D^{H}_{\mathsf{Coh}}(Y)}$${D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)}$${D^{H}_{\mathsf{Coh}}(X)}$$\scriptstyle{\xi_{Y}}$$\scriptstyle{f^{*}}$$\scriptstyle{\bar{f}^{*}}$$\scriptstyle{\xi_{X}}$

Moreover, $\bar{f}^{*}$ is left adjoint to $\bar{f}_{*}$.

###### Proof.

Since $f_{*}$ is an exact functor of coherent sheaves, part (1) is clear. For
part (2), note that the assumption on the existence of a bounded free
resolution of $\Bbbk[X]$ implies that $f^{*}$ takes values in
$D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)$ rather than
$D^{-}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)$.

Let us show that $\bar{f}_{*}$ admits a left adjoint, i.e., that for $M\in
D^{H}_{\mathsf{Coh}}(Y)$, the functor $\operatorname{Hom}(M,\bar{f}_{*}({-}))$
is representable. We first claim that this property is preserved under taking
cones. That is, if $M_{1}\xrightarrow{\phi}M_{2}\to M_{3}\to$ is a
distinguished triangle in $D^{H}_{\mathsf{Coh}}(Y)$, and if
$\operatorname{Hom}(M_{i},\bar{f}_{*}({-}))$ is representable for $i=1,2$, we
claim that it is also representable for $i=3$. Let $N_{1},N_{2}\in
D^{H}_{\mathsf{Coh}}(X)$ be such that we have natural isomorphisms

$\operatorname{Hom}(M_{i},\bar{f}_{*}({-}))\cong\operatorname{Hom}(N_{i},{-})\qquad\text{for
$i=1,2$.}$

These isomorphisms give rise to maps $\eta_{i}:M_{i}\to\bar{f}_{*}N_{i}$ for
$i=1,2$. Moreover, the natural transformation
$\operatorname{Hom}(M_{2},\bar{f}_{*}({-}))\to\operatorname{Hom}(M_{1},\bar{f}_{*}({-}))$
induced by $\phi$ gives rise to a map $\tilde{\phi}:N_{1}\to N_{2}$ making the
following diagram commute:

${M_{1}}$${M_{2}}$${\bar{f}_{*}N_{1}}$${\bar{f}_{*}N_{2}}$$\scriptstyle{\phi}$$\scriptstyle{\eta_{1}}$$\scriptstyle{\eta_{2}}$$\scriptstyle{\bar{f}_{*}\tilde{\phi}}$

Extend $\tilde{\phi}:N_{1}\to N_{2}$ to a distinguished triangle
$N_{1}\xrightarrow{\tilde{\phi}}N_{2}\to N_{3}\to$, and then extend the
diagram above to a morphism of distinguished triangles

${M_{1}}$${M_{2}}$${M_{3}}$${\bar{f}_{*}N_{1}}$${\bar{f}_{*}N_{2}}$${\bar{f}_{*}N_{3}}$$\scriptstyle{\phi}$$\scriptstyle{\eta_{1}}$$\scriptstyle{\eta_{2}}$$\scriptstyle{\eta_{3}}$$\scriptstyle{\bar{f}_{*}\tilde{\phi}}$

The new map $\eta_{3}:M_{3}\to\bar{f}_{*}N_{3}$ gives rise to a natural
transformation

$\operatorname{Hom}(N_{3},{-})\to\operatorname{Hom}(M_{3},\bar{f}_{*}({-})),$

and then a five-lemma argument shows that this map is an isomorphism. Thus,
$\operatorname{Hom}(N_{i},{-})$ is representable.

We now return to the main problem of showing that
$\operatorname{Hom}(M,\bar{f}_{*}({-}))$ is representable. Since
$D^{H}_{\mathsf{Coh}}(Y)$ is generated by the image of $\xi_{Y}$, the previous
paragraph tells us that it is enough to show this when $M=\xi_{Y}(M^{\prime})$
for some $M^{\prime}\in
D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(Y)$. The
adjunction map $M^{\prime}\to f_{*}f^{*}M^{\prime}$ gives rise to a map
$\xi_{Y}(M^{\prime})\to\xi_{Y}(f_{*}f^{*}M^{\prime})\cong\bar{f}_{*}\xi_{X}(f^{*}M^{\prime})$,
and this map induces a natural transformation

$\operatorname{Hom}(\xi_{X}(f^{*}M^{\prime}),{-})\to\operatorname{Hom}(M,\bar{f}_{*}({-})).$

This map is an isomorphism on objects in the image of $\xi_{X}$, and hence an
isomorphism in general. We have shown that $\bar{f}_{*}$ admits a left adjoint
$\bar{f}^{*}$, and that if $M=\xi_{Y}(M^{\prime})$, then
$\bar{f}^{*}M\cong\xi_{X}(f^{*}M^{\prime})$. ∎

We conclude with a lemma on change of equivariance. If $K\subset H$ is a
closed subgroup, there is a forgetful functor
$\operatorname{Res}_{K}^{H}:D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)\to
D^{\mathrm{b}}\mathsf{Coh}^{K\times\mathbb{G}_{\mathrm{m}}}$. If $H/K$ is a
projective variety, then this functor has a right adjoint
$R\operatorname{Ind}_{K}^{H}:D^{\mathrm{b}}\mathsf{Coh}^{K\times\mathbb{G}_{\mathrm{m}}}(X)\to
D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)$, defined as
the composition of the equivalence

$D^{\mathrm{b}}\mathsf{Coh}^{K\times\mathbb{G}_{\mathrm{m}}}(X)\cong
D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(H\times^{K}X)$

with push-forward along the map $\sigma:H\times^{K}X\to X$ given by
$\sigma(h,x)=h\cdot x$. (The assumption that $H/K$ is projective implies that
$\sigma$ is proper, so that $\sigma_{*}$ takes values in
$D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)$ rather than
in $D^{+}\mathsf{QCoh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)$.)

######  Lemma A.4.

Let $K\subset H$ be a closed subgroup such that $H/K$ is projective. Let $X$
be an $H$-variety. There are functors
$R\operatorname{Ind}_{K}^{H}:D^{K}_{\mathsf{Coh}}(X)\to
D^{H}_{\mathsf{Coh}}(X)$ and
$\operatorname{Res}_{K}^{H}:D^{H}_{\mathsf{Coh}}(X)\to
D^{K}_{\mathsf{Coh}}(X)$ such that the following diagrams commute:

${D^{\mathrm{b}}\mathsf{Coh}^{K\times\mathbb{G}_{\mathrm{m}}}(X)}$${D^{K}_{\mathsf{Coh}}(X)}$${D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)}$${D^{H}_{\mathsf{Coh}}(X)}$$\scriptstyle{\xi_{X}}$$\scriptstyle{R\operatorname{Ind}_{K}^{H}}$$\scriptstyle{R\operatorname{Ind}_{K}^{H}}$$\scriptstyle{\xi_{X}}$
${D^{\mathrm{b}}\mathsf{Coh}^{H\times\mathbb{G}_{\mathrm{m}}}(X)}$${D^{H}_{\mathsf{Coh}}(X)}$${D^{\mathrm{b}}\mathsf{Coh}^{K\times\mathbb{G}_{\mathrm{m}}}(X)}$${D^{K}_{\mathsf{Coh}}(X)}$$\scriptstyle{\xi_{X}}$$\scriptstyle{\operatorname{Res}_{K}^{H}}$$\scriptstyle{\operatorname{Res}_{K}^{H}}$$\scriptstyle{\xi_{X}}$

Moreover:

  1. (1)

The functor $\operatorname{Res}_{K}^{H}$ is left adjoint to
$R\operatorname{Ind}_{K}^{H}$.

  2. (2)

Let $d=\dim H-\dim K$. The functor
$\operatorname{Res}_{K}^{H}({-})\otimes\bigwedge^{d}(\mathrm{Lie}(H)/\mathrm{Lie}(K))^{*}[d]$
is right adjoint to $R\operatorname{Ind}_{K}^{H}$.

###### Proof.

The existence of functors $R\operatorname{Ind}_{K}^{H}$ and
$\operatorname{Res}_{K}^{H}$ in the dg setting making these diagrams commute
is straightforward (using $K$-injective resolutions for the former, and
exactness for the latter). In the coherent sheaf setting, we noted above that
$\operatorname{Res}^{K}_{H}$ is left adjoint to $R\operatorname{Ind}^{K}_{H}$.
In the dg setting, there is at least an obvious natural transformation
$\epsilon:\operatorname{Res}^{K}_{H}R\operatorname{Ind}^{K}_{H}(M)\to M$ for
any $K$-injective object $M$. This natural transformation gives rise to a
natural map

$\operatorname{Hom}(N,R\operatorname{Ind}^{K}_{H}(M))\to\operatorname{Hom}(\operatorname{Res}^{K}_{H}(N),M).$

This map is at least an isomorphism when $N$ and $M$ lie in the essential
image of $\xi_{X}$ (by the coherent sheaf version of the adjunction), so it is
in fact an isomorphism for all $N$ and $M$.

For the last assertion in the lemma, the dg version again follows from the
coherent sheaf version. The coherent version is a well-known consequence of
Serre–Grothendieck duality. Let us briefly explain how to obtain it. Let
$p:H\times^{K}X\to X$ be the map given by $p(h,x)=h\cdot x$. Then $p$ is a
smooth, projective bundle over $X$ with fibers isomorphic to $H/K$.
Serre–Grothendieck duality says that for $\mathcal{F}\in
D^{\mathrm{b}}\mathsf{Coh}^{H}(H\times^{K}X)$ and $\mathcal{G}\in
D^{\mathrm{b}}\mathsf{Coh}^{H}(X)$, there is a natural isomorphism

(A.3)
$\operatorname{Hom}(\mathcal{F},p^{*}\mathcal{G}\mathbin{\mathop{\otimes}^{L}}\omega_{p}[d])\cong\operatorname{Hom}(p_{*}\mathcal{F},\mathcal{G}),$

where $\omega_{p}$ is the relative canonical bundle. (See [Ht, Theorem
III.11.1] for the nonequivariant version of this statement, and see [AB,
Example 2.16] for a discussion of how to deduce the equivariant version.) If
we identify $\mathsf{Coh}^{H}(H\times^{K}X)$ with $\mathsf{Coh}^{K}(X)$ (cf.
(3.1)), then $p_{*}$ and $p^{*}$ are identified with
$R\operatorname{Ind}_{H}^{K}$ and $\operatorname{Res}_{H}^{K}$, respectively,
while $\omega_{p}$ corresponds to the canonical bundle of $H/K$, i.e., line
bundle whose fiber over $1K\in K/H$ is identified with
$\bigwedge^{d}(\mathrm{Lie}(H)/\mathrm{Lie}(K))^{*}$. Thus, (A.3) yields the
desired adjunction. ∎

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