11institutetext: University of Nebraska-Lincoln, Lincoln, NE, USA
22institutetext: Royal Holloway University London, United Kingdom
33institutetext: Max-Planck-Institute for Physics, Munich, Germany
44institutetext: SLAC National Accelerator Laboratory, Menlo Park, CA, USA
55institutetext: CP3-Origins, University of Southern Denmark, Denmark
66institutetext: Sanford Underground Research Facility (SURF), Lead, SD, USA
77institutetext: University of Sheffield, United Kingdom 88institutetext:
Fermi National Accelerator Laboratory (Fermilab), Batavia, IL, USA

# Climate impacts of particle physics

Kenneth Bloom Contact author, 11<EMAIL_ADDRESS>Veronique Boisvert Contact
author, 22<EMAIL_ADDRESS>Daniel Britzger 33 Micah Buuck 44
Astrid Eichhorn 55 Michael Headley 66 Kristin Lohwasser 77 Petra Merkel 88

###### Abstract

The pursuit of particle physics requires a stable and prosperous society.
Today, our society is increasingly threatened by global climate change. Human-
influenced climate change has already impacted weather patterns, and global
warming will only increase unless deep reductions in emissions of CO2 and
other greenhouse gases are achieved. Current and future activities in particle
physics need to be considered in this context, either on the moral ground that
we have a responsibility to leave a habitable planet to future generations, or
on the more practical ground that, because of their scale, particle physics
projects and activities will be under scrutiny for their impact on the
climate. In this white paper for the U.S. Particle Physics Community Planning
Exercise (“Snowmass”), we examine several contexts in which the practice of
particle physics has impacts on the climate. These include the construction of
facilities, the design and operation of particle detectors, the use of large-
scale computing, and the research activities of scientists. We offer
recommendations on establishing climate-aware practices in particle physics,
with the goal of reducing our impact on the climate. We invite members of the
community to show their support for a sustainable particle physics field
bib:indico .

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

##  1 Introduction

Global climate change, and how to mitigate it, is one of the most crucial
issues facing humanity today bib:UNoals . The Intergovernmental Panel on
Climate Change (IPCC) has stated, “It is unequivocal that human influence has
warmed the atmosphere, ocean and land. Widespread and rapid changes in the
atmosphere, ocean, cryosphere and biosphere have occurred” bib:IPCC6 . This
change in climate has already affected weather patterns and climate extremes
in regions across the globe, including in the U.S. The IPCC further states
that “Global warming of 1.5∘C and 2∘C will be exceeded during the 21st century
unless deep reductions in CO2 and other greenhouse gas emissions occur in the
coming decades.” To limit the amount of warming, we will have to achieve
significant reductions in CO2 emissions, to the point of net-zero emissions,
along with reductions in the production of other greenhouse gases (GHGs), in
line with the goals of the Paris Agreement bib:Paris . Each 1000 gigatons of
cumulative CO2 emissions is likely to lead to 0.27∘C to 0.63∘C increases in
global surface temperature, implying that limiting temperature increases means
adhering to a “carbon budget”. The 6th IPCC assessment report estimates a
total budget of 300 gigatons CO2e emissions for an 83% chance to limit global
warming to below 1.5 degrees Celsius bib:IPCC6 . This amounts to about 1.1 t
CO2e111Carbon dioxide equivalent or CO2e means the number of metric tons of
CO2 emissions with the same global warming potential as one metric ton of
another greenhouse gas, and is calculated using Equation A-1 in 40 CFR Part 98
bib:co2e_def . emissions per capita per year until 2050. This should be
compared to current per capita per year emission rates of 14.2 tCO2 in the
United States bib:pcCO2 . Significant reductions in carbon emissions must be
achieved for the future health of the planet, and measures must be taken now
rather than later to begin to limit the long-term impacts.

The U.S is a crucial player in the climate agenda. It is the top producer and
consumer of both oil and natural gas, it has the worldâs second largest number
of coal-fired power plants, and fossil fuels contributed 63% to its overall
electricity generation CarbonBriefProfile . On the other hand, it also has the
largest nuclear and second largest renewable capacity in the world. Not only
is the U.S. the second largest emitter of GHGs, but over the course of history
it has cumulatively produced more than any other country bib:histemissions .
Its citizens have emissions footprints that are roughly three times the global
average. The U.S also has a large impact on climate policy. For example, the
success of the 2015 Paris agreement was due in significant part to the
leadership demonstrated by the U.S. at that time Obama . More recently, the
current administration has submitted an updated Nationally Determined
Contribution to cut greenhouse gas emissions 50-52% below 2005 levels by 2030
NDC  and has pledged to achieve net-zero emissions by âno later than 2050â
netzero .

Current and future activities in particle physics need to be considered in
this context. The pursuit of particle physics requires substantial
construction projects; the consumption of electricity in the operation of
accelerators, detectors, and computing; the use of GHGs in particle detectors;
and in some cases significant amounts of travel. All of these lead to the
potential for particle physicists to have a carbon impact well above that of
typical citizens, and thus particle physicists should be paying attention to
the impacts of the discipline on the planet and seeking to reduce them. This
could be argued from a moral point of view – that we have responsibilities to
future generations to leave behind a habitable planet. However, one could also
justify it on more practical grounds. Particle physics takes place on such a
scale that future major projects, such as a new facility, will be subject to
scrutiny for their climate impacts, and thus the field must be prepared for
that scrutiny. Should society decide that a price on carbon is necessary, then
future projects will have to be prepared to pay that price. It may be possible
to argue that because of its societal benefits, some exception to constraints
on carbon emission should be made for scientific activities. However, even
today emissions rates around the world tend to be above those necessary to
avert a significant temperature rise. Also, asking for such exceptions can
lead to difficult discussions of the relative size and type of societal
benefits from different areas of science, especially for fundamental research
areas such as particle physics, where the ultimate benefits cannot be known,
or questions about whether other sectors of the economy (e.g. health care or
agriculture) provide similar benefits. Instead, just as our field currently
demonstrates world leadership in international cooperation towards common
goals, we can also demonstrate world leadership in this critical area that
impacts the future of society.

In this white paper for the U.S. Particle Physics Community Planning Exercise
(“Snowmass"), we examine several contexts in which the practice of particle
physics has impacts on the climate. These include the construction of large-
scale experimental facilities, the design and operation of particle detectors
that make use of GHGs, the operation of computing facilities, and the common
research activities of physicists, including long-distance travel. We conclude
with a set of recommendations for how we as a field can begin to reduce our
impact on the climate. Our work builds on that of the European Strategy
Update, which made a recommendation on studying and significantly reducing the
environmental impact of particle physics activities bib:ESU . We hope that
this work stimulates conversations within the U.S. particle physics community
about how we can pursue the science that we love in a sustainable fashion.

##  2 Impacts of facility construction

A key goal of the Snowmass process is to identify promising opportunities to
address the most important questions in particle physics. We expect that these
opportunities will require the construction of new, large-scale experimental
facilities. The building construction industry currently contributes 10% of
the world’s total carbon emissions bib:globalABC . If we assume that the
electric grid is successfully de-carbonized by 2040, a goal of many climate
plans (for example the U.S. “has set a goal to reach 100 percent carbon
pollution-free electricity by 2035"NDC ), then construction, rather than
operations, may well dominate the climate impact of a new particle physics
facility. Here we consider the climate impact of the construction of a new
accelerator facility and put it in its global context.

One potential new energy-frontier accelerator facility foreseen by the
particle physics community is the Future Circular Collider (FCC) FCC . This
accelerator would likely first collide electrons and positrons to make
precision measurements and later could accelerate hadrons with a center-of-
mass energy of 100 TeV, allowing for the next stage of discovery following the
HL-LHC era. The FCC-ee project bib:FCC-ee  would operate in the era of de-
carbonized electricity. The tunnel for the accelerator would be one of the
longest tunnels in the world, projected at 97.75 km circumference in the
conceptual design report. In addition, many bypass tunnels, access shafts,
large experimental caverns, and new surface sites are planned. It is estimated
that 7 million cubic meters of spoil, a mixture of marls and sandstone, would
need to be excavated.222This projected amount of excavated material is smaller
than the 9 million cubic meters described in the conceptual design report,
after further optimization, and possible re-use of this material is being
explored bib:miningthefuture .

We can attempt a rough estimate of the carbon impact of the main tunnel alone.
A bottom-up calculation is driven by the construction parameters of the
tunnel. It is expected to have an inner diameter of 5.5 m, and an excavation
diameter of 6.3 m. We assume that the region between those two diameters is
filled with concrete, and that concrete is composed 15% of cement (and we
neglect the necessary steel tunnel reinforcement). The production of cement is
a significant contributor to greenhouse gas emissions, releasing a ton of CO2
per ton of cement created bib:CBcement .333If world production of cement were
a country, it would be the third largest emitter. We thus calculate that the
main tunnel of FCC-ee alone would lead to the release of 237 ktons of CO2.

Alternatively, one could make a top-down estimate of the impact using rules of
thumb from studies of previous road tunnel construction bib:tunnelCO2 . These
studies attempt to obtain a complete accounting of emissions. For example,
they consider the impacts of fuel and electricity used in tunnel construction,
along with those of the construction materials and of the possible release of
methane inherent to the excavation process. The carbon impacts depend very
much on the “rock mass quality" of the excavation site, with a range between
5,000 and 10,000 kg CO2 per meter of tunnel length. This leads to estimates of
489 to 978 kton of CO2 for the main FCC-ee tunnel, providing an order-of-
magnitude agreement with the bottom-up estimate. Using 500 kton of CO2 as a
conservative estimate and dividing by the rough number of about 6000
physicists that could be contributing to this project, this amounts to about
80 t of emissions per physicist, to be compared with the target of reaching
1.1 t of emissions per person per year introduced in Section 1. Alternatively,
we estimate that 6 million trees would need to be planted to absorb this
amount of CO2bib:trees .

How does this compare to other forms of civil construction? For context, we
can compare with the carbon impact of typical buildings. A study bib:buildings
estimates that the “embodied" carbon emissions during building construction is
500-600 kg of CO2e per square meter; we use 550 kg CO2e as a working value. As
a sample building we take New York City’s 1 World Trade Center, a prominent
recently-built skyscraper, which is 94 stories tall and 3.5 Mft2. The embodied
carbon in 1 WTC is thus 197 ktons, meaning that the FCC-ee main tunnel alone
has a carbon impact several times as large as one of the most significant
building projects in the U.S. in recent years.

When the complete FCC-ee, or any similar-scale particle physics facility, is
considered – the full tunnel system, the additional buildings on the site, the
materials for the accelerator and detectors – we expect that the project will
have a carbon impact similar to that of the redevelopment of a neighborhood of
a major city. This implies that the environmental impact of a future facility
is going to receive the same scrutiny as that of a major urban construction
project. Our field needs to be prepared for this scrutiny, by preparing to
collect and analyze data on carbon impacts, and also for taking reasonable
measures for the reduction of climate impacts through the development and use
of low-carbon materials, with a prioritized use of reused and recycled
materials. We can already begin investments in R&D on how to reduce our carbon
impact to prepare for future environmental reviews.

Some efforts on this, beyond the matter of facility construction, are already
in progress within the accelerator community. For example there is a definite
focus on the energy efficiency and power reduction of future large
accelerators. This ranges for example from active research on energy recovery
schemes bib:ERL ; bib:Plasma , to using the heat produced by accelerator
operations for district heating bib:districtheating .

Any facility construction project will have a measurable impact on climate
change, and as responsible citizens we must consider what we can do to reduce
the potential harms to future generations by our actions.

##  3 Impacts of detector gases

According to CERN’s environmental reports bib:CERNenvurl , the dominant source
of CO2e emissions at the laboratory are from greenhouse gases used for
detectors and for cooling. As can be seen in Figure 1, whether the LHC
experiments are in operation or in a shut down, gas emissions dominate over
the emissions stemming from CERN’s electricity usage. Figure 2 shows the
amount of emissions from various gases during 2017-18, when the LHC
experiments were in operation, prior to Long Shutdown 2 (LS2). In general SF6,
hydrofluorocarbon (HFC) and perfluorocarbons (PFC) gases are used in particle
detection. HFCs and PFCs are also used for detector cooling, HFCs are used in
air conditioning systems, and SF6 is also used for electrical insulation in
power supply systems. All these gases are subject to the UN Kyoto protocol
bib:Kyoto  and their usage shall fade out according to the Kigali amendment of
the Montreal protocol bib:Kigali . Overall, out of emissions specifically from
particle detectors, 78% of emissions come from the gas C2H2F4 (R134a) which
has a global warming potential (GWP) of 1300 times the warming potential of
CO2 (over a 100-year time scale) bib:GWP , while SF6 has a GWP of 23500 and
contributes 8% of total particle detection emissions bib:BeatriceTalk . The
types of detectors that make use of those gases include Resistive Plate
Chambers (RPC), which are pairs of parallel plastic plates at an electric
potential difference, separated by a gas volume. The choice of gases allows
for a relatively low operating voltage, non-flammability, low cost and an
adequate plateau for safe avalanche operations. Another detector is Cathode
Strip Chambers, which usually includes the CF4 gas (GWP of 6630). CF4 gas is
commonly used in particle detectors owing to its properties that prevent
ageing, enhance timing resolution or because of its scintillation emission
bib:BeatriceTalk .

Figure 1: CERN Scope 1 and 2 emissions from 2017 to 2020. The different Scopes
are explained in Section 5 bib:CERNenvurl . Figure 2: Breakdown of Scope 1
emissions by gas types for 2017-2018 bib:CERNenv1sturl .

Due to their very high GWP, F-gases are under regulation in the EU bib:EUreg
and mandatory reporting in the U.S. bib:USreg . Consequently, their continued
procurement and price for the whole duration of the LHC program is under
threat. CERN has put together various strategies to mitigate the emissions
from those gases bib:CERNenvurl . For example, gas re-circulation is used for
all gas systems across the LHC experiments. In addition, gas recuperation is
also in use in some areas. During LS2, an extensive campaign of fixing leaks
has occurred bib:BeatriceTalk . Leaks tend to be concentrated in the gas
inlets, the polycarbonate gas connectors and the polyethylene pipes, so having
access to the chambers is necessary to fix those leaks.

In the longer term, both for current and future detectors, finding alternative
gases with lower GWP would be very beneficial, and studies are currently
ongoing along those lines (e.g. bib:gas1 ; bib:gas2 ; bib:gas3 ). Although new
liquids and gases have been developed for industry as refrigerants and high
voltage insulating media, those are not necessarily appropriate for detector
operation, especially taking into account the constraints of having to operate
the current detectors. For example, there cannot be changes to the high
voltage system or to the front-end electronics. Finding replacement gases
needs to take into account several factors: their safety (non-flammable and
low toxicity) and their environmental impact (low GWP) while maintaining their
detector performance (including preventing the ageing of the detectors,
ensuring good quenching and being radiation-hard) bib:BeatriceTalk .

Looking to the long term future, these results highlight the crucial need to
design future detectors (including cooling systems) with gas GWP in mind. For
example, the proposed FCC-ee detector ideas mention using RPCs for the muon
detectors and different gaseous detectors containing gas mixtures which
include 10% iC4H10 (isobutane, GWP of about 3) bib:FCC-ee . Over the next few
years, it will be imperative to perform R&D aimed at reducing the GHG
emissions of future detectors and cooling systems as much as possible.

##  4 Impacts of computing

High-performance computing (HPC) is an essential part of physics research. It
is also a growing source of greenhouse gas emissions, primarily due to the
large amount of electricity used by computation itself. Across all sectors,
data centers and computing already contribute approximately 2-4% of global GHG
emissions bib:lannelongue_1 , and that fraction is only predicted to grow in
the next 10 years. While the environmental costs of HPC are not unknown, they
are often not prioritized, or even discussed, when planning and scheduling
computational projects. We believe the particle physics community should
improve its accounting of this issue. We provide several specific suggestions
below, many of which are inspired by Lannelongue et al. bib:lannelongue_1

The best way to reduce the carbon emissions of high-performance computing is
to ensure that compute facilities are powered by carbon-free energy sources.
In the early planning stages of a compute facility, when siting decisions are
being made, the carbon intensity of the electricity used to power the facility
should be a major factor into the decision making process. In many places,
renewable electricity is already cheaper than fossil-fueled electricity,
making this a cost-effective decision as well. Local installation of renewable
energy generating capacity can also reduce the carbon impact of both future
and existing computing facilities, most easily with rooftop solar panels.
Other options may also be available depending on the location of the compute
center. These strategies have a direct and immediate impact on the carbon
emissions associated with a compute center, and do not rely on the cooperation
of users as several of our other suggestions do, making them likely more
effective.

Reductions in GHG emissions can also be achieved on the demand side with
current technology, through a combination of optimization of resource use and
careful planning of timing and siting of computation. These suggestions give
individual users tools to directly reduce their own carbon emissions impact.
However, it is often not obvious how to successfully follow them, since the
information required is unavailable or difficult to access. The website Green-
algorithms.org bib:lannelongue_2  can provide a useful estimate of the cost of
running a particular algorithm, but it requires knowledge of the specific
hardware a computation will be using, and also the specific energy mix the
compute center uses, to calculate the most accurate emissions estimate. This
information could be provided by compute centers, and while the former
frequently is, the latter often is not. If they aren’t already doing so,
compute centers should provide information on the GHG intensity of their
computation, ideally in a standardized format to facilitate easy comparisons.
This information could be integrated into the green-algorithms website, or
else made available in a standalone website, and statistics from cloud
computing services could also be included if that information is available.

Going a step further, compute centers, and the scientific collaborations that
make use of them, could even provide a simple tool to estimate the GHG
emissions impact of a particular job request, perhaps as a simple command-line
utility. A precise estimate would be difficult to achieve, but with knowledge
of the hardware, energy mix, and particulars of the job request, a reasonably
accurate range of possible GHG emissions should be possible to produce, likely
ranging from zero to full resource utilization. Compute centers could then
report their CO2e intensity to a common location, ideally a centralized
website of some sort. A version of this already exists as the Green500, an
offshoot of the better-known TOP500 bib:green500 . However, the Green500
compares the power efficiency of various systems, which is related to, but not
the same as, the CO2e intensity. For example, while HiPerGator AI at the
University of Florida has a slightly better power efficiency rating than
Perlmutter at the National Energy Research Supercomputing Center (NERSC,
Berkeley, CA), 80.3% of electricity consumption in Florida in 2019 was fossil
fueled bib:fl_energy  while only 35.9% California’s electricity consumption
over the same period was fossil fueled bib:ca_energy , meaning that the per-
FLOP CO2e emissions impact of Perlmutter is likely less than that of
HiPerGator AI.444This is only an example to illustrate the concept; obviously
the exact energy mix for each compute center is likely not identical to the
statewide average.

While obtaining better information about compute center GHG emissions can be
very useful for long-term planning, such as an experiment deciding where to do
the bulk of their computing, it is less useful for individual users who often
do not have the ability to choose where to do their large-scale computing.
Users do usually have the ability to improve the efficiency of their code,
which can lead to substantial improvements in emissions if done properly.
However, frequently users are writing code that makes use of large libraries,
such as Geant4 or ROOT, that they do not have control over, which can limit
potential improvements from optimization. Developers of these libraries should
be sure to provide information on how to minimize GHG emissions, such as
optimal hardware, and scaling of memory utilization. Tracking the GHG
emissions efficiency over software releases with some kind of standard
benchmark would provide an incentive to minimize emissions.

Even if one makes optimal choices to minimize the GHG emissions of their
computation, some emissions will still inherently result from the electricity
used. Even if the electricity is produced from renewable sources, the life
cycle emissions of those power sources, and also of the compute center itself,
are nonzero. Purchasing carbon offsets, while never a true substitute for
emissions reductions, can help to mitigate the remaining unavoidable
emissions. Incentivizing the purchasing of carbon offsets for unavoidable
emissions is something funding agencies could do. Some Department of Energy
compute centers, such as NERSC, use an allocation-and-charging framework for
distributing computational resources. Carbon offsets could be folded into the
cost of batch jobs by the compute center, drawn directly from the user’s
allocation. This would incentivize experiments to choose greener compute
centers, and users to ensure that their code runs as efficiently as possible.
At minimum, compute centers could provide links to legitimate offset
companies, or other agencies that rate and verify them.

Finally, compute centers could coordinate their power loads with their
electricity suppliers, scheduling more jobs when electricity is cheap and
relatively clean (e.g. midday or nighttime), and reducing their loads when
electricity is expensive and relatively dirty (e.g. late afternoon through the
early evening). For the greatest impact, this would require real-time
information from the electricity supplier, but if the compute center is a
sufficiently large electricity customer, the supplier would likely be happy to
provide this information since it would improve their demand response. Ahmed
et al. have completed a study of this possibility. bib:demand_response

##  5 Impacts of scientists’ research activities

###  5.1 Overview of GHG emissions from particle physics laboratories

The previous sections demonstrate that particle physics laboratories are
expected to have emissions associated both with their research activities, as
well as with the other work-related activities of their employees. GHG
emissions fall under three scopes widely used in the reporting of emissions
bib:ghgprotocol : Scope 1 refers to direct emissions from the organization,
Scope 2 includes indirect emissions, most notably from electricity generation,
heating, etc., and Scope 3 includes all other indirect emissions, upstream and
downstream of the organization, including e.g. business travel, personnel
commutes, catering, etc.

For example, Figure 1 in Section 3 showed the emissions associated with CERN’s
Scope 1 and Scope 2 activities since 2017. During the time periods when the
LHC is operating, the accelerator complex is the largest contributor to Scope
2 emissions. CERN obtains its electricity mainly from France and uses an
emission calculation which is location-based from the “EDF Bilans des
Ãmissions de GES". These emissions benefit from the fact that France mainly
gets its electricity from nuclear power and as such it is largely carbon free
(88% in 2017) bib:CERNenv1sturl . Consequently, as explained in Section 3, the
largest overall emissions from CERN are from GHG gases used for detectors and
cooling (part of Scope 1). In CERN’s second environment report, covering the
years 2019 and 2020, CERN included its first estimate of Scope 3 emissions as
shown in Figure 3. These do not yet include emissions from procurement, which
is expected to be the largest source. CERN plans to include those in future
reports. Most of the business travel emissions are from air travel, especially
from long distance flights and only include the business travel of CERN
employees, not from the very large number of CERN users. In a similar way,
only the electricity used for on-site computing is included in the Scope 2
numbers, not the computing done on Grid sites located outside of CERN.

Figure 3: CERN’s Scope 3 emissions from 2019 and 2020 bib:CERNenvurl .

In the U.S, many particle physics laboratories are mandated to report on their
environmental impact, including GHG emissions, every year. As an example,
Fermilab’s annual Site Environmental Reports bib:fnal_env_rprts  include this
information. Fermilab is committed to assist the U.S. Department of Energy in
meeting reduction goals for all three scopes. Emissions are calculated based
on the MRR (Greenhouse Gas Mandatory Reporting Rule US_GHG_MRR ), and many of
Fermilabâs greenhouse gas emission sources are exempt from reporting (such as
emergency back-up power generators). In 2013, Fermilab was registered as a
ROSS site (Registration of Smaller Sources ROSS ), which corresponds to the
adherence to certain standards and limits, e.g. limits on combined pollutants
such as carbon monoxide, nitrogen oxides, sulfur dioxide, limits on mercury or
lead air emissions. Table 1 summarizes emissions at Fermilab over the last
four years and compared to the reference year of 2008, split into the
different emission scopes. It has to be noted that 2020 and 2021 were highly
unusual years with reduced on-site operations and occupancy during the
COVID-19 pandemic. Fermilab’s dominant emissions are from its electricity
consumption. Fermilab is part of the PJM balancing authority, itself part of
the wider Eastern Interconnection electricity grid. In 2016, the carbon
intensity of PJM was 351 kg/MWh (using Emissions Factors: mass of pollutant
per unit of electrical energy), which is about the average for the whole of
the U.S. bib:PJMcarbon  and, as mentioned in Section 1, about 37% of the
national electricity grid is carbon free. Since 2011, Fermilab uses Renewable
Energy Certificates (RECs) based on the lab’s purchased power consumption as a
primary mechanism to reduce those Scope 2 emissions. As an example, in 2013,
Fermilab offset 29,000 MWh by purchasing RECs generated from new biomass
plants that help reduce the environmental impact of the paper industry in
Louisiana bib:FNALToday . In 2019 Fermilab purchased RECs to offset 11% of its
facility GHG emissions bib:fnal_env_rprts .

| 2008 | 2018 | 2019 | 2020 | 2021
---|---|---|---|---|---
Scope 1+2 | 384,666 | 128,304 | 144,013 | 106,961 | 163,818
Scope 3 | 29,503 | 16,495 | 14,468 | 6,516 | 17,456
Table 1: Summary of Fermilab GHG emission data from 2008 (reference year) and
2018 - 2021. Emissions are divided into the three scope areas and given in
CO2e metric tons bib:fnal_env_rprts .

Past and projected electricity usage, Fermilab’s major single contributor to
GHG emissions, is listed in Figure 4. Significant changes to Fermilabâs
projected energy consumption are anticipated beginning 2024, when a planned
2-year accelerator shutdown begins to bring the new PIP-II accelerator complex
online. During the shutdown, the Long Baseline Neutrino Facility
infrastructure will tie into the existing accelerator complex. The addition of
the PIP-II accelerator and beam requirements for LBNF are expected to increase
Fermilabâs overall energy consumption by 30% over historic peak levels.

Figure 4: Past and projected future electricity consumption at Fermilab split
by the different accelerator projects, computing and site base operation,
adapted from bib:FNAL_sustainability .

###  5.2 Overview of GHG emissions from the academic system

Numerous activities in the academic system result in GHG emissions, from
campus operations at universities, to construction and operation of large-
scale research infrastructure, to academic travel for various activities such
as conferences, committee meetings, and funding reviews. To provide an
impression of the amount of GHG associated to these aspects of academia, we
discuss several examples of activities and compare their associated GHG
emissions to the personal budget of 1.1 t per year mentioned in Section 1.

In Europe, GHG emissions from campus operations can amount to 33-35 t CO2e
emissions per employee per year, of which more than 75% can be Scope 3
emissions, including air travel and the supply chain (including consumables
and equipment for research) bib:Borgermann . The estimates of many
universities are significantly lower than this, because Scope 3 emissions are
not comprehensively accounted for. For example, ETH Zürich quotes 2.8 (3.0)
tCO2e per FTE employee in 2012 (2018) but notes that data for Scope 3 is only
available for selected domains bib:ETH . In the U.S., numbers tend to be
higher. For instance, MIT reports about 193 kt CO2e per year as on-campus
emissions, which amounts to about about 12 t CO2e per employee bib:MIT . This
is exclusive of all Scope 3 emissions. At UCLA, Scope 1 and 2 emissions
together with commuting and business travel, i.e., with a very incomplete
accounting of Scope 3, come to about 7 t CO2e per year per full-time-
equivalent bib:UCLA . At academic institutions, there is significant
heterogeneity in the reporting of GHG emissions from flying
bib:KreilStauffacher2021 . For example, often only trips paid by the
institutions are accounted for, and so trips on external grants are not
included in any institution’s accounting.

Similar numbers arise in the related research field of astronomy, which also
relies on large-scale infrastructure as well as computing. Jahnke bib:Jahnke
estimates about 18 t CO2e emissions per year per researcher at the Max-Planck-
Institute for Astronomy in Germany, with 8.5 t for air travel and roughly 5 t
for computing. In Australia (see discussion in bib:Jahnke , based on
bib:Stevens ), where less electricity comes from renewable sources and more
air miles are flown, the total emissions are more than twice as high.

It is worth considering the structure of future collaborations and
requirements to be at the experimental site in person. As air travel has
become cheaper and funds to bring along family members for extended stays to
particle physics laboratories have become considerably smaller, regular
commutes of working group conveners to the labs have become the new normal.
This is particularly the case for positions in the LHC collaborations that are
expected to have “a significant presence at CERN". While this has been largely
waived during the COVID pandemic, it is not yet clear how this policy will
evolve in future. Similarly, there are detector operations and data taking
shifts that might be moved to remote locations. A number of institutes created
remote control rooms before the start of the LHC, but the extent to which they
have been used and the emissions saved is undocumented. Generally, as the size
of the collaborations is expected to grow in size and in geographical
diversity, regional centers may be established as alternative hubs to the main
experimental site, as Fermilab can already be considered today for CMS. If
successful, such a system might reduce the travel-related carbon emissions by
more than half. For example, according to the carbon-offset provider
atmosfair, a return flight from San Francisco to Geneva has four times the
carbon emissions compared to a return flight to Chicago only.

There are additional sources of GHG emissions in the academic system which are
also relevant in particle physics. One example is travel for committee
meetings, funding reviews and the like. Data on this is scarce, but an example
exists from the European Research Council bib:Bousema , which estimates more
than 1 t CO2e emissions per review, due to travel of the international panel
as well as the international pool of applicants. Similar examples could be
found in the U.S., such as funding agency reviews of construction projects or
operations programs. When these reviews are held in person, at least 25 people
need to travel from around the country to a single location, which is
typically not chosen with a goal of minimizing climate impacts.

Finally, conference travel is a much-debated source of GHG emissions in the
academic system bib:Bjorkdahl , and the term “conference tourism" bib:Hoyer
has been coined, relating to conferences at exotic locations. Across various
disciplines, the systematic study in bib:Spinellis  estimates that typically
about 1 t CO2e are emitted per participant, and calculations from various
individual examples result in similar average emissions bib:Jaeckle ;
bib:Burtscher . Reviews of the International Conference on High Energy Physics
(ICHEP) estimate about 2 t CO2e per participant using the methodology employed
in bib:agu , which accounts not only for CO2 emissions but also for the impact
due to contrails, ozone formation and other effects. The notable exception was
the 2012 edition in Melbourne, causing per participant emissions of about four
times the average due to its remote location, highlighting to some extent the
tension between geographical inclusion and reducing travel emissions.

Likely more relevant than the average emissions is the distribution of
emissions. Often, a smaller fraction of participants produces the largest
share of emissions, e.g., in bib:Burtscher , 10% of the flights taken resulted
in over 50% of the emissions. Thus, long-haul flights are a particular
challenge. Reductions of anywhere between 94% and 98% of CO2e emissions have
been estimated for fully virtual meetings. Similarly, hub-based formats, which
eliminate long-haul travel by collecting participants at regional hubs and
link those hubs virtually, are expected to result in significant reductions
bib:agu . When applied to ICHEP, the emissions could be reduced to 15-35% of
those of a traditional one-hub ICHEP conference.

Even for fully in-person meetings, reductions of 20% of GHG emissions have
been found by optimizing the meeting location bib:Stroud ; with some even
finding variations of up to a factor of two bib:Jaeckle .

Calls for the reduction (not necessarily complete elimination) of air travel
to conferences are often met with the worry as to whether this will result in
a reduced amount of networking, dissemination of results and similar. It has
long been assumed that there is a direct correlation (or even causation)
between the amount of air travel by a scientist and their scientific impact.
Quantitative studies are now starting to question this correlation. bib:Wynes
and bib:Ciers  do not find a correlation between the amount of air travel and
the h-index in their studies of researchers across multiple fields.

Nevertheless, a reduction of air travel in science would have to be
implemented carefully, in order to avoid unintended consequences. This could
include early-career researchers finding it more challenging to build their
networks, or that researchers from the Global South, who typically have to
travel further to meet colleagues or attend conferences, are cut off from
scientific exchanges. Both can be avoided by carefully evaluating for which
purposes and by whom air travel may be scientifically valuable, when in other
cases it could be substituted by a virtual exchange.

Moreover, holding meetings in hybrid or virtual form can have important co-
benefits, in particular for inclusivity, strengthening the case for
hybrid/virtual meetings. For instance bib:Sarabipour  shows a significant
increase in participation from countries outside North America, including in
Latin America and Africa, when the APS April meeting shifted to online format.
Further, bib:Skiles  found an increase in female participation (and abstract
submission from female researchers), when conferences shifted from in-person
to online format. Climate sustainability and (global) inclusivity can thus be
achieved together.

Additionally, GHG emissions at conferences are related to accommodation as
well as catering. The effect of offering fully vegetarian/vegan catering
during a four-day conferences was estimated in bib:Jaeckle  to result in about
16 (vegetarian) or 23 (vegan) t CO2e emissions for a conference series with an
average of about 1600 participants. For comparison, the GHG emissions from air
travel, which depended on the choice of location, varied between 900 and 1800
t CO2e.

##  6 Recommendations

We offer the following recommendations on how we as a field can reduce our
impact on the climate and moderate the ongoing trends of global climate
change.

  * •

New experiments and facility construction projects should report on their
planned emissions and energy usage as part of their environmental assessment,
which will be part of their evaluation criteria. These reports should be
inclusive of all aspects of activities, including construction, detector
operations, computing, and researcher activities.

  * •

U.S. laboratories should be involved in a review across all international
laboratories to ascertain whether emissions are reported clearly and in a
standardized way. This will also allow other U.S. particle physics research
centers (including universities) to use those standards for calculating their
emissions across all scopes.

  * •

Using the reported information as a guide, all participants in particle
physics – laboratories, experiments, universities, and individual researchers
– should take steps to mitigate their impact on climate change by setting
concrete reduction goals and defining pathways to reaching them by means of an
open and transparent process involving all relevant members of the community.
This may include spending a portion of research time on directly tackling
challenges related to climate change in the context of particle physics.

  * •

U.S. laboratories should invest in the development and affordable deployment
of next-generation digital meeting spaces in order to minimize the travel
emissions of their users. Moreover the particle physics community should
actively promote hybrid or virtual research meetings and travel should be more
fairly distributed between junior and senior members of the community. For in-
person meetings, the meeting location should be chosen carefully such as to
minimize the number of long-distance flights and avoid layovers.

  * •

Long-term projects should consider the evolving social and economic context,
such as the expectation of de-carbonized electricity production by 2040, and
the possibility of carbon pricing that will have an impact on total project
costs.

  * •

All U.S. particle physics researchers should actively engage in learning about
the climate emergency and about the climate impact of particle-physics
research.

  * •

The U.S. particle physics community should promote and publicize their actions
surrounding the climate emergency to the general public and other scientific
communities.

  * •

The U.S. particle physics community and funding agencies should engage with
the broader international community to collectively reduce emissions.

##  7 Acknowledgments

The authors would like to thank many colleagues for crucial input and
discussions and in particular Roberto Guida, Johannes Gutleber, Sonja Kleiner,
Karen Kosky, Beatrice Mandelli, Eric Mieland, and Marlene Turner.

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# Limits via relations

Sergei O. Ivanov<EMAIL_ADDRESS><EMAIL_ADDRESS>Beijing
Yanqi Lake Institute of Applied Mathematics (BIMSA)  ,  Roman Mikhailov  Saint
Petersburg University, 7/9 Universitetskaya nab., St. Petersburg, 199034
Russia  and  Fedor Pavutnitskiy  Beijing Yanqi Lake Institute of Applied
Mathematics (BIMSA)

###### Abstract.

In this paper, we study operations on functors in the category of abelian
groups simplar to the derivation in the sense of Dold-Puppe. They are defined
as derived limits of a functor applied to the relation subgroup over a
category of free presentations of the group. The integral homology of the
Eilenberg-Maclane space $K(\mathbb{Z},3)$ appears as a part of description of
these operations applied to symmetric powers.

The work of the second named author was performed at the Saint Petersburg
Leonhard Euler International Mathematical Institute and supported by the
Ministry of Science and Higher Education of the Russian Federation (agreement
no. 075–15–2022–287).

##  1\. Introduction

Theory of limits and colimits in the category of presentations is studied in
details in the series of papers. Speaking informally, theory of (derived)
(co)limits is a way to design functors and natural transformations in
algebraic categories with enough projective objects.

In this paper, we introduce an operation on functors in the category of
abelian groups, which looks like the derivation in the sense of Dold-Puppe. In
particular, the short exact sequences of functors give rise to a long exact
sequence of these new “derived” ones.

For an abelian group $A,$ consider the category of free presentations ${\sf
Pres}(A),$ with object being free groups $F$ with epimorphisms $F\to A.$
Morphisms are group homomorphisms over $A$. For any functor from the category
of presentations $\mathcal{F}:{\sf Pres}(A)\to{\sf Ab}$ to the category of
abelian groups, one can consider the derived limits ${\sf
lim}^{i}\mathcal{F},\ i\geq 0.$ These limits depend only on $A.$ Moreover, if
we denote by ${\sf Pres}_{\sf Ab}$ the category, whose objects are
epimorphisms from free abelian groups to abelian groups $F\twoheadrightarrow
A,$ and morphisms $(F\twoheadrightarrow A)\to(F^{\prime}\twoheadrightarrow
A^{\prime})$ are commutative squares, then any functor $\mathcal{F}:{\sf
Pres}\to{\sf Ab}$ defines a functor ${\sf Ab}\to{\sf Ab}$ given by
$A\mapsto\mathsf{lim}_{{\sf Pres}(A)}\mathcal{F}.$

Let $F$ be an endofunctor in the category of abelian groups. Consider an
object of ${\sf Pres}(A),$ $0\to R\to F\to A\to 0,$ i.e. denote by $R$ the
kernel of an epimorphism $F\to A.$ Our operation is the following:

$F\mapsto{\sf lim}^{i}FR_{A}:={\sf lim}^{i}F(R),\ i\geq 0.$

Let’s show how it works on the main examples. Denote by $\otimes^{n},{\sf
S}^{n},\Lambda^{n},\Gamma^{n},\ n\geq 0,$ the tensor, symmetric, exterior and
divided powers respectively. There are the following natural isomorphisms

$\displaystyle{\sf lim}^{i}\otimes^{n}R_{A}=L_{n-i}\otimes^{n}(A),$
$\displaystyle{\sf lim}^{i}\Lambda^{n}R_{A}=L_{n-i}{\sf S}^{n}(A),$
$\displaystyle{\sf lim}^{i}\Gamma^{n}R_{A}=L_{n-i}\Lambda^{n}(A),$

for $i=0,\dots,n.$ Here $L_{i}$ are derived functors in the sense of Dold-
Puppe. The proof follows from the Koszul-type sequences and properties of
limits, see [6, Th. 8.1].

The case of functors ${\sf lim}^{i}{\sf S}^{n}R_{A},$ is more complicated and
is the main subject of this paper. We show that, for a free abelian group $A,$
there is the following description

$\displaystyle\mathsf{lim}^{2}{\sf S}^{2}R_{A}=\tilde{\otimes}^{2}(A)$
$\displaystyle\mathsf{lim}^{3}{\sf S}^{3}R_{A}=\tilde{\otimes}^{3}(A)$
$\displaystyle\mathsf{lim}^{2}{\sf S}^{3}R_{A}=A\otimes\mathbb{Z}/3.$

with all other limits to be zero for ${\sf S}^{2},{\sf S}^{3}.$ Here
$\tilde{\otimes}^{2},\tilde{\otimes}^{3}$ are anty-symmetric tensor square and
cube respectively.

The structure of functors ${\sf lim}^{i}{\sf S}^{n}(R)$ is complicated. We
show that, for $A=\mathbb{Z},$ there is an isomorphism

$H_{n}(K(\mathbb{Z},3),\mathbb{Z})\cong\bigoplus_{d\geq
2}\mathsf{lim}^{n-2d+1}S^{d}R_{\mathbb{Z}}.$

Here $H_{n}(K(\mathbb{Z},3),\mathbb{Z})$ is the $n$th integral homology group
of the Eilenberg-MacLane space $K(\mathbb{Z},3).$ This description follows
from the following statement (see Corollary 5.3): for a free finitely
generated abelian group $A$ there are non-natural isomorphisms of abelian
groups

$\mathsf{lim}^{i}{\sf
S}^{d}R_{A}\cong\begin{cases}L_{i-1}\Gamma^{d}(A,1),&i<d,\\\
L_{d-1}\Gamma^{d}(A,1)\oplus L_{d}\Gamma^{d}(A,1),&i=d,\\\ 0,&i>d.\end{cases}$

Here $L_{i}\Gamma^{d}(A,1)$ are derived functors in the sense of Dold-Puppe.
The situation is interesting and rare. Usually when we have two complicated
graded functors with the property that they are non-naturally isomorphic,
there seems to be an either hidden natural isomorphism or a problem of non-
splitting sequences raises in some way. However in this case, there is no
natural isomorphism. The functorial description of the derived functors
$L_{i}\Gamma^{n}(A,1)$ for free abelian groups $A$ and well as of the integral
homology groups $H_{i}(K(A,3),\mathbb{Z})$ is given in the paper [1]. All
functors which appear in that description are known and there are no anty-
symmetric powers.

The paper is organized as follows. In Section 2 we recall needed facts on
Dold-Kan correspondence and cosimplicial groups. In Section 3, We show that,
for a polynomial functor $\Phi$ of degree $d\geq 1,$ ${\sf lim}^{i}\Phi
R_{A}=0$ for $i>d.$ Observe that the derived functors in the sense of Dold-
Puppe have the same property. The key point of the proof is based on
cosimplicial models constructed for arbitrary functors which give a way to
compute derived limits. In Section 4 we recall Kuhn duality for functors and
apply it in the context of limits. In particlular, we show how to describe the
limits via shifted devided functors of the dual functors (see Corollary 4.2).
In Section 5 we prove the mentioned above results on ${\sf lim}^{i}{\sf
S}R_{A}.$

##  2\. Cosimplicial modules

###  2.1. Reminder on Dold–Kan correspondence

In this subsection we remind some information of the Dold-Kan correspondence
that can be found in [8].

Let $\mathcal{V}$ be an abelian category. For a simplicial object $V$ of an
abelian category $\mathcal{V}$ the non-normalised complex
$\mathsf{C}_{\bullet}V$ is a chain complex whose components are
$\mathsf{C}_{n}V=V_{n}$ and the differential is defined by the alternating sum
of face maps. The normalised complex $\mathsf{N}_{\bullet}V$ can be defined in
two ways which are equivalent up to natural isomorphism: as a subcomplex of
$\mathsf{C}_{\bullet}V$ and as a quotient complex of $\mathsf{C}_{\bullet}V$
[8, Lemma 8.3.8]. For our purposes it is more convenient to define it as a
quotient complex, whose components are

(2.1) $\mathsf{N}_{n}V={\sf Coker}((s_{0},\dots,s_{n-1}):V_{n-1}^{\oplus n}\to
V_{n}).$

The map $\mathsf{C}_{\bullet}V\twoheadrightarrow\mathsf{N}_{\bullet}V$ is a
chain homotopy equivalence [8, Th.8.3.8]. If we denote by $\mathsf{D}V$ the
kernel of the map $\mathsf{C}V\to\mathsf{N}V,$ then there is a natural
splitting of the short exact sequence
$\mathsf{D}V\rightarrowtail\mathsf{C}V\twoheadrightarrow\mathsf{N}V$

(2.2) $\mathsf{C}V\cong\mathsf{N}V\oplus\mathsf{D}V,$

where $\mathsf{D}V$ is a chain contractible complex. The homology of
$\mathsf{N}V$ and $\mathsf{C}V$ is called homotopy groups of the simplicial
object

(2.3) $\pi_{*}V=H_{*}(\mathsf{N}_{\bullet}V)\cong
H_{*}(\mathsf{C}_{\bullet}V).$

The construction of the normalised complex defines a equivalence between the
category of simplicial objects and the category of non-negatively graded chain
complexes, whose inverse functor is denoted by $\mathsf{DK}$

(2.4) $\mathsf{N}_{\bullet}:\mathcal{V}^{\Delta^{\sf op}}\leftrightarrows{\sf
Ch}_{\geq 0}(\mathcal{V}):{\sf DK}_{\bullet}.$

The functor $\mathsf{DK}_{\bullet}$ can be constructed as follows. For a chain
complex $U$ we define a simplicial object $\mathsf{DK}_{\bullet}U,$ whose
components are

(2.5) $\mathsf{DK}_{n}U=\bigoplus_{\sigma:[n]\twoheadrightarrow[k]}U_{k},$

where the summation is taken by all surjective order-preserving maps
$\sigma:[n]\twoheadrightarrow[k],$ where $0\leq k\leq n$. If $f:[m]\to[n]$ is
an order preserving map, then the map
$f^{*}:\mathsf{DK}_{n}U\to\mathsf{DK}_{m}U$ is defined so that its component
$(f^{*})_{\tau,\sigma}:U_{k}\to U_{l}$ from the direct summand indexed by
$\sigma:[n]\twoheadrightarrow[k]$ to the direct summand indexed by
$\tau:[m]\twoheadrightarrow[l]$ has the following form

(2.6) $(f^{*})_{\tau,\sigma}=\begin{cases}{\sf id}_{U_{k}},&\sigma f=\tau,\\\
\partial^{U}_{k},&\sigma f=d^{k}\tau,\\\ 0,&\text{else.}\end{cases}$

Note that the construction of $\mathsf{DK}$ commutes with additive functors
i.e. for any additive functor between abelian categories
$\Phi:\mathcal{V}\to\mathcal{V}^{\prime}$ there is an natural isomorphism

(2.7) $\mathsf{DK}_{\bullet}(\Phi(U))\cong\Phi(\mathsf{DK}_{\bullet}(U)).$

###  2.2. Dold-Kan correspondence for cosimplicial modules

Since the dual of an abelian category is also abelian, there is a dual version
of this picture [8, Cor./Def. 8.4.3]. The non-normalised cochain complex
$\mathsf{C}^{\bullet}V$ of a cosimplicial object $V$ is a cochain complex,
whose components are $\mathsf{C}^{n}V=V^{n},$ and the differential is defined
as the alternating sum of coface maps. The normalised cochain complex
$\mathsf{N}^{\bullet}V$ is a subcomplex of $\mathsf{C}^{\bullet}V$ whose
components are defined as

(2.8) $\mathsf{N}^{n}V={\sf
Ker}((s^{0},\dots,s^{n-1})^{T}:V^{n}\to(V^{n-1})^{\oplus n+1}).$

The monomorphiam $\mathsf{N}^{\bullet}V\to\mathsf{C}^{\bullet}V$ is a
quasiisomorphism and cohomotopy groups of $V$ are defined as

(2.9) $\pi^{*}V=H^{*}(\mathsf{N}^{\bullet}V)\cong
H^{*}(\mathsf{C}^{\bullet}V).$

The Dold-Kan correspondence is an equivalence between the category of
cosimplicial objects and the category of non-negatively graded cochain
complexes

(2.10) $\mathsf{N}^{\bullet}:\mathcal{V}^{\Delta}\leftrightarrows{\sf
CoCh}^{\geq 0}(\mathcal{V}):{\sf DK}^{\bullet}.$

######  Proposition 2.1.

Let $V$ be a cosimplicial module over a ring $\mathbb{K}.$ Then the components
of the normalized cochain complex $\mathsf{N}V$ are intersections of kernels
of codegeneracy maps

(2.11) $\mathsf{N}^{n}V=\bigcap_{i=0}^{n}{\sf Ker}(s^{i}:V^{n}\to V^{n-1}),$

and the differential on $\mathsf{N}V$ is given by the restriction of the
alternating sum of cofaces $\sum(-1)^{i}d^{i}.$

###### Proof.

It follows from the fact that $\mathsf{N}^{\bullet}V$ is a subcomplex of
$\mathsf{C}^{\bullet}V$ and the formula (2.8). ∎

######  Proposition 2.2.

If we treat a $\mathbb{K}$-homomorphism $f:U^{0}\to U^{1}$ as a cochain
complex concentrated in degrees $0$ and $1,$ we obtain that the cosimplicial
module $\mathsf{DK}^{\bullet}(f)$ has components

(2.12) $\mathsf{DK}^{n}(f)=(U^{1})^{\oplus n}\oplus U^{0}.$

Its coface and codegeneracy maps are defined by

(2.13)
$\begin{split}d^{i}_{\mathsf{DK}^{\bullet}(f)}(x_{0},\dots,x_{n-2},y)&=\begin{cases}(0,x_{0},\dots,x_{n-2},y),&i=0\\\
(x_{0},\dots,x_{i-1},x_{i-1},\dots,x_{n-2},y),&1\leq i\leq n-1\\\
(x_{0},\dots,x_{n-1},f(y),y),&i=n\end{cases}\\\
s^{i}_{\mathsf{DK}^{\bullet}(f)}(x_{0},\dots,x_{n},y)&=(x_{0},\dots,\hat{x}_{i},\dots,x_{n},y).\end{split}$

###### Proof.

It is easy to check that the described cosimplicial module is well defined and
that its normalized complex is $U^{0}\to U^{1}.$ ∎

##  3\. Cochain complex of crossed effects computing higher limits

###  3.1. Cross effects of functors

Let $\mathbb{K}$ be a ring and $A_{1},\dots,A_{n}$ is a collection of
$\mathbb{K}$-modules. For $1\leq j\leq n$ we denote by

(3.1) ${\sf pr}^{j}:\bigoplus_{i=1}^{n}A_{i}\longrightarrow\bigoplus_{i\neq
j}A_{i},\hskip 28.45274pt{\sf em}^{j}:\bigoplus_{i\neq
j}A_{i}\longrightarrow\bigoplus_{i=1}^{n}A_{i}$

the canonical projection and the canonical embedding.

Now assume that $\Phi:{\sf Mod}(\mathbb{K})\to{\sf Mod}(\mathbb{K})$ is a
functor to an abelian category such that $\Phi(0)=0.$ For $n\geq 0$ the $n$-th
crossed effect of $\Phi$ is a functor ${\sf
Mod}(\mathbb{K})^{n}\to\mathcal{\sf Mod}(\mathbb{K})$ given by

(3.2)
$\Phi(A_{1}|\dots|A_{n})=\mathsf{Ker}\left(\Phi\left(\bigoplus_{i=1}^{n}A_{i}\right)\longrightarrow\bigoplus_{i=1}^{n}\Phi\left(\bigoplus_{j\neq
i}A_{j}\right)\right),$

where the homomorphisms are induced by the canonical projections. Note that if
$A_{i}=0$ for some $i,$ then the crossed effect vanishes

(3.3) $\Phi(A_{1}|\dots|0|\dots|A_{n})=0.$

The cross effect is a direct summand of $\Phi(\bigoplus_{i=1}^{n}A_{i})$ and
there is a decomposition (see [4, p.2], [5, p.1149], [2, p.18])

(3.4) $\Phi(\bigoplus_{i=1}^{n}A_{i})=\bigoplus_{s=0}^{n}\ \bigoplus_{1\leq
j_{1}<\dots<j_{s}\leq n}\Phi(A_{j_{1}}|\dots|A_{j_{s}}).$

A functor $\Phi$ is called polynomial (in the sense of Eilenberg-Mac Lane) of
degree $\leq d$ if $\Phi_{d+1}=0.$

######  Lemma 3.1.

There are equations

(3.5) $\begin{split}\mathsf{Ker}(\Phi({\sf pr}^{j}))&=\bigoplus_{s=1}^{n}\
\bigoplus_{j\in\\{j_{1}<\dots<j_{s}\\}}\Phi(A_{j_{1}}|\dots|A_{j_{s}})\\\
\mathsf{Im}(\Phi({\sf em}^{j}))&=\bigoplus_{s=1}^{n}\
\bigoplus_{j\notin\\{j_{1}<\dots<j_{s}\\}}\Phi(A_{j_{1}}|\dots|A_{j_{s}})\end{split}$

Moreover, we have

(3.6) $\bigcap_{j=1}^{n-1}\mathsf{Ker}(\Phi({\sf
pr}^{j}))=\Phi(A_{1}|\dots|A_{n-1})\oplus\Phi(A_{1}|\dots|A_{n}).$

###### Proof.

We can think about the sum $\bigoplus_{i\neq j}A_{i}$ as about the sum
$\bigoplus_{i=1}^{n}B_{i},$ where $B_{i}=A_{i}$ for $i\neq j$ and $B_{j}=0.$
Then ${\sf pr}^{j}$ can be redefined as ${\sf
pr}^{j}=1_{A_{1}\oplus\dots\oplus A_{j-1}}\oplus 0_{A_{i},0}\oplus
1_{A_{j+1}\oplus\dots\oplus A_{n}}$ and ${\sf em}^{j}$ can be redefined as
${\sf em}^{j}=1_{A_{1}\oplus\dots\oplus A_{j-1}}\oplus 0_{0,A_{i}}\oplus
1_{A_{j+1}\oplus\dots\oplus A_{n}}.$ Then, using the functoriality of the
crossed effect and (3.3), we obtain (3.5). The formula (3.6) follows from
(3.5). ∎

###  3.2. A chain complex of crossed effects associated with a morphism

For a homomorphism $\varphi:B\to A$ and $0\leq i\leq n$ we consider the
following maps

(3.7) $\Delta^{i,n}_{\varphi}:A^{\oplus n}\oplus B\longrightarrow A^{\oplus
n+1}\oplus B$

defined by

(3.8)
$\Delta^{i,n}_{\varphi}(a_{1},\dots,a_{n},b)=\begin{cases}(a_{1},\dots,a_{i},a_{i},\dots,a_{n},b),&i<n\\\
(a_{1},\dots,a_{n},\varphi(b),b),&i=n\end{cases}$

We also use the following notation for the canonical projections and
embeddings

(3.9) ${\sf pr}^{n+1}_{\varphi}:A^{\oplus n}\oplus B\longrightarrow A^{\oplus
n},\hskip 28.45274pt{\sf em}^{n+1}_{\varphi}:A^{\oplus n}\longrightarrow
A^{\oplus n}\oplus B,$

and set $\Delta^{i,n}_{A}:=\Delta^{i,n}_{1_{A}},$ ${\sf pr}^{n}_{A}:={\sf
pr}^{n}_{1_{A}},$ ${\sf em}^{n}_{A}={\sf em}^{n}_{1_{A}}.$

######  Lemma 3.2.

The following equations are satisfied

  1. (1)

Cosimplicial equations
$\Delta^{j,n+1}_{\varphi}\Delta^{i,n}_{\varphi}=\Delta^{i,n+1}_{\varphi}\Delta^{j-1,n}_{\varphi},$
where $i<j;$

  2. (2)

${\sf pr}^{n+1}_{\varphi}\Delta^{i,n}_{\varphi}=\Delta^{i,n-1}_{A}{\sf
pr}^{n}_{\varphi},$ for $i\neq n;$

  3. (3)

${\sf pr}^{n+1}_{\varphi}\Delta^{n,n}_{\varphi}=1_{A^{\oplus
n}}\oplus\varphi.$

  4. (4)

$\Delta^{i,n}_{\varphi}{\sf em}^{n}_{\varphi}={\sf
em}^{n+1}_{\varphi}\Delta^{i,n-1}_{A},$ for $i\neq n;$

  5. (5)

$\Delta^{n,n}_{\varphi}{\sf em}^{n}_{\varphi}={\sf em}^{n+1}_{\varphi}{\sf
em}^{n}_{A}$

  6. (6)

${\sf pr}^{n+1}_{\varphi}\>\Delta_{\varphi}^{i,n}\>{\sf
em}^{n}_{\varphi}=\Delta_{A}^{i,n-1}$ for $i\neq n;$

  7. (7)

${\sf pr}^{n+1}_{\varphi}\>\Delta_{\varphi}^{n,n}\>{\sf em}^{n}_{\varphi}={\sf
em}^{n}_{A}.$

###### Proof.

Direct computation. ∎

One can say that we obtain a coaugmented semi-co-simplicial module

(3.10) ${B}$${A\oplus B}$${A^{\oplus 2}\oplus
B}$${\dots}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

Further we set

(3.11) $\Phi_{[n|1]}(A,B)=\Phi(A|\dots|A|B)$

and define morphisms $h^{i}_{\varphi}:\Phi_{[n|1]}(A,B)\to\Phi_{[n+1|1]}(A,B)$
as compositions

(3.12) ${\Phi_{[n|1]}(A,B)}$${\Phi_{[n+1|1]}(A,B)}$${\Phi(A^{\oplus n}\oplus
V)}$${\Phi(A^{n+1}\oplus
B)}$$\leftarrow$$\rightarrow$$\scriptstyle{h^{i}_{\varphi}}$$\leftarrowtail$$\rightarrow$$\leftarrow$$\rightarrow$$\scriptstyle{\Phi(\Delta^{i}_{\varphi})}$$\leftarrow$$\twoheadrightarrow$

Cosimplicial identities for $\Delta^{i}_{\varphi}$ imply cosimplicial
identities for $h^{i}_{\varphi}$:
$h^{j}_{\varphi}h^{i}_{\varphi}=h^{i}_{\varphi}h^{j-1}_{\varphi}$ where $i<j$.
So we can consider a cochain complex $C_{\Phi}(\varphi),$ whose components are
$C_{\Phi}(\varphi)^{n}=\Phi_{[n|1]}(A,B)$ and the differential is defined as
the alternating sum of $h_{i}$

(3.13) $C_{\Phi}(\varphi):\hskip
14.22636pt\Phi(B)\overset{h^{0}_{\varphi}}{\longrightarrow}\Phi(A|B)\overset{h^{0}_{\varphi}-h^{1}}{\longrightarrow}\Phi(A|A|B)\overset{h^{0}_{\varphi}-h^{1}_{\varphi}-h^{2}_{\varphi}}{\longrightarrow}\dots$

We will also use the notation $C_{\Phi}(A)=C_{\Phi}({\sf id}_{A}).$

It is easy to see that the construction of the complex is natural by the
morphism $\varphi$ is natural by the morphism in the following sense. Any
commutative square

(3.14)
${B}$${A}$${B^{\prime}}$${A^{\prime}.}$$\leftarrow$$\rightarrow$$\scriptstyle{\alpha}$$\leftarrow$$\rightarrow$$\scriptstyle{\psi}$$\leftarrow$$\rightarrow$$\scriptstyle{\beta}$$\leftarrow$$\rightarrow$$\scriptstyle{\psi}$

induces a morphism

(3.15) $C_{\Phi}(\alpha,\beta):C_{\Phi}(\varphi)\longrightarrow
C_{\Phi}(\psi).$

###  3.3. The standard complex for higher limits

For any object $c$ of any category with pairwise coproducts $\mathsf{C}$ (not
necessarily with an initial object) we denote by $c^{\sqcup n}$ the coproduct
$\coprod_{i=0}^{n-1}c$ with embeddings $\alpha_{i}:c\to c^{\sqcup n}$ indexed
by $0\leq i\leq n-1.$ Any morphism $f:c^{\sqcup n}\to d$ will be written as
$f=(f_{0},\dots,f_{n-1}),$ where $f_{i}=f\alpha_{i}.$ We consider the
cosimplicial object $c^{\bullet}$ whose components are defined by

(3.16) $(c^{\bullet})^{n}=c^{\sqcup n+1}$

the coface maps $d^{i}:c^{\sqcup n}\to c^{\sqcup n+1}$ and degeneracy maps
$s^{i}:c^{\sqcup n+2}\to c^{\sqcup n+1}$ are defined by

(3.17) $d^{i}=(\alpha_{0},\dots,\hat{\alpha}_{i},\dots,\alpha_{n}),\hskip
28.45274pts^{i}=(\alpha_{0},\dots,\alpha_{i},\alpha_{i},\dots,\alpha_{n})$

for $0\leq i\leq n.$

######  Theorem 3.3 ([7, Th. 2.12]).

Let $\mathsf{C}$ be a strongly connected category with pairwise coproducts,
$\mathbb{K}$ be a ring and $\Phi:\mathsf{C}\to{\sf Mod}(\mathbb{K})$ be a
functor. Then for any object $c$ of $\mathsf{C}$ and any $i\geq 0$ we have an
isomorphism

(3.18) $\mathsf{lim}^{i}\>\Phi\cong\pi^{i}\Phi(c^{\bullet}).$

In particular, the right-hand side of the isomorphism is independent of $c.$

Further we will use the notation for the chain complex

(3.19) ${\sf Lim}_{(c)}\Phi=N^{\bullet}(\Phi(c^{\bullet}))$

######  Remark 3.4.

The Theorem 3.3 was proved in [7, Th. 2.12] only for the case
$\mathbb{K}=\mathbb{Z},$ but the proof for any $\mathbb{K}$ can be done
without any changes.

###  3.4. The relation functor

Further we consider a $\mathbb{K}$-module $A$ and the category of its
presentations $\mathsf{Pres}(A).$ It is easy to check that this category is
strongly connected and has pairwise coproducts $p\sqcup p^{\prime}:F\oplus
F^{\prime}\twoheadrightarrow A$. This category is a full subcategory of the
comma category ${\sf Mod}(\mathbb{K})\downarrow A$, whose objects are
epimorphisms from free modules $p:F\twoheadrightarrow A.$ The functor

(3.20) ${\sf R}_{A}:\mathsf{Pres}(A)\longrightarrow{\sf
Mod}(\mathbb{K}),\hskip 28.45274pt{\sf R}_{A}(p)={\sf Ker}(p)$

will be called relation functor. For a giver presentation
$p:F\twoheadrightarrow A$ we will always use the notation $R={\sf R}_{A}(p).$

######  Proposition 3.5.

For any presentation $p:F\twoheadrightarrow A,$ if we denote by
$\varphi_{p}:R\to F$ the embedding of the kernel, there is a natural
isomorphism of cosimplicial modules

(3.21) ${\sf R}_{A}(p^{\bullet})\cong\mathsf{DK}^{\bullet}(\varphi_{p})$

(see Proposition 2.2).

###### Proof.

The coproduct in the category $\mathsf{Pres}(A)$ is defined by $p\sqcup
p^{\prime}:F\oplus F^{\prime}\to A.$ Therefore $p^{\sqcup n+1}:F^{\oplus
n+1}\to A.$ The kernel of $p^{\sqcup n+1}$ consists of tuples
$(a_{0},\dots,a_{n})$ such that $\sum_{i=0}^{n}a_{i}\in R.$ Consider a map

(3.22) $\theta_{n}:{\sf Ker}(p^{\sqcup n+1})\longrightarrow F^{\oplus n}\oplus
R,$ (3.23) $\theta_{n}(a_{0},\dots,a_{n})=(a_{0},\ a_{0}+a_{1},\
a_{0}+a_{1}+a_{2},\ \dots,\ \sum_{i=0}^{n}a_{i}).$

The homomorphism $\theta_{n}$ is an isomorphism with the inverse given by

(3.24) $\theta_{n}^{-1}(a_{0},\dots,a_{n-1},r)=(a_{0},\ a_{1}-a_{0},\ \dots,\
a_{n-1}-a_{n-2},\ r-a_{n-1}).$

By the definition (3.17) the cofaces and codeneracy maps of ${\sf
R}_{A}(p^{\bullet})$ are defined by the formulas

(3.25) $\begin{split}d^{i}_{{\sf
R}_{A}(p^{\bullet})}(a_{0},\dots,a_{n})&=(a_{0},\dots,a_{i-1},0,a_{i},\dots,a_{n}),\\\
s^{i}_{{\sf
R}_{A}(p^{\bullet})}(a_{0},\dots,a_{n})&=(a_{0},\dots,a_{i}+a_{i+1},\dots,a_{n}).\end{split}$

A direct computation shows that $\theta_{n}$ respects the face and degeneracy
maps. ∎

###  3.5. Limits of compositions with the relation functor

Let $\mathbb{K}$ be a commutative ring and

(3.26) $\Phi:{\sf Mod}(\mathbb{K})\longrightarrow{\sf Mod}(\mathbb{K})$

be a functor. For any $\mathbb{K}$-module $A$ we are interested in the higher
limits of the composition with the relation functor

(3.27) $\Phi{\sf R}_{A}:{\sf Pres}(A)\longrightarrow{\sf Mod}(\mathbb{K}).$

######  Theorem 3.6.

Let $\Phi:{\sf Mod}(\mathbb{K})\to{\sf Mod}(\mathbb{K})$ be a functor such
that $\Phi(0)=0$ and $p:F\twoheadrightarrow A$ be any presentation. Denote by
$\varphi_{p}:R\to F$ the embedding of the kernel. Then there are natural
isomorphisms

(3.28) $\begin{split}{\sf Lim}_{(p)}\Phi{\sf R}_{A}&\cong{\sf
Cone}(C_{\Phi}(1_{F},\varphi_{p}):C_{\Phi}(\varphi)\to
C_{\Phi}(F))[-1].\end{split}$

###### Proof.

Set $N:={\sf Lim}_{(p)}\Phi{\sf R}_{A}.$ Since ${\sf
R}^{\bullet}(p)^{n}=F^{n}\oplus R,$ and $s^{i}_{{\sf R}^{\bullet}(p)}={\sf
pr}^{i}:F^{n}\oplus R\to F^{n-1}\oplus R$ are projections, Lemma 3.1 implies
that components of the normalised complex can be described as

(3.29) $N^{n}=\Phi_{[n|1]}(F,R)\oplus\Phi_{[n]}(F).$

So now we need to describe the differentials $\partial^{n}_{N},$ which are
restrictions of the alternating sum $\sum_{i=0}^{n-1}(-1)^{i}d^{i}_{\Phi({\sf
R}^{\bullet}(p))},$ and prove that they coincide with the differential for the
cone.

Further in this prove we will use the following notations $d^{i}=d^{i}_{{\sf
R}^{\bullet}(p)};$ $d^{i}_{\Phi}=d^{i}_{\Phi({\sf
R}^{\bullet}(p))}=\Phi(d^{i}_{{\sf R}^{\bullet}(p)});$

(3.30) $\pi:\Phi(F^{\oplus n}\oplus R)\to\Phi_{[n|1]}(F,R),\hskip
28.45274pt\pi^{\prime}:\Phi(F^{\oplus n}\oplus R)\to\Phi_{[n]}(F)$

are the projections;

(3.31) $\rho:\Phi_{[n|1]}(F,R)\to\Phi(F^{\oplus n}\oplus R),\hskip
28.45274pt\rho^{\prime}:\Phi_{[n]}(F)\to\Phi(F^{\oplus n}\oplus R)$

are the embeddings. Therefore, $\partial^{n}$ can be described as the
alternating sum of matrices

(3.32) $M_{i}\coloneq\left(\begin{matrix}\pi d^{i}_{\Phi}\rho&\pi
d^{i}_{\Phi}\rho^{\prime}\\\
\pi^{\prime}d^{i}_{\Phi}\rho&\pi^{\prime}d^{i}_{\Phi}\rho^{\prime}\end{matrix}\right)$

for $0\leq i\leq n+1.$ We will also consider the maps

(3.33) $\tilde{\pi}:\Phi(F^{\oplus n})\longrightarrow\Phi_{[n]}(F),\hskip
28.45274pt\tilde{\rho}:\Phi_{[n]}(F)\longrightarrow\Phi(F^{\oplus n}).$

and the maps

(3.34) ${\sf em}^{n}_{\varphi}:F^{\oplus n}\longrightarrow F^{\oplus n}\oplus
R,\hskip 28.45274pt{\sf pr}^{n}_{\varphi}:F^{\oplus n}\oplus R\longrightarrow
F^{\oplus n}.$

Then we have

(3.35) $\pi^{\prime}=\tilde{\pi}\Phi({\sf pr}^{n}_{\varphi}),\hskip
28.45274pt\rho^{\prime}=\Phi({\sf em}^{n}_{\varphi})\tilde{\rho}.$

The map $d^{0}$ is the embedding $F^{n}\oplus R\to F^{n+1}\oplus R,$ which is
trivially mapped to the first summand. Therefore the image of $d^{0}_{\Phi}$
is the direct sum of all crossed effects, where the first summand does not
appear (Lemma 3.1). Therefore $\pi d^{0}_{\Phi}=0$ and
$\pi^{\prime}d^{0}_{\Phi}=0,$ and hence $M_{0}=0.$

Further we will freely use the equations from Lemma 3.2 for the computations.
Now assume that $0\leq i\leq n.$ Then $d^{i}=\Delta^{i-1}_{\varphi}.$ So we
have

(3.36) $\pi
d^{i}_{\Phi}\rho=\pi\Phi(\Delta^{i-1}_{\varphi})\rho=h^{i-1}_{\varphi},$

and

(3.37) $\pi^{\prime}d^{i}_{\Phi}\rho=\tilde{\pi}\Phi({\sf
pr}^{n+1}_{\varphi}\>\Delta^{i-1}_{\varphi})\rho=\tilde{\pi}\Phi(\Delta^{i-1}_{F}\>{\sf
pr}^{n}_{\varphi})\rho=0$

because by Lemma 3.1 $\Phi({\sf pr}^{n}_{\varphi})\rho=0.$ Further we have

(3.38) $\pi d^{i}_{\Phi}\rho^{\prime}=\pi\Phi(\Delta_{\varphi}^{i-1}{\sf
em}^{n}_{\varphi})\tilde{\rho}=\pi\Phi({\sf
em}^{n+1}_{\varphi}\Delta_{F}^{i-1})\tilde{\rho}=0$

because by Lemma 3.1 $\pi\Phi({\sf em}_{\varphi}^{n+1})=0$. Finally we have

(3.39) $\pi^{\prime}d^{i}_{\Phi}\rho^{\prime}=\tilde{\pi}\Phi({\sf
pr}^{n+1}_{\varphi}\Delta^{i-1}_{\phi}{\sf
em}^{n}_{\varphi})\tilde{\rho}=\tilde{\pi}\Phi(\Delta^{i-1}_{F})\tilde{\rho}=h^{i-1}_{F}.$

Therefore for $1\leq i\leq n$ we have

(3.40) $M_{i}=\left(\begin{matrix}h^{n}_{\varphi}&0\\\
0&h^{i-1}_{F}\end{matrix}\right).$

Now assume that $i=n+1.$ Then $d^{n+1}=\Delta^{n}_{\varphi}.$ Similarly to the
previous case we obtain

(3.41) $\pi d^{i}_{\Phi}\rho=h^{n}_{\varphi}.$

Further we have

(3.42) $\pi^{\prime}d^{n+1}_{\Phi}\rho=\tilde{\pi}\Phi({\sf
pr}^{n+1}_{\varphi}\Delta^{n}_{\varphi})\rho=\tilde{\pi}\Phi(1\oplus\varphi)\rho=\Phi_{[n|1]}(1,\varphi),$

and

(3.43) $\pi d^{n+1}_{\Phi}\rho^{\prime}=\pi\Phi(\Delta_{\varphi}^{n}{\sf
em}^{n}_{\varphi})\tilde{\rho}=\pi\Phi({\sf em}^{n+1}_{\varphi}{\sf
em}^{n}_{F})\tilde{\rho}=0,$

because $\pi\Phi({\sf em}^{n+1}_{\varphi})=0.$ Finally we have

(3.44) $\pi^{\prime}d^{n+1}_{\Phi}\rho^{\prime}=\tilde{\pi}\Phi({\sf
pr}^{n+1}_{\varphi}\Delta^{n}_{\varphi}{\sf
em}^{n}_{\varphi})\tilde{\rho}=\tilde{\pi}\Phi({\sf
em}_{F}^{n})\tilde{\rho}=0,$

because $\tilde{\pi}\Phi({\sf em}_{F}^{n})=0.$ Therefore

(3.45) $M_{n+1}=\left(\begin{matrix}h^{n}_{\varphi}&0\\\
\Phi_{[n|1]}(1,\varphi)&0\end{matrix}\right).$

Therefore the differential will be given by

(3.46)
$\partial^{n}_{N}=\left(\begin{matrix}-\sum_{i=0}^{n}(-1)^{i}h^{i}_{\varphi}&0\\\
(-1)^{n+1}\Phi_{[n|1]}(1,\varphi)&-\sum_{i=0}^{n-1}(-1)^{i}h^{i}_{F}\end{matrix}\right).$

It follows that

(3.47)
$\partial^{n}_{N[1]}=\left(\begin{matrix}\partial^{n+1}_{C_{\Phi}(\varphi)}&0\\\
(-1)^{n}\Phi_{[n+1|1]}(1,\varphi)&\partial^{n}_{C_{\Phi}(F)}\end{matrix}\right).$

Consider an automorphism $\theta^{n}:N^{n}\to N^{n}$ defined by a matrix
$\left(\begin{smallmatrix}(-1)^{n}&0\\\ 0&1\end{smallmatrix}\right)$ Then

(3.48)
$(\theta^{n+1})^{-1}\partial^{n}_{N[1]}\theta^{n}=\left(\begin{matrix}-\partial^{n+1}_{C_{\Phi}(\varphi)}&0\\\
\Phi_{[n+1|1]}(1,\varphi)&\partial^{n}_{C_{\Phi}(F)}\end{matrix}\right).$

The right-hand side of the above formula is the formula for the differential
of the cone. Therefore, $\theta^{n}$ defines an isomorphism between $N[1]$ and
${\sf Cone}(C_{\Phi}(1_{F},\varphi):C_{\Phi}(\varphi)\to C_{\Phi}(F)).$ ∎

######  Corollary 3.7.

Under the assumption of Theorem 3.6, if $A$ is a free $\mathbb{K}$-module, we
obtain

(3.49) ${\sf Lim}_{({\sf id})}\Phi{\sf R}_{A}\cong C_{\Phi}(A)[-1]$

######  Corollary 3.8.

If $\Phi$ is a polynomial functor of degree $d,$ then

(3.50) $\mathsf{lim}^{i}\Phi{\sf R}_{A}=0,\hskip 28.45274pti>d.$

##  4\. Kuhn duality and higher limits

In this section we assume that $\mathbb{K}$ is a principal ideal domain. In
this case, for a finitely generated module $A$ and a finitely generated
presentation $R\rightarrowtail F\twoheadrightarrow A$ the module $R$ is also
free and finitely generated.

For a module $A$ we set $A^{\vee}={\sf Hom}_{\mathbb{K}}(A,\mathbb{K}).$ For a
functor $\Phi:{\sf Mod}(\mathbb{K})\to{\sf Mod}(\mathbb{K})$ we denote by
$\Phi^{\\#}:{\sf Mod}(\mathbb{K})\to{\sf Mod}(\mathbb{K})$ the functor defined
by the formula

(4.1) $\Phi^{\\#}(A):=\Phi(A^{\vee})^{\vee}.$

The functor $\Phi^{\\#}$ is known as Khun dual to the functor $\Phi.$

######  Proposition 4.1.

Let $A$ be a finitely generated $\mathbb{K}$-module, and
$p:F\twoheadrightarrow A$ be a finitely generated presentation, then there is
an isomorphism

(4.2) $({\sf Lim}_{(p)}\Phi{\sf R}_{A})^{\vee}\cong
N_{\bullet}(\Phi^{\\#}({\sf DK}_{\bullet}(F^{\vee}\to R^{\vee}))).$

If we also assume that $\Phi$ takes finitely generated free modules to
finitely generated free modules, we obtain

(4.3) ${\sf Lim}_{(p)}\Phi{\sf R}_{A}\cong(N_{\bullet}(\Phi^{\\#}({\sf
DK}_{\bullet}(F^{\vee}\to R^{\vee}))))^{\vee}.$

###### Proof.

Since ${\sf R}_{A}(p^{\bullet})\cong{\sf DK}^{\bullet}(R\rightarrowtail F)$ by
Proposition 3.5, using (2.7) we obtain ${\sf R}^{\bullet}(p)^{\vee}={\sf
DK}_{\bullet}(F^{\vee}\to R^{\vee}).$ Since for a finitely generated module
$F$ we have a natural isomorphism $(F^{\vee})^{\vee}\cong F,$ we obtain

(4.4) $\Phi({\sf R}^{\bullet}(p))^{\vee}=\Phi^{\\#}({\sf
R}^{\bullet}(p)^{\vee})=\Phi^{\\#}({\sf DK}_{\bullet}(F^{\vee}\to R^{\vee})).$

The first isomorphism of the statement follows from Proposition 3.5. The
second isomorphism follows from the fact that, if $\Phi(F^{\oplus n})$ and
$\Phi(F^{\oplus n}\oplus R)$ are free finitely generated, then the cross
effects $\Phi_{[n]}(F),$ $\Phi_{[n|1]}(F,R)$ are also free finitely generated
modules (because they are direct summands and $\mathbb{K}$ is a principal
ideal domain), and hence, by Theorem 3.6, components of ${\sf
Lim}_{(p)}\Phi{\sf R}_{A}$ are also free finitely generated. ∎

Further we set

(4.5) $A^{\diamond}={\sf Ext}^{1}_{\mathbb{K}}(A,\mathbb{K}).$

######  Corollary 4.2.

Assume that $A$ is a free finitely generated $\mathbb{K}$-module and $\Phi$
takes free finitely generated modules to free finitely generated modules. Then
there is an isomorphism of chain complexes

(4.6) ${\sf Lim}_{({\sf id})}\Phi{\sf
R}_{A}\cong(N_{\bullet}(\Phi^{\\#}(K(A,1))))^{\vee}$

and for each $i$ we have a short exact sequence

(4.7)
$0\to(L_{i-1}\Phi^{\\#}(A^{\vee},1))^{\diamond}\to\mathsf{lim}^{i}\Phi{\sf
R}_{A}\to(L_{i}\Phi^{\\#}(A^{\vee},1))^{\vee}\to 0$

###### Proof.

It follows from isomorphism (4.3), the fact that ${\sf
DK}_{\bullet}(A^{\vee}\to 0)=K(A^{\vee},1)$ and from the universal coefficient
theorem for chain complexes. ∎

######  Corollary 4.3.

Assume that $A$ is a torsion finitely generated $\mathbb{K}$-module and $\Phi$
takes free finitely generated modules to free finitely generated modules. Then
for each $i$ there is a short exact sequence

(4.8)
$0\to(L_{i-1}\Phi^{\\#}(A^{\diamond},0))^{\diamond}\to\mathsf{lim}^{i}\Phi{\sf
R}_{A}\to(L_{i}\Phi^{\\#}(A^{\diamond},0))^{\vee}\to 0$

###### Proof.

Since $A$ is torsion module, $A^{\vee}=0.$ Therefore, $F^{\vee}\to R^{\vee}$
is a free resolution of $A^{\diamond}.$ It follows that ${\sf
DK}_{\bullet}(F^{\vee}\to R^{\vee})$ is a free simplicial resolution of
$K(A^{\diamond},0),$ and hence $\pi_{i}(\Phi^{\\#}({\sf
DK}_{\bullet}(F^{\vee}\to R^{\vee})))=L_{i}\Phi^{\\#}(A^{\diamond},0).$ Then
the assertion follows from the isomorphism (4.3) and the universal coefficient
theorem for chain complexes. ∎

##  5\. Higher limits for symmetric powers

In this section we assume that $\mathbb{K}=\mathbb{Z}$ and study
$\mathsf{lim}^{i}S^{d}{\sf R}_{A},$ where $S^{d}$ is $d$-th symmetric power
and $A$ is a free finitely generated abelian group. In this description we
will use results of [1] about the description of the non-additive derived
functors of the functors of divided powers. Namely we will use that for a
finitely generated free abelian group $A$ we have

(5.1) $L_{i}\Gamma^{d}(A,1)=\begin{cases}\Lambda^{d}(A),&i=d,\\\ \text{a
finite abelian group},&0<i<d,\\\ 0,&i=0\text{ or }i>d.\end{cases}$

(see [1, Theorem 6.3], and the beginning of §7 in [1]).

For an abelian group $A$ we denote by $\otimes(A)$ the tensor algebra. Then
the symmetric algebra ${\sf S}(A)$ is its quotient by the relations $ab=ba$
for $a,b\in A,$ and the exterior algebra $\Lambda(A)$ is its quotient by the
relations $a^{2}=0$ for $a\in A.$ We will also consider the anti-symmetric
algebra $\tilde{\otimes}(A)$ which is a quotient of $\otimes(A)$ by the
relations $ab=-ba$ for $a,b\in A.$ Since the relation $ab=-ba$ is satisfied in
$\Lambda(A),$ we have a natural epimorphism
$\tilde{\otimes}(A)\twoheadrightarrow\Lambda(A).$ Consider the kernel

(5.2) $ASK(A):={\sf Ker}(\tilde{\otimes}(A)\twoheadrightarrow\Lambda(A)).$

The ideal $ASK(A)$ is generated by elements $a^{2}$ for $a\in A.$ The graded
abelian group $ASK(A)$ is a $2$-torsion group, because $a^{2}=-a^{2}$ in
$\tilde{\otimes}(A).$ Since $\tilde{\otimes}(A)$ and $\Lambda(A)$ are
naturally graded we can consider the homogeneous components
$\Lambda^{d}(A),\tilde{\otimes}^{d}(A),ASK^{d}(A).$

Note that for a finite abelian group $T$ we have

(5.3) $T^{\diamond}\cong{\sf Hom}(T,\mathbb{Q}/\mathbb{Z})$

and the functor $(-)^{\diamond}$ induces a self-duality on the category of
finite abelian groups, as well as the functor $(-)^{\vee}$ is a self-duality
on the category of free finitely generated abelian groups.

######  Theorem 5.1.

For a free finitely generated abelian group $A$ there is a natural isomorphism

(5.4) $\mathsf{lim}^{i}S^{d}{\sf
R}_{A}\cong(L_{i-1}\Gamma^{d}(A^{\vee},1))^{\diamond},\hskip
28.45274pt\text{for }i<d,$

a natural short exact sequence

(5.5)
$0\longrightarrow(L_{d-1}\Gamma^{d}(A^{\vee},1))^{\diamond}\longrightarrow\mathsf{lim}^{d}S^{d}{\sf
R}_{A}\longrightarrow\Lambda^{d}(A)\longrightarrow 0$

and $\mathsf{lim}^{i}S^{d}{\sf R}_{A}=0$ for $i>d.$ Moreover, for a free
abelian group $A$ (not necessarily finitely generated) there are natural
isomorphisms

(5.6) $\mathsf{lim}^{d}S^{d}{\sf R}_{A}\cong\tilde{\otimes}^{d}(A),$

for $d\geq 1,$ and $\mathsf{lim}^{1}S^{d}{\sf R}_{A}=0$ for $d\geq 2.$

###### Proof.

First we note that for a free finitely generated abelian group $A$ we have
$\Gamma^{d}(A)=(S^{d})^{\\#}(A)$ and $\Lambda^{d}(A)=(\Lambda^{d})^{\\#}(A)$
(see [1, §2.3]). The equation (5.1) implies that

(5.7)
$L_{i}\Gamma^{d}(A^{\vee},1)^{\vee}=\begin{cases}(\Lambda^{d})^{\\#}(A)=\Lambda^{d}(A),&i=d,\\\
0,&i\neq d.\end{cases}$

Then the isomorphism 5.4 and the short exact sequence 5.5 follow from the
short exact sequence (4.7).

Let us prove the isomorphism (5.6). Here we will use Corollary 3.7. Let’s
compute cross effects of $S^{d}$ and the maps $h_{i}$ for them. The symmetric
algebra is an exponential functor $S(A\oplus B)=S(A)\otimes S(B),$ with the
isomorphism defined by

(5.8)
$(a_{1}+b_{1})\cdot{\dots}\cdot(a_{n}+b_{n})\mapsto\sum_{\\{i_{1}<\dots<i_{k}\\}\sqcup\\{j_{1}<\dots<j_{l}\\}=\\{1,\dots,n\\}}(a_{i_{1}}\dots
a_{i_{k}})\otimes(b_{j_{1}}\dots b_{j_{l}}).$

If we take $A=B$ and compose it with the diagonal map $\Delta:A\to A\oplus A,$
we obtain the map $S(A)\to S(A)\otimes S(A)$ defined by

(5.9) $a_{1}\cdot{\dots}\cdot
a_{n}\mapsto\sum_{\\{i_{1}<\dots<i_{k}\\}\sqcup\\{j_{1}<\dots<j_{l}\\}=\\{1,\dots,n\\}}(a_{i_{1}}\dots
a_{i_{k}})\otimes(a_{j_{1}}\dots a_{j_{l}}).$

We obtain $S(A_{1}\oplus\dots\oplus A_{n})=S(A_{1})\otimes\dots\otimes
S(A_{n})$ for any sequence of abelian groups $A_{1},\dots,A_{n}.$ Therefore

(5.10) $S^{d}(A_{1}\oplus\dots\oplus A_{n})=\bigoplus_{d_{1}+\dots+d_{n}=d,\
d\geq 0}S^{d_{1}}(A_{1})\otimes\dots\otimes S^{d_{n}}(A_{n}).$

It follows that

(5.11) $S^{d}(A_{1}|\dots|A_{n})=\bigoplus_{d_{1}+\dots+d_{n}=d,\ d_{i}\geq
1}S^{d_{1}}(A_{1})\otimes\dots\otimes S^{d_{n}}(A_{n}).$

In particular for $n=d-1,d$ and $A=A_{1}=\dots=A_{d}$ we obtain

(5.12) $\begin{split}S^{d}_{[d]}(A)&=A^{\otimes d}.\\\
S^{d}_{[d-1]}(A)&=\bigoplus_{i=1}^{d-1}A^{\otimes i-1}\otimes S^{2}(A)\otimes
A^{\otimes d-i-1}.\end{split}$

Now we need to compute $h_{i}:S^{d}_{[d-1]}(A)\to S^{d}_{[d]}(A).$ Looking on
the formula 5.9 we see that $h_{i}$ is trivial on all the summands $A^{\otimes
j-1}\otimes S^{2}(A)\otimes A^{\otimes d-j-1}$ for $j\neq i$ and the
restriction of of $h_{i}$ to $A^{\otimes i-1}\otimes S^{2}(A)\otimes
A^{\otimes d-i-1}$ is given by the formula

(5.13) $\begin{split}&h_{i}(a_{1}\otimes\dots\otimes
a_{i}a_{i+1}\otimes\dots\otimes a_{d})=\\\ &=a_{1}\otimes\dots\otimes
a_{i}\otimes a_{i+1}\otimes\dots\otimes a_{d}+a_{1}\otimes\dots\otimes
a_{i+1}\otimes a_{i}\otimes\dots\otimes a_{d}\end{split}$

It follows that the cokernel of the map $S^{d}_{[d-1]}(A)\to S^{d}_{[d]}(A)$
is the quotient of the tensor power $A^{\otimes d}$ by the relations

(5.14) $a_{1}\otimes\dots\otimes a_{i}\otimes a_{i+1}\otimes\dots\otimes
a_{d}=-a_{1}\otimes\dots\otimes a_{i+1}\otimes a_{i}\otimes\dots\otimes
a_{d},$

which is equal to $\tilde{\otimes}^{d}(A).$ It follows that
$\mathsf{lim}^{d}S^{d}{\sf R}_{A}\cong
H^{d-1}(C_{S^{d}}(A))\cong\tilde{\otimes}^{d}(A).$

For finitely generated free abelian groups $A$ the isomorphism (5.4) implies
$\mathsf{lim}^{1}S^{d}{\sf R}_{A}\cong
L_{0}\Gamma^{d}(A^{\vee},1)^{\diamond}=0.$ For all free abelian groups the
isomorphism $\mathsf{lim}^{1}S^{d}{\sf R}_{A}=0$ follows from the fact that
all the constructions
$S^{d}(A),S^{d}_{[d]}(A),C_{S^{d}}(A),H^{*}(C_{S^{d}}(A))$ commute with
filtered colimits of abelian groups, and that any free abelian group can be
presented as a filtered colimits of its finitely generated subgroups, which
are also free. ∎

######  Corollary 5.2.

For a free finitely generated abelian group $A$ there is a natural isomorphism

(5.15) $L_{d-1}\Gamma^{d}(A,1)\cong(ASK^{d}(A^{\vee}))^{\diamond}.$

###### Proof.

The short exact sequence 5.5 implies that the torsion subgroup of
$\mathsf{lim}^{d}S^{d}{\sf R}_{A}$ is naturally isomorphic to
$(L_{d-1}\Gamma^{d}(A^{\vee},1))^{\diamond}.$ The isomorphism 5.6 implies that
the torsion group of $\mathsf{lim}^{d}S^{d}{\sf R}_{A}$ is naturally
isomorphic to $ASK^{d}(A).$ Therefore
$(L_{d-1}\Gamma^{d}(A^{\vee},1))^{\diamond}\cong ASK^{d}(A).$ Using that
$(-)^{\diamond}$ is a self-duality for finite abelian groups and $(-)^{\vee}$
is a self-duality for free finitely generated abelian groups, we obtain the
required isomorphism. ∎

######  Corollary 5.3.

For a free finitely generated abelian group $A$ there are non-natural
isomorphisms of abelian groups

(5.16) $\mathsf{lim}^{i}S^{d}{\sf
R}_{A}\cong\begin{cases}L_{i-1}\Gamma^{d}(A,1),&i<d,\\\
L_{d-1}\Gamma^{d}(A,1)\oplus L_{d}\Gamma^{d}(A,1),&i=d,\\\ 0,&i>d.\end{cases}$

###### Proof.

For any finitely generated free abelian group $A$ we have an non-natural
isomorphism $A\cong A^{\vee}.$ For any finite group $T$ we have a non-natural
isomorphism $T\cong{\sf Hom}(T,\mathbb{Q}/\mathbb{Z})\cong T^{\diamond}.$
Therefore, using (5.1) for $i\neq d$ we have a non-natural isomorphism
$L_{i}\Gamma^{d}(A^{\vee},1)^{\diamond}\cong L_{i}\Gamma^{d}(A,1).$ Using
(5.4) we obtain the required isomorphism for $i<d.$ Since
$L_{d}\Gamma^{d}(A,1)\cong\Lambda^{d}(A)$ is free abelian, we obtain that the
short exact sequence (5.5) non-naturally splits. The isomorphism for $i=d$
follows. ∎

######  Example 5.4.

Theorem 5.1 implies that for a free abelian group $A$ we have
$\mathsf{lim}^{1}S^{2}{\sf R}_{A}=0$ and $\mathsf{lim}^{2}S^{2}{\sf
R}_{A}\cong\tilde{\otimes}^{2}(A).$ For $S^{3}$ we similarly have
$\mathsf{lim}^{1}S^{3}{\sf R}_{A}=0$ and $\mathsf{lim}^{3}S^{3}{\sf
R}_{A}=\tilde{\otimes}^{3}(A).$ We also claim that

(5.17) $\mathsf{lim}^{2}S^{3}{\sf R}_{A}\cong A\otimes\mathbb{Z}/3.$

For a finitely generated free abelian group $A$ it follows from the fact that
$L_{1}\Gamma^{3}(A,1)\cong A\otimes\mathbb{Z}/3$ (see [1, (8-20)]) and that
$(A^{\vee}\otimes\mathbb{Z}/3)^{\diamond}\cong A\otimes\mathbb{Z}/3.$ In order
to prove it for arbitrary free abelian group $A$ we need to use that all the
constructions commute with filtered colimits. A free abelian group $A$ can be
presented as a filtered colimit of its finitely generated subgroups
$A=\underset{B\subseteq_{\sf f.g.}A}{\sf colim}B,$ and using the fact that all
the constructions that we use here commute with filtered colimits, we obtain
$\mathsf{lim}^{2}S^{3}{\sf R}_{A}\cong H^{1}(C_{S^{3}}(A))\cong$ ${\sf
colim}_{B}H^{1}(C_{S^{3}}(B))$ $\cong{\sf colim}_{B}B\otimes\mathbb{Z}/3\cong
A\otimes\mathbb{Z}/3.$

######  Proposition 5.5.

For $n\geq 4$ we have an isomorphism of abelian groups

(5.18) $H_{n}(K(\mathbb{Z},3),\mathbb{Z})\cong\bigoplus_{d\geq
2}\mathsf{lim}^{n-2d+1}S^{d}{\sf R}_{\mathbb{Z}}$

(here we assume that $\mathsf{lim}^{i}=0$ for $i<0$).

###### Proof.

In [1, (B-3)] (see also [3, Satz 4.16]) in is proved that for a free finitely
generated abelian group $A$ and any $n\geq 0$ there is an isomorphism

(5.19) $H_{n}(K(A,3),\mathbb{Z})\cong\bigoplus_{d}L_{n-2d}\Gamma^{d}(A,1).$

Since $L_{i}\Gamma^{d}(A,1)=0$ for $i>d,$ we can assume that the summation is
taken over all $d$ such that $n-2d\leq d.$ Equivalently we can rewrite this as
$d\geq n/3.$ In particular, if $n\geq 4,$ we can assume that the summation is
taken over indexes $d\geq 2.$ Since $L_{d}\Gamma^{d}(A,1)=\Lambda^{d}(A),$ we
see that for $A=\mathbb{Z}$ and $d\geq 2$ we have
$L_{d}\Gamma^{d}(\mathbb{Z},1)=0.$ Therefore by Corollary 5.3 for $d\geq 2$ we
have an isomorphism

(5.20) $\mathsf{lim}^{i}S^{d}{\sf R}_{\mathbb{Z}}\cong
L_{i-1}\Gamma^{d}(\mathbb{Z},1).$

The assertion follows. ∎

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