11institutetext: Utrecht University, Utrecht, the Netherlands
11email: {v.steindani, i.m.beerepoot<EMAIL_ADDRESS>
# Turning Logs into Lumber:
Preprocessing Tasks in Process Mining

Ying Liu 11 Vinicius Stein Dani 11 Iris Beerepoot 11 Xixi Lu 11

###### Abstract

Event logs are invaluable for conducting process mining projects, offering
insights into process improvement and data-driven decision-making. However,
data quality issues affect the correctness and trustworthiness of these
insights, making preprocessing tasks a necessity. Despite the recognized
importance, the execution of preprocessing tasks remains ad-hoc, lacking
support. This paper presents a systematic literature review that establishes a
comprehensive repository of preprocessing tasks and their usage in case
studies. We identify six high-level and 20 low-level preprocessing tasks in
case studies. Log filtering, transformation, and abstraction are commonly
used, while log enriching, integration, and reduction are less frequent. These
results can be considered a first step in contributing to more structured,
transparent event log preprocessing, enhancing process mining reliability.

###### Keywords:

Log preprocessing Process mining Event log

##  1 Introduction

In the landscape of data-driven decision-making, event logs stand as
invaluable assets, capturing the execution of activities of processes and
their interactions within diverse operational systems. The potential insights
that can be obtained from these logs are immense, spanning process
improvement, anomaly detection, performance evaluation, and strategic planning
[1]. However, the axiom “garbage in, garbage out” holds particularly true in
this context [85]. The presence of data quality issues underscores the vital
importance of preprocessing techniques. Without proper preprocessing, the very
foundation of analysis is compromised.

The importance of data quality and preprocessing in the field of process
mining has been acknowledged, as evidenced by the growing attention dedicated
to these subjects [85, 97]. Despite the acknowledgment, the execution of log
preprocessing seems to remain ad-hoc. Moreover, little support has been
provided on which preprocessing tasks are possible and how to select them.
Although a few process mining methodologies sketched potential preprocessing
tasks, a comprehensive overview of these tasks has been notably absent.
Furthermore, the way these preprocessing tasks are used in real-life has
remained unclear.

Existing systematic literature reviews (SLRs) have attempted to tackle
specific tasks of log preprocessing, such as event abstraction techniques [97]
and data extraction [83]. However, a comprehensive review that covers diverse
preprocessing tasks and their practical applications in real-world scenarios
is lacking.

In this paper, we perform a systematic literature review to establish an
initial, comprehensive overview of the preprocessing tasks and their
utilization in process mining case studies. By undertaking this endeavor, we
aim to create a repository of log preprocessing tasks that may provide
guidance and support for researchers and practitioners.

We identified six high-level preprocessing tasks, and for four of these tasks,
we observed 20 low-level preprocessing tasks described in the case studies.
The results show that log filtering, transformation, and abstraction have been
more frequently used in case studies, while log enriching, integration, and
reduction (e.g., sampling) are much less frequently performed.

The remainder of this paper is organized as follows. In Section 2, we discuss
related work. Next, we explain the methodology followed in Section 3 and
present the results in Section 4. Finally, we conclude the paper in Section 5.

##  2 Related Work

In this section, we discuss the related work, based on which we synthesized an
initial set of six high-level preprocessing tasks: (a) _log integration_ , (b)
_log transformation_ , (c) _log reduction_ , (d) _log abstraction_ , (e) _log
filtering_ , and (f) _log enriching_ , see Fig. 1.

Figure 1: Initial result of high-level log preprocessing tasks and techniques
in the related work.

###  2.1 Taxonomy of Log Preprocessing Tasks

Han et al. [36] propose four categories of data preprocessing techniques: data
cleaning, data integration, data transformation, and data reduction. Data
cleaning focuses on handling missing values, identifying noise or outliers,
and repairing errors. Since these subtasks are not interesting (e.g.,
identifying missing variable values) or not directly applicable to process
mining (e.g., identifying noise/outliers in a distribution), we omit this task
and decide to focus on the latter three tasks. For each, we create a
corresponding log preprocessing task: _log integration_ , _log transformation_
, and _log reduction_.

Van Eck et al. [26] listed four tasks in the preprocessing stage, which are
specifically tailored towards event logs: creating views, aggregating events,
filtering logs, and enriching logs. We exclude the task “creating views”
because this task assumes that there is no event log yet, while we assume we
have a raw event log as input. We match the task “aggregating events” to _log
abstraction_ , also known as _event abstraction_ that has been already
surveyed [97]. The filtering of event logs (“filtering logs”) is also
considered a log preprocessing task within our scope, which we refer to as
_log filtering_. Finally, the preprocessing task “enriching logs” is mapped to
_log enriching_. As for log enriching and log integration, we consider _log
integration_ as creating a new event log by integrating one or more external
data sources, while _log enriching_ focuses on using the information within
the event log to derive additional attributes.

Fahland [29] indicated that there are three basic preprocessing operations on
event logs, which are: selection, projection, and aggregation. We consider the
“selection” and “projection” as a part of the _log filtering_ task, while the
aggregation operation is considered as part of the _log abstraction_.

Regarding _log filtering_ , _log abstraction_ , and _log reduction_ , both
_log filtering_ and _log abstraction_ can reduce the size of the logs, but we
consider the following subtle differences in comparison to log reduction here.
Log filtering tends to focus on the quality issues of the original data. It
obtains higher-quality logs by filtering out incorrect, incomplete,
inconsistent, and irrelevant data. Log abstraction focuses on the complexity
and granularity of the original data. It groups the events through
aggregation, defining event classes, and clustering to reduce the complexity
of logs. Log reduction is due to the data volume of the original data. It
reduces the amount of data processed in a single analysis by random sampling,
dividing, or cutting, but still makes the data representative.

###  2.2 Literature Review in Event Log Preprocessing

To the best of our knowledge, there is only one literature review focusing on
the log preprocessing tasks: Marin-Castro and Tello-Leal [56] reviewed 70
related papers that were published from the years 2005 to 2020 and explicitly
mentioned event log preprocessing or cleaning. This literature review grouped
preprocessing techniques into two types of _techniques_ : transformation
techniques and detection-visualization techniques. _Transformation techniques_
mark modifications made toward the original structure of the event log, while
the events or traces that can lead to issues with data quality are identified,
grouped, and isolated using _detection-visualization techniques_. In this
paper, we cover six high-level _preprocessing tasks_ , instead of the
techniques. We include log enriching, log integration, and log reduction,
which have not been discussed.

Van Zelst et al. [97] conducted a review and presented a taxonomy of event
abstraction techniques. While valuable and detailed insights are provided into
the event abstraction techniques, no insights are provided into their usage in
practice, and no overview is provided for other preprocessing tasks.
Similarly, Stein Dani et al. [83] report that preprocessing, on a high level
represented by filtering-related tasks, is still a manual effort in the event
log preparation phase of a process mining project. However, they mainly focus
on data extraction tasks and do not provide an overview of the preprocessing
tasks, including automated ones, and their usage in real-life case studies.

Currently, there is no clear overview of log preprocessing tasks and how
frequently are these preprocessing tasks being used in process mining
projects. Using the six high-level tasks as our scope, we conduct an SLR in
order to provide insights into the usage of log preprocessing techniques in
process mining case studies.

##  3 Systematic Literature Review

To arrive at an initial selection of relevant papers, and inspired by
Kitchenham and Petersen [44, 64], we applied the following search string on
Scopus: (“process mining”) AND (“case study” OR “case studies”) within the
article title, abstract, and keywords. As of December 20, 2022, we initially
found 4565 papers. Fig. 2 shows an overview of the paper screening process we
followed. Next, we applied the exclusion and inclusion criteria in order to
narrow down the scope of the review. The following exclusion criteria were
defined and applied directly via the search engine: (1) the paper is published
in 2021 and 2022; (2) the paper is published in conferences or journals under
peer-review; (3) the paper explicitly mentions “process mining” in the
keywords; and, (4) the paper is written in English. As we are particularly
interested in the current trend in case studies that use process mining as the
core technique, this is our inclusion criteria. Therefore, only papers meeting
these criteria were selected to be further analyzed.

Figure 2: Paper screening procedure.

After applying our exclusion and inclusion criteria, we obtained 355 papers.
Because our focus is on log preprocessing tasks applied in real-world
settings, we then read the abstracts of all these papers and filtered out the
ones that did not mention collecting data from a real-world scenario.
Thereafter, we obtained 159 papers to go through the full paper screening.
These papers were downloaded and imported into the software
Nvivo111https://lumivero.com/products/nvivo/ for further analysis. During the
full paper analysis stage, the papers that did not mention any data
preprocessing steps were discarded and, finally, 86 papers were obtained as
relevant papers to go through the coding stage of our work.

The following codes were defined for the analysis: high-level category, low-
level category, and data domain. Next, we discuss what each one of them
entails. High-level categories were defined based on related work and used to
deductively categorize the papers. The six high-level categories are (1) log
integration, (2) log reduction, (3) log abstraction, (4) log filtering, (5)
log enriching, and (6) log transformation. Several Low-level categories within
the high-level categories were inductively defined from the studied papers.
Finally, in addition, we also coded the data domain (e.g., healthcare,
education, manufacture, etc.), the analysis purpose, the PM task, and the
year. Due to space limits, we do not discuss these results in this paper. The
initial result of the categorization process is presented in Fig. 1.

##  4 Results

In this section, we present the results of coding 86 papers. The results are
discussed for each high-level category. A complete overview of the results and
the detailed coding can be found online, see the Google sheet file. We also
include the overview table listed in Appendix LABEL:app:overview 222The
Appendix will be published via Zenodo (https://zenodo.org/) for the purpose of
reviewing.

###  4.1 Log filtering

Log filtering is the most commonly performed preprocessing task, with 55 out
of 86 papers performing this preprocessing task. These 55 papers mentioned
filtering different objects, such as noise, outliers, redundant, duplicated
cases and events, missing values, useless values, blank values, irrelevant
values, and so on. Using the objects mentioned in these papers, the category
log filtering is subdivided into 9 detailed low-level categories.

Filtering irrelevant data We observed that 29 out of 55 papers mentioned
filtering “irrelevant” data. After analyzing these papers, we define
_irrelevant data_ as _those resources, activities, attributes, events, and
traces that are not relevant or not important for the specific analysis to be
conducted_.

Whether the data is relevant to the analysis task seems to be mostly
determined by experts or analysts based on their domain knowledge and analysis
requirements. For example, in [16], the analysis only focused on the students
who participated in the class (resource), so the events generated by other
resources were defined as irrelevant data and filtered. In [32], the authors
intended to analyze the activities of Ph.D. students and improve their
journeys. So after a discussion by analysts and stakeholders, a filter is
applied to retain the traces of full-time students who completed their Ph.D.
and who withdrew (case status). The term _useless data_ is also used in some
of the papers to describe irrelevant data. For example, in [86], the authors
mentioned “ _filtering useless information such as links and marker symbols_
”, since the links and marker symbols (attributes) cannot make any
contribution to the intended analysis and are regarded as useless data.

Filtering incomplete data In 16 out of 55 papers, the authors mentioned
filtering incomplete data. Incomplete data can be divided into _incomplete
events_ and _incomplete cases_. _Incomplete events_ usually refer to events
having missing values or missing attributes. Incomplete events include missing
case id [50], missing timestamps [23, 32, 63], and missing activities [20,
23], missing other attribute values that are relevant to this analysis [31].

The incompleteness of a case is usually described as cases that are not
completed or do not represent the end-to-end process. It means that the cases
lack some events, for example, “ _remove any record that may create only one
event per case as it will not depict the sequence of activities and hinder the
performance analysis of the model_ ” [67] and “ _removing cases that did not
cover the whole steps_ ” [20].

Filtering infrequent data We use _infrequent data_ to refer to the infrequent
case variant. In 13 papers, the authors mentioned that they performed the
infrequent case variants filtering as a preprocessing task. Filtering
infrequent data is done to “ _prevent the PM tool from returning
incomprehensible or inaccurate results_ ” [75], and “ _to improve the quality
of results, and to avoid low precision and highly complex results_ ” [89].

Filtering inconsistent data A simple example of inconsistent data is that the
values are recorded in different formats, e.g., “2023-01-01” and “2023/01/01”
as the attribute timestamps. This inconsistency in data format may be due to
recording errors or caused by manual input. It may also be that different data
sources have used different data formats. Inconsistent event labels make it
difficult to assign clear semantics to the activities of a discovered process
model [1], and may also bring about a dimensional explosion of the process
model.

Filtering incorrect data Incorrect data is erroneous or unreliable data that
violates the logic of reality. For example, in the real process, activity $A$
should be executed earlier than activity $B$, but in the log, the timestamp of
$A$ in a specific case is later than activity $B$ [63].

Filtering duplicates Duplicates refer to repeated data. In process mining, the
case ID needs to be a unique identifier, and the traces represented by
different case IDs must be different, so as to ensure the accuracy of the
data. However, in real life, duplicate data is usually generated due to system
bugs or other reasons. For example, in [22], repeated events with the same
Call-ID were excluded.

Filtering redundant data Only two papers mentioned redundant data [19, 20]. In
[20], redundant events were included in data error: “ _we conducted some data
preprocessing, including handling data error (e.g., removing redundant events
and eliminating multiple yield values)_ ”, while there was no further
definition and explanation in [19].

Filtering outliers In [13, 14, 52], the authors only mentioned “ _removing
outliers_ ” without any further explanation or definition. In [18], the
authors mention “ _we noticed the existence of outliers, i.e., cases that take
too long, or incomplete_ ”; so, too long trace and incomplete data are
considered outliers. In [74], “ _if lecture activities in the short semester
are included, it will be an outlier because it has activities that are far
more than short than activities in the semester in general_ ”; thus, traces
that are too short are also considered outliers. It seems that process
analysts use the distributions of a case or event-attribute to define
outliers, e.g., the number of events per case, the case duration per case,
etc.

Filtering noise Noise is an overused word. Data that is not conducive to the
analysis task is often defined as noise. An interesting point is that among
the 86 papers, more than one paper mentioned noise, but only one paper
described what noise is and how to filter it, “ _In the original log the noisy
activities were conveniently named ‘Noise’, so they were removed using a
filter on the activity name_ ” [51].

###  4.2 Log transformation

In 38 of the 86 papers, the authors described that they performed a _log
transformation_ task. The coding resulted in four data objects that are being
transformed, which we use to further divide the high-level category.

Transforming format Among the format transformations, the transformation of
the log format from CSV to XES was mentioned the most (14 out of 25 papers),
such that the event logs can be used in the PM tool. This is because the log
format after extraction is usually CSV, and PM tools require the log format to
be imported as XES. The remaining format transformation is related to
determining which columns are the key columns (such as case ID, activity name,
and timestamps) after importing the log into PM tools.

Transforming values The difference between transforming values and
transforming format is that transforming values means the change of one or
more specific values in an event. For example, replacing infrequent values
with the value ‘other’ to avoid dimension explosion, replacing missing values,
replacing NaN values with ‘zero’, capturing data, and encrypting data.

Reordering Reordering is the process of sorting the log by a particular
timestamp. When the original log is out of order, it is essential to reorder
it so that the process model displays the activities’ proper execution
sequence.

Transition matrices and encoding In particular, transition matrices and trace
encoding are used as a preprocessing for predictive process monitoring. Given
that the trace encoding is a subfield itself and was not included in the
search, we consider this category outside of our scope. We found two case
studies mentioning this preprocessing task and coded them without further
analysis.

###  4.3 Log enriching

In 16 out of 86 papers, the log-enriching techniques were applied. Log
enriching is split into four categories. Three of them are shown using an
example in Fig. 3.

Figure 3: Examples of three log integration tasks versus log enriching.

Adding calculation metrics In this low-level category, the calculation metrics
are computed from existing attributes in the log. For example, in [22], call
center processes of a company were examined. In the original event log, each
call only had attributes Start and Call Duration, but process analysis
required the end time of the call. Therefore, the attribute End was obtained
by adding Call Duration to Start.

Labelling Labeling is the task of assigning a tag or a class to an event or a
trace. In [87], “ _the cases are labeled as either successful or failed,
depending on how they have been executed and their outcome_ ”, to further
divide the log into two logs. In [62], for recording differences over time
between the intended operation and the actual execution, a label was assigned
to each event to indicate if the event was carried out on time or not.

Adding case id Case id is a unique attribute in event logs. The data collected
in some case studies did not have the attribute of case id, then the case id
was created artificially in the data preprocessing stage. For example, in
[84], “the caseid is created by combining the three-digit client number
(MANDT) with a ten-digit document number and a five-digit item number”.

Adding noise Adding noise is not a typical preprocessing task, as just one
publication described it. [81] evaluated privacy assurance of healthcare
metadata. Noise-adding plugins in the tool ProM were used to make the original
event logs more privacy-preserving [60].

###  4.4 Log reduction

In 11 out of 86 papers, the authors used log reduction to do log
preprocessing. Examples of the three log reduction tasks are shown in Fig. 4.

Dividing into sub-logs In the example presented in Fig. 4, the original log is
divided into two logs by the date in timestamp. In [51, 28], IoT logs were
collected in a smart house and the aim was to explore human habits. They
firstly divided logs into smaller pieces by timestamps to analyse the time
distribution of the activities (user habits) within a day [51].

Resource could also be a common attribute for division. The authors of [70]
divided the traces into subsets to model different profiles of users. Dividing
original logs according to specific attributes is usually for more in-depth
analysis [32].

In addition, in order to test the proposed algorithm or approach, the log was
divided into training data and test data according to a certain proportion
[20, 37].

Sampling The most notable characteristic of sampling is randomness. The
reduction here is to reduce the trace; that is to say, the data processing
needs to be in the unit of a trace. In the example shown in Fig. 4, there are
four traces $[\langle A,B,C,D,E\rangle,\langle A\rangle,\langle
A,C\rangle,\langle B\rangle]$. After randomly sampling 50% of the traces, the
log $[\langle A\rangle,\langle A,C\rangle]$ in the lower right corner is
obtained.

Cutting traces In the example in Fig. 4, compared to other traces, the trace
$\langle A,B,C,D,E\rangle$ is obviously longer and contains more events.
Cutting off the event at the end of the trace will get the processed log in
the lower left corner. The purpose of this technique is to avoid bias from
very long traces [30].

Figure 4: A simple example of log reduction.

###  4.5 Log integration

Among the 86 papers, 14 papers used log integration to combine multiple data
tables. No objects of interest are repetitively mentioned, nor have we
observed obvious low-level tasks. Therefore, the log integration task has not
been further divided.

Fig. 3 shows an example where a new event log is created by matching two data
tables using the shared attribute “ _student_id_ ”. It is worth mentioning
that some papers mention that additional data was added to the original event
data without indicating the source, but we believe that the combination of
these data is realized by log integration. According to [38], “ _Besides the
attributes shown in Table 4, we included the educational level of the nurses
executing the activity, as well as their nursing experience/organisational
role, the hospital shift and weekday on which the activities were performed,
and the ward in which the shift took place_ ”. It is reasonable to speculate
that this additional information actually comes from a separate data table
that stores information about all nurses.

###  4.6 Log abstraction

In 37 out of the 86 papers, the authors used preprocessing techniques in log
abstraction, which is the most widely performed task after log filtering and
log transformation among the six preprocessing tasks. In [97], a review and
taxonomy of event abstraction were presented. Therefore, we will not focus on
this category here.

###  4.7 Discussion

The _log filtering_ task emerges as the most commonly performed preprocessing
task, with over $63\%$ of the case studies mentioning that some filtering is
performed. However, it’s worth noting that the specifics of the log filtering
tasks appear to heavily rely on domain knowledge. Moreover, more than 30
papers use somewhat ambiguous terminologies such as ‘irrelevant’ or ‘noise’.
The _log transformation_ task ranks as the second most frequently employed,
accounting for $44\%$. Currently, the majority of subtasks in the log
transformation focus on fixing format-related and data-quality issues. This
highlights the importance of data quality in process mining and suggests that
efforts to enhance data quality should continue to be a focal point in log
preprocessing.

In contrast, log enriching ($18\%$), log integration ($16\%$), and log
reduction ($12\%$) tasks are notably less commonly performed. One plausible
explanation is the limited support for these tasks in both academic and
commercial tools. Furthermore, the relatively uncommon use of log reduction
can be attributed to the fact that many filtering techniques inherently reduce
the log size.

Table 1: Category citation details of 86 papers. High-level category | Low-level category |  References
---|---|---
Log filtering (55) | Filtering irrelevant data (29) |  [2, 6, 9, 14, 16, 17, 18, 21, 25, 32, 33, 35, 40, 41, 43, 53, 65, 66, 69, 72, 76, 77, 78, 79, 84, 86, 90, 91, 95]
Filtering incomplete data (16) |  [8, 20, 23, 25, 31, 32, 37, 43, 50, 63, 65, 67, 75, 76, 77, 90]
Filtering infrequent data (13) |  [7, 11, 19, 25, 38, 40, 42, 50, 67, 75, 88, 87, 89]
Filtering duplicates (8) |  [14, 22, 23, 25, 30, 67, 76, 78]
Filtering outliers (5) |  [13, 14, 18, 52, 74]
Filtering incorrect data (4) |  [31, 48, 63, 82]
Filtering redundant data (2) |  [19, 20]
Filtering inconsistent data (1) |  [17]
Filtering noise (3) |  [17, 51, 70]
Log transformation (38) | Transforming format (25) |  [6, 10, 12, 14, 16, 17, 68, 23, 25, 27, 34, 35, 39, 40, 43, 46, 57, 62, 63, 67, 69, 74, 92, 93, 94]
Transforming values (12) |  [20, 25, 30, 48, 50, 58, 59, 62, 65, 76, 78, 79]
Reordering (5) |  [9, 23, 25, 53, 63]
Transition matrices and encoding (2) |  [25, 96]
Log abstraction (37) | - |  [3, 4, 11, 13, 16, 19, 18, 21, 68, 24, 25, 27, 28, 33, 39, 45, 47, 48, 50, 51, 53, 54, 55, 57, 58, 59, 63, 66, 71, 73, 78, 86, 88, 87, 92, 95, 96]
Log enriching (16) | Adding calculation metrics (9) |  [22, 24, 37, 42, 45, 58, 61, 73, 80]
Labelling (4) |  [5, 41, 62, 87]
Adding case id (2) |  [74, 84]
Adding noise (1) |  [81]
Log integration (14) | - |  [15, 21, 68, 22, 27, 32, 38, 49, 59, 61, 67, 74, 78, 80]
Log reduction (11) | Dividing into sub-logs (9) |  [20, 28, 32, 37, 46, 51, 70, 80, 91]
Sampling (2) |  [30, 82]
Cutting traces (1) |  [30]

##  5 Conclusion

In this paper, we conducted a systematic literature review, examining the use
of log preprocessing tasks in process mining case studies and presented the
results. We identified six high-level tasks that were synthesized from the
related work discussion and 20 low-level tasks inducted from the reported case
studies. The log filtering task emerges as the most frequently used
preprocessing task, featured in over 63% of the case studies reviewed. The log
transformation task follows closely behind, accounting for 44% of the cases.
Conversely, log enriching, integration, and reduction tasks are less commonly
performed, possibly due to limited tool support. Future research can delve
into these preprocessing tasks, providing operational guidance.
Standardization in reporting practices and greater support for less common
preprocessing tasks are valuable for improving traceability and advancing the
reliability of process mining results.

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$\delta$ and consider the Koszul complex888We use the notation and standard
properties of the Koszul complex as in [BH93, §1.6]. $K_{A}(\mathbf{x})$. It
is a graded-commutative differential graded $A$-algebra whose homology modules
are $B$-modules. Let $H_{\delta}(K_{A}(\mathbf{x}))_{1}$ be the submodule of
$H_{\delta}(K_{A}(\mathbf{x}))$ generated by products of elements of
$H_{1}(K_{A}(\mathbf{x}))$. The Koszul complex is functorial for ring
homomorphisms, so we have a map $\pi_{A,*}:H_{*}(K_{A}(\mathbf{x}))\to
H_{*}(K_{\mathcal{O}}(\overline{\mathbf{x}}))$, where $\overline{\mathbf{x}}$
denotes the image of the sequence $\mathbf{x}$ in $\mathcal{O}$. However, all
terms of this sequence are $0$, so
$H_{*}(K_{\mathcal{O}}(\overline{\mathbf{x}}))$ is the exterior algebra in
$\delta$ generators (in homological degree $1$). In particular,
$H_{\delta}(K_{\mathcal{O}}(\overline{\mathbf{x}}))\cong{\mathcal{O}}$.

######  Definition (CI).

$c_{1}(B):=\ell(\pi_{A,*}(H_{\delta}(K_{A}(\mathbf{x})))/\pi_{A,*}(H_{\delta}(K_{A}(\mathbf{x}))_{1}))$.

We see that this is finite by localising at the prime ideal corresponding to
the kernel of $\pi_{A}$ and observing that this localisation map factors
through $\pi_{A}$.

From the definition of the Koszul complex, it follows that
$H_{\delta}(K(\mathbf{x}))$ is the annihilator of the ideal $I$ generated by
the sequence $\mathbf{x}$. The $A$-submodule of $H_{\delta}(K(\mathbf{x}))$
generated by products of elements of $H_{1}(K(\mathbf{x}))$ is precisely the
Fitting ideal of $I$ (sitting inside its annihilator). It follows that

$c_{1}(B)=\ell(\pi_{A}(\operatorname{Ann}(\ker(\phi_{B})))/\pi_{A}(\operatorname{Fitt}(\ker(\phi_{B})))).$
(A.1)

We now show that $c_{1}(B)$ is independent of all choices. For a fixed $A$ as
above, the Koszul complex only depends on the number of generators of the
kernel. Moreover, adding more elements in the kernel to the sequence of
generators has the effect of tensoring the Koszul complex with an exterior
algebra in which case it is easy to see that $c_{1}$ does not change.

To show that it is independent of the choice of $\phi_{B}:A\to B$, we will
need the following elementary lemma.

######  Lemma (CI).

Let $\mathcal{O}$ be any commutative ring, $A_{1},A_{2},B$ be local Noetherian
$\mathcal{O}$-algebras and $\phi_{i}:A_{i}\to B$, $i=1,2$ surjections of
$\mathcal{O}$-algebras. Then

  1. 1.

$A:=A_{1}\times_{B}A_{2}$ is also a local Noetherian $\mathcal{O}$-algebra and
$\dim(A)=\max\\{\dim(A_{1}),\dim(A_{2})\\}$.

  2. 2.

If $A_{1}$ and $A_{2}$ are complete then so is $A$.

  3. 3.

Let $P$ be any prime ideal in $B$, $P_{i}=\phi_{i}^{-1}(P)$ the corresponding
prime ideals of $A_{i}$ and $P_{A}=\phi^{-1}(P)$ that of $A$ (where $\phi:A\to
B$ is the surjection induced by $\phi_{i}$). Then
$A_{P_{A}}=(A_{1})_{P_{1}}\times_{B_{P}}(A_{1})_{P_{2}}$.

###### Proof.

We have $A=\\{(a_{1},a_{2})\in A_{1}\times
A_{2}:\phi_{1}(a_{1})=\phi_{2}(a_{2})\\}$. The ideal $m_{A}$ of $A$ consisting
of all pairs $(a_{1},a_{2})$ with $a_{i}\in m_{A_{i}}$ is the unique maximal
ideal of $A$, since the surjectivity of $\phi_{1},\phi_{2}$ implies that the
complement consists of invertible elements, so $A$ is local. The two
projections induce surjections $p_{i}:A\to A_{i}$. If $I$ is an ideal of $A$
then $p_{1}(I)$ is an ideal of $A_{1}$. The kernel of the map $I\to p_{1}(I)$
is naturally an ideal of $A_{2}$. Since $A_{1}$ and $A_{2}$ are Noetherian,
this implies that $A$ is Noetherian.

Now since $A$ is a subring of $A_{1}\times A_{2}$ which is finite as an
$A$-module (it is generated by $(1,0)$ and $(0,1)$), it follows from the
going-up theorem [Mat80, Theorem 5, (i), (ii) and (iii)] that
$\dim(A)=\dim(A_{1}\times A_{2})=\max\\{\dim(A_{1}),\dim(A_{2})\\}$.

Suppose $A_{1}$ and $A_{2}$ are complete. To show that $A$ is complete it
suffices to prove that the $m_{A}$-adic topology on $A$ is the same as the
topology induced from the inclusion of $A$ in $A_{1}\times A_{2}$. Since
$m_{A}^{n}\subset m_{A_{1}}^{n}\times m_{A_{2}}^{n}$ for all $n>0$, we only
need to show that given any $n^{\prime}>0$, $(m_{A_{1}}^{n}\times
m_{A_{2}}^{n})\cap A\subset m_{A}^{n^{\prime}}$ for all $n\gg 0$. This follows
immediately by applying the Artin–Rees lemma [Mat80, Theorem 15], with
$I=m_{A}$, $M=A_{1}\times A_{2}$ and $N=A$, since $I^{n}M=m_{A_{1}}^{n}\times
m_{A_{2}}^{n}$.

We will use the following elementary fact whose simple proof we skip: If $A$
is any commutative ring, $S$ any multiplicative subset of $A$, $M_{1}$,
$M_{2}$ and $N$ any $A$-modules with maps $M_{i}\to N$, $i=1,2$, then the
natural map $M_{1}\times_{N}M_{2}\to(M_{2})_{S}\times_{N_{S}}(M_{2})_{S}$ of
$A$-modules induces an isomorphism
$(M_{1}\times_{N}M_{2})_{S}\to(M_{1})_{S}\times_{N_{S}}(M_{2})_{S}$. The
statement (3) follows from this by taking $M_{i}$ to be $A_{i}$, $N$ to be $B$
and $S=A\setminus P_{A}$ and by observing that
$A_{i}\otimes_{A}A_{P_{A}}=(A_{i})_{P_{i}}$, $i=1,2$ and
$B\otimes_{A}A_{P_{A}}=B_{P}$.

∎

It follows from Lemma (CI) and the prime avoidance lemma already used earlier,
that if $A_{i}$ are complete intersections of the same dimension with
surjections to $B$, then both of them may be dominated by a complete
intersection $A^{\prime}$ of the same dimension. The condition on the
finiteness of the cotangent space can also be preserved by (3) of Lemma (CI).
For the independence of the choice of $A$ in the definition of $c_{1}(B)$ we
will also need:

######  Lemma (CI).

Let $f:A^{\prime}\to A$ be a surjection of (complete) complete intersection
local rings and let $\phi_{B}:A\to B$ be any surjection of rings. Let
$\mathbf{z}$ be any finite sequence of generators of $\ker(f)$, $\mathbf{x}$
any sequence of generators of $\ker(\phi_{B})$, and $\mathbf{x}^{\prime}$ a
lift of $\mathbf{x}$ to $A^{\prime}$. Then
$H_{*}(K_{A^{\prime}}((\mathbf{z},\mathbf{x}^{\prime}))$ is isomorphic to
$H_{*}(K_{A}(\mathbf{x}))$ tensored with an exterior algebra over $A$ with
$|\mathbf{z}|+\dim(A)-\dim(A^{\prime})$ free generators.

###### Proof.

Let $g:S\to A^{\prime}$ be a surjection from a regular local ring $S$ (which
exists because $A^{\prime}$ is complete), so both $\ker(g)$ and $\ker(fg)$ are
generated by regular sequences. Choose a sequence of generators ${\mathbf{y}}$
of $\ker(\phi_{B}fg)$ by first choosing a regular sequence of generators
$\mathbf{w}$ of $\ker(g)$ and then adding lifts $\tilde{\mathbf{z}}$ of
elements of $\mathbf{z}$ and lifts $\tilde{\mathbf{x}}^{\prime}$ of lifts
$\mathbf{x}^{\prime}$ in $A^{\prime}$ of elements of $\mathbf{x}$. We then set
$\mathbf{y}=(\mathbf{w},\tilde{\mathbf{x}}^{\prime},\tilde{\mathbf{z}})$ and
consider $K_{S}(\mathbf{y})$. Since the Koszul complex of a regular sequence
is a resolution of the corresponding quotient ring, by applying this to
$\mathbf{w}$ we see that $K_{S}(\mathbf{y})$ is quasi-isomorphic (as a
differential graded $S$-algebra) to
$K_{S}((\tilde{\mathbf{z}},\tilde{\mathbf{x}}^{\prime}))\otimes_{S}A^{\prime}$,
i.e., $K_{A^{\prime}}((\mathbf{z},\mathbf{x}^{\prime}))$. On the other hand,
since $A$ is also complete intersection ring, by choosing a minimal generating
set of $\ker(fg)$ from among the elements of
$(\mathbf{w},\tilde{\mathbf{z}})$, one sees that $K_{S}(\mathbf{y})$ is quasi-
isomorphic to $K_{A}(\mathbf{x})$ tensored with an exterior algebra (since
$gf(\tilde{\mathbf{x}}^{\prime})=\mathbf{x}$ and the remaining elements of
$(\mathbf{w},\tilde{\mathbf{z}})$ become $0$ in $A$). On taking homology we
see that $H_{*}(K_{A^{\prime}}((\mathbf{z},\mathbf{x}^{\prime}))$ is
isomorphic to $H_{*}(K_{A}(\mathbf{x}))$ tensored with an exterior algebra.
The number of free generators of this exterior algebra is easily seen to be
$|\mathbf{z}|+\dim(A)-\dim(A^{\prime})$ since
$|\mathbf{w}|=\dim(S)-\dim(A^{\prime})$.

∎

######  Lemma (CI).

The invariant $c_{1}(B)$ is well-defined.

###### Proof.

By Lemma (CI) and the remarks following it, it suffices to show that if
$\phi_{B}:A\to B$ is as above and we have a surjection $f:A^{\prime}\to A$
such that $\phi_{B}^{\prime}:=\phi_{B}f$ also satisfies the conditions
analogous to those imposed on $\phi_{B}$, then the numbers defined using
$\phi_{B}$ and $\phi_{B}^{\prime}$ are equal.

Let $\mathbf{x}$ be a sequence of generators of $\ker(\phi_{B})$,
$\mathbf{x}^{\prime}$ a lift of this sequence to $A^{\prime}$ and $\mathbf{z}$
a sequence of generators of $\ker(f)$. Let
$\mathbf{w}=(\mathbf{z},\mathbf{x}^{\prime})$, so $\mathbf{w}$ is a sequence
of generators of $\ker(\phi_{B}^{\prime})$. Thus, $\delta=|\mathbf{x}|$ and
$\delta^{\prime}$, the corresponding number of generators for
$\ker(\phi_{B}^{\prime})$, equals $|\mathbf{w}|=\delta+|\mathbf{z}|$. Note
that $H_{\delta^{\prime}}(K_{A^{\prime}}((\mathbf{z},\mathbf{x}^{\prime}))$ is
canonically isomorphic to $\operatorname{Ann}(\ker(\phi_{B}^{\prime}))$ and
$H_{\delta}(K_{A}(\mathbf{x}))$ is canonically isomorphic to
$\operatorname{Ann}(\ker(\phi_{B}))$. From the result of Lemma (CI)
(specialised to the case $\dim(A)=\dim(A^{\prime})$) that
$H_{*}(K_{A^{\prime}}((\mathbf{z},\mathbf{x}^{\prime}))$ is isomorphic to
$H_{*}(K_{A}(\mathbf{x}))$ tensored with an exterior algebra over $A$ with
$|\mathbf{z}|$ free generators, it follows that there is an isomorphism of
$A$-modules
$\alpha:\operatorname{Ann}(\ker(\phi_{B}^{\prime}))\to\operatorname{Ann}(\ker(\phi_{B}))$
such that
$\alpha(\operatorname{Fitt}(\ker(\phi_{B}^{\prime})))=\operatorname{Fitt}(\ker(\phi_{B}))$.

Now we use the finite cotangent space assumption on $A$ and $A^{\prime}$. This
implies that $\pi_{A}(\operatorname{Ann}(\ker(\phi_{B})))\subset\mathcal{O}$
is nonzero and equal to the image of
$\operatorname{Ann}(\ker(\phi_{B}))\otimes_{A}\mathcal{O}$ in
$A\otimes_{A}\mathcal{O}=\mathcal{O}$ (and similarly for $A^{\prime}$ and also
for the Fitting ideals). The $\mathcal{O}$-module
$\operatorname{Ann}(\ker(\phi_{B}))\otimes_{A}\mathcal{O}$ modulo its torsion
is free of rank one (and similarly for $A^{\prime}$) so the lemma follows from
(A.1) and the above by using the isomorphism $\alpha\otimes_{A}\mathcal{O}$.

∎

For any map of rings $R_{1}\to R_{2}$, an $R_{2}$-module $M$ and $i\geq 0$, we
denote by $\operatorname{Der}^{i}_{R_{1}}(R_{2},M)$ the $i$-th André–Quillen
cohomology group of $R_{2}$ with coefficients in $M$. Let $E$ denote the
quotient field of $\mathcal{O}$ viewed as a $B$-module via $\pi_{B}$.

The invariants $c_{0}(B)$ and $c_{1}(B)$ defined above are linked by the
following proposition, which may be viewed as a derived version of Wiles’s
formula for complete intersections [Wil95], [Len95], [FKR21, §A]; a variant of
this formula was first discovered by A. Venkatesh [Ven16].

######  Proposition (CI).

$c_{0}(B)-c_{1}(B)=\ell(\operatorname{Der}^{0}_{\mathcal{O}}(B,E/{\mathcal{O}}))-\ell(\operatorname{Der}^{1}_{\mathcal{O}}(B,E/{\mathcal{O}})).$
(A.2)

###### Proof.

We denote by $J$ the ideal $\ker(\phi_{B})$ with $\phi_{B}:A\to B$ as above.
The sequence of maps ${\mathcal{O}}\to A\to B$ gives rise to an exact sequence
of André–Quillen cohomology

$0\to\operatorname{Der}^{0}_{\mathcal{O}}(B,E/{\mathcal{O}})\to\operatorname{Der}^{0}_{\mathcal{O}}(A,E/{\mathcal{O}})\to\mathrm{Hom}_{A}(J/J^{2},E/{\mathcal{O}})\to\operatorname{Der}^{1}_{\mathcal{O}}(B,E/{\mathcal{O}})\to
0.$ (A.3)

The $0$ on the left comes from the fact that
$\operatorname{Der}^{0}_{A}(B,E/{\mathcal{O}})=0$ since $\phi_{B}$ is
surjective (which also gives that $\mathrm{Hom}_{A}(J/J^{2},E/{\mathcal{O}})$
is equal to $\operatorname{Der}^{1}_{A}(B,E/{\mathcal{O}})$). The $0$ on the
right comes from the fact that
$\operatorname{Der}^{1}_{\mathcal{O}}(A,E/{\mathcal{O}})=\operatorname{Der}^{2}_{\mathcal{O}}(A,{\mathcal{O}})=0$,
where the first equality is because
$\operatorname{Der}^{i}_{\mathcal{O}}(A,E)=0$ for all $i$ (a consequence of
the finite tangent space condition on $\pi_{A}$) and the second follows from
[Avr99, (1.2) Theorem] because $A$ is a complete intersection, $\mathcal{O}$
is regular, and we have a surjection from $S$ onto $A$.

We claim that $\mathrm{Hom}_{A}(J/J^{2},E/{\mathcal{O}})$ and
$\operatorname{Der}^{0}_{\mathcal{O}}(A,E/{\mathcal{O}})$ are finite length
${\mathcal{O}}$-modules and that we have equalities
$\ell(\mathrm{Hom}_{A}(J/J^{2},E/{\mathcal{O}}))=\ell({\mathcal{O}}/\pi_{A}(\operatorname{Fitt}(J)))$
and
$\ell(\operatorname{Der}^{0}_{\mathcal{O}}(A,E/{\mathcal{O}}))=\ell({\mathcal{O}}/\pi_{A}(\operatorname{Fitt}(\ker(\pi_{A}))))$.
Assuming the claim, from sequence (A.3) we deduce

$\ell(\operatorname{Der}^{0}_{\mathcal{O}}(B,E/{\mathcal{O}}))-\ell(\operatorname{Der}^{1}_{\mathcal{O}}(B,E/{\mathcal{O}}))=\ell({\mathcal{O}}/\pi_{A}(\operatorname{Fitt}(\ker(\pi_{A}))))-\ell({\mathcal{O}}/\pi_{A}(\operatorname{Fitt}(J))).$
(A.4)

By definition
$c_{0}(B)=\ell({\mathcal{O}}/\pi_{C}(\operatorname{Ann}(\ker(\pi_{C}))))$,
$c_{1}(B)=\ell(\pi_{A}(\operatorname{Ann}(\ker(\phi_{B})))/\pi_{A}(\operatorname{Fitt}(\ker(\phi_{B}))))$
by (A.1), and Lemma (CI) below implies that

$\ell(\mathcal{O}/\pi_{A}(\operatorname{Ann}_{A}(\ker(\pi_{A}))))=\ell(\mathcal{O}/\pi_{A}(\operatorname{Ann}_{A}(\ker(\phi_{B}))))+\ell(\mathcal{O}/\pi_{C}(\operatorname{Ann}_{C}(\ker(\pi_{C})))).$

Recalling that $J=\ker(\phi_{B})$ and
$\operatorname{Fitt}(\ker(\pi_{A}))=\operatorname{Ann}_{A}(\ker(\pi_{A}))$
(since $A$ is a complete intersection), the proposition follows by inserting
these three equalities in (A.4).

We now prove the claim made above: For the first part, note that
$E/{\mathcal{O}}$ is an $A$-module via $\pi_{A}$, so that
$\mathrm{Hom}_{A}(J/J^{2},E/{\mathcal{O}})\cong\operatorname{Hom}_{\mathcal{O}}(J/J^{2}\otimes_{A}{\mathcal{O}},E/{\mathcal{O}})$,
where ${\mathcal{O}}$ is an $A$-module via $\pi_{A}$, and it suffices to show
that $J/J^{2}\otimes_{A}{\mathcal{O}}$ is a finite length
${\mathcal{O}}$-module. The module $J/J^{2}\otimes_{A}{\mathcal{O}}$ is of
finite type over ${\mathcal{O}}$ because $J/J^{2}$ is of finite type over $A$,
and so we need to show that $J/J^{2}[1/\varpi]\otimes_{A[1/\varpi]}E$
vanishes. Now the map $\phi[1/\varpi]:A[1/\varpi]\to B[1/\varpi]$ is a map of
finite-dimensional $E$-algebras and the (compatible) augmentations to
$\pi_{A}[1/\varpi]$ and $\pi_{B}[1/\varpi]$ give rise to isomorphisms of a
single factor with $E$, i.e., $\pi_{A}[1/\varpi]\otimes_{A[1/\varpi]}E$ is an
isomorphism and $J[1/\varpi]\otimes_{A[1/\varpi]}E=0$ because $J[1/\varpi]$
must then be supported on the other factors, and hence
$J/J^{2}[1/\varpi]\otimes_{A[1/\varpi]}E=0$. For the second part we apply the
conormal sequence to ${\mathcal{O}}\to A\to{\mathcal{O}}$ which gives the
isomorphism
$\ker(\pi_{A})/\ker(\pi_{A})^{2}\cong\Omega_{A/{\mathcal{O}}}\otimes_{A}{\mathcal{O}}$
due to the splitting of $A\to{\mathcal{O}}$. By construction the right hand
term in the isomorphism is of finite ${\mathcal{O}}$-length, and the second
part now follows from
$\operatorname{Der}^{0}_{\mathcal{O}}(A,E/{\mathcal{O}})\cong\operatorname{Hom}_{A}(\Omega_{A/{\mathcal{O}}},E/{\mathcal{O}})\cong\operatorname{Hom}_{\mathcal{O}}(\Omega_{A/{\mathcal{O}}}\otimes_{A}{\mathcal{O}},E/{\mathcal{O}})$.

For the first assertion on lengths we need to show that
$J/J^{2}\otimes_{A}{\mathcal{O}}$ and
${\mathcal{O}}/\pi_{A}(\operatorname{Fitt}(J))$ have the same lengths. Because
$\pi_{A}(J)=0$, the image of $J^{2}\otimes_{A}{\mathcal{O}}$ in
$J\otimes_{A}{\mathcal{O}}$ is zero, and hence
$J/J^{2}\otimes_{A}{\mathcal{O}}\cong J\otimes_{A}{\mathcal{O}}$. Next observe
that
$\pi_{A}(\operatorname{Fitt}(J))=\operatorname{Fitt}(J\otimes_{A}{\mathcal{O}})$,
as follows from the definition of the Fitting ideal. The equality of length
now follows because for a finite length ${\mathcal{O}}$-module over the dvr
${\mathcal{O}}$ the theory of elementary divisors gives
$\ell(M)=\ell({\mathcal{O}}/\operatorname{Fitt}(M))$. The argument for the
second length equality proceeds in the same way. One reduces the equality to
showing that
$\ker(\pi_{A})/\ker(\pi_{A})^{2}\cong\ker(\pi_{A})\otimes_{A}{\mathcal{O}}$
and ${\mathcal{O}}/\operatorname{Fitt}(\ker(\pi_{A})\otimes_{A}{\mathcal{O}})$
have the same length. ∎

######  Remark (CI).

The above proof shows in particular, that the terms
$\operatorname{Der}^{1}_{\mathcal{O}}(B,E/{\mathcal{O}})$ and
${\mathcal{O}}/\pi_{A}(\operatorname{Fitt}(J))$ are of finite
${\mathcal{O}}$-length.

######  Remark (CI).

If $B$ is a complete intersection in Proposition (CI) we may take $A=B$, so
$c_{1}(B)=0$, $c_{0}(B)=\ell({\mathcal{O}}/\eta_{B})$ and (A.4) shows that
Proposition (CI) reduces to Wiles’s formula. The proposition shows once again
that $c_{1}(B)$ is independent of all choices since all the other terms in the
formula are clearly so.

The following lemma was used in the proof of Proposition (CI).

######  Lemma (CI).

Let $A$ be a Gorenstein local ring with an augmentation
$\pi_{A}:A\to\mathcal{O}$ such that $\ell(\Phi_{A})<\infty$. Assume that
$\pi_{A}$ factors through a surjective ring homomorphism $\phi_{B}:A\to B$ and
let $C$ be the largest quotient of $B$ which is Cohen–Macaulay, so there are
surjections $\phi_{C}:A\to C$, $\pi_{B}:B\to\mathcal{O}$ and
$\pi_{C}:C\to\mathcal{O}$. Then

$\pi_{A}(\operatorname{Ann}_{A}(\ker(\pi_{A})))=\pi_{A}(\operatorname{Ann}_{A}(\ker(\phi_{B})))\,\pi_{C}(\operatorname{Ann}_{C}(\ker(\pi_{C}))).$

###### Proof.

We may apply Lemma A.10 of [FKR21] to the map $\phi_{C}$, since $C$ is
Cohen–Macaulay, to deduce that

$\pi_{A}(\operatorname{Ann}_{A}(\ker(\pi_{A})))=\pi_{A}(\operatorname{Ann}_{A}(\ker(\phi_{C})))\,\pi_{C}(\operatorname{Ann}_{C}(\ker(\pi_{C}))),$

so it suffices to to prove that
$\operatorname{Ann}_{A}(\ker(\phi_{C}))=\operatorname{Ann}_{A}(\ker(\phi_{B}))$.
We have $\ker(\phi_{B})\subset\ker(\phi_{C})$ and the quotient is a finite
length $A$-module by the definition of $C$. The quotient map

$\operatorname{Ann}_{A}(\ker(\phi_{B}))\ker(\phi_{C})\to\operatorname{Ann}_{A}(\ker(\phi_{B}))(\ker(\phi_{C})/\ker(\phi_{B}))$

is an isomporphism since
$\operatorname{Ann}_{A}(\ker(\phi_{B}))\ker(\phi_{B})=(0)$, so
$\operatorname{Ann}_{A}(\ker(\phi_{B}))\ker(\phi_{C})$, being a submodule of a
finite length $A$-module, is also of finite length. On the other hand, it is a
submodule of $A$ and ${\rm depth}(A)=1$, so it must be $(0)$. Thus,
$\operatorname{Ann}_{A}(\ker(\phi_{C}))=\operatorname{Ann}_{A}(\ker(\phi_{B}))$.
∎

### A.1

We briefly explain how the formula (A.2) can be viewed as a derived version of
Wiles’s formula:

Suppose we have a presentation
$B=\mathcal{O}[[x_{1},x_{2},\dots,x_{n}]]/(f_{1},f_{2},\dots,f_{n+\delta})$
with $\delta\geq 0$. We may use this to construct a “derived” ring

$\mathcal{B}=\mathcal{O}[[x_{1},x_{2},\dots,x_{n}]]\
{\otimes}_{\mathcal{O}[[y_{1},y_{2},\dots,y_{n},\dots,y_{n+\delta}]]}\
\mathcal{O},$

where the tensor product is defined as in [GV18, Definition 3.3]. Here the
$x_{i},y_{j}$ are in “degree 0” and the map from
$\mathcal{O}[[y_{1},y_{2},\dots,y_{n},\dots,y_{n+\delta}]]$ to
$\mathcal{O}[[x_{1},x_{2},\dots,x_{n}]]$ is given by $y_{j}\mapsto f_{j}$ and
to $\mathcal{O}$ by $y_{j}\mapsto 0$.

If we assume that
$A=\mathcal{O}[[x_{1},\dots,x_{n}]]/(f_{1},f_{2},\dots,f_{n})$ is a one
dimensional complete intersection, then the derived ring has “defect” equal to
$\delta$. The invariant $c_{1}(B)$ may then be viewed as coming from
$\pi_{*}(\mathcal{B})$, since this may be computed in terms of a Koszul
complex. Venkatesh views (A.2) as an analogue of Wiles’s formula for the
derived ring $\mathcal{B}$, which is a “derived complete intersection”.
(However, as we have shown, all the terms in the formula only depend on
$B=\pi_{0}(\mathcal{B})$, so it may also be viewed as a generalisation of
Wiles’s formula to rings which are not necessarily complete intersections.)

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Address of authors:

(GB) Interdisciplinary Center for Scientific Computing, Universität
Heidelberg, Heidelberg, Germany. Email address<EMAIL_ADDRESS>
(CK) Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA. Email
address<EMAIL_ADDRESS>
(JM) Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA. Email
address<EMAIL_ADDRESS>