# Three-stream network for enriched Action Recognition

Ivaxi Sheth
Imagination Technologies
<EMAIL_ADDRESS>
###### Abstract

Understanding accurate information on human’s behaviours is one of the most
important tasks in machine intelligence. Human Activity Recognition that aims
to understand human activities from a video is a challenging task due to
various problems including background, camera motion and dataset variations.
This paper proposes two CNN based architectures with three streams which allow
the model to exploit the dataset under different settings. The three pathways
are differentiated in frame rates. The single pathway, operates at a single
frame rate captures spatial information, the slow pathway operates at low
frame rates captures the spatial information and the fast pathway operates at
high frame rates that capture fine temporal information. Post CNN encoders, we
add bidirectional LSTM and attention heads respectively to capture the context
and temporal features. By experimenting with various algorithms on UCF-101,
Kinetics-600 and AVA dataset, we observe that the proposed models achieve
state-of-art performance for human action recognition task.

##  1 Introduction

As imaging systems become ubiquitous, the ability to recognize human actions
is becoming increasingly important. Activity Recognition is an elemental task
in the computer vision that realises human actions. These actions are detected
post complete action execution in a video. Through this approach, action and
its purpose can be identified. Applications of AR are becoming highly
recognisable within surveillance, video retrieval, human robot interactions
and self driving vehicles.

From a group of videos $S$ and its corresponding action labels $L$, each video
$V\in S$ contains one or more actions $l_{V}$. Thus the aim of activity
recognition problem is to predict labels $l_{V}$ based on video understanding
$V$. In case of action recognition, the video is generally segmented to
contain only one execution of a human action. In more general cases, the video
may contain multiple actions and the goal of action detection is not just to
recognize the actions being performed in the video but also determine the
spatio-temporal boundaries of them.

One of the reasons why the transition from 2D images to 3D videos hasn’t been
smooth is lack of temporal understanding. This paper proposes network that
enhances not only temporal features but also spatial features. The spatial
dimensions X and Y in 2D images are considered with equal importance.[5] paper
introduces that time dimension, T in a video is not as important as the
spatial dimensions, X and Y. Hence our paper exploits this property to perform
action recognition from videos.

Figure 1: An overview of our three stream network with attention head. The
network architecture for our baseline i.e. three stream network with bi-LSTM
follows similar architecture with bidirectional LSTM block instead of
Attention block

Our method proposes a three lane network where each pathway is differentiated
in frame rates. The single pathway, operates at single frame rate captures
spatial information, the slow pathway operates at low frame rates captures the
spatial information and the fast pathway operates at high frame rates that
captures fine temporal information. Inspired from textual tasks, we propose
two networks one with bi-directional LSTM head (we call it baseline) and other
with attention mechanism to achieve state of art accuracy. We test our models
on multiple datasets, UCF-101 [20], Kinetics-600 [2] and Atomic Visual Actions
(AVA) [7]. AVA is a particularly difficult dataset as it requires detecting
multiple people in videos semi-densely in time, and recognizing multiple basic
actions. With three stream network and bi-directional LSTM head we achieve
state of art accuracy for UCF-101 and Kinetics dataset while the three stream
network with attention head outperforms all methods for all of 3 datasets.

##  2 Related Works

Traditionally, for video action recognition, human-created features, like
Histogram of Oriented Gradients (HOG)[3] and Histogram of Optical Flow
(HOF)[14] were used by the research community. These features can be either
sparsely[14] and densely[27] sampled. These early methods consider independent
interest points across frames.

In recent times, deep learning has brought remarkable improvements to image
understanding[12]. The high performance from image classification networks has
propelled these networks to inspire networks for video understanding [11] with
minimal modifications. One of the methods, extracts features independently
from each frame of the video, applies image classification CNN and finally
pools predictions across all the frames of the video. The drawback of this
method is that it ignores the temporal structure example: models can’t
conceivably recognize opening an entryway from shutting an entryway. Adding a
recurrent layer to CNN can encode state although unrolling RNNs.

An interesting and practical model was proposed by[19]. Here a 2D ConvNet was
used on short temporal snapshots of videos. The output from single RGB frame
and stack of 10 optical flow frames were averaged, after passing them through
2D ConvNet. This model appeared to get extremely superior on existing datasets
and computationally efficient as well.

3D Convolution Networks are important to video modeling, and are similar
standard convolutional networks, but with spatio-temporal filters. 3D CNNs
have an key attribute: ability to generate hierarchical representations of the
input video data. Temporal Segment Network [29] is based on the idea of long-
range temporal structure modelling. It was successful at tackling limited
information to temporal context, as the other networks operated on unit frames
or a single stack of multiple frames. Instead of working on short snippets,
TSN work on short snippets sparsely sampled from the entire video.

###  2.1 Two stream networks

Two stream networks were introduced by [19] for action recognition in videos.
They used two CNNs for each spatial and temporal information in videos. In
this architecture, RGB images are extracted from frames of video and optical
flow from consecutive frames are fed into different streams. This work was
then expanded to give inflated two stream 3D network. This work [13] proposed
to use neural networks along with shallow local features. [19] proposed a
spatiotemporal architecture and explored various layer fusion schemes. They
claimed that fusing last convolution layers spatially increases accuracy. [29]
introduced a neural network performs pixel level action recognition and
segmentation by adding a two-stream network along with temporal aggregation.
[5] proposes a two stream network where CNN on each stream works on different
frame rates.

###  2.2 Temporal Understanding

2D image models have also been extended to videos as seen in [2]. Long-term
filtering and pooling using temporal strides [26] were then used. Video
compression methods were also used [33] for a faster network. Extending on 3D
CNNs, some works have separated 2D spatial and 1D temporal filters [24].
Eventually 2D image classification networks were inflated to 3D extended to
action recognition. Our work proposes to perform temporal filtering at
different rates along with adding extras blocks to understand the temporal
structure.

###  2.3 Bidirectional LSTM

Since RNNs can encode long-term sequence information in a data, they have
recently been added into action recognition tasks. Bidirectional LSTMs add a
hidden layer that allows information to flow in background direction.
Therefore output layer receives information from past and future states
simultaneously. Thus Bi-LSTM [6] increases the amount of input data that can
be ingested by network by the network. CNNs have restrictions on the input
information adaptability, as they require fixed input data. A normal RNN also
has restriction as upcoming input data can not be attained from the present
state. On the contrast, Bi-LSTM don’t require the fixed input data and their
future input information can be accessed from the current state.

In literature, LSTMs have been used seldom. [36] connects LSTM cells to CNN
feature maps for action recognition. [25] proposes to use bi-LSTM along with a
convolutional encoder.

###  2.4 Attention models

The concept of attention was introduced by Bahdanau, Cho, and Bengio (2014)
for the objective of machine translation. Attention mechanism is based on
concept that the nwural network learns how admissible are some feature maps
with regarding to output state. These values of importance are specified as
weights of attention and are generally calculated at the same time as other
model parameters trained for a specific goal.

Attention was used in first person action recognition by having a joint
learning of gaze and actions [15], by using object-centric for ego centric
activity recognition [21] and by event modulated attention [10]. Attention was
used in to extract spatial information by generating spatial masks by traning
on video labels[37]. Temporal attention was used for action recognition by
detecting change in gaze [18].

##  3 Methodology

Since prior works weren’t able to extract temporal information effectively, we
explore the network under multiple temporal settings namely bidirectional LSTM
and Attention. For better spatial understanding we have a combined three
stream CNN based network as encoder. We have our baseline (three stream
network with bi-LSTM) perform well over UCF-101 and Kinetics dataset. To
further improve performance, we use with attention mechanism that gives state-
of-art performance over all three datasets.

###  3.1 Three stream

Section 2 discusses works relating to two stream networks. These (and current)
methods cannot unfortunately show decent performance on a difficult dataset
like AVA. Hence we suggest a novel architecture with three streams.

####  3.1.1 Single pathway

The main aim of single pathway is extract spatial features. The network can
have any 2D architecture. We see [11] that single 2D net can extract a great
deal of spatial feature. This pathways is essentially equivalent to having 3D
network with temporal stride of $\theta_{1}$, where $\theta_{1}$ is the
equivalent to the speed at which the video is processed. Hence for 30fps
video, the stride is 30, therefore ingesting only first frame. This is a very
light weight stream with much lower MACs as compared to other streams.

####  3.1.2 Slow pathway

The main aim is to extract spatio-temporal features. Here the temporal stride,
$\theta_{2}$ is smaller than $\theta_{1}$. After experimenting as mentioned in
section 7, the best value for $\theta_{2}$ is 16, also as mentioned in paper
[5]. This means that approx. 2 frames in a 30fps input video are processed.

####  3.1.3 Fast pathway

Similar to slow pathway, this is a 3d CNN but with much smaller temporal
stride to particularly encode for more temporal info. To understand finer
representations, we select alternate frames, making stride 2. Hence
$\theta_{3}$ ¡ $\theta_{2}$ ¡ $\theta_{1}$. one of the key features of three
stream network is different values of $\theta$ that works on different
temporal speeds, and thus drives the expertise of the two subnets
instantiating the two pathways. Our Fast pathway also distinguishes with
existing models in that it can use significantly lower channel capacity. The
low channel capacity can also be interpreted as a weaker ability of
representing spatial semantics.

###  3.2 Bi-directional LSTM

In bi-directional LSTM [25], the output at time $t$ is not only dependent on
the previous frames in the sequence, but also on the upcoming frames. In our
work, we use multiple LSTM layers, so our scheme has two LSTM layers for both
forward pass and backward pass. Figure 2 shows the overall concept of
bidirectional LSTM used in the proposed method. The input data is fed to the
bidirectional RNN, and the hidden states of forward pass and backward pass are
combined in the output layer. Both forward and backward consist of two LSTM
cells, making our model a deeper. The proposed method outperforms other state-
of-the-art methods due to its mechanism of computing the output. The output of
a frame at time $t$ is calculated from the previous frame at time $t-1$ and
the upcoming frame at time $t+1$ because layers are performing processing in
both directions.

Figure 2: Our bi-directional LSTM head

###  3.3 Attention head

We use the self mechanism explained in [16] for our attention head. Our self-
attention head computes correlation between arbitrary positions of a sequence
input. An attention function consists of a query $A_{Q}$, keys $A_{K}$ and
values $A_{V}$. The query and keys have same vector dimension $d_{k}$, and
values and outputs have same size, $d_{v}$. The output is computed as a
weighted sum of the values and its weight computed by scaled dot-product of
query and keys. The attention function is defined as

$Attention(Q,K,V)=softmax(\frac{QK^{T}}{\sqrt{d_{k}}})$

$\sqrt{d_{k}}$ is a scaling factor.

The equation computes scaled dot-product attention and the network computes
the attention multiple times in parallel (multi-head) to extract different
correlation information. The multi-head attention outputs are concatenated and
transformed to the same vector dimension the input sequence. A residual
connection is adopted to take the input and output of the multi-head self
attention layer and a layer normalization is applied to the summed output. A
fully-connected feed-forward network with a residual connection is applied to
the normalized self-attention output.

##  4 Network Architecture

A major advantage of this skeleton is that one can use any network
architecture for each of the streams. We use inflated ResNet [9] as the 3D
backbone network, for its promising performance on various datasets [8]. We
have studied performance difference for various other network backbones to
three stream in Ablation study. Meanwhile, original ResNet-52 serves as our 3D
backbone. We use the output features of res2, res3, res4, res5 to build our
network, where they are spatially downsampled respectively.

Figure 3: Overview of ResNet-52 for different pathways of the network encoder

###  4.1 Pathway link

Similar to [19], [35], we attach one lateral connection between the three
pathways for every stage. Specifically for ResNets , these connections are
right after pool1, res2, res3, and res4. The two pathways have different
temporal dimensions, so the lateral connections perform a transformation to
match them. We use unidirectional connections that fuse features of the Fast
pathway into the Slow pathway and finally Slow pathway to Single pathway. To
aggregate all of features, and to ensure the compatibility of the addition
between consecutive features, during aggregation a down/up-sampling operation,
we multiply by a factor.

##  5 Experiments

We evaluate our approach on four video recognition datasets using standard
evaluation protocols. For the action classification experiments, presented in
this section we consider the widely used Kinetics-600 [1] and UCF-101 [20].
For action detection experiments in Sec. 5, we use the challenging AVA [7]
dataset.

###  5.1 Dataset

In this subsection, we discuss about the different datasets we use to evaluate
the performance of our networks.

####  5.1.1 UCF-101

UCF [20] dataset is an open source dataset collected from videos on YouTube.
It contains 101 action classes with over 13000 videos and 27 complete hours of
data. The dataset consists of pragmatic videos consisting camera motion and
untidy and uneven background uploaded by the user. The videos are recorded
unconstrained environments and typically includes camera movement, different
lighting environments, partial obstructions, low quality clips. The action
categories are divided into five types - Human Object actions, Body-Motion
actions, Human Human actions, Playing Musical Instruments actions, Sports
based action. Categories like sports has multiple actions, where actions
performed with similar background, like greenery in most cases. Some clips
were captured with different illuminations, poses, and from different
viewpoints. A crucial challenge of this collection of data is actions are
performed in real life and are very realistic, which is significant compared
to other datasets where an actor is used to perform the actions.

Figure 4: Performance of Bi-LSTM based three stream network on UCF-101
dataset, here shown as confusion matrix.

####  5.1.2 Kinetics

Kinetics is a large open source dataset by Google that around seven hundred
action classes. Each class contains atleast 600 clips. Length of each video is
10s and is taken from a different YouTube video. The actions in video are
human centric and cover a wide assortment of classes including human object
relations such as playing instruments, human human interactions such as
hugging. The videos have a variable resolution and frame rate.

####  5.1.3 AVA

The AVA dataset densely annotates 80 atomic visual actions in 430 15-minute
video clips, where actions are localized in space and time, resulting in 1.58M
action labels with multiple labels per person occurring frequently. The
dataset is sourced from the 15th to 30th minute time intervals of 430
different movies, which given the 1 Hz sampling frequency gives us nearly 900
keyframes for each movie. Training/validation/test sets are split at the video
level, so that all segments of one video appear only in one split. There are
211k training and 57k validation video segments. The performance metric is
mean Average Precision (mAP) over 80 classes, using a frame-level IoU
threshold of 0.5.

###  5.2 Training

Following the setting in [5], the input frames are sampled at the $\theta_{3}$
and $\theta_{3}$ frames from the video frames. We randomly crop 224×224. A
dropout of 0.5 are adopted to reduce overfitting. BatchNorm is not frozen. We
use a momentum of 0.9, a weight decay of 0.00001 and a synchronized SGD
training over 8 GPUs. Each GPU has a batch-size of 8, resulting in a mini-
batch of 64 in total.

####  5.2.1 AVA dataset

The AVA [7] dataset focuses on spatiotemporal localization of human actions.
Hence for this dataset, unlike the other datasets, we need a detection
architecture. The architecture we use is inspired from and is similar to
Faster RCNN [17] and [5]. We use three stream network backbones along with
modified RCNNs. The fifth residual block in our implementation for AVA, we set
the stride as 1 instead of 2 and dilate its filters by 2. Therefore region-of-
interest (ROI) is extracted via the features of last(fifth) residual block.
Like [1], we extend 2D ROI to 3D ROI by replicating it along temporal
dimension. These features are fed through max pooling and a per-class sigmiod
based classifier.

###  5.3 Inference

We uniformly sample 10 clips from a video along its temporal axis. Following
[31] and [5], we scale the shorter spatial side to 256 pixels and take 3 crops
of 256×256 to cover the spatial dimensions, as an approximation of fully-
convolutional testing. We average the softmax scores for prediction.

##  6 Results

Figure 5: Bar chart for each action on AVA dataset, per category AP

We evaluate our method on baseline method and attention based model along with
current state of art models on Kinetics, AVA and UCF dataset in Table 1. We
observe that our model outperforms the state of art networks with attention
based model performing the best. It is worth noting that we obtain this
performance only using raw RGB frames as input, while prior works use RGB,
flow, and in some cases audio as well.

###  6.1 Baseline model for UCF-101 and Kinetics

Table 1 shows comparison with various network architechures along with our
implementation. In comparison with previous state of art our model provides
3.1$\%$ higher top-1 accuracy for Kinetics dataset and 1.4$\%$ UCF-101 dataset
and. Note that our results are better even without Image-net pre training. In
particular our model is 6$\%$ per better than best result of previous kind. We
had experimented with ImageNet pre-training and observed that there is less
than 1 percent improvement when compared to training from scratch.

Model | Kinetics-600 | UCF-101
---|---|---
| top-1 | top-5 |
Two stream [19] | 63.2 | 79.9 | 88
I3D [2] | 72.1 | 89.9 | 94
Two stream I3D [2] | 75.7 | 90.1 | 95.6
S3D [34] | 69.4 | 88.0 | 95.9
Nonlocal [30] | 77.3 | 92.6 | 93.2
R(2+1)D [24] | 73.8 | 88.5 | 96.8
STC [4] | 68.2 | 88.4 | 85.3
ARTNet [28] | 69.2 | 88.0 | 89.7
ECO [38] | 70.0 | 89.4 | 90.1
SlowFast [5] | 77.0 | 92.6 | 96.8
CoViAR [32] | 65.4 | 75.6 | 86.4
3S+Bi-LSTM(ours) | 80.1 | 94.0 | 98.2
3S+Attention(ours) | 82.3 | 95.5 | 99.0
Table 1: A comparison of different methods incl. current state-of-art with our
proposed methods on UCF-101 and Kinetics-600. Both of our methods evidently
outperform the networks proposed by current literature.

###  6.2 Attention based model for UCF-101 and Kinetics

As we observe in Section 6.1 that our baseline demonstrates best performance,
we note that attention based three stream network outperforms that as well.
This is particularly due to the ability of attention mechanism that can
generate more effective temporal understandings. Our attention based model
outperforms the current state of art by 5.3 $\%$. Although attention based
model has enhanced temporal understandings, spatial features are not
compromised as observed by lack of improvement when using ImageNet pre-trained
weights.

###  6.3 Results on AVA dataset

We compare the performance of both of our methods along with previous results.
We observe that the 3 stream with Bi-LSTM has shown improvement by +0.1 mAP
and our method with attention mechanism shows improvement by +6.1 mAP. It is
important to note that the methods with flow stream as well can double
computational cost. We observe that Kinetics pre-training shows a great
improvement in accuracy too with +6.8 mAP.

Model(+our version RCNN) | AVA (mAP %)
---|---
Two stream [19] | 7.4
I3D [2] | 14.5
Two stream I3D [2] | 15.8
S3D [34] | 17.2
Nonlocal [30] | 20.0
R(2+1)D [24] | 21.1
STC [4] | 15.6
ARTNet [28] | 13.2
ECO [38] | 12.8
SlowFast [5] | 26.3
CoViAR [32] | 21.9
3S+bi-LSTM(ours) | 26.4
3S+Attention(ours) | 32.2
Table 2: A comparison of different methods incl. current state-of-art with our
proposed methods on AVA. While our baseline model shows very small improvement
over AVA dataset, the attention head shows significant improvement from the
current state-of-art.

##  7 Ablation Study

All of our models have used class agnostic regression data augmentation (and
Kinetics pre-training for AVA models) techniques we observed early on to be
critical for good performance.

A key intuition for designing the different pathways was to use lower channel
capacity for Fast pathway. We experimented with different values of
$\theta_{2}$ ranging from 4 to 32. As expected the accuracy (measured on
Kinetics) decreased with the values of $\theta_{2}$. But this also increased
the GFLOPs. Hence to balance, we choose 16 frames.

$\theta_{2}$ value | top-1 ($\%$) | top-5 ($\%$)
---|---|---
4 | 79.9 | 94.8
6 | 80.2 | 94.9
12 | 80.3 | 95.6
16 | 80.0 | 95.3
32 | 80.1 | 95.0
Table 3: Different methods to laterally fuse

We also compared the effectiveness of adding temporal block i.e. Bi-LSTM and
attention blocks to our three stream network on the AVA dataset. Temporal
blocks improves performance on 74 out of 80 actions as compared to vanilla
three stream networks. The categories that showed major gains were: run/jog
(21.2 AP), hand wave (15.7 AP), hand clap (24.6 AP), eat (13.5 AP). These are
categories where modeling dynamics are of vital importance.

Our three stream CNN encoder can have any architecture type. We experimented
with various networks. We observe that Res-152 performs best as the backbone
as compared to InceptionV3[22], Res-101 [9], Res-52, EfficientNet B5 [23], and
EfficientNet B7. But we select ResNet-52 as trade-off against the GFLOPs
trade-off.

Figure 6: Performance of Attention based three stream network with respective
to the number of bounding boxes in the clip.  Figure 7: Performance of
Attention based three stream network with respective to the area covered by
bounding boxes of the clip.

##  8 Conclusion

In this paper we proposed two network architechures that signifies improvement
in spatio-temporal understanding of actions in a video. It achieves state-of-
the-art accuracy for video action classification and detection. We hope three
stream network will foster further research in action and video understanding.

##  9 Acknowledgements

We would like to thank Dr. Carlo Ciliberto for his very helpful insights and
Imperial College London for hardware support in initial project phase.

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$\mathcal{F}$ on $X_{\bullet,\operatorname{lis-et}}$ is quasi-coherent if for
each $i$ the sheaf $\mathcal{F}_{i}$ is a quasicoherent
$\mathscr{O}_{X_{i}}$-module and if for each $\varphi:[i]\to[j]$ the induced
maps

$f^{-1}_{\varphi}\mathcal{F}_{i}\otimes_{f^{-1}\mathscr{O}_{X_{i}}}\mathscr{O}_{X_{j}}\to\mathcal{F}_{j}$

are isomorphisms.

####  A.3.3. The descent theorem

For $X_{\bullet}$ a very smooth hypercover of $\mathcal{X}$, let
$a_{i}:X_{i}\to\mathcal{X}$ denote the given (smooth) morphism of algebraic
stacks.

The morphism $X_{0}\to\mathcal{X}$ induces an augmentation of
$\operatorname{Le}(X_{\bullet})$ towards $\operatorname{Le}(\mathcal{X})$ in
the sense of [Stacks, Tag 0D6Z]. By [Stacks, Tag 0D70] we get a morphism of
topoi

(64) $a:X_{\bullet,\operatorname{lis-et}}\to\mathcal{X}_{\operatorname{lis-
et}}$

such that $a^{-1}\mathcal{F}$ is given by the system with
$(a^{-1}\mathcal{F})_{i}:=a_{i}^{-1}\mathcal{F}$ and the natural transition
maps (they are all isomorphisms), and $a_{*}\mathcal{G}$ is given by the
equalizer of the two maps $a_{0*}\mathcal{G}_{0}\to a_{1*}\mathcal{G}_{1}$.

Using the maps $a_{i}^{-1}\mathscr{O}_{\mathcal{X}}\to\mathscr{O}_{X_{i}}$, we
get a morphism
$a^{-1}\mathscr{O}_{\mathcal{X}}\to\mathscr{O}_{X_{\bullet,\operatorname{lis-
et}}}$ that makes $a$ a morphism of ringed topoi. Define
$a^{*}:\mathrm{Mod}(\mathscr{O}_{\mathcal{X}})\to\mathrm{Mod}(\mathscr{O}_{X_{\bullet,\operatorname{lis-
et}}})$ by

$a^{*}\mathcal{F}:=a^{-1}\mathcal{F}\otimes_{a^{-1}\mathscr{O}_{\mathcal{X}}}\mathscr{O}_{X_{\bullet,\operatorname{lis-
et}}}$

It is clear that $a^{*}$ is exact and sends
$\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-et}})$ to
$\operatorname{QCoh}(X_{\bullet,\operatorname{lis-et}})$.

######  Proposition A.3.12.

Let $X_{\bullet}\to\mathcal{X}$ be a very smooth hypercover. Then

(65) $a^{*}:\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-
et}})\to\operatorname{QCoh}(X_{\bullet,\operatorname{lis-et}})$

is an equivalence of categories with quasi-inverse $a_{*}$. Moreover, the
functors $\mathsf{R}a_{*}$ and $a^{*}$ are inverse equivalences of
$\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$ and
$\operatorname{D_{qc}}(X_{\bullet,\operatorname{lis-et}})$.

###### Proof.

We first show that $a^{*}$ is an equivalence of categories of quasi-coherent
sheaves with quasi-inverse $a_{*}$. Let $U\to\mathcal{X}$ be a smooth map from
an algebraic space $U$, and let $U^{+}_{\bullet,\mathrm{et}}$ be the strictly
simplicial étale topos defined in Section A.2. We apply Construction A.2.1 to
the category $I=\Delta\times\Delta^{+}$. For $(i,j)\in\Delta\times\Delta^{+}$
we set
$\mathcal{C}_{i,j}=\operatorname{QCoh}\big{(}(X_{i}\times_{\mathcal{X}}U_{j})_{\mathrm{et}}\big{)}$
(observe that the fiber product is an algebraic space) and we let
$\phi^{*}:(X_{i}\times_{\mathcal{X}}U_{j})_{\mathrm{et}}\to(X_{k}\times_{\mathcal{X}}U_{\ell})_{\mathrm{et}}$
be the usual pullback of quasi-coherent sheaves. Let
$\operatorname{QCoh}((X_{\bullet}\times_{\mathcal{X}}U_{\bullet}^{+})_{\mathrm{et}})$
denote the resulting category of Cartesian systems.

Let $U_{X_{i}}=X_{i}\times_{\mathcal{X}}U$. By viewing
$\operatorname{QCoh}(X_{\bullet,\operatorname{lis-et}})$ and
$\operatorname{QCoh}((X_{\bullet}\times_{\mathcal{X}}U_{\bullet}^{+})_{\mathrm{et}})$
both as categories of systems with $I=\Delta$, we define functors

${\operatorname{QCoh}(X_{\bullet,\operatorname{lis-
et}})}$${\operatorname{QCoh}((X_{\bullet}\times_{\mathcal{X}}U_{\bullet}^{+})_{\mathrm{et}})}$$\scriptstyle{\varpi_{\bullet}^{*}}$$\scriptstyle{\varpi_{\bullet,*}}$

induced via Remark A.2.2 by the inverse equivalences

${\operatorname{QCoh}(X_{i,\operatorname{lis-
et}})}$${\operatorname{QCoh}(U^{+}_{X_{i},\bullet,\mathrm{et}})}$$\scriptstyle{\varpi^{*}}$$\scriptstyle{\varpi_{*}}$

of Proposition A.2.3. The rules $\varpi^{*}_{\bullet}$ and
$\varpi_{*,\bullet}$ are indeed functors of categories of systems by Remark
A.2.5, and one checks that they are inverse equivalences.

Similarly, let $X_{U_{i},\bullet}$ be the pullback of the hypercover
$X_{\bullet}\to\mathcal{X}$ to $U_{i}$ as in Remark A.3.7. By viewing
$U^{+}_{\bullet,\mathrm{et}}$ and
$\operatorname{QCoh}((X_{\bullet}\times_{\mathcal{X}}U_{\bullet}^{+})_{\mathrm{et}})$
as categories of systems with $I=\Delta^{+}$ we define functors

${\operatorname{QCoh}(U^{+}_{\bullet,\mathrm{et}})}$${\operatorname{QCoh}((X_{\bullet}\times_{\mathcal{X}}U_{\bullet}^{+})_{\mathrm{et}})}$$\scriptstyle{a_{\bullet}^{*}}$$\scriptstyle{a_{\bullet,*}}$

induced via Remark A.2.2 by the functors

(66)
${\operatorname{QCoh}(U_{i,\mathrm{et}})}$${\operatorname{QCoh}(X_{U_{i},\bullet,\mathrm{et}})}$$\scriptstyle{a^{*}}$$\scriptstyle{a_{*}}$

defined in analogy with (64) above. The functors $a_{*}$ and $a^{*}$ in (66)
are inverse equivalences by [Stacks, Tag 0DHD].

We have constructed a diagram

${\operatorname{QCoh}(X_{\bullet,\operatorname{lis-
et}})}$${\operatorname{QCoh}((X_{\bullet}\times_{\mathcal{X}}U_{\bullet}^{+})_{\mathrm{et}})}$${\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-
et}})}$${\operatorname{QCoh}(U^{+}_{\bullet,\mathrm{et}})}$$\scriptstyle{\varpi^{*}_{\bullet}}$$\scriptstyle{a_{*}}$$\scriptstyle{\varpi_{\bullet,*}}$$\scriptstyle{a_{\bullet,*}}$$\scriptstyle{a^{*}}$$\scriptstyle{\varpi^{*}}$$\scriptstyle{\varpi_{*}}$$\scriptstyle{a_{\bullet}^{*}}$

where three of the four pairs of morphisms are known to be inverse
equivalences. It follows from Remark A.1.1.5 that
$\varpi^{*}_{\bullet}a^{*}=a_{\bullet}^{*}\varpi^{*}$, so $a^{*}$ is an
equivalence with inverse $\varpi_{*}a_{\bullet,*}\varpi^{*}_{\bullet}$. Using
Remark A.1.1.5 and the fact that $\varpi^{*}$ is exact one can check that
$a_{\bullet,*}\varpi^{*}_{\bullet}=\varpi^{*}a_{*}$, so
$a_{*}=\varpi_{*}a_{\bullet,*}\varpi^{*}_{\bullet}$. This shows that $a^{*}$
and $a_{*}$ are inverse equivalences of quasi-coherent sheaves.

To finish the proof of the Proposition, we use [Stacks, Tag 0D7V]. To do so we
must verify its five hypotheses. The category
$\operatorname{QCoh}({X_{\bullet,\operatorname{lis-et}}})$ is a weak Serre
subcategory of $\mathrm{Mod}(\mathscr{O}_{X_{\bullet,\operatorname{lis-et}}})$
and conditions (1), (4), and (5) of [Stacks, Tag 0D7V] hold as in the proof of
[Stacks, Tag 0DHF]. Condition (2) is the inverse equivalence of $a^{*}$ and
$a_{*}$ that we just proved. The final condition, number (3), is the statement
that for $\mathcal{F}\in\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-
et}})$, the unit $\mathcal{F}\to\mathsf{R}a_{*}a^{*}\mathcal{F}$ is an
isomorphism. Since we already know $\mathcal{F}\to a_{*}a^{*}\mathcal{F}$ is
an isomorphism, it suffices to show $\mathsf{R}^{n}a_{*}a^{*}\mathcal{F}=0$
for $n>0$.

For any smooth map $m:U\to\mathcal{X}$ from a scheme $U$, let $U_{\bullet}\to
U$ be the very smooth hypercover equal to the pullback of $X_{\bullet}$. We
have a diagram of morphisms of topoi

${U_{\bullet,\mathrm{et}}}$${U_{\bullet,\operatorname{lis-
et}}}$${X_{\bullet,\operatorname{lis-
et}}}$${U_{\mathrm{et}}}$${U_{\operatorname{lis-
et}}}$${\mathcal{X}_{\operatorname{lis-
et}}}$$\scriptstyle{a^{\prime\prime}}$$\scriptstyle{m_{\bullet,\operatorname{lis-
et}}}$$\scriptstyle{a^{\prime}}$$\scriptstyle{a}$$\scriptstyle{m_{\operatorname{lis-
et}}}$

where the site $U_{\bullet,\mathrm{et}}$ is constructed with [Stacks, Tag
09WC] and $a^{\prime\prime}$ and $a^{\prime}$ are defined as in (64). The top
horizontal morphisms come from [Stacks, Tag 0DH0]. It follows from [73,
V.5.1(1)], [Ols07, Lem 3.5], and Lemma A.3.13 that
$\mathsf{R}^{n}a_{*}a^{*}\mathcal{F}$ is the sheafification of the presheaf
that associates to an smooth map $m:U\to\mathcal{X}$ from a scheme $U$ the
group $H^{n}(U_{\bullet,\operatorname{lis-
et}},m^{-1}_{\bullet,\operatorname{lis-et}}a^{*}\mathcal{F})$. Since
restriction to the étale site is exact and preserves injectives, this is equal
to $H^{i}(U_{\bullet,\mathrm{et}},m_{\bullet}^{*}a^{*}\mathcal{F})$ where
$m_{\bullet}:U_{\bullet,\mathrm{et}}\to X_{\bullet,\operatorname{lis-et}}$ is
defined as in Remark A.1.1. Finally, by Remark A.1.1.5 (since $\mathcal{F}$ is
quasi-coherent) this equals
$H^{n}(U_{\bullet,\mathrm{et}},a^{\prime\prime*}m^{*}\mathcal{F}).$

On the other hand, it follows from [73, V.5.1(1)] and Lemma A.3.13 that
$\mathsf{R}^{n}a^{\prime\prime}_{*}a^{\prime\prime*}(m^{*}\mathcal{F})$ is the
sheafification of the presheaf that associates to an étale map $f:V\to U$ from
a scheme $V$ the group
$H^{n}(V_{\bullet,\mathrm{et}},a^{\prime\prime*}m^{*}\mathcal{F})$. This group
is equal to $H^{i}(V_{\bullet,\mathrm{et}},a^{\prime\prime*}_{V}(m\circ
f)^{*}\mathcal{F})$, where $a^{\prime\prime}_{V}:V_{\bullet,\mathrm{et}}\to
V_{\mathrm{et}}$ is the usual morphism (64). It follows that if
$m:U\to\mathcal{X}$ is a smooth cover by a scheme, the étale sheaves
$m^{*}(\mathsf{R}^{n}a_{*}a^{*}\mathcal{F})$ and
$\mathsf{R}^{n}a^{\prime\prime}_{*}a^{\prime\prime*}(m^{*}\mathcal{F})$ are
the sheafification of the same presheaf, hence isomorphic. But it follows from
[Stacks, Tag 0DHE] that
$\mathsf{R}^{n}a^{\prime\prime}_{*}a^{\prime\prime*}(m^{*}\mathcal{F})=0$. ∎

######  Lemma A.3.13.

Consider a fiber square of algebraic stacks

${U_{X}}$${\mathcal{X}}$${U}$${\mathcal{Y}}$$\scriptstyle{b}$$\scriptstyle{m}$

such that $U$ is an algebraic space.

  1. (1)

If $m$ and $b$ are smooth and representable and $\tilde{U}$ is the sheaf
represented by $U$ on $\mathcal{X}_{\operatorname{lis-et}}$, then
$b^{-1}\tilde{U}$ is represented by $U_{X}$.

  2. (2)

If $\mathcal{Y}$ is representable, $m$ is étale, and $b$ is representable, and
if $\tilde{U}$ is the sheaf represented by $U$ on $\mathcal{X}_{\mathrm{et}}$,
then the étale sheaf $b^{-1}\tilde{U}$ is represented by $U_{X}$.

###### Proof.

We first sketch the proof of (1). The sheaf $b^{-1}\tilde{U}$ is the
sheafification of the presheaf that assigns to a scheme $T$ with a smooth map
$g:T\to\mathcal{X}$ the set $\operatorname{colim}\operatorname{Hom}_{Y}(W,U)$,
where the colimit is taken over schemes $W$ fitting into diagrams

(67)
${T}$${\mathcal{X}}$${W}$${\mathcal{Y}}$$\scriptstyle{g}$$\scriptstyle{b}$smooth

Composition induces a map

(68)
$\operatorname{colim}\operatorname{Hom}_{\mathcal{Y}}(W,U)\to\operatorname{Hom}_{\mathcal{Y}}(T,U)$

which is an isomorphism since $T\to Y$ is smooth and hence defines the final
object in the category over which we take the colimit. Finally we note that
$\operatorname{Hom}_{Y}(T,U)=\operatorname{Hom}_{Y}(T,U_{X})$. For (2), the
map $T\to\mathcal{X}$ is now étale and the colimit is over diagrams (67) with
$W\to\mathcal{Y}$ étale, so $T\to\mathcal{Y}$ is not an object of the colimit
category. However, the map (68) is still surjective. It is injective as well
because an element of $\operatorname{Hom}_{\mathcal{Y}}(W,U)$ must be étale,
so if we have elements of $\operatorname{Hom}_{\mathcal{Y}}(W_{1},U)$ and
$\operatorname{Hom}_{\mathcal{Y}}(W_{2},U)$ that yield the same map $T\to U$,
we may compare them via the étale $U$-scheme $W_{1}\times_{U}W_{2}$. ∎

######  Remark A.3.14.

Let $X_{\bullet}\to\mathcal{X}$ and $Y_{\bullet}\to\mathcal{Y}$ be very smooth
hypercovers of algebraic stacks $\mathcal{X}$ and $\mathcal{Y}$, and suppose
we are given a morphism $f_{\bullet}:X_{\bullet}\to Y_{\bullet}$ of simplicial
algebraic stacks and $f:\mathcal{X}\to\mathcal{Y}$ such that these maps
commute with the augmentations. Then for
$\mathcal{F}\in\operatorname{D_{qc}}(\mathcal{Y}_{\operatorname{lis-et}})$ we
have

$(a^{*}\mathsf{L}f^{*}\mathcal{F})|_{X_{n}}=\mathsf{L}f_{n}^{*}(a^{*}\mathcal{F})|_{X_{n}},$

and if $f$ is concentrated and
$\mathcal{G}\in\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$
then

$(a^{*}\mathsf{R}f_{*}\mathcal{G})|_{Y_{n}}=\mathsf{R}f_{n*}(a^{*}\mathcal{G})|_{Y_{n}},$

where the functor $\mathsf{L}f_{n}^{*}$ (resp. $\mathsf{R}f_{n*}$) is the
usual pullback functor (resp. direct image) between
$\operatorname{D_{qc}}(Y_{n,\operatorname{lis-et}})$ and
$\operatorname{D_{qc}}(X_{n,\operatorname{lis-et}})$. Indeed, the functor
$(a^{*}-)|_{X_{n}}$ is just $a^{*}_{n}(-)$, so the desired equalities are
equivalent to

$a_{n}^{*}\mathsf{L}f^{*}\mathcal{F}=\mathsf{L}f_{n}^{*}a_{n}^{*}\mathcal{F}\quad\quad\text{and}\quad\quad
a_{n}^{*}\mathsf{R}f_{*}\mathcal{G}=\mathsf{R}f_{n*}a_{n}^{*}\mathcal{G}.$

These follow from naturality of derived pullback and [HR17, Cor 4.13],
respectivley.

##  Appendix B Functoriality of the Fundamental Theorem

In this section we prove Lemmas 4.2.2 and 4.2.3.

###  B.1. Categories of algebra extensions

In this section, $\mathscr{S}$ is a site with $A\rightarrow B$ a morphism of
sheaves of rings on $\mathscr{S}$, and $I$ be a sheaf of $B$-modules.

####  B.1.1. Categories

The Picard category $\underline{\mathrm{Exal}}_{A}(B,I)$ was defined in
[Ill71, Sec III.1.1.2.3]: an object is a surjective $A$-algebra map
$E\rightarrow B$ whose kernel is (1) square-zero as an ideal of $E$, and (2)
isomorphic to $I$ as a $B$-module. We write these objects as short exact
sequences of abelian sheaves

(69) $0\rightarrow I\rightarrow E\rightarrow B\rightarrow 0.$

A morphism in $\underline{\mathrm{Exal}}_{A}(B,I)$ is a commuting diagram

${0}$${I}$${E}$${B}$${0}$${0}$${I}$${E^{\prime}}$${B}$${0}$$\scriptstyle{g}$

where $f$ is a morphism of $B$-modules and $g,h$ are morphisms of
$A$-algebras.

####  B.1.2. Functors

If $I\rightarrow I^{\prime}$ is a morphism of $B$-modules,
$B^{\prime}\rightarrow B$ is a morphism of $A$-algebras, and
$A^{\prime}\rightarrow A$ is a morphism of rings, then we have natural
functors

(70)
$\displaystyle\underline{\mathrm{Exal}}_{A}(B,I)\rightarrow\underline{\mathrm{Exal}}_{A}(B,I^{\prime})$
(71)
$\displaystyle\underline{\mathrm{Exal}}_{A}(B,I)\rightarrow\underline{\mathrm{Exal}}_{A}(B^{\prime},I_{B^{\prime}})$
(72)
$\displaystyle\underline{\mathrm{Exal}}_{A}(B,I)\rightarrow\underline{\mathrm{Exal}}_{A^{\prime}}(B,I)$

defined in [Ill71, Equ III.1.1.5.2], [Ill71, Equ III.1.1.5.3], and [Ill71, Equ
III.1.1.5.4], respectively. Here, $I_{B^{\prime}}$ denotes the sheaf $I$
considered as a $B^{\prime}$-module. Let
$\mathscr{S}^{\prime}\rightarrow\mathscr{S}$ be a continuous morphism of sites
inducing a morphism of topoi $(p^{-1},p_{*})$. Then we have an induced
morphism

(73)
$\underline{\mathrm{Exal}}_{A}(B,I)\rightarrow\underline{\mathrm{Exal}}_{p^{-1}A}(p^{-1}B,p^{-1}I)$

sending (69) to its image under $p^{-1}$. We are using that $p^{-1}$ is an
exact functor.

######  Lemma B.1.1.

The morphisms (70), (71), (72), and (73) commute pairwise.

###### Proof.

The most involved pair to check is (70) and (71). We work it out in detail and
offer a few words about the remaining pairs at the end of the proof. When we
say (70) and (71) commute, we mean that if $B^{\prime}\rightarrow B$ is a
morphism of rings and $I\rightarrow I^{\prime}$ is a morphism of $B$-modules,
then the diagram

(74)
${\underline{\mathrm{Exal}}_{A}(B,I)}$${\underline{\mathrm{Exal}}_{A}(B^{\prime},I_{B^{\prime}})}$${\underline{\mathrm{Exal}}_{A}(B,I^{\prime})}$${\underline{\mathrm{Exal}}_{A}(B^{\prime},I^{\prime}_{B^{\prime}})}$(71)(70)(70)(71)

commutes up to a natural transformation.

Given an element (69) of $\underline{\mathrm{Exal}}_{A}(B,I)$, we have a
diagram

(75)
${I^{\prime}}$${P}$${0}$${I}$${E}$${B}$${0}$${F}$${B^{\prime}}$$\scriptstyle{a^{\prime}}$$\scriptstyle{(0,b)}$$\scriptstyle{a}$$\scriptstyle{(a,0)}$$\scriptstyle{\iota_{E}}$$\scriptstyle{b}$$\scriptstyle{p_{E}}$$\scriptstyle{b^{\prime}}$

where $P=I^{\prime}\oplus_{I}E$ and $F=E\times_{B}B^{\prime}$: as abelian
groups, $P$ and $F$ are the colimit and limit of the usual diagrams, while the
ring structures are described in [GD67, $0_{\mathrm{IV}}$.18.2.8] and [GD67,
$0_{\mathrm{IV}}$.18.1.5]. Set $Q=I^{\prime}\oplus_{I}F$ and
$G=P\times_{B}B^{\prime}$. Then

$0\rightarrow I^{\prime}\rightarrow Q\rightarrow B^{\prime}\rightarrow 0$

is the image of (69) under the composition ${}^{\rightarrow}\downarrow$ in
(74), and likewise $G$ defines the image under the composition
$\downarrow{}_{\rightarrow}$. An arrow from $Q$ to $G$ in the groupoid
$\underline{\mathrm{Exal}}_{A}(B^{\prime},I^{\prime})$ is given by four dashed
arrows so that this diagram commutes:

(76) ${I}$${I^{\prime}}$${P}$${F}$${B^{\prime}}$${B}$

(To check commutativity, it suffices to check that the quadrilaterals
$I^{\prime}IFP$ and $I^{\prime}IFB^{\prime}$ and the perimeter commute.) The
required collection of dotted arrows is given by
$a^{\prime}:I^{\prime}\rightarrow P$, $0:I^{\prime}\rightarrow B^{\prime}$,
$\iota_{E}\circ p_{E}:F\rightarrow P$, and $b^{\prime}:F\rightarrow
B^{\prime}$.

To show that the resulting arrows in
$\underline{\mathrm{Exal}}_{A}(B^{\prime},I^{\prime})$ define a natural
transformation (in this groupoid), suppose we are given an arrow

${0}$${I}$${E_{1}}$${B}$${0}$${E_{2}}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{2}}$$\scriptstyle{f}$$\scriptstyle{b_{1}}$$\scriptstyle{b_{2}}$

in $\underline{\mathrm{Exal}}_{A}(B,I)$. Let $f_{P}:P_{1}\rightarrow P_{2}$
and $f_{F}:F_{1}\rightarrow F_{2}$ be the maps induced by $f$, where $P_{i}$
and $F_{i}$ are defined as in (75). Likewise let $Q_{i}$ and $G_{i}$ be the
images of $E_{i}$ in $\underline{\mathrm{Exal}}_{A}(B^{\prime},I^{\prime})$
under the maps in (74). We must compare two maps from $Q_{1}$ to $G_{2}$ in
$\underline{\mathrm{Exal}}_{A}(B^{\prime},I^{\prime})$. Such maps are given by
diagrams of the form (76) with $F$ replaced by $F_{1}$ and $P$ replaced by
$P_{2}$. In the situation at hand, one of the maps from $Q_{1}$ to $G_{2}$ is
given by the diagram

${I^{\prime}}$${P_{2}}$${F_{1}}$${B^{\prime}}$$\scriptstyle{f_{P}\circ
a_{1}^{\prime}}$$\scriptstyle{0}$$\scriptstyle{f_{P}\circ\iota_{E_{1}}\circ
p_{E_{1}}}$$\scriptstyle{b_{1}^{\prime}}$

and the other is given by the diagram

${I^{\prime}}$${P_{2}}$${F_{1}}$${B^{\prime}}$$\scriptstyle{a_{2}^{\prime}}$$\scriptstyle{0}$$\scriptstyle{\iota_{E_{2}}\circ
p_{E_{2}}\circ f_{F}}$$\scriptstyle{b_{2}^{\prime}\circ f_{F}}$

These are easily seen to consist of the same morphisms.

This completes the proof that (71) and (70) commute. Of the remaining pairs,
most of the checks are trivial (in particular, the analog of (74) is strictly
commutative). Only the pairs ((70), (73)) and ((71), (73)) are nontrivial. For
these, one uses that $p^{-1}$ is exact and hence preserves finite limits and
colimits. ∎

###  B.2. Illusie’s theorem

####  B.2.1. Statement

In this section, $\mathscr{S}$ is a site with $A\rightarrow B$ a morphism of
sheaves of rings on $\mathscr{S}$, and $I$ be a sheaf of $B$-modules.

######  Theorem B.2.1 ([Ols06, Thm A.7], [Ill71, Sec III.1.2.2]).

There is a canonical isomorphism

(77)
$\beta:\underline{\mathrm{Exal}}_{A}(B,I)\rightarrow\mathrm{Ext}^{0/-1}_{B}(\mathbb{L}_{B/A},I[1]),$

where the right hand side was defined in (39).

###### Proof.

Since the isomophism in [Ols06, Thm A.7] is defined on groupoid fibers, we may
use the same definition for our morphism (77) (written out in the proof of
Lemma B.2.4), and the argument in [Ols06, Thm A.7] shows that it is an
isomorphism. Note that when $\mathscr{S}$ has a final object $S$, the map (77)
is the value on $S$ of the isomorphism in [Ols06, Thm A.7]. ∎

####  B.2.2. Functoriality

We will show that (77) is compatible with the functors defined in Section
B.1.2. We will use the following instances of Situation 2.1.1.

######  Example B.2.2.

The following is an example of Situation 2.1.1. Let $\mathscr{S}$ be a site
and $B$ a sheaf of rings on $\mathscr{S}$. Then $\operatorname{D}(B)$ is a
closed symmetric monoidal category with product
$\overset{\mathsf{L}}{\otimes}_{B}$ and internal hom
$\mathsf{R}\mathcal{H}om_{B}$. If $B^{\prime}\rightarrow B$ is a flat morphism
of sheaves of rings, then extension of scalars
$-\otimes_{B^{\prime}}B:B^{\prime}\mathrm{-mod}\rightarrow B\mathrm{-mod}$ is
strong monoidal and exact with an exact right adjoint $(-)_{B^{\prime}}$ given
by restriction of scalars.

Let $\mathscr{C}=\operatorname{D}(B^{\prime})$ and let
$\mathscr{D}=\operatorname{D}(B)$. By [Stacks, Tag 0DVC] the functors
$-\otimes_{B^{\prime}}$ and $(-)_{B^{\prime}}$ extend to an adjoint pair for
$\operatorname{D}(B^{\prime})$ and $\operatorname{D}(B)$, and by [Stacks, Tags
07A4, 08I6] the functor
$-\otimes_{B^{\prime}}B:\operatorname{D}(B^{\prime})\rightarrow\operatorname{D}(B)$
is still strong monoidal.

In addition, it follows from [Stacks, Tags 08J9, 0A90, 0A5Y] that if
$M^{\bullet}$ is $K$-flat and $N^{\bullet}$ is injective, the counit
$\mathsf{R}\mathcal{H}om_{B^{\prime}}(M,N)\overset{\mathsf{L}}{\otimes}_{B^{\prime}}M\rightarrow
N$ is given in degree $n$ by a product over $p+q+r=n$ of the sheaf maps

$\mathcal{H}om_{B^{\prime}}(M^{-p},N^{q})\otimes_{B^{\prime}}M^{r}\rightarrow
N^{n},$

where this map is equal to the usual evaluation map if $q=n$ and it is zero
otherwise. We will give an explicit description of (7) in the proof of Lemma
B.2.5.

######  Example B.2.3.

If
$p:(\mathscr{C},\mathscr{O}_{\mathscr{C}})\rightarrow(\mathscr{D},\mathscr{O}_{\mathscr{D}})$
is a flat morphism of ringed topoi given by an adjoint pair $(p^{-1},p_{*})$,
then $p^{*}$ is exact and hence defines a strong monoidal functor
$\operatorname{D}(\mathscr{O}_{\mathscr{D}})\rightarrow\operatorname{D}(\mathscr{O}_{\mathscr{C}})$
with a right adjoint $\mathsf{R}p_{*}$. We will give an explicit description
of (8) in the proof of Lemma B.2.6.

######  Lemma B.2.4.

The isomorphism (77) is functorial as follows.

  1. (1)

Let $A\rightarrow B$ be a map of sheaves of rings on $\mathscr{S}$. If
$I\rightarrow J$ is a morphism of $B$-modules, there is a commuting diagram

${\underline{\mathrm{Exal}}_{A}(B,I)}$${\mathrm{Ext}^{0/-1}_{B}(\mathbb{L}_{B/A},I[1])}$${\underline{\mathrm{Exal}}_{A}(B,J)}$${\mathrm{Ext}^{0/-1}_{B}(\mathbb{L}_{B/A},J[1])}$$\scriptstyle{\beta}$(70)$\scriptstyle{\beta}$

  2. (2)

If there is a commuting square of rings with $B^{\prime}\rightarrow B$ flat

${A^{\prime}}$${A}$${B^{\prime}}$${B}$

then the canonical map
$\mathbb{L}_{B^{\prime}/A^{\prime}}\overset{\mathsf{L}}{\otimes}_{B^{\prime}}B\rightarrow\mathbb{L}_{B/A}$
induces a commuting square

(78)
${\underline{\mathrm{Exal}}_{A}(B,I)}$${\mathrm{Ext}^{0/-1}_{B}(\mathbb{L}_{B/A},I[1])}$${\underline{\mathrm{Exal}}_{A^{\prime}}(B^{\prime},I_{B^{\prime}})}$${\mathrm{Ext}^{0/-1}_{B^{\prime}}(\mathbb{L}_{B^{\prime}/A^{\prime}},I_{B^{\prime}}[1])}$${\mathrm{Ext}^{0/-1}_{B}(\mathbb{L}_{B^{\prime}/A^{\prime}}\otimes_{B^{\prime}}B,I[1])}$$\scriptstyle{\text{\eqref{eq:changeB}}\circ\text{\eqref{eq:changeA}}}$$\scriptstyle{\beta}$$\scriptstyle{\beta}$(6)

where (6) is defined in the context of Example B.2.2.

  3. (3)

Let
$(\mathscr{S},\mathscr{O}_{\mathscr{S}})\rightarrow(\mathscr{S}^{\prime},\mathscr{O}_{\mathscr{S}^{\prime}})$
be a continuous morphism of ringed sites inducing a flat morphism of topoi
$(p^{-1},p_{*})$. Let $A$ and $B=\mathscr{O}_{\mathscr{S}}$ be sheaves of
rings on $\mathscr{S}$. Then if $I$ is a sheaf of $B$-modules, there is a
commuting diagram

(79)
${\underline{\mathrm{Exal}}_{A}(B,I)}$${\mathrm{Ext}^{0/-1}_{B}(\mathbb{L}_{B/A},I[1])}$${\underline{\mathrm{Exal}}_{p^{-1}A}(p^{-1}B,p^{-1}I)}$${\mathrm{Ext}^{0/-1}_{p^{-1}B}(p^{-1}\mathbb{L}_{B/A},p^{-1}I[1])}$${\mathrm{Ext}^{0/-1}_{p^{*}B}(p^{*}\mathbb{L}_{B/A},p^{*}I[1])}$$\scriptstyle{\beta}$(73)(8)$\scriptstyle{\beta}$(8)

where the horizontal instance of (8) is defined in the context of Example
B.2.2 and the vertical instance of (8) is defined in the context of Example
B.2.3, and we have suppressed an isomorphism induced by
$\mathbb{L}_{p^{-1}B/p^{-1}A}\simeq p^{-1}\mathbb{L}_{B/A}$.

###### Proof.

We summarize the definition of $\beta$; see [Ols06, Thm A.7] for more details.
Let $P_{\bullet}$ be the simplicial $A$-algebra given by the standard free
resolution of the $A$-algebra $B$ [Stacks, Tag 08SR]. The morphism $\beta$ is
defined to be the composition

$\underline{\mathrm{Exal}}_{A}(B,I)\xrightarrow{\beta_{1}}\underline{\mathrm{Exal}}_{A}(P_{\bullet},I)\xrightarrow{\beta_{2}}\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}/A},I)\xrightarrow{\beta_{3}}\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}/A}\otimes
B,I)\xrightarrow{\beta_{4}}\mathrm{Ext}^{0/-1}(\mathbb{L}_{B/A},I[1])$

Here, $\underline{\mathrm{Ext}}(\Omega_{\bullet},I)$ denotes the Picard
category of simplicial $\mathscr{O}_{\mathcal{C}}$-module extensions of
$\Omega_{\bullet}$ by $I$ (viewed as a simplicial module); see [Ols06, Sec
A.1]. The map $\beta_{1}$ is given by the map (71) applied to the augmentation
$P_{\bullet}\rightarrow B$, the morphism $\beta_{2}$ is given by taking
differentials, $\beta_{3}$ is given by tensoring with $B$, and $\beta_{4}$ is
the functorial isomorphism in [Ols06, Prop A.3].

Proof of (1). The desired functoriality follows from a commuting diagram

${\underline{\mathrm{Exal}}_{A}(B,I)}$${\underline{\mathrm{Exal}}_{A}(P_{\bullet},I)}$${\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}/A},I)}$${\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}/A}\otimes
B,I)}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{B/A},I[1])}$${\underline{\mathrm{Exal}}_{A}(B,J)}$${\underline{\mathrm{Exal}}_{A}(P_{\bullet},J)}$${\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}/A},J)}$${\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}/A}\otimes
B,J)}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{B/A},J[1])}$$\scriptstyle{\beta_{1}}$(70)$\scriptstyle{\beta_{2}}$(70)$\scriptstyle{\beta_{3}}$$\scriptstyle{\beta_{4}}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\beta_{3}}$$\scriptstyle{\beta_{4}}$

The square with $\beta_{1}$ commutes by Lemma B.1.1. The square with
$\beta_{2}$ commutes because differentials commute with colimits [Stacks, Tag
031G]. The square with $\beta_{3}$ commutes because tensor product is a left
adjoint and so commutes with colimits, and the square with $\beta_{4}$
commutes by the naturality in [Ols06, Prop A.3].

Proof of (2).  The desired functoriality follows from two commuting diagrams.
First we have

(80)
${\underline{\mathrm{Exal}}_{A}(B,I)}$${\underline{\mathrm{Exal}}_{A}(P_{\bullet},I)}$${\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}/A},I)}$${\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}/A}\otimes
B,I)}$${\underline{\mathrm{Exal}}_{A^{\prime}}(B^{\prime},I_{B^{\prime}})}$${\underline{\mathrm{Exal}}_{A^{\prime}}(P_{\bullet}^{\prime},I_{B^{\prime}})}$${\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}^{\prime}/A^{\prime}},I_{P^{\prime}_{\bullet}})}$${\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}^{\prime}/A^{\prime}}\otimes
B^{\prime},I_{B^{\prime}})}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\beta_{3}}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\beta_{3}}$

which we claim commutes. Here $P^{\prime}_{\bullet}$ is the simplicial
$A^{\prime}$-algebra that is the standard resolution of $B^{\prime}$. The two
left vertical arrows are given by (72) and (71); the next two vertical arrows
are given by the analog of (71) for the $\underline{\mathrm{Ext}}$ categories.
The first square commutes by Lemma B.1.1. The commutativity of the squares
with $\beta_{2}$ and $\beta_{3}$ may be checked with the same type of
computation used in Lemma B.1.1 and we will be brief here.

For the square with $\beta_{2}$, if $0\rightarrow I\rightarrow
E_{\bullet}\rightarrow P_{\bullet}\rightarrow 0$ is an object of
$\underline{\mathrm{Exal}}_{A}(P_{\bullet},I)$, then the natural
transformation is given on this object by the (iso)morphism

$\Omega_{E_{\bullet}\times_{P_{\bullet}}P^{\prime}_{\bullet}/A^{\prime}}\rightarrow\Omega_{E_{\bullet}/A}\times_{\Omega_{P_{\bullet}/A}}\Omega_{P^{\prime}_{\bullet}/A^{\prime}}$

induced by the commuting cube

${E_{\bullet}}$${P_{\bullet}}$${E_{\bullet}\times_{P_{\bullet}}P^{\prime}_{\bullet}}$${P^{\prime}_{\bullet}}$${A}$${A}$${A^{\prime}}$${A^{\prime}}$

For the square with $\beta_{3}$, if $0\rightarrow I\rightarrow
E_{\bullet}\rightarrow\Omega_{P_{\bullet}/A}\rightarrow 0$ is an object of
$\underline{\mathrm{Ext}}_{P_{\bullet}}(\Omega_{P_{\bullet}/A},I)$, then the
natural transformation is given on this object by the (iso)morphism of
$B^{\prime}$-modules

$(E_{\bullet}\times_{\Omega_{P_{\bullet}/A}}\Omega_{P^{\prime}_{\bullet}/A^{\prime}})\otimes_{P^{\prime}_{\bullet}}B^{\prime}\rightarrow(X\otimes_{P}B)\times_{\Omega_{P_{\bullet}/A}\otimes_{P_{\bullet}}B}(\Omega_{P^{\prime}_{\bullet}/A^{\prime}}\otimes_{P^{\prime}_{\bullet}}B^{\prime})$

induced by the natural map of $P^{\prime}_{\bullet}$-modules

$E_{\bullet}\times_{\Omega_{P_{\bullet}/A}}\Omega_{P^{\prime}_{\bullet}/A^{\prime}}\rightarrow(E_{\bullet}\otimes_{P_{\bullet}}B)\times_{\Omega_{P_{\bullet}/A}\otimes_{P_{\bullet}}B}(\Omega_{P^{\prime}_{\bullet}/A^{\prime}}\otimes_{P^{\prime}_{\bullet}}B^{\prime}).$

The second diagram comprising (78) is as follows.

${\underline{\mathrm{Ext}}_{B}(\Omega_{P_{\bullet}/A}\otimes
B,I)}$${\mathrm{Ext}^{0/-1}_{B}(\mathbb{L}_{B/A},I[1])}$${\underline{\mathrm{Ext}}_{B^{\prime}}((\Omega_{P_{\bullet}/A}\otimes
B)_{B^{\prime}},I_{B^{\prime}})}$${\mathrm{Ext}^{0/-1}_{B^{\prime}}((\mathbb{L}_{B/A})_{B^{\prime}},I_{B^{\prime}}[1])}$${\mathrm{Ext}^{0/-1}_{B}((\mathbb{L}_{B/A})_{B^{\prime}}\otimes_{B^{\prime}}B,I[1])}$${\underline{\mathrm{Ext}}_{B^{\prime}}(\Omega_{(P_{\bullet})^{\prime}/A^{\prime}}\otimes
B^{\prime},I_{B^{\prime}})}$${\mathrm{Ext}^{0/-1}_{B^{\prime}}(\mathbb{L}_{B^{\prime}/A^{\prime}},I_{B^{\prime}}[1])}$${\mathrm{Ext}^{0/-1}_{B}(\mathbb{L}_{B^{\prime}/A^{\prime}}\otimes_{B^{\prime}}B,I[1])}$$\scriptstyle{\beta_{4}}$$\scriptstyle{\rho}$(7)$\scriptstyle{\epsilon_{\mathbb{L}_{B/A}}}$$\scriptstyle{\beta_{4}}$(71)(6)$\scriptstyle{\beta_{4}}$(6)

The arrow labeled $\rho$ sends an extension of $B$-modules to the extension of
$B^{\prime}$-modules obtained by restriction of scalars (an exact functor).
One may check directly that the composition of the left vertical arrows is
equal to the right vertical arrow in (80). The map labeled (7) is in the
context of Example B.2.2 and the triangle commutes by definition, while the
top left square commutes by Lemma B.2.5 below. The unlabeled vertical maps are
induced by the canonical map
$(\mathbb{L}_{B/A})_{B^{\prime}}\rightarrow\mathbb{L}_{B^{\prime}/A^{\prime}}$,
so the bottom squares commute by functoriality of $\beta_{4}$ and (6).

Proof of (3). Let $A^{\prime}=p^{-1}A$, let $B^{\prime}=p^{-1}B$, and let
$P^{\prime}_{\bullet}$ denote the standard resolution of $B^{\prime}$ as an
$A^{\prime}$-algebra. The desired commuting square comes from two commuting
diagrams. First, we have

(81)
${\underline{\mathrm{Exal}}_{A}(B,I)}$${\underline{\mathrm{Exal}}_{A}(P_{\bullet},I)}$${\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}/A},I)}$${\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}/A}\otimes
B,I)}$${\underline{\mathrm{Exal}}_{A^{\prime}}(B^{\prime},a^{-1}I)}$${\underline{\mathrm{Exal}}_{A^{\prime}}(P^{\prime}_{\bullet},p^{-1}I)}$${\underline{\mathrm{Ext}}(\Omega_{P^{\prime}_{\bullet}/A^{\prime}},p^{-1}I)}$${\underline{\mathrm{Ext}}(\Omega_{P^{\prime}_{\bullet}/A^{\prime}}\otimes
B^{\prime},p^{-1}I)}$$\scriptstyle{\beta_{1}}$$\scriptstyle{p^{-1}}$$\scriptstyle{\beta_{2}}$$\scriptstyle{p^{-1}}$$\scriptstyle{\beta_{3}}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\beta_{3}}$

The first square commutes by Lemma B.1.1. The third vertical map is induced by
$p^{-1}$ and the isomorphism
$p^{-1}\Omega_{P_{\bullet}/A}\simeq\Omega_{p^{-1}P_{\bullet}/p^{-1}A}$
([Stacks, Tag 08TQ]), and the fourth is induced by $p^{-1}$ and the
isomorphism $p^{-1}(\Omega_{P_{\bullet}/A}\otimes
B)\simeq\Omega_{p^{-1}P_{\bullet}/p^{-1}A}\otimes p^{-1}B$ ([Stacks, Tag
03EL]), and the squares commute by functoriality of the same isomorphisms.
Second, we have

${\underline{\mathrm{Ext}}(\Omega_{P_{\bullet}/A}\otimes
B,I)}$${\mathrm{Ext}^{0/-1}_{B}(\mathbb{L}_{B/A},I[1])}$${\underline{\mathrm{Ext}}(p^{-1}(\Omega_{P_{\bullet}/A}\otimes
B),p^{-1}I)}$${\mathrm{Ext}^{0/-1}_{p^{-1}B}(p^{-1}\mathbb{L}_{B/A},p^{-1}I[1])}$${\mathrm{Ext}^{0/-1}_{p^{*}B}(p^{*}\mathbb{L}_{B/A},p^{*}I[1])}$${\underline{\mathrm{Ext}}(\Omega_{P^{\prime}_{\bullet}/A^{\prime}}\otimes
B^{\prime},p^{-1}I)}$${p_{*}\mathrm{Ext}^{0/-1}_{p^{-1}B}(\mathbb{L}_{B^{\prime}/A^{\prime}},p^{-1}I[1])}$$\scriptstyle{\beta_{4}}$$\scriptstyle{p^{-1}}$(8)(8)(8)$\scriptstyle{\beta_{4}}$

The composition of the left vertical arrows is equal to the right vertical
arrow in (81). The middle horizontal arrow is comprised of $\beta_{4}$ and an
isomorphism (see Lemma B.2.6), and the top left square commutes by Lemma
B.2.6. The bottom square commutes by functoriality of $\beta_{4}$, and the
triangle commutes by functoriality of (8) in the functors. ∎

######  Lemma B.2.5.

Let $(\mathscr{S},\mathscr{O}_{\mathscr{S}})$ be a ringed site, let
$\Omega_{\bullet}$ be a simplicial $\Omega_{\mathscr{S}}$-module, and let $I$
be an $\Omega_{\mathscr{S}}$-module. Let
$\mathscr{O}_{\mathscr{S}}^{\prime}\rightarrow\mathscr{O}_{\mathscr{S}}$ be a
flat morphism of rings. There is a commuting diagram

(82)
${\underline{\mathrm{Ext}}_{\mathscr{O}_{\mathscr{S}}}(\Omega_{\bullet},I)}$${\mathrm{Ext}^{0/-1}_{\mathscr{O}_{\mathscr{S}}}(N(\Omega_{\bullet}),I[1])}$${\underline{\mathrm{Ext}}_{\mathscr{O}_{\mathscr{S}}^{\prime}}((\Omega_{\bullet})_{\mathscr{O}_{\mathscr{S}}^{\prime}},I_{\mathscr{O}_{\mathscr{S}}^{\prime}})}$${\mathrm{Ext}^{0/-1}_{\mathscr{O}^{\prime}_{\mathscr{S}}}(N((\Omega_{\bullet})_{\mathscr{O}_{\mathscr{S}}^{\prime}}),(I)_{\mathscr{O}_{\mathscr{S}}^{\prime}}[1])}$${\mathrm{Ext}^{0/-1}_{\mathscr{O}^{\prime}_{\mathscr{S}}}((N(\Omega_{\bullet}))_{\mathscr{O}_{\mathscr{S}}^{\prime}},(I)_{\mathscr{O}_{\mathscr{S}}^{\prime}}[1])}$$\scriptstyle{\rho}$$\scriptstyle{\beta_{4}}$(7)$\scriptstyle{\beta_{4}}$

where $N(\Omega_{\bullet})$ is the normalization of the Moore complex
associated to $\Omega_{\bullet}$ (see [Stacks, Tag 0194]), $\beta_{4}$ is the
isomorphism of [Ols07, Prop A.3], $\rho$ applies restriction of scalars to an
exact sequence, and (7) is in the context of Example B.2.2.

###### Proof.

By [Stacks, Tag 05NI, 05T7] there is a quasi-isomorphism $N\rightarrow
N(\Omega_{\bullet})$ from a complex $N\in
D^{[-\infty,0]}(\mathscr{O}_{\mathscr{S}})$ of flat
$\mathscr{O}_{\mathscr{S}}$-modules. We enlarge (82) on its right side by
composing with the square induced by $N\rightarrow N(\Omega_{\bullet})$ and
show that the perimeter of the new diagram commutes. From the definition of
$\beta_{4}$ we may assume $I$ is injective. To simplify notation let
$B=\mathscr{O}_{\mathscr{S}}$ and
$B^{\prime}=\mathscr{O}_{\mathscr{S}}^{\prime}$.

Most of the work is to describe (7) explicitly. To this end we first recall
the definition of (6): in the notation of Section 2.1, it is the image of the
composition

(83) $f^{*}\operatorname{Hom}(X,f_{*}Y)\otimes
f^{*}X\xrightarrow[\sim]{\text{\eqref{eq:strong-
monoidal}}}f^{*}(\operatorname{Hom}(X,f_{*}Y)\otimes
X)\xrightarrow{f^{*}(\epsilon^{\otimes}_{f_{*}Y})}f^{*}f_{*}Y\xrightarrow{\epsilon^{f^{*}}_{Y}}Y$

under the $(\otimes,\operatorname{Hom})$ adjunction and then the
$(f^{*},f_{*})$ adjunction. One sees using the description of
$\epsilon^{\otimes}$ in Example B.2.2 that with
$M\in\operatorname{D}^{[-\infty,0]}(B^{\prime})$ a complex of flat
$B^{\prime}$-modules and $I$ as in the previous paragraph, the morphism
$(\mathsf{R}\mathcal{H}om_{B^{\prime}}(M,(I)_{B^{\prime}}[1])\otimes_{B^{\prime}}B)\overset{\mathsf{L}}{\otimes}_{B}(M\otimes_{B^{\prime}}B)\rightarrow
I[1]$ of (83) is given by a product over $p+r=0$ of the canonical sheaf maps

$(\mathcal{H}om_{B^{\prime}}(M^{-p},(I)_{B^{\prime}})\otimes_{B^{\prime}}B)\otimes_{B}(M^{r}\otimes_{B^{\prime}}B)\rightarrow
I.$

We are using the fact that $(-)_{B^{\prime}}$ preserves injectives (since it
has an exact left adjoint) and hence $(I)_{B^{\prime}}$ is injective. We see
that (6):
$\mathsf{R}\mathcal{H}om_{B^{\prime}}(M,(I)_{B^{\prime}}[1])\to(\mathsf{R}\mathcal{H}om_{B}(M\otimes_{B^{\prime}}B,I[1])_{B^{\prime}})$
is given in degree $n$ by the canonical sheaf map

$\mathcal{H}om_{B^{\prime}}(M^{-n-1},(I)_{B^{\prime}})\to(\mathcal{H}om_{B}(M^{-n-1}\otimes_{B^{\prime}}B,I)_{B^{\prime}}).$

To compute (7), given $N$ as at the beginning of this proof, we note that
$(N)_{B^{\prime}}\in\operatorname{D}^{[-\infty,0]}$ is a complex of flat
$B^{\prime}$-modules by [Stacks, Tag 00HC], so our previous description of (6)
applies with $M=(N)_{B^{\prime}}$. From the definition of

(84) $\text{\eqref{eq:sheafy-
push}}:(\mathsf{R}\mathcal{H}om_{B}(N,I[1]))_{B^{\prime}}\rightarrow\mathsf{R}\mathcal{H}om_{B^{\prime}}((N)_{B^{\prime}},(I)_{B^{\prime}}[1])$

we see that it is given in degree $n$ by the usual sheaf map

$\mathcal{H}om_{B}(N^{-n-1},I)\to\mathcal{H}om_{B^{\prime}}((N^{n-1})_{B^{\prime}},(I)_{B^{\prime}}).$

We are interested in the cases $n=0$ and $n=-1$. Given $U\in\mathscr{S}$ and a
section $f:N^{n-1}|_{U}\rightarrow I|_{U}$ of the left hand side, i.e. a
morphism of $B$-modules, this map sends $f$ to the corresponding morphism of
$B^{\prime}$-modules. (One may verify this claim by unwinding the definitions
and ultimately appealing to Example 2.1.2.) Now applying $\mathsf{R}\Gamma$ to
(84) is straightforward since both complexes are complexes of injectives by
[Stacks, Tag 0A96].

This gives a completely explicit description of the morphism labeled (7) in
(82). With this in hand it is easy to check that (82) commutes.

∎

######  Lemma B.2.6.

Let $(\mathscr{S},\mathscr{O}_{\mathscr{S}})$ be a ringed site, let
$\Omega_{\bullet}$ be a simplicial $\Omega_{\mathscr{S}}$-module, and let $I$
be an $\Omega_{\mathscr{S}}$-module. Let
$(\mathscr{S},\mathscr{O}_{\mathscr{S}})\rightarrow(\mathscr{S}^{\prime},\mathscr{O}_{\mathscr{S}^{\prime}})$
be a continuous morphism of ringed sites inducing a flat morphism of topoi
$(p^{-1},p_{*})$ such that
$p^{-1}\mathscr{O}_{\mathscr{S}}=\mathscr{O}_{\mathscr{S}^{\prime}}$. Then
there is a commuting diagram

(85)
${\underline{\mathrm{Ext}}_{\mathscr{O}_{\mathscr{S}}}(\Omega_{\bullet},I)}$${\mathrm{Ext}^{0/-1}_{\mathscr{O}_{\mathscr{S}}}(N(\Omega_{\bullet}),I[1])}$${\underline{\mathrm{Ext}}_{\mathscr{O}_{\mathscr{S}^{\prime}}}(p^{-1}(\Omega_{\bullet}),p^{-1}(I))}$${\mathrm{Ext}^{0/-1}_{\mathscr{O}_{\mathscr{S}^{\prime}}}(N(p^{-1}\Omega_{\bullet}),p^{-1}I[1])}$${\mathrm{Ext}^{0/-1}_{\mathscr{O}_{\mathscr{S}^{\prime}}}(p^{-1}N(\Omega_{\bullet}),p^{-1}I[1])}$$\scriptstyle{\beta_{4}}$(8)$\scriptstyle{\beta_{4}}$

where $N(\Omega_{\bullet})$ is the normalization of the simplical module,
$\beta_{4}$ is the isomorphism of [Ols07, Prop A.3], (8) is in the context of
Example B.2.3, and the left vertical arrow applies $p^{-1}$ to an exact
sequence.

###### Proof.

By [Stacks, Tag 05NI, 05T7] there is a quasi-isomorphism $N\rightarrow
N(\Omega_{\bullet})$ from a complex $N\in
D^{[-\infty,0]}(\mathscr{O}_{\mathscr{S}})$ of flat
$\mathscr{O}_{\mathscr{S}}$-modules. We enlarge (85) on its right side by
composing with the square induced by $N\rightarrow N(\Omega_{\bullet})$ and
show that the perimeter of the new diagram commutes. From the definition of
$\beta_{4}$ we may assume $I$ is injective. To simplify notation let
$B=\mathscr{O}_{\mathscr{S}}$ and
$B^{\prime}=\mathscr{O}_{\mathscr{S}}^{\prime}$.

Most of the work is to describe (8) explicitly. To this end, we first note
that (in the notation of Section 2.1) (8) is equal to the image of the
composition

(86) $f^{*}\operatorname{Hom}(X,Y)\otimes
f^{*}X\xrightarrow[\sim]{\text{\eqref{eq:strong-
monoidal}}}f^{*}(\operatorname{Hom}(X,Y)\otimes
X)\xrightarrow{f^{*}(\epsilon^{\otimes}_{Y})}f^{*}Y$

under the $(\otimes,\operatorname{Hom})$ adjunction and the $(f^{*},f_{*})$
adjunction. (To see this, use (83) and the triangle identity
$\epsilon^{f^{*}}_{f^{*}Y}\circ f^{*}\eta^{f^{*}}_{Y}=1_{f^{*}Y}$.) One sees
using the description of $\epsilon^{\otimes}$ in Example B.2.2 that with $N$
and $I$ as in the previous paragraph, the morphism (86):
$p^{-1}\mathsf{R}\mathcal{H}om_{B}(N,I[1])\overset{\mathsf{L}}{\otimes}_{B^{\prime}}p^{-1}N\rightarrow
p^{-1}I[1]$ is given by the product over $r\in\mathbb{Z}$ of the usual sheaf
maps

$p^{-1}\mathcal{H}om_{B}(N^{r},I)\otimes_{B^{\prime}}p^{-1}N^{r}\rightarrow
p^{-1}I.$

To compute the $(\otimes,\operatorname{Hom})$ adjunction we must take an
injective resolution $p^{-1}I[1]\rightarrow J$ of $p^{-1}I[1]$. Given this,
one checks that

(87) $\text{\eqref{eq:sheafy-
pullback}}:\mathsf{R}\mathcal{H}om_{B}(N,I[1])\rightarrow\mathsf{R}p_{*}\mathsf{R}\mathcal{H}om_{B^{\prime}}(p^{-1}N,J)$

is given in degree $n$ by the composition of the usual sheaf maps

$\mathcal{H}om_{B}(N^{-n-1},I)\rightarrow
p_{*}\mathcal{H}om_{B}^{\prime}(p^{-1}N^{-n-1},p^{-1}I)\rightarrow
p_{*}\mathcal{H}om_{B^{\prime}}(p^{-1}N^{-n-1},J^{-1})$

We have used [Stacks, Tag 0A96] to conclude that
$\mathsf{R}\mathcal{H}om_{B^{\prime}}(p^{-1}N,J)$ is a complex of injectives
so its pushforward can be computed termwise. We are interested in the cases
$n=0,-1$. Given $U\in\mathscr{S}$ and a section $f:N^{-n-1}|_{U}\rightarrow
I|_{U}$ of the left hand side, this map sends $f$ to
$p^{-1}f:p^{-1}N^{-n-1}|_{U}\rightarrow p^{-1}I|_{U}\to J^{-1}|_{U}$ (it is an
exercise to check that the “usual sheaf map” does this). Now applying
$\mathsf{R}\Gamma$ to (87) is straightforward since both complexes are
complexes of injectives by [Stacks, Tag 0730, 0A96].

This gives a completely explicit description of (8) in (85). With this in hand
one may check directly that (85) commutes.

∎

###  B.3. Description of (42)

To define (42), we use another example of Situation 2.1.1.

######  Example B.3.1.

Let $\mathcal{X}$ be an algebraic stack and let $X\rightarrow\mathcal{X}$ be a
smooth cover by an algebraic space $X$. By Proposition A.2.3, the morphism
$\varpi^{*}:\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-
et}})\to\operatorname{D_{qc}}(U^{+}_{\bullet,\mathrm{et}})$ is an equivalence
of categories. In fact, it follows from the construction of $\varpi^{*}$ that
it is a strong monoidal equivalence of symmetric monoidal categories. A
standard argument shows that the inverse equivalence $\mathsf{R}\varpi_{*}$ is
also strong monoidal.

Let $f:\mathcal{X}\rightarrow\mathcal{Y}$ be a representable morphism of
algebraic stacks. Let $Y\rightarrow\mathcal{Y}$ be a smooth cover by a scheme
with $Y^{+}_{\bullet}\rightarrow\mathcal{Y}$ the associated strictly
simplicial algebraic space and $\varpi:X^{+}_{\bullet}\rightarrow\mathcal{X}$
its pullback to $\mathcal{X}$. We will use $\varpi^{*}$ and
$\mathsf{R}\varpi_{*}$ to denote the functors in Example B.3.1. We recall that
the cotangent complex $\mathbb{L}_{\mathcal{X}/\mathcal{Y}}$ is defined to be
the object in $\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$
corresponding, under the equivalence $\mathsf{R}\varpi_{*}$ of Example B.3.1,
to the cotangent complex of the morphism of topoi
$X^{+}_{\bullet,\mathrm{et}}\rightarrow Y^{+}_{\bullet,\mathrm{et}}$.

######  Definition B.3.2 ([Ols06]).

Let $I\in\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-et}})$. The
isomorphism (42) is defined to be the following composition of morphisms of
Picard categories on $\mathcal{X}_{\operatorname{lis-et}}$:

$\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)\xrightarrow{\alpha}\underline{\mathrm{Exal}}_{f^{-1}\mathscr{O}_{Y^{+}_{\bullet}}}(\mathscr{O}_{X^{+}_{\bullet}},\varpi^{*}I)\xrightarrow{\varpi_{*}\beta}\mathrm{Ext}^{0/-1}(\mathbb{L}_{X^{+}_{\bullet}/Y^{+}_{\bullet}},\varpi^{*}I[1])\xleftarrow{\gamma}\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{X}/\mathcal{Y}},I[1]).$

The objects and maps in this composition are defined as follows.

  * •

The Picard category
$\underline{\mathrm{Exal}}_{f^{-1}\mathscr{O}_{Y^{+}_{\bullet}}}(\mathscr{O}_{X^{+}_{\bullet}},\varpi^{*}I)$
is defined as in Section B.2 on the site $X^{+}_{\bullet,\mathrm{et}}$

  * •

The map $\alpha$ is the composition of [Ols06, (2.8.1),(2.20.1)]: it sends an
extension $\mathcal{X}\rightarrow\mathcal{X}^{\prime}$ by $I$ to the exact
sequence of $f^{-1}\mathscr{O}_{Y^{+}_{\bullet}}$-modules

$0\rightarrow\varpi^{*}I\rightarrow\mathscr{O}_{\mathcal{X}^{\prime+}_{\bullet}}\rightarrow\mathscr{O}_{\mathcal{X}^{+}_{\bullet}}\rightarrow
0.$

It is an isomorphism by [Ols06, Prop 2.9, Lem 2.21].

  * •

The map $\beta$ was defined in (77).

  * •

The arrow $\gamma$ is induced by applying $ch\circ\tau_{\leq
0}\circ\mathsf{R}\Gamma$ to (8) in the context of Example B.3.1. It is an
isomorphism since $\varpi^{*}$ is fully faithful (in fact, an equivalence of
categories).

######  Remark B.3.3.

The proof of Lemma 4.2.2 shows that (42) is independent of the choice of cover
$Y\rightarrow\mathcal{Y}$.

###  B.4. Proofs of Lemmas 4.2.2 and 4.2.3

We describe an amalgamation (88) of the three diagrams in Lemma B.2.4 that
will be used to prove both functoriality lemmas. Let
$(\mathscr{S},\mathscr{O}_{\mathscr{S}})\rightarrow(\mathscr{S}^{\prime},\mathscr{O}_{\mathscr{S}^{\prime}})$
be a continuous morphism of ringed sites inducing a flat morphism of topoi
$(p^{-1},p_{*})$. Let $A$ and $B:=\mathscr{O}_{\mathscr{S}}$ be sheaves of
rings on $\mathscr{S}$, let $I$ be a sheaf of $B$-modules, and let
$A^{\prime}$ be a sheaf of rings on $\mathscr{S}^{\prime}$ such that there is
a commuting diagram as follows (note that
$p^{*}B=\mathscr{O}_{\mathscr{S}^{\prime}}$):

${p^{-1}A}$${A^{\prime}}$${p^{-1}B}$${p^{*}B}$

Then we obtain the following commuting diagram.

(88)
${\mathrm{Ext}^{0/-1}_{B}(\mathbb{L}_{B/A},I[1])}$${\mathrm{Ext}^{0/-1}_{p^{*}B}(p^{*}\mathbb{L}_{B/A},p^{*}I[1])}$${\mathrm{Ext}^{0/-1}_{p^{*}B}(\mathbb{L}_{p^{*}B/A^{\prime}},p^{*}I[1])}$${\mathrm{Ext}^{0/-1}_{p^{-1}B}(p^{-1}\mathbb{L}_{B/A},p^{-1}I[1])}$${\mathrm{Ext}^{0/-1}_{p^{-1}B}(p^{-1}\mathbb{L}_{B/A},(p^{*}I[1])_{p^{-1}B})}$${\underline{\mathrm{Exal}}_{A}(B,I)}$${\underline{\mathrm{Exal}}_{p^{-1}A}(p^{-1}B,p^{-1}I)}$${\underline{\mathrm{Exal}}_{p^{-1}A}(p^{-1}B,(p^{*}I)_{p^{-1}B})}$${\underline{\mathrm{Exal}}_{A^{\prime}}(p^{*}B,p^{*}I)}$(8)(6)$\scriptstyle{\sim}$$\scriptstyle{\beta}$$\scriptstyle{\beta}$$\scriptstyle{\beta}$$\scriptstyle{\beta}$

Here, the left square is Lemma B.2.4 (3) and the right square is Lemma B.2.4
(2). The middle square is Lemma B.2.4 (1), using the unit
$p^{-1}I\rightarrow(p^{*}I)_{p^{-1}B}$ of the $(\otimes,\operatorname{Hom})$
adjunction. The triangle commutes by definition of the maps involved.

###### Proof of Lemma 4.2.2.

Construct a diagram

${U}$${V}$${Z}$${\mathcal{W}\times_{\mathcal{Y}}Y}$${Y}$${\mathcal{Z}}$${\mathcal{W}}$${\mathcal{Y}}$$\scriptstyle{\rho}$$\scriptstyle{r}$$\scriptstyle{\varpi}$

where $U,V,Z$, and $Y$ are algebraic spaces with $Y\rightarrow\mathcal{Y}$ and
$V\rightarrow\mathcal{W}\times_{\mathcal{Y}}Y$ smooth and surjective and all
squares are fibered. Let $q$ denote the map $Z\rightarrow Y$ and let
$\varpi^{\prime}=\varpi\circ\rho$, and use the same letters to denote induced
morphisms of (simplicial) topoi. Then commutativity of (43) is equivalent to
commutativity of the following diagram.

(89)
${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{Z}/\mathcal{W}},I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{Z}/\mathcal{Y}},I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{U^{+}_{\bullet}/{V^{+}_{\bullet}}},\varpi^{\prime*}I[1])}$${\mathrm{Ext}^{0/-1}(\rho^{*}\mathbb{L}_{Z^{+}_{\bullet}/{Y^{+}_{\bullet}}},\varpi^{\prime*}I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{Z^{+}_{\bullet}/{Y^{+}_{\bullet}}},\varpi^{*}I[1])}$${\underline{\mathrm{Exal}}_{r^{-1}\mathscr{O}_{V^{+}_{\bullet}}}(\mathscr{O}_{U^{+}_{\bullet}},\varpi^{\prime*}I)}$${\underline{\mathrm{Exal}}_{\rho^{-1}q^{-1}\mathscr{O}_{Y^{+}_{\bullet}}}(\rho^{-1}\mathscr{O}_{Z^{+}_{\bullet}},\varpi^{\prime*}I)}$${\underline{\mathrm{Exal}}_{q^{-1}\mathscr{O}_{Y^{+}_{\bullet}}}(\mathscr{O}_{Z^{+}_{\bullet}},\varpi^{*}I)}$${\underline{\mathrm{Exal}}_{\mathcal{W}}(\mathcal{Z},I)}$${\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{Z},I)}$$\scriptstyle{\gamma}$$\scriptstyle{\sim}$$\scriptstyle{A}$$\scriptstyle{\gamma}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\beta}$$\scriptstyle{\beta}$$\scriptstyle{B}$$\scriptstyle{\alpha}$$\scriptstyle{\alpha}$

In the triangle, all of the maps are equal to (8), and the triangle commutes
by the functoriality of (8) in the adjoint pair. The arrow

$\mathrm{Ext}^{0/-1}(\rho^{*}\mathbb{L}_{Z^{\bullet}/{Y^{\bullet}}},\varpi^{\prime*}I[1])\leftarrow\mathrm{Ext}^{0/-1}(\mathbb{L}_{Z^{\bullet}/{Y^{\bullet}}},\varpi^{*}I[1])$

is an equivalence (as claimed in the diagram) because
$\rho^{*}:\operatorname{D_{qc}}(Z^{+}_{\bullet,\mathrm{et}})\rightarrow\operatorname{D_{qc}}(U^{+}_{\bullet,\mathrm{et}})$
is fully faithful. The trapezoid commutes by definition of the canonical map
$\mathbb{L}_{\mathcal{Z}/\mathcal{Y}}\rightarrow\mathbb{L}_{\mathcal{Z}/\mathcal{W}}$
(one can produce an explicit description for $\gamma$ by the same argument as
was used in the proof of Lemma B.2.6). The commutativity of the middle square
is (88), reflected left-to-right, with $p=\rho$,
$A=q^{-1}\mathscr{O}_{Y^{+}_{\bullet}}$,
$A^{\prime}=r^{-1}\mathscr{O}_{V^{+}_{\bullet}}$, and
$B=\mathscr{O}_{Z^{+}_{\bullet}}$.

It remains to check that the bottom square commutes. We do this by direct
computation. Let $i:\mathcal{Z}\hookrightarrow\mathcal{Z}^{\prime}$ be an
element of $\underline{\mathrm{Exal}}_{\mathcal{W}}(\mathcal{Z},I)$. We have
the following commuting diagram, where all squares are fibered:

(90)
${U}$${U^{\prime}}$${V}$${Z}$${Z^{\prime}}$${\mathcal{W}\times_{\mathcal{Y}}Y}$${Y}$${\mathcal{Z}}$${\mathcal{Z}^{\prime}}$${\mathcal{W}}$${\mathcal{Y}}$$\scriptstyle{i}$$\scriptstyle{\rho}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{i}$$\scriptstyle{\varpi}$$\scriptstyle{i}$

The map $\alpha$ sends $i:\mathcal{Z}\hookrightarrow\mathcal{Z}^{\prime}$ to
the extension

$0\rightarrow\varpi^{\prime}{}^{*}I\xrightarrow{m}i^{-1}\mathscr{O}_{U^{\prime}{}^{+}_{\bullet}}\rightarrow\mathscr{O}_{U^{+}_{\bullet}}\rightarrow
0$

of $r^{-1}\mathscr{O}_{V^{+}_{\bullet}}$-modules, and the maps (72) and (71)
send this to the extension

(91)
$0\rightarrow\varpi^{\prime}{}^{*}I\xrightarrow{(m,0)}i^{-1}\mathscr{O}_{U^{\prime}{}^{+}_{\bullet}}\times_{\mathscr{O}_{U^{+}_{\bullet}}}\rho^{-1}\mathscr{O}_{Z^{+}_{\bullet}}\rightarrow\rho^{-1}\mathscr{O}_{Z^{+}_{\bullet}}\rightarrow
0$

of $\rho^{-1}q^{-1}\mathscr{O}_{Y^{+}_{\bullet}}$-modules.

On the other hand, the map $B$ sends
$i:\mathcal{Z}\hookrightarrow\mathcal{Z}^{\prime}$ to the same extension, now
as an element of $\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{Z},I)$. The
image of this under $\rho^{-1}\circ\alpha$ is

$0\rightarrow\rho^{-1}\varpi^{*}I\rightarrow\rho^{-1}i^{-1}\mathscr{O}_{Z^{\prime}{}^{+}_{\bullet}}\xrightarrow{n}\rho^{-1}\mathscr{O}_{Z^{+}_{\bullet}}\rightarrow
0$

an extension of $\rho^{-1}q^{-1}\mathscr{O}_{Y^{\bullet}}$-modules. Here $n$
is part of the data of the morphism of ringed topoi associated to
$i:Z\rightarrow Z^{\prime}$. Finally the map (70) sends this extension to

(92)
$0\rightarrow\varpi^{\prime}{}^{*}I\rightarrow\varpi^{\prime}{}^{*}I\oplus_{\rho^{-1}\varpi^{*}I}\rho^{-1}i^{-1}\mathscr{O}_{Z^{\prime}{}^{+}_{\bullet}}\xrightarrow{(0,n)}\rho^{-1}\mathscr{O}_{Z^{+}_{\bullet}}\rightarrow
0$

also an extension of $\rho^{-1}q^{-1}\mathscr{O}_{Y^{+}_{\bullet}}$-modules.

A morphism from (92) to (91) in the groupoid
$\underline{\mathrm{Exal}}_{\rho^{-1}q^{-1}\mathscr{O}_{Y^{+}_{\bullet}}}(\rho^{-1}\mathscr{O}_{Z^{+}_{\bullet}},\varpi^{\prime}{}^{*}I)$
is given by a collection of dotted arrows making the following diagram
commute.

(93)
${\rho^{-1}\varpi^{*}I}$${\varpi^{\prime}{}^{*}I}$${i^{-1}\mathscr{O}_{U^{\prime}{}^{+}_{\bullet}}}$${\rho^{-1}i^{-1}\mathscr{O}_{Z^{\prime}{}^{+}_{\bullet}}}$${\rho^{-1}\mathscr{O}_{Z^{+}_{\bullet}}}$${\mathscr{O}_{U^{+}_{\bullet}}}$

We choose arrows as follows (note they are compatible with restriction).

$\displaystyle m:\varpi^{\prime}{}^{*}I\rightarrow
i^{-1}\mathscr{O}_{U^{\prime+}_{\bullet}}$ $\displaystyle
n:\rho^{-1}i^{-1}\mathscr{O}_{Z^{\prime}{}^{+}_{\bullet}}\rightarrow\rho^{-1}\mathscr{O}_{Z^{+}_{\bullet}}$
$\displaystyle
0:\varpi^{\prime}{}^{*}I\rightarrow\rho^{-1}\mathscr{O}_{Z^{+}_{\bullet}}$
$\displaystyle
k:\rho^{-1}i^{-1}\mathscr{O}_{Z^{\prime}{}^{+}_{\bullet}}\rightarrow
i^{-1}\mathscr{O}_{U^{\prime+}_{\bullet}}.$

where $k$ is equal to $i^{-1}$ applied to the canonical morphism
$\rho^{-1}\mathscr{O}_{Z^{\prime+}_{\bullet}}\rightarrow\mathscr{O}_{U^{\prime+}_{\bullet}}$.
Commutativity of the resulting diagram follows from commutativity of (90).

We claim that this morphism is natural for arrows coming from
$\underline{\mathrm{Exal}}_{\mathcal{W}}(\mathcal{Z},I)$. If we are given an
arrow $f$ from $i_{1}:\mathcal{Z}\rightarrow\mathcal{Z}_{1}$ to
$i_{2}:\mathcal{Z}\rightarrow\mathcal{Z}_{2}$ inducing maps
$f_{U}:i_{1}^{-1}\mathscr{O}_{U^{+}_{1,\bullet}}\rightarrow
i_{2}^{-1}\mathscr{O}_{U^{+}_{2,\bullet}}$ and
$f_{Z}:\rho^{-1}i_{1}^{-1}\mathscr{O}_{Z^{+}_{1,\bullet}}\rightarrow\rho^{-1}i_{1}^{-1}\mathscr{O}_{Z^{+}_{2,\bullet}}$,
then this naturality is equivalent to the fact that the maps in the following
two criss-cross diagrams coincide.

${\varpi^{\prime}{}^{*}I}$${i^{-1}_{2}\mathscr{O}_{U{}^{+}_{2,\bullet}}}$${\rho^{-1}i^{-1}_{1}\mathscr{O}_{Z{}^{+}_{1,\bullet}}}$${\rho^{-1}\mathscr{O}_{Z^{+}_{\bullet}}}$$\scriptstyle{m_{2}}$$\scriptstyle{0}$$\scriptstyle{k_{2}\circ
f_{Z}}$$\scriptstyle{n_{2}\circ
f_{Z}}$${\varpi^{\prime}{}^{*}I}$${i^{-1}_{2}\mathscr{O}_{U{}^{+}_{2,\bullet}}}$${\rho^{-1}i^{-1}_{1}\mathscr{O}_{Z{}^{+}_{1,\bullet}}}$${\rho^{-1}\mathscr{O}_{Z^{+}_{\bullet}}}$$\scriptstyle{f_{U}\circ
m_{1}}$$\scriptstyle{0}$$\scriptstyle{f_{U}\circ k_{1}}$$\scriptstyle{n_{1}}$

∎

###### Proof of Lemma 4.2.3.

Construct a fiber diagram

${Z}$${X\times_{\mathcal{X}}\mathcal{Z}}$${X}$${W}$${Y\times_{\mathcal{Y}}\mathcal{W}}$${Y}$$\scriptstyle{r}$$\scriptstyle{q}$

where $Y\rightarrow\mathcal{Y}$ is a smooth cover by a scheme,
$X=Y\times_{\mathcal{Y}}\mathcal{X}$, and $W\rightarrow
Y\times_{\mathcal{Y}}\mathcal{W}$ is a smooth cover by a scheme. Use $p$ to
denote the map $Z\rightarrow X$. Then commutativity of (46) is equivalent to
commutativity of the diagram below.

(94)
${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{Z}/\mathcal{W}},I[1])}$${\mathrm{Ext}^{0/-1}(p^{*}\mathbb{L}_{\mathcal{X}/\mathcal{Y}},p^{*}I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{Z}/\mathcal{W}},p^{*}I[1])}$${\mathrm{Ext}^{0/-1}_{\mathscr{O}_{X^{+}_{\bullet}}}(\mathbb{L}_{X^{+}_{\bullet}/Y^{+}_{\bullet}},\varpi^{*}I[1])}$${\mathrm{Ext}^{0/-1}_{p^{*}\mathscr{O}_{X^{+}_{\bullet}}}(p^{*}\mathbb{L}_{X^{+}_{\bullet}/Y^{+}_{\bullet}},p^{*}\varpi^{*}I[1])}$${\mathrm{Ext}^{0/-1}_{\mathscr{O}_{Z^{+}_{\bullet}}}(\mathbb{L}_{Z^{+}_{\bullet}/W^{+}_{\bullet}},\varpi^{*}p^{*}I[1])}$${\underline{\mathrm{Exal}}_{q^{-1}\mathscr{O}_{Y^{+}_{\bullet}}}(\mathscr{O}_{X^{+}_{\bullet}},\varpi^{*}I)}$${\underline{\mathrm{Exal}}_{r^{-1}\mathscr{O}_{W^{+}_{\bullet}}}(\mathscr{O}_{Z^{+}_{\bullet}},\varpi^{*}p^{*}I)}$${\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)}$${\underline{\mathrm{Exal}}_{\mathcal{W}}(\mathcal{Z},p^{*}I)}$(8)$\scriptstyle{C}$$\scriptstyle{\gamma}$$\scriptstyle{\sim}$(8)$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{D}$$\scriptstyle{\gamma}$$\scriptstyle{\sim}$(8)$\scriptstyle{\sim}$$\scriptstyle{\beta}$$\scriptstyle{\beta}$$\scriptstyle{E}$$\scriptstyle{\alpha}$$\scriptstyle{\alpha}$

The vertical instance of (8) is an isomorphism since $\varpi^{*}$ is fully
faithful. This implies that the unnamed arrow in the top right square of (94)
is an isomorphism (it is already labeled as such), since the other three maps
in the square are. The commutativity of the top left square uses functoriality
of (8) in the adjoint functors. The top right square commutes by definition of
the canonical map of cotangent complexes. The middle rectangle is (88) with
$p$ the map $Z\rightarrow X$, $A=q^{-1}\mathscr{O}_{Y^{+}_{\bullet}}$,
$B=\mathscr{O}_{X^{+}_{\bullet}}$, and
$A^{\prime}=r^{-1}\mathscr{O}_{W^{+}_{\bullet}}$. We have suppressed various
squares commuting the maps $p$ and $\varpi$.

It remains to check that the bottom square of (94) commutes. This we do by
direct computation, using (88) to factor the map

$\underline{\mathrm{Exal}}_{q^{-1}\mathscr{O}_{Y^{+}_{\bullet}}}(\mathscr{O}_{X^{+}_{\bullet}},\varpi^{*}I)\rightarrow\underline{\mathrm{Exal}}_{r^{-1}\mathscr{O}_{W^{+}_{\bullet}}}(\mathscr{O}_{Z^{+}_{\bullet}},\varpi^{*}p^{*}I).$

Let $i:\mathcal{X}\hookrightarrow\mathcal{X}^{\prime}$ be an element of
$\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)$. Then we have a
commuting diagram

${Z}$${Z^{\prime}}$${W}$${X}$${X^{\prime}}$${Y^{\prime}}$${\mathcal{Z}}$${\mathcal{Z}^{\prime}}$${\mathcal{W}}$${\mathcal{X}}$${\mathcal{X}^{\prime}}$${\mathcal{Y}}$

where the front, bottom, and back squares are fibered (six squares in all).
The map $p^{-1}\circ\alpha$ sends
$i:\mathcal{X}\hookrightarrow\mathcal{X}^{\prime}$ to the extension

$0\rightarrow p^{-1}\varpi^{*}I\rightarrow
p^{-1}i^{-1}\mathscr{O}_{X^{\prime}{}^{+}_{\bullet}}\rightarrow
p^{-1}\mathscr{O}_{X^{+}_{\bullet}}\rightarrow 0$

of $p^{-1}q^{-1}\mathscr{O}_{Y^{+}_{\bullet}}$-algebras, and the map (70)
sends this extension to the extension

(95) $0\rightarrow p^{*}\varpi^{*}I\rightarrow
p^{*}\varpi^{*}I\oplus_{z^{-1}\varpi^{*}I}p^{-1}i^{-1}\mathscr{O}_{X^{+}_{\bullet}}\rightarrow
p^{-1}\mathscr{O}_{X^{+}_{\bullet}}\rightarrow 0.$

On the other hand, the map $E$ sends
$i:\mathcal{X}\hookrightarrow\mathcal{X}^{\prime}$ to
$\mathcal{Z}\hookrightarrow\mathcal{Z}^{\prime}$, which under $\alpha$
corresponds to the extension

$0\rightarrow\varpi^{*}p^{*}I\rightarrow
i^{-1}\mathscr{O}_{Z^{\prime}{}^{+}_{\bullet}}\rightarrow\mathscr{O}_{Z^{+}_{\bullet}}\rightarrow
0$

of $r^{-1}\mathscr{O}_{W^{+}_{\bullet}}$-algebras. After applying
$\varpi^{*}p^{*}=p^{*}\varpi^{*}$ and the morphisms (72) and (71), this
becomes the extension

(96) $0\rightarrow p^{*}\varpi^{*}\rightarrow
i^{-1}\mathscr{O}_{Z^{\prime}{}^{+}_{\bullet}}\times_{\mathscr{O}_{Z^{+}_{\bullet}}}p^{-1}\mathscr{O}_{X^{+}_{\bullet}}\rightarrow
p^{-1}\mathscr{O}_{X^{+}_{\bullet}}\rightarrow 0$

of $p^{-1}q^{-1}\mathscr{O}_{Y^{+}_{\bullet}}$-algebras. As in the proof of
Lemma 4.2.2, one can write down a functorial (necessarily iso)morphism between
(95) and (96), and check that it is compatible with restrictions.

∎

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