Knowledge Distillation in Iterative Generative Models for Improved Sampling
Speed

Eric Luhman***Equal contribution Troy Luhman11footnotemark: 1
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

[] Iterative generative models, such as noise conditional score networks and
denoising diffusion probabilistic models, produce high quality samples by
gradually denoising an initial noise vector. However, their denoising process
has many steps, making them 2-3 orders of magnitude slower than other
generative models such as GANs and VAEs. In this paper, we establish a novel
connection between knowledge distillation and image generation with a
technique that distills a multi-step denoising process into a single step,
resulting in a sampling speed similar to other single-step generative models.
Our Denoising Student generates high quality samples comparable to GANs on the
CIFAR-10 and CelebA datasets, without adversarial training. We demonstrate
that our method scales to higher resolutions through experiments on $256\times
256$ LSUN. Code and checkpoints are available at
https://github.com/tcl9876/Denoising_Student

##  1 Introduction

Image Generation is an important and well studied problem in computer vision.
There are a variety of approaches to image generation which yield high quality
results, such as Generative Adversarial Networks (GANs, [10]), Variational
Autoencoders (VAEs, [19]), and energy-based models ([21]). GANs generally have
been able to generate the highest quality images, especially at higher
resolutions ([15]; [16]; [1]). However, their adversarial training procedure
makes them more difficult to train than other methods of generative modeling.

Two increasingly popular methods of generative modeling are Denoising score-
matching ([39]; [31]) and Denoising Diffusion Probabilistic Models (DDPMs,
[29]; [13]), both of which model data by gradually reversing a noise-adding
process. Denoising score-matching methods use a neural network such as a Noise
conditional score network (NCSN) to estimate the score, or gradient of the
logarithmic data density. DDPMs are trained to reverse each step of the noise-
adding (a.k.a. diffusion) process, and can also be parameterized to implicitly
estimate scores ([13]; [33]b). We refer to both models as score-based
generative models. Both approcahes use MCMC methods similar to Langevin
dynamics during sampling time. Score based models are capable of producing
samples that rival even the best GAN methods, without adversarial training
([13]; [33]b).

A major downside to score-based generative models is that they require
performing expensive MCMC sampling, often with a thousand steps or more. As a
result, they can be up to three orders of magnitude slower than GANs, which
only require a single network evaluation. To address this issue, Denoising
Diffusion Implicit Models, or DDIMs, have been proposed ([30]a). DDIMs use a
generative Markov chain to reverse a non-Markovian inference process, and are
a type of implicit probabilistic model ([24]). Unlike DDPMs and NCSNs, they
have a deterministic generative process that only depends on an initial latent
variable. Nevertheless, DDIMs are still rather inefficient, requiring 20-100
function evaluations to produce good samples.

In this work, we propose a method of approximating a multi-step generative
process with only a single function evaluation, through the use of knowledge
distillation ([2]; [12]), a technique of compressing a computationally
expensive teacher into a smaller student model that learns to approximate the
teacher’s output distribution. Our Denoising Student synthesizes data directly
from Gaussian noise without any intermediate denoising steps, and is trained
to predict the output of a DDIM given the same initial noise vector.

Our approach has a number of desirable properties. Firstly, it is far more
efficient ($20\times$ to $1000\times$) than existing score-based approaches,
making it similar to GANs and VAEs in sampling speed. Secondly, it has a
simple, stable objective that does not involve any adversarial training or
surrogate losses (e.g. the encoder in VAEs). Thirdly, it can be easily applied
to any iterative model with a deterministic generative process, as it involves
no additional architectural considerations or hyperparameters (e.g. the noise
schedule, the stepsize of Langevin dynamics). Lastly, it retains the abilities
of other implicit models, such as semantically meaningful interpolations
through the latent space.

Despite these advantages, it produces high fidelity samples comparable to GANs
on lower resolution datasets such as CIFAR-10 and CelebA $64\times 64$. It
also scales to higher resolutions, generating decent quality samples of size
$256\times 256$. To the best of our knowledge, Denoising Student is the first
non-adversarial model that can produce $256\times 256$ LSUN images in only a
single step.

##  2 Knowledge Distillation in Deterministic Generative Models

###  2.1 Knowledge Distillation

Knowledge distillation ([2]; [12]) involves compressing an expensive but high-
performing teacher model into a smaller student model. The student model is
trained to minimize the cross-entropy between its output distribution and the
output distribution of the teacher, which results in better performance than
if it was trained normally. In supervised learning tasks, the motivation for
this approach comes from the information hidden in the teacher’s output, which
isn’t present in a one-hot label. For instance, an image with two objects
could be plausibly classified as either one, which is reflected in the
teacher’s output distribution, but not the original label.

One condition of knowledge distillation is that the function to be learned
must be deterministic, otherwise, it will be difficult, if not impossible to
learn. In supervised learning tasks, this is a rather trivial condition, since
models produce the same output given the same input. In generative modeling,
however, this condition is no longer trivial. Score-based and energy-based
models use a stochastic MCMC procedure such as Langevin dynamics, making their
usage as a teacher infeasible. We could apply knowledge distillation to a GAN,
which is deterministic given the same latent variable, but we would gain only
a minor speedup over an already fast model.

Recently, [30] (ddima) proposed a new class of generative models called
denoising diffusion implicit models (DDIMs), which are similar to other
iterative generative models in terms of sample quality but have somewhat
faster sampling speed. Unlike other iterative models111An even more recent
work ([33]b) proposed a neural probability flow ODE that also has a
deterministic, multi-step generative process. However, pretrained models are
unavailable as of writing, and we wanted to avoid retraining the teacher.,
their generative process is deterministic, which motivates their usage as a
teacher model. We will discuss them below. For further explanation and proofs
related to DDIMs, the reader is referred to [30] (ddima).

###  2.2 Denoising Diffusion Implicit Models

Consider an inference process with latent variables
$\textbf{x}_{1},\ldots,\textbf{x}_{T}$ of the same dimensionality as the data
$\textbf{x}_{0}\sim q(\textbf{x}_{0})$, parameterized by a decreasing sequence
of constants $\alpha_{1:T}\in(0,1]^{T}$. The inference process
$q(\textbf{x}_{1:T}|\textbf{x}_{0})$ is defined as:

$\displaystyle q(\textbf{x}_{1:T}|\textbf{x}_{0})\coloneqq
q(\textbf{x}_{T}|\textbf{x}_{0})\prod_{t=2}^{T}q(\textbf{x}_{t-1}|\textbf{x}_{t},\textbf{x}_{0}),\text{
with}$ (1) $\displaystyle
q(\textbf{x}_{T}|\textbf{x}_{0})=\mathcal{N}(\textbf{x}_{T};\sqrt{\alpha_{T}}\textbf{x}_{0},(1-\alpha_{T})\textbf{I}),\text{
and }$ (2) $\displaystyle
q(\textbf{x}_{t-1}|\textbf{x}_{t},\textbf{x}_{0})=\delta(\textbf{x}_{t-1}-\bm{\mu}(\textbf{x}_{t},\textbf{x}_{0}))$
(3)

where $\delta(\cdot)$ is the Dirac delta function,
$\bm{\mu}(\textbf{x}_{t},\textbf{x}_{0})=\sqrt{\alpha_{t-1}}\textbf{x}_{0}+\sqrt{\frac{1-\alpha_{t-1}}{1-\alpha_{t}}}(\textbf{x}_{t}-\sqrt{\alpha_{t}}\textbf{x}_{0})$.
This inference process is not trainable, unlike in other probabilistic models
such as VAEs ([19]). One notable property of this inference process is that
for any $t$,
$q(\textbf{x}_{t}|\textbf{x}_{0})=\mathcal{N}(\textbf{x}_{t};\sqrt{\alpha_{t}}\textbf{x}_{0},(1-\alpha_{t})\textbf{I})$,
so we can write:

$\textbf{x}_{t}=\sqrt{\alpha_{t}}\textbf{x}_{0}+\sqrt{1-\alpha_{t}}\epsilon,\epsilon\sim\mathcal{N}(\textbf{0},\textbf{I})$
(4)

A denoising diffusion implicit model ([30]a) is a latent variable model whose
generative process reverses the above inference process. Formally, it is
defined as:

$p_{\theta}(\textbf{x}_{0})\coloneqq\int
p_{\theta}(\textbf{x}_{0:T})\text{d}\textbf{x}_{1:T},\text{ with
}p_{\theta}(\textbf{x}_{0:T})\coloneqq
p(\textbf{x}_{T})\prod_{t=1}^{T}p_{\theta}(\textbf{x}_{t-1}|\textbf{x}_{t})$
(5)

where $p_{\theta}(\textbf{x}_{0:T})$ is the generative process, with prior
$p(\textbf{x}_{T})=\mathcal{N}(\textbf{x}_{T};\textbf{0},\textbf{I})$. Each
distribution $p_{\theta}(\textbf{x}_{t-1}|\textbf{x}_{t})$ in the generative
process estimates the corresponding inference distribution
$q(\textbf{x}_{t-1}|\textbf{x}_{t},\textbf{x}_{0})$. However, sampling from
$q(\textbf{x}_{t-1}|\textbf{x}_{t},\textbf{x}_{0})$ requires knowledge of
$\textbf{x}_{0}$, which is unavailable during the generative process. As a
result, we make a prediction $f_{\theta}(\textbf{x}_{t})$ of $\textbf{x}_{0}$,
and use this in our estimate
$q(\textbf{x}_{t-1}|\textbf{x}_{t},f_{\theta}(\textbf{x}_{t}))$. Putting it
together, we get

$\displaystyle
p_{\theta}(\textbf{x}_{t-1}|\textbf{x}_{t})=\begin{cases}q(\textbf{x}_{t-1}|\textbf{x}_{t},f_{\theta}(\textbf{x}_{t}))&\text{if
$t>1$}\\\ \delta(\textbf{x}_{0}-f_{\theta}(\textbf{x}_{1}))&\text{if
$t=1$}\end{cases}$ (6)

where $f_{\theta}(\textbf{x}_{t})$ is found by rewriting Eq. (4), to get
$f_{\theta}(\textbf{x}_{t})=\frac{1}{\sqrt{\alpha_{t}}}(\textbf{x}_{t}-\sqrt{1-\alpha_{t}}\epsilon_{\theta})$.
To sample from a DDIM, we first sample a noise vector $\textbf{x}_{T}$ from a
standard normal, then gradually denoise it over time by iteratively computing
$\textbf{x}_{T-1},\textbf{x}_{T-2},\ldots,\textbf{x}_{1},\textbf{x}_{0}$ with:

$\sqrt{\frac{1}{\alpha_{t-1}}}\textbf{x}_{t-1}=\sqrt{\frac{1}{\alpha_{t}}}\textbf{x}_{t}-\left(\sqrt{\frac{1-\alpha_{t}}{\alpha_{t}}}-\sqrt{\frac{1-\alpha_{t-1}}{\alpha_{t-1}}}\right)\epsilon_{\theta}(\textbf{x}_{t}),\text{
or }$ (7)
$\textbf{x}_{t-1}=\sqrt{\frac{\alpha_{t-1}}{\alpha_{t}}}(\textbf{x}_{t}-\sqrt{1-\alpha_{t}}\epsilon_{\theta}(\textbf{x}_{t}))+\sqrt{1-\alpha_{t-1}}\epsilon_{\theta}(\textbf{x}_{t})$
(8)

The generative model is trained by maximizing the variational lower bound:
$\text{ELBO}=\mathbb{E}_{q}[\log p_{\theta}(\textbf{x}_{0:T})-\log
q(\textbf{x}_{1:T}|\textbf{x}_{0})]$. Here, we treat
$p_{\theta}(\textbf{x}_{t-1}|\textbf{x}_{t})$ and each
$q(\textbf{x}_{t-1}|\textbf{x}_{t},\textbf{x}_{0})$ as isotropic Gaussians
with very small variance, to ensure that the generative and inference
processes are nonzero everywhere.222Their original definitions were chosen to
emphasize their non-stochasticity. This objective can then be simplified to a
closed form expression:

$L=\sum_{t=1}^{T}\gamma_{t}\mathbb{E}_{\textbf{x}_{0},\epsilon_{t}}\left[\left\lVert\epsilon_{t}-\epsilon_{\theta}(\sqrt{\alpha_{t}}\textbf{x}_{0}+\sqrt{1-\alpha_{t}}\epsilon_{t},t)\right\rVert_{2}^{2}\right]+C,$
(9)

where $\gamma_{t}$ is a positive constant333In practice, [13] (ddpm) found it
beneficial to set $\gamma_{t}=1$ for all $t$ related to $\alpha_{t}$ and
$\alpha_{t-1}$, and $\epsilon_{\theta}$ is a trainable function (i.e. neural
network) that approximates $\epsilon_{t}$ given a noisy input $\textbf{x}_{t}$
and timestep $t$. This is the same training objective used by DDPMs ([13]), so
one can use a trained DDPM as a DDIM by changing only the sampling procedure.

From Eq. (8), we see that for a given $\epsilon_{\theta}$, $\textbf{x}_{t-1}$
is known and fixed given $\textbf{x}_{t}$. In other words, each individual
step of the sampling procedure is deterministic. As a result, the entire
generative process from $\textbf{x}_{T-1}$ to $\textbf{x}_{0}$ is
deterministic as well, dependent only on the initial latent $\textbf{x}_{T}$.
So, one could choose to ignore the intermediate latents $\textbf{x}_{1:T-1}$,
since knowledge of them is not necessary to predict $\textbf{x}_{0}$. We can
therefore directly model the data $\textbf{x}_{0}$ with a conditional
distribution $p_{\theta}(\textbf{x}_{0}|\textbf{x}_{T})$.

Figure 1: A graphical representation of our method.

From this perspective, we could think of a DDIM as an implicit probabilistic
model that transforms a latent variable $\textbf{x}_{T}$ with a deterministic
function to model data. Unfortunately, evaluating this function requires T
forward passes through a neural network, making it very computationally
expensive. This motivates our usage of knowledge distillation, where we
distill the knowledge in this “teacher” function into a much faster “student”
network that uses only a single network evaluation. The student’s output
distribution $p_{\text{student}}(\textbf{x}_{0}|\textbf{x}_{T})$ approximates
the teacher’s output distribution
$p_{\text{teacher}}(\textbf{x}_{0}|\textbf{x}_{T})$, using only
$\textbf{x}_{T}$. We illustrate this concept in Figure 1.

##  3 Generative Modeling with Denoising Student

###  3.1 Training Objective

As previously mentioned, the student’s conditional distribution
$p_{\text{student}}(\textbf{x}_{0}|\textbf{x}_{T})$ should be as close as
possible close to the teacher’s conditional distribution
$p_{\text{teacher}}(\textbf{x}_{0}|\textbf{x}_{T})$. We therefore train the
student by minimizing the following:

$L_{\text{student}}=\mathbb{E}_{\textbf{x}_{T}}\left[D_{\text{KL}}(p_{\text{teacher}}(\textbf{x}_{0}|\textbf{x}_{T})\
\|\ p_{\text{student}}(\textbf{x}_{0}|\textbf{x}_{T}))\right]$ (10)

This training objective is very similar to that in a supervised learning task,
where models are trained minimize the KL Divergence between two categorical
distributions over labels.

We parameterize the student as a simple Gaussian model with trainable mean
$\mathcal{F}_{\text{student}}(\textbf{x}_{T})$ and unit variance. Since the
teacher is a deterministic function $\mathcal{F}_{\text{teacher}}$ of
$\textbf{x}_{T}$, we add some Gaussian noise to its output to ensure that
$p_{\text{teacher}}(\textbf{x}_{0}|\textbf{x}_{T})$ is nonzero everywhere.
Since the KL term in Eq. (10) is between two Gaussians, the equation
simplifies to:

$L_{\text{student}}=\frac{1}{2}\mathbb{E}_{\textbf{x}_{T}}[\left\lVert\mathcal{F}_{\text{student}}(\textbf{x}_{T})-\mathcal{F}_{\text{teacher}}(\textbf{x}_{T})\right\rVert_{2}^{2}]+C$
(11)

Training of the student is done by sampling random $\textbf{x}_{T}$ from the
prior $p(\textbf{x}_{T})$ and computing the corresponding
$\mathcal{F}_{\text{teacher}}(\textbf{x}_{T})$ with a DDIM, then minimizing
Eq. (11). Unlike other generative models, our method does not require the
joint training of two networks (e.g. discriminator in GANs, inference model in
VAEs). In theory, one could train the student on as many unique examples as
desired, since both the prior and the teacher model are known. In our
experiments however, we use only 1.024 million synthetic examples for each
dataset and iterate over these same ones multiple times.

###  3.2 Model Architecture

Unlike normalizing flows and autoregressive models, our method does not have
any significant architectural restrictions. So, one could use an arbitrary
neural network as the student, then randomly initialize weights. But we
observe that the entirety of the teacher’s knowledge is contained in a much
smaller neural network $\epsilon_{\theta}$ that is reused many times. As such,
we initialize the student to have the same architecture and weights as
$\epsilon_{\theta}$. This allows the student to inherit knowledge from the
teacher, which speeds up training.

In the remaining sections, we use the term “teacher” to refer to the DDIM’s
sampling procedure from $\textbf{x}_{T}$ to $\textbf{x}_{0}$, and the term
“teacher network” to refer to the neural network
$\epsilon_{\theta}(\textbf{x}_{t},t)$ that parameterizes this sampling
procedure.

There are a couple minor concerns with this approach that we will address
here. Firstly, the student is trained to model the data $\textbf{x}_{0}$, so
it may not be ideal to initialize it to a teacher network that models the
noise $\epsilon_{t}$. To resolve this, we have the student predict the noise
as well, and subtract the predicted noise from the input noise
$\textbf{x}_{T}$ to produce samples and compute loss. Secondly, the teacher
network (and therefore the student as well) is conditioned on time, so we
condition the student on timestep T, corresponding to the highest noise level.

##  4 Experiments

In this section, we examine the proposed method through multiple experiments.
We consider four datasets: CIFAR-10 $32\times 32$ [20], CelebA $64\times 64$
[22], LSUN [45] Bedroom $256\times 256$, and LSUN Church $256\times 256$. We
use the pretrained models from [13] (ddpm) as our teacher network, except for
CelebA, where we use the one from [30] (ddima). For our experiments, we use a
100-step DDIM as our teacher for CIFAR-10 and CelebA, and a 50-step DDIM for
LSUN. Further details can be found in Appendix B.

###  4.1 Image Generation

We show random samples from our CIFAR-10 and CelebA model in Figures 2 and 3
respectively. The samples are both high quality and diverse, demonstrating the
efficacy of our method. Note that unlike VAEs, we do not reduce the
temperature (standard deviation) of our prior when showing qualitative
samples, which hinders diversity.

For quantitative evaluation, we show FID [11] and Inception scores [27] in
Table 2 for various methods on CIFAR-10, as well as the number of steps
(network evaluations) each needs. On CIFAR-10, our model achieves an FID of
9.36, which is lower than several GANs and far lower than NVAE [38], a state
of the art VAE.444While [4] (vdvae) has slightly lower negative log-likelihood
than NVAE (2.87 vs 2.91), we do not believe that this improvement would fully
account for the disparity between our model and NVAE. Our CIFAR-10 scores are
also similar to state-of-the-art EBMs, even though our method uses only a
single network evaluation. On CelebA, we obtain a competitive FID of 10.68.

Table 1: Results for CIFAR-10. ”Steps” refers to the number of neural network evaluations needed to generate a sample, and a model labeled with ”cond.” is class-conditional. Scores marked with an asterisk were computed by us and are not official. Model |  FID $\downarrow$ |  IS $\uparrow$ |  Steps $\downarrow$
---|---|---|---
Single-Step |  |  |
Denoising Student (Ours) | 9.36 | 8.36 | 1
NVAE [38] | 51.67 | 5.51 | 1
MoLM [25] | 18.9 | 7.90 | 1
Single-Step, GAN |  |  |
SNGAN [23] | 21.7 | 8.22 | 1
BigGAN (cond.) [1] | 14.73 | 9.22 | 1
PPOGAN [41] | 10.87 | 8.69 | 1
StyleGAN2+ADA [16] | 2.92 | 9.83 | 1
StyleGAN2+ADA (cond.) [16] | 2.42 | 10.14 | 1
Many-Step |  |  |
DDIM (100 step, Teacher) | 4.16 | 8.96* | 100
EBM [5] | 38.2 | 6.78 | 60
VAEBM [42] | 12.19 | 8.43 | 16
EBM+recovery likelihood [8] | 9.60 | 8.58 | 180
NCSNv2 [32] | 10.87 | 8.40 | 1160
DDPM [13] | 3.17 | 9.46 | 1000
NCSN++ (8 blocks/res) [33] | 2.20 | 9.89 | 2000

Table 2: Time (s) to generate 50k CIFAR-10 images Model | Time
---|---
Denoising Student | 51.5
NVAE | 146.5
DDIM (Teacher) | $4.96\times 10^{3}$
DDPM | $5.02\times 10^{4}$

To see how our method scales to higher resolutions, we evaluated it on the
$256\times 256$ LSUN Bedroom and Church datasets. We show random samples in
Figure 4 (Bedroom) and 5 (Church). We find that our model learns structure and
color quite well, as well as larger details such as lamps and windows in
Bedroom, and clouds and trees in Church. We do find, however, that our model
produces rather blurry samples, without very defined textures. We believe this
bluriness arises from the choice of objective. While GANs are designed to
produce images that fool the discriminator, our model has to replicate the
teacher’s output down to the pixel, which is difficult for higher dimensional
data.

Figure 2: Uncurated samples from our CIFAR-10 model Figure 3: Uncurated
samples from our CelebA model Figure 4: Uncurated samples from our LSUN
Bedroom model Figure 5: Uncurated samples from our LSUN Church-Outdoor model

To assess the sampling speed of Denoising Student, we measured the time it
took to generate 50k CIFAR-10 images, and compared it in Table 2. Notably, it
is about $100\times$ faster than the teacher, and $1000\times$ faster than a
DDPM, indicating that for the same architecture, sampling time is proportional
to number of steps. When compared with NVAE, we find ours is slightly faster,
showing that our model is indeed similar to other single-step models in speed.
Timing experiments were done with a batch size of 250 on a single Nvidia V100
GPU.

###  4.2 Latent Space Manipulation

[30] (ddima) observed that in DDIMs, the latent variable $\textbf{x}_{T}$ is
an effective representation of the data, where interpolations in the latent
space result in meaningful image interpolations. Our model was capable of
learning the latent mapping of a DDIM, so we hypothesize it would be able to
produce meaningful interpolations as well. To test this hypothesis, we perform
spherical interpolation between 2 random $\textbf{x}_{T}$, and show the result
in Figure 6. See Figure 12 for more.

To gain some further insight into the learned latent representation of
Denoising Student, we decided to apply it to different sized images than it
was trained on, by simply changing the size of the latent variable.
Remarkably, it still manages to replicate the teacher quite well, producing
coherent images of a different size (Figure 13 in Appendix). This would
indicate that the student retains the teacher’s advanced generalization
capabilities.

Figure 6: Interpolation results on the CelebA dataset.

##  5 Related Work

Knowledge distillation ([2]; [12]) has been explored extensively in supervised
learning tasks such as image classification [26, 46, 44, 7, 36, 43], language
modeling [35, 37, 28, 14, 34], and speech recognition [3, 6, 40, 17, 9]. Most
existing work on knowledge distillation is concerned with finding the best way
to transfer knowledge (e.g. best loss function) from the teacher to the
student. Additionally, knowledge distillation is usually concerned with
reducing the size (parameters) of a network, instead of the number of
evaluations.

Our work is also based on denoising score-based models such as NCSNs ([31])
and DDPMs ([13]), which estimate the score of the data density and use a
denoising auto-encoder objective. [33] (sdeb) observed that an NCSN / DDPM’s
generative process could be seen as solving a reverse time stochastic
differential equation that reverses a noise-adding SDE. A DDIM’s sampling
procedure, on the other hand, more closely resembles integrating an ODE (see.
Eq. 7). From this perspective, our student model could loosely be seen as
approximating this integral with a function, instead of a more accurate but
expensive numerical method (the teacher).

##  6 Conclusion

In this work, we presented a simple method to vastly reduce the sampling time
of certain iterative generative models, with only a minor degradation in
performance. Our knowledge distillation technique can be easily applied to any
trained model with a deterministic generative process, since it has a stable
training objective and requires no extra architectural considerations. Despite
its simplicity and stability, our model is capable of producing significantly
better samples than other non-adversarial, single-step methods such as VAEs.
It also learns meaningful latent representations, allowing for easy data
manipulation through the latent space. Future work includes bridging the gap
between the teacher and student through more advanced distillation techniques,
and producing sharper images at high resolutions.

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##  Appendix A Samples

We show additional samples in Figure 7 (CIFAR-10), 8 (CelebA), 9 (Bedroom),
and 10 (Church). Additional interpolation results can be seen in 12, and
different size images can be seen in Figure 13.

Since our model was not trained on real data, but instead on the teacher’s
output, we do not expect it to memorize training examples. To demonstrate
this, we include nearest neighbor visualizations in Figure 11.

Figure 7: Uncurated Samples from our CIFAR-10 model Figure 8: Uncurated
Samples from our CelebA model Figure 9: Uncurated Samples from our LSUN
Bedroom model Figure 10: Uncurated Samples from our LSUN Church-Outdoor model.
Figure 11: Inception feature space nearest neighbors for CIFAR-10. Images
generated from our model are in the leftmost column, and to the right of the
black line are the nearest neighbors in the training set. Figure 12: Extended
Interpolation results. Rows 1 and 2: CelebA images, Rows 3 and 4: LSUN
Bedroom, Rows 5 and 6: LSUN church Figure 13: Uncurated $256\times 384$ images
from a 50 step DDIM (teacher, left) and our model (right). Neither model has
ever seen a $256\times 384$ image in training, yet both produce decent
samples, demonstrating that both the teacher and student generalize well.

##  Appendix B Experimental Details

A list of the hyperparameters we used can be seen in Table 3. We used the Adam
([18]) optimizer. We linearly increased the learning rate for a number of
warmup steps, and used no decay. We did not use dropout or any other
regularization, and used test loss evaluated on unseen images for early
stopping. We made some interesting observations when training, and list them
as follows:

  * •

For CIFAR-10, we found that training on L1 distance yielded better results
than L2.

  * •

For LSUN Bedroom, we initially clipped the values of the gradients to $\pm
1.0$, but observed that the gradients were very large (norm consistently
$>10^{5}$). We found that not clipping by value improved stability.

  * •

For LSUN Bedroom, we initially chose $\beta_{1}=0.9,lr=2\times 10^{-5}$, but
found that increasing $\beta_{1}$ to 0.98 and decreasing the learning rate to
$5\times 10^{-6}$ helped stabilize training. We applied these to Church
without sweeping.

Table 3: Hyperparameters used in our experiments. Model | CIFAR-10 | CelebA | LSUN Bedroom | LSUN Church
---|---|---|---|---
Parameters (M) | 35.7 | 78.7 | 114 | 114
Batch size | 512 | 512 | 32 | 32
Learning Rate | $2\times 10^{-4}$ | $5\times 10^{-5}$ | $5\times 10^{-6}$ | $5\times 10^{-6}$
Adam $\beta_{1}$ | 0.9 | 0.9 | 0.98 | 0.98
Adam $\beta_{2}$ | 0.98 | 0.98 | 0.999 | 0.999
EMA decay rate | 0.995 | 0.995 | 0.9995 | 0.9995
Warmup steps | 5000 | 5000 | 1000 | 1000
Max training iterations | 50k | 64k | 320k | 256k


# The $\ell$-adic Dualizing Complex on an Excellent Surface with Rational
Singularities

Ting Li  Department of Mathematics
Sichuan University
Chengdu 610064
P. R. China<EMAIL_ADDRESS>
###### Abstract.

In this article, we show that if $X$ is an excellent surface with rational
singularities, the constant sheaf $\mathbb{Q}_{\ell}$ is a dualizing complex.
In coefficient $\mathbb{Z}_{\ell}$, we also prove that the obstruction for
$\mathbb{Z}_{\ell}$ to become a dualizing complex lying on the divisor class
groups at singular points. As applications, we study the perverse sheaves and
the weights of $\ell$-adic cohomology groups on such surfaces.

###### Key words and phrases:

Poincaré Duality, rational singularity, $\ell$-adic cohomology

††Mathematics Subject Classification 2000: 14F20, 14J17††This work is
supported by NSFC(10626036)

## Introduction

In [9, I], the dualizing complexes in étale cohomology was considered by
Grothendieck. O. Gabber made a breakthrough on this fields. He prove that
every excellent schemes admits a dualizing complex; and on a regular excellent
scheme, the constant sheaf $\mathbb{Z}_{\ell}$ is a dualizing complex. See
[12] for the detail.

In this paper, we study the properties of dualizing complexes on surfaces with
rational singularities. In [11], J. Lipman prove that a two-dimensional normal
local ring $R$ has a rational singularity if and only if $R$ has a finite
divisor class group. This makes me think that rational singularities only
affect the torsion part of the étale cohomology with coefficient
$\mathbb{Z}_{\ell}$, but leave the free part invariant. As an evidence, we
prove that on an excellent surface $X$ with at most rational singularities,
the potential dualizing complex $\mathscr{K}_{X}$ (determinated by the
dimension function $x\mapsto 2-\dim\mathcal{O}_{X,x}$) concentrates on the
entries $-2$ and $-4$; in detail,
$\mathrm{H}^{-4}(\mathscr{K}_{X})=\mathbb{Z}_{\ell}(2)$ and
$\mathrm{H}^{-2}(\mathscr{K}_{X})(-1)$ is the $\ell$-torsion part of the
divisor class groups at singular points (see Theorem 3.2). In particular,
$\mathbb{Q}_{\ell}$ is a dualizing complex of such surface in coefficient
$\mathbb{Q}_{\ell}$.

The paper is organized as follows. In §1, we briefly review the theory of
“potential dualizing complex” in [12]. Because the dualizing complexes on a
scheme are not unique, they vary by tensor products with invertible complexes.
So we prefer the _potential dualizing complex_ which is a dualizing complex
unique determinated by the dimension function. Basing on this theory, we
reiterate the étale homology in [10]. Since lacking of suitable references
related to étale homology both in adic coefficients and over arbitrary base
schemes, it is necessary to spend a little more space on rewriting it. In §2,
we calculate the étale homology on curves.

In §3, we prove the main results. First we calculate the étale homology on
arbitrary surfaces. Basing on it, we obtain the main results about the
dualizing complexes on an excellent surface with rational singularities; and
prove the Poincaré Duality for such surfaces. In §4, we give two applications
of above theory. First we study perverse sheaves on a surface with rational
singularities. Basing on it, we prove that the cohomologies of smooth sheaves
of punctually pure weights on such surfaces, are also punctually pure of
weights.

##  1\. Review of dualizing sheaves and étale homology

Let $\ell$ be a prime number, $\Lambda$ the integral closure of
$\mathbb{Z}_{\ell}$ in a finite extension field of $\mathbb{Q}_{\ell}$.

We consider the following conditions related to a scheme $X$:

  * (†)

$X$ is Noetherian, excellent, of finite Krull dimension, $\ell$ is invertible
on $X$ and $\mathrm{cd}_{\ell}(X)<\infty$.

From the Gabber’s finitenes theorem for étale cohomology in [7], we may
construct an adic formalism
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\Lambda)$ together
with Grothendieck’s six operations on schemes satisfying (†). See also [4] for
details. Moreover if $X$ is a scheme satisfying (†), then any scheme of finite
type over $X$ satisfies (†).

Next, we introduces the “potential dualizing complex” defined in [12].

######  Definition 1.1 ([12], 2.2).

Let $X$ be a scheme satisfying (†), $\delta\colon X\to\mathbb{Z}$ a dimension
function. A _potential dualizing complex_ (in coefficient $\Lambda$) on $X$
consisting of

  1. (1)

an object $\mathscr{K}$ in
$\mathbf{D}_{\mathrm{c}}^{\textbf{+}}(X_{\textup{\'{e}t}},\Lambda)$,

  2. (2)

an isomorphism
$\mathbf{R}\varGamma_{x}(\mathscr{K})\xrightarrow{\,\sim\,}\Lambda\bigl{(}\delta(x)\bigr{)}\bigl{[}2\delta(x)\bigr{]}$
in $\mathbf{D}_{\mathrm{c}}^{\textbf{+}}(x_{\textup{\'{e}t}},\Lambda)$ for
every point $x$ on $X$;

and these data satisfy that: for every immediate specialization
$\bar{y}\to\bar{x}$ on $X$, the following diagram commutes,

$\textstyle{\mathbf{R}\varGamma_{\bar{y}}(\mathscr{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{sp}^{X}_{\bar{y}\to\bar{x}}}$$\scriptstyle{\cong}$$\textstyle{\mathbf{R}\varGamma_{\bar{x}}(\mathscr{K})(1)[2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{R\bigl{(}\delta(y)\bigr{)}\bigl{[}2\delta(y)\bigr{]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\bigl{(}\delta(x)+1\bigr{)}\bigl{[}2(\delta(x)+1)\bigr{]}}$

where $\mathrm{sp}^{X}_{\bar{y}\to\bar{x}}$ is the transition morphism of
codimension $1$ defined in [12, §1].

The potential dualizing complexes have the following properties.

######  Proposition 1.2 ([12], 4.1 & 5.1).

Every scheme $X$ satisfying (†) equipped with a dimension function $\delta$
has a potential dualizing complex, unique up to unique isomorphism, which we
denoted by $\mathscr{K}_{X,\delta}$ or simply by $\mathscr{K}_{X}$. Moreover
the potential dualizing complex is a dualizing complex.

######  Proposition 1.3 ([12], 2.8).

Let $X$ be a regular scheme satisfying (†). Then
$\delta(x)\mathrel{\mathop{:}}=-\dim\mathcal{O}_{X,x}$ is a dimension function
on $X$ and we have $\mathscr{K}_{X,\delta}=\Lambda$.

######  Proposition 1.4 ([12], 4.3).

Let $f\colon X\to Y$ be a compactifiable morphism of schemes satisfying (†).
Let $\delta_{Y}$ be a dimension function on $Y$ and equip $X$ with the
dimension function

$\delta_{X}(x)\mathrel{\mathop{:}}=\delta_{Y}\bigl{(}f(x)\bigr{)}+\operatorname{tr.d.}k(x)\big{/}k\bigl{(}f(x)\bigr{)}\,.$

Then we have
$\mathscr{K}_{X,\delta_{X}}=\mathbf{R}f^{!}\mathscr{K}_{Y,\delta_{Y}}$.

######  Proposition 1.5 ([12], 4.2).

Let $f\colon X\to Y$ be a regular morphism of schemes satisfying (†). Let
$\delta_{Y}$ be a dimension function on $Y$ and equip $X$ with the dimension
function

$\delta_{X}(x)\mathrel{\mathop{:}}=\delta_{Y}(y)-\dim\mathcal{O}_{X_{y},x}\qquad\bigl{(}y\mathrel{\mathop{:}}=f(x)\bigr{)}$

Then we have $\mathscr{K}_{X,\delta_{X}}=f^{\ast}\mathscr{K}_{Y,\delta_{Y}}$.

Basing on above theory, we may generalize the étale homology in [10] to
arbitrary schemes satisfying (†). Since most proof are almost the same with
[10], we only give these which meed special care.

######  Definition 1.6.

Let $X$ be a scheme satisfying (†) equipped with a dimension function
$\delta$. For each $n\in\mathbb{Z}$, we define

$\mathrm{H}_{n}(X,\delta,\Lambda)\mathrel{\mathop{:}}=\mathrm{H}^{-n}(X_{\textup{\'{e}t}},\mathscr{K}_{X,\delta})=\operatorname{Hom}_{\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\Lambda)}\bigl{(}\Lambda_{X},\mathscr{K}_{X,\delta}[-n]\bigr{)}\,.$

We also use $\mathrm{H}_{n}(X,\delta)$ or $\mathrm{H}_{n}(X,\Lambda)$ or
$\mathrm{H}_{n}(X)$ to denote $\mathrm{H}_{n}(X,\delta,\Lambda)$, if no
confusion arise.

Let $f\colon X\to Y$ be a compactifiable morphism of schemes satisfying (†),
$\delta$ a dimension function on $Y$, and $\delta^{\prime}$ the dimension
function on $X$ induced by $\delta$ as in section 1. Then we also use
$\mathrm{H}_{n}(X,\delta,\Lambda)$ or $\mathrm{H}_{n}(X,\delta)$ to denote
$\mathrm{H}_{n}(X,\delta^{\prime},\Lambda)$.

If $f\colon X\to Y$ is a proper morphism of schemes satisfying (†), $\delta$ a
dimension function on $Y$. For each $n\in\mathbb{Z}$, we define a homomorphism
of $\Lambda$-modules

$f_{\ast}\colon\mathrm{H}_{n}(X,\delta)\to\mathrm{H}_{n}(Y,\delta)$

as follows. For each $\alpha\in\mathrm{H}_{n}(X,\delta)$, regarding
$\alpha\colon\Lambda_{X}\to\mathscr{K}_{X,\delta}[-n]$ as a morphism in
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\Lambda)$, then
$f_{\ast}(\alpha)$ is defined to be the composition

$\Lambda_{Y}\to\mathbf{R}f_{\ast}\Lambda_{X}\xrightarrow{\mathbf{R}f_{\ast}(\alpha)}\mathbf{R}f_{\ast}\mathscr{K}_{X,\delta}[-n]\xrightarrow{\,\sim\,}\mathbf{R}f_{\ast}\circ
f^{\ast}\mathscr{K}_{Y,\delta}[-n]\to\mathscr{K}_{Y,\delta}[-n]\,,$

where the first and the last morphisms are induced by the adjunctions.

It is easy to verify that if $f\colon X\to Y$ and $g\colon Y\to Z$ are two
proper morphisms, then $(g\circ f)_{\ast}=g_{\ast}\circ f_{\ast}$.

If $Y$ is the spectrum of a separably closed field, as
$\mathrm{H}_{0}(Y)=\Lambda$, we may write

$\deg\mathrel{\mathop{:}}=f_{\ast}\colon\mathrm{H}_{0}(X)\to\Lambda\,.$

######  Proposition 1.7.

Let $X$ be a scheme satisfying (†) equipped with a dimension function
$\delta$, $Y$ a closed subscheme of $X$ and $U\mathrel{\mathop{:}}=X\setminus
Y$. Then there is a long exact sequence of $\Lambda$-modules:

$\cdots\mathrm{H}_{n+1}(U,\delta)\to\mathrm{H}_{n}(Y,\delta)\to\mathrm{H}_{n}(X,\delta)\to\mathrm{H}_{n}(U,\delta)\to\mathrm{H}_{n-1}(Y,\delta)\to\cdots\,.$

######  Proposition 1.8 (Mayer-Vietoris Sequence).

Let $X$ be a scheme satisfying (†) equipped with a dimension function
$\delta$, $X_{1}$ and $X_{2}$ two closed subschemes of $X$ such that
$X=X_{1}\cup X_{2}$ (as sets). Then we have a long exact sequence

$\cdots\to\mathrm{H}_{n}(X_{1}\cap
X_{2},\,\delta)\to\mathrm{H}_{n}(X_{1},\delta)\oplus\mathrm{H}_{n}(X_{2},\delta)\to\mathrm{H}_{n}(X,\delta)\to\mathrm{H}_{n-1}(X_{1}\cap
X_{2},\,\delta)\to\cdots\,.$

######  Notation 1.9.

Let $X$ be a Noetherian scheme, $\delta$ a dimension function on $X$. Then we
define

$\dim_{\delta}(X)\mathrel{\mathop{:}}=\sup_{x\in X}\delta(x)\,.$

If $\xi_{1},\xi_{2},\ldots,\xi_{r}$ are all generic points of irreducible
components of $X$, then

$\dim_{\delta}(X)=\max_{1\leqslant i\leqslant r}\delta(\xi_{i})<+\infty\,.$

######  Proposition 1.10 (Vanishing).

Let $X$ be a scheme satisfying (†) equipped with a dimension function
$\delta$. Then $\mathrm{H}_{n}(X,\delta)=0$ for all $n>2\dim_{\delta}(X)$.

######  Corollary 1.11.

Let $X$ be a scheme satisfying (†) equipped with a dimension function
$\delta$, $Y$ a closed subscheme of $X$,
$X^{\prime}\mathrel{\mathop{:}}=X\setminus Y$. Then for each integer
$n>2\dim_{\delta}(Y)+1$, there is a canonical isomorphism
$\mathrm{H}_{n}(X,\delta)\xrightarrow{\,\sim\,}\mathrm{H}_{n}(X^{\prime},\delta)$
of $\Lambda$-modules.

###### Proof.

We have only to apply section 1. ∎

We define the pull-back maps as follows. Let $Y$ be a scheme satisfying (†)
equipped with a dimension function $\delta$, $f\colon X\to Y$ is a
compactifiable flat morphism of relative dimension $d$. For each
$n\in\mathbb{Z}$, we define a homomorphism of $\Lambda$-modules

$f^{\ast}\colon\mathrm{H}_{n}(Y,\delta)\to\mathrm{H}_{n+2d}(X,\delta)(-d)$

as follows. For each $\beta\in\mathrm{H}_{n}(Y,\delta)$, $f^{\ast}(\beta)$ is
defined to be the composition

$\Lambda_{X}\xrightarrow{\mathrm{t}_{f}}\mathbf{R}f^{!}\Lambda_{X}[-2d](-d)\xrightarrow{\mathbf{R}f^{!}(\beta)}\mathbf{R}f^{!}\mathscr{K}_{Y}[-n-2d](-d)\xrightarrow{\,\sim\,}\mathscr{K}_{X}[-n-2d](-d)\,,$

where $\mathrm{t}_{f}$ is the morphism dual to the trace morphism
$\mathrm{Tr}_{f}$ (see [8, XVIII (3.2.1.2)]).

It is easy to verify that if $f\colon X\to Y$ and $g\colon Y\to Z$ be two
compactifiable flat morphisms of relative dimension $d$ and $e$ respectively,
then

$(g\circ f)^{\ast}=f^{\ast}\circ
g^{\ast}\colon\mathrm{H}_{n}(Z)\to\mathrm{H}_{n+2d+2e}(X)(-d-e)\,.$

Moreover the maps $f_{\ast}$ and $f^{\ast}$ commute in Cartesian squares.

Now we define the cycle map for étale homology.

######  Notation 1.12.

Let $X$ be a scheme satisfying (†) equipped with a dimension function
$\delta$, $d\mathrel{\mathop{:}}=\dim_{\delta}(X)$.

  1. (1)

First we assume that $X$ is regular and integral. Then we use
$\mathrm{t}_{X}\colon\Lambda\xrightarrow{\,\sim\,}\mathscr{K}_{X,\delta}(-d)[-2d]$
to denote the canonical morphism defined by section 1, and use
$\mathrm{cl}(X)$ to denote the element in $\mathrm{H}_{2d}(X,\delta)(-d)$
corresponding to $\mathrm{t}_{X}$.

  2. (2)

Second we only assume that $X$ is integral. As $X$ is excellent,
$U\mathrel{\mathop{:}}=X_{\mathrm{reg}}$ is an open dense subset of $X$.
Obviously $\dim_{\delta}(X\setminus U)<d$. By section 1, there is a canonical
isomorphism

$\mathrm{H}_{2d}(X,\delta)(-d)\xrightarrow{\,\sim\,}\mathrm{H}_{2d}(U,\delta)(-d)\,.$

Let $\mathrm{cl}(X)$ be the inverse image of $\mathrm{cl}(U)$ under above
isomorphism.

  3. (3)

In general case, we let $X_{1},X_{2},\ldots,X_{r}$ be all irreducible
components of $X$ with $\dim_{\delta}(X_{i})=d$. For each $i$, regard $X_{i}$
as a reduced subscheme of $X$, let $\iota_{i}\colon X_{i}\hookrightarrow X$ be
the inclusion and let $\xi_{i}$ be the generic point of $X_{i}$. Then we
define

$\mathrm{cl}(X)\mathrel{\mathop{:}}=\sum^{r}_{i=1}\mathrm{length}(\mathcal{O}_{X,\xi_{i}})\cdot\iota_{i,\ast}\bigl{(}\mathrm{cl}(X_{i})\bigr{)}\in\mathrm{H}_{2d}(X,\delta)(-d)\,.$

We also use
$\mathrm{t}_{X}\colon\Lambda\xrightarrow{\,\sim\,}\mathscr{K}_{X,\delta}(-d)[-2d]$
to denote the morphism in $\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X,\Lambda)$
corresponding to $\mathrm{cl}(X)$.

######  Notation 1.13.

Let $X$ be a scheme satisfying (†) equipped with a dimension function
$\delta$, $Y$ a closed subscheme of $X$,
$d\mathrel{\mathop{:}}=\dim_{\delta}(Y)$. Then we define

$\mathrm{cl}_{X}(Y)\mathrel{\mathop{:}}=\iota_{\ast}\bigl{(}\mathrm{cl}(Y)\bigr{)}\in\mathrm{H}_{2d}(X,\delta)(-d)\,,$

where $\iota\colon Y\hookrightarrow X$ is the inclusion.

Next, we study the étale homology under birational morphisms. First we need
two lemmas.

######  Lemma 1.14.

Let $X$ be a scheme satisfying (†), $Y$ a closed subscheme of $X$,
$U\mathrel{\mathop{:}}=X\setminus Y$, $i\colon Y\hookrightarrow X$ and
$j\colon U\hookrightarrow X$ the inclusions. Then for each object
$\mathscr{F}$ in
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\Lambda)$, there are
two distinguished triangles in
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\Lambda)$:

  1. (1)

$j_{!}j^{\ast}\mathscr{F}\to\mathscr{F}\to i_{\ast}i^{\ast}\mathscr{F}\to
j_{!}j^{\ast}\mathscr{F}[1]$ ,

  2. (2)

$i_{\ast}\mathbf{R}i^{!}\mathscr{F}\to\mathscr{F}\to\mathbf{R}j_{\ast}j^{\ast}\mathscr{F}\to
i_{\ast}\mathbf{R}i^{!}\mathscr{F}[1]$ .

######  Lemma 1.15.

Let $X$ be a scheme satisfying (†), $Y$ a closed subscheme of $X$,
$U\mathrel{\mathop{:}}=X\setminus Y$, $i\colon Y\hookrightarrow X$ and
$j\colon U\hookrightarrow X$ the inclusions,
$\mathscr{F}^{\prime}\xrightarrow{\varphi}\mathscr{F}\xrightarrow{\psi}\mathscr{F}^{\prime\prime}$
a sequence of objects in
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\Lambda)$. Assume
that one of the following two conditions holds:

  1. (1)

$j^{\ast}(\varphi)\colon
j^{\ast}\mathscr{F}^{\prime}\xrightarrow{\,\sim\,}j^{\ast}\mathscr{F}$ is an
isomorphism, $j^{\ast}\mathscr{F}^{\prime\prime}=0$, and for all
$n\in\mathbb{Z}$,

$0\to\mathrm{H}^{n}(i^{\ast}\mathscr{F}^{\prime})\to\mathrm{H}^{n}(i^{\ast}\mathscr{F})\to\mathrm{H}^{n}(i^{\ast}\mathscr{F}^{\prime\prime})\to
0$

is a short exact sequence;

  2. (2)

$j^{\ast}(\psi)\colon
j^{\ast}\mathscr{F}\xrightarrow{\,\sim\,}j^{\ast}\mathscr{F}^{\prime\prime}$
is an isomorphism, $j^{\ast}\mathscr{F}^{\prime}=0$, and for all
$n\in\mathbb{Z}$,

$0\to\mathrm{H}^{n}(\mathbf{R}i^{!}\mathscr{F}^{\prime})\to\mathrm{H}^{n}(\mathbf{R}i^{!}\mathscr{F})\to\mathrm{H}^{n}(\mathbf{R}i^{!}\mathscr{F}^{\prime\prime})\to
0$

is a short exact sequence.

Then there exists a morphism
$\delta\colon\mathscr{F}^{\prime\prime}\to\mathscr{F}^{\prime}[1]$ which makes

$\mathscr{F}^{\prime}\xrightarrow{\varphi}\mathscr{F}\xrightarrow{\psi}\mathscr{F}^{\prime\prime}\xrightarrow{\delta}\mathscr{F}^{\prime}[1]$

a distinguished triangle in
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\Lambda)$.

###### Proof.

The morphism $\varphi$ extends to a distinguished triangle

$\mathscr{F}^{\prime}\xrightarrow{\varphi}\mathscr{F}\xrightarrow{\phi}\mathscr{G}\xrightarrow{\rho}\mathscr{F}^{\prime}[1]$

in $\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\Lambda)$. Note
that
$0\to\mathscr{F}^{\prime\prime}\xrightarrow{\mathrm{id}}\mathscr{F}^{\prime\prime}\to
0[1]$ is also a distinguished triangle. Hence there exists a morphism
$\alpha\colon\mathscr{G}\to\mathscr{F}^{\prime\prime}$ such that the triple
$(0,\psi,\alpha)$ is a morphism of distinguished triangles, i.e., the
following diagram commutes.

$\textstyle{\mathscr{F}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{0}$$\textstyle{\mathscr{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$$\textstyle{\mathscr{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\alpha}$$\textstyle{\mathscr{F}^{\prime}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{F}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}}$$\textstyle{\mathscr{F}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0[1]}$

Now if (1) holds, then both $j^{\ast}(\alpha)$ and $i^{\ast}(\alpha)$ are
isomorphic; and if (2) holds, then both $j^{\ast}(\alpha)$ and
$\mathbf{R}i^{!}(\alpha)$ are isomorphic. In either case, by section 1, the
morphism
$\alpha\colon\mathscr{G}\xrightarrow{\,\sim\,}\mathscr{F}^{\prime\prime}$ is
an isomorphism. Therefore we may let
$\delta\mathrel{\mathop{:}}=\rho\circ\alpha^{-1}$. ∎

Basing on above lemma, we have the following two propositions.

######  Proposition 1.16.

Let

$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\scriptstyle{q}$$\textstyle{\square}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{X}$

be a Cartesian square of schemes satisfying (†), $r\mathrel{\mathop{:}}=p\circ
i^{\prime}=i\circ q$. Assume that $p$ is proper, $i$ is a closed immersion;
and there exists an open subset $U$ of $X$ such that $i(Y)=X\setminus U$ and
$p$ induces an isomorphism $p^{-1}(U)\xrightarrow{\,\sim\,}U$. Then for each
object $\mathscr{F}$ in
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\Lambda)$, there are
two distinguished triangles:

$\displaystyle\mathscr{F}\xrightarrow{\delta_{i}\oplus\delta_{p}}i_{\ast}\circ
i^{\ast}\mathscr{F}\oplus\mathbf{R}p_{\ast}\circ
p^{\ast}\mathscr{F}\xrightarrow{\delta_{q}\circ\mathrm{pr}_{1}-\delta_{i^{\prime}}\circ\mathrm{pr}_{2}}\mathbf{R}r_{\ast}\circ
r^{\ast}\mathscr{F}\to\mathscr{F}[1]\,,$
$\displaystyle\mathbf{R}r_{\ast}\circ\mathbf{R}r^{!}\mathscr{F}\xrightarrow{\vartheta_{q}\oplus\vartheta_{i^{\prime}}}i_{\ast}\circ\mathbf{R}i^{!}\mathscr{F}\oplus\mathbf{R}p_{\ast}\circ\mathbf{R}p^{!}\mathscr{F}\xrightarrow{-\vartheta_{i}\circ\mathrm{pr}_{1}+\vartheta_{p}\circ\mathrm{pr}_{2}}\mathscr{F}\to(\mathbf{R}r_{\ast}\circ\mathbf{R}r^{!}\mathscr{F})[1]\,,$

in $\mathbf{D}_{\mathrm{c}}^{\textbf{+}}(X_{\textup{\'{e}t}},\Lambda)$, where
$\delta$ and $\vartheta$ are induced by the adjunctions.

Apply the second triangle in above proposition to $\mathscr{K}_{X}$, we obtain
the following proposition.

######  Proposition 1.17.

Let the assumptions and the notations be as in section 1; and let $\delta$ be
a dimension function on $X$. Then there is a long exact sequence of
$\Lambda$-modules:

$\cdots\to\mathrm{H}_{n+1}(X,\delta)\to\mathrm{H}_{n}(Y^{\prime},\delta)\xrightarrow{q_{\ast}\oplus
i^{\prime}_{\ast}}\mathrm{H}_{n}(Y,\delta)\oplus\mathrm{H}_{n}(X^{\prime},\delta)\xrightarrow{-i_{\ast}\circ\mathrm{pr}_{1}+p_{\ast}\circ\mathrm{pr}_{2}}\mathrm{H}_{n}(X,\delta)\to\cdots\,.$

##  2\. Calculation of étale homology on the exceptional divisor

In this section, we calculate the étale homology of exceptional curves.

######  Proposition 2.1.

Let $C$ be a $1$-equidimensional proper algebraic scheme over a separably
closed field $k$. Let $C^{(1)},C^{(2)},\ldots,C^{(r)}$ be all connected
components of $C$; and let $C_{1},C_{2},\ldots,C_{n}$ be all irreducible
components of $C$. Assume that $\mathrm{H}^{1}(C,\mathcal{O}_{C})=0$. Then we
have

  1. (1)

As to homology, we have

    1. (a)

There is a canonical isomorphism:

$\mathrm{H}_{0}(C)=\bigoplus^{r}_{i=1}\mathrm{H}_{0}(C^{(i)})\xrightarrow[\,\sim\,]{\oplus\deg}\Lambda^{\oplus
r}\,.$

    2. (b)

$\mathrm{H}_{1}(C)=0$.

    3. (c)

$\mathrm{H}_{2}(C)(-1)$ is a free $\Lambda$-module with basis
$\mathrm{cl}(C_{1}),\ldots,\mathrm{cl}(C_{n})$.

  2. (2)

As to cohomology, we have

    1. (a)

There is a canonical isomorphism:

$\mathrm{H}^{0}(C,\Lambda)=\bigoplus^{r}_{i=1}\mathrm{H}^{0}(C^{(i)})\xrightarrow{\,\sim\,}\Lambda^{\oplus
r}\,.$

    2. (b)

$\mathrm{H}^{1}(C,\Lambda)=0$.

    3. (c)

There is a canonical isomorphism
$\mathrm{H}^{2}\bigl{(}C,\Lambda(1)\bigr{)}\xrightarrow{\,\sim\,}\Lambda^{\oplus
n}$ which sends $c_{1}(\mathscr{L})$ to
$\bigl{(}\deg(\mathscr{L}|_{C_{1}}),\ldots,\deg(\mathscr{L}|_{C_{r}})\bigr{)}$
for each invertible $\mathcal{O}_{C}$-module $\mathscr{L}$.

###### Proof.

(1). Let $\bar{k}$ be the algebraic closure of $k$. After replacing $k$ with
$\bar{k}$ and $X$ with $(X\otimes_{k}\bar{k})_{\mathrm{red}}$, we may assume
that $k$ is algebraically closed and $X$ is reduced. Now we use induction on
the number $n$ of irreducible components of $C$.

First consider the case $n=1$. In this case, $C$ is integral. Let
$\widetilde{C}$ be the normalization of $C$. Then there is a long exact
sequence

$0\to\varGamma(C,\mathcal{O}_{C})\to\varGamma(C,\mathcal{O}_{\widetilde{C}})\to\varGamma(C,\mathcal{O}_{\widetilde{C}}/\mathcal{O}_{C})\to\mathrm{H}^{1}(C,\mathcal{O}_{C})\,.$

Note that $\mathrm{H}^{1}(C,\mathcal{O}_{C})=0$,
$\varGamma(C,\mathcal{O}_{C})=k$ and
$\varGamma(C,\mathcal{O}_{\widetilde{C}})=\varGamma(\widetilde{C},\mathcal{O}_{\widetilde{C}})=k$.
Hence $\varGamma(C,\mathcal{O}_{\widetilde{C}}/\mathcal{O}_{C})=0$. Since
$\mathcal{O}_{\widetilde{C}}/\mathcal{O}_{C}$ is a coherent
$\mathcal{O}_{C}$-module whose support is of dimension $\leqslant 0$, we have
$\mathcal{O}_{\widetilde{C}}/\mathcal{O}_{C}=0$, i.e., $C=\widetilde{C}$ is
normal. So we have $C=\mathrm{P}_{k}^{1}$.

Assume that (1) is valid for all integers $<n$. Let $C^{\prime}$ be an
irreducible component $C$ and let $C^{\prime\prime}$ be the union of all other
irreducible components. Regard $C^{\prime}$ and $C^{\prime\prime}$ as reduced
closed subschemes of $C$. Then there is a short exact sequence:

$0\to\mathcal{O}_{C}\xrightarrow{c\mapsto(\bar{c},\bar{c})}\mathcal{O}_{C^{\prime}}\oplus\mathcal{O}_{C^{\prime\prime}}\xrightarrow{(a,b)\mapsto\bar{a}-\bar{b}}\mathcal{O}_{C^{\prime}\cap
C^{\prime\prime}}\to 0\,,$

which induces a long exact sequence:

$\displaystyle
0\to\varGamma(C,\mathcal{O}_{C})\to\varGamma(C,\mathcal{O}_{C^{\prime}})\oplus\varGamma(C,\mathcal{O}_{C^{\prime\prime}})\to\varGamma(C,\mathcal{O}_{C^{\prime}\cap
C^{\prime}})$
$\displaystyle\to\mathrm{H}^{1}(C,\mathcal{O}_{C})\to\mathrm{H}^{1}(C,\mathcal{O}_{C^{\prime}})\oplus\mathrm{H}^{1}(C,\mathcal{O}_{C^{\prime\prime}})\to
0\,.$

Let $r$ and $s$ be the numbers of connected components of $C$ and
$C^{\prime\prime}$ respectively; and let $Z_{1},Z_{2},\ldots,Z_{m}$ be all
connected components of $C^{\prime\prime}$ which intersect with $C^{\prime}$.
Then $m=s-r+1$. On the other hand, we have

$\dim_{k}\varGamma(C,\mathcal{O}_{C})=r,\quad\dim_{k}\varGamma(C,\mathcal{O}_{C^{\prime}})=1,\quad\dim_{k}\varGamma(C,\mathcal{O}_{C^{\prime\prime}})=s.$

As $\mathrm{H}^{1}(C,\mathcal{O}_{C})=0$, we have
$\mathrm{H}^{1}(C,\mathcal{O}_{C^{\prime}})=\mathrm{H}^{1}(C,\mathcal{O}_{C^{\prime\prime}})=0$,
and

$\dim_{k}\varGamma(C,\mathcal{O}_{C^{\prime}\cap C^{\prime}})=s-r+1=m\,.$

This shows that for each $i$, $Z_{i}\cap C^{\prime}$ contains only one point
$P_{i}$, and $C^{\prime}\cap C^{\prime\prime}=\\{P_{1},P_{2},\ldots,P_{m}\\}$.

Using induction on $Z_{i}$, $C^{\prime}$ and $C^{\prime\prime}$, we obtain
that (1) is valid on all $Z_{i}$, $C^{\prime}$ and $C^{\prime\prime}$.
Applying section 1, we obtain an isomorphism
$\mathrm{H}_{2}(C^{\prime})\oplus\mathrm{H}_{2}(C^{\prime\prime})\xrightarrow{\,\sim\,}\mathrm{H}_{2}(C)$
and an exact sequence

$0\to\mathrm{H}_{1}(C)\to\mathrm{H}_{0}(C^{\prime}\cap
C^{\prime\prime})\xrightarrow{l}\mathrm{H}_{0}(C^{\prime})\oplus\mathrm{H}_{0}(C^{\prime\prime})\to\mathrm{H}_{0}(C)\to
0\,.$

Note that the map $l$ has an inverse which is defined by the composition

$\mathrm{H}_{0}(C^{\prime})\oplus\mathrm{H}_{0}(C^{\prime\prime})\xrightarrow{\text{pr}}\mathrm{H}_{0}(C^{\prime\prime})\xrightarrow{\text{pr}}\bigoplus^{m}_{i=1}\mathrm{H}_{0}(Z_{i})\xrightarrow[\,\sim\,]{\oplus\varphi_{i}}\bigoplus^{m}_{i=1}\mathrm{H}_{0}(P_{i})=\mathrm{H}_{0}(C^{\prime}\cap
C^{\prime\prime})\,,$

where the isomorphism $\varphi_{i}$, which maps $\mathrm{cl}_{Z_{i}}(P_{i})$
to $\mathrm{cl}(P_{i})$, is obtained by applying (a) on $Z_{i}$. Therefore (1)
are valid for $C$.

(2) Let $\pi\colon C\to\operatorname{Spec}k$ be the structural morphism. Note
that

$\mathrm{H}^{q}(C,\Lambda)=\mathrm{H}^{q}(\mathbf{R}\pi_{\ast}\Lambda_{C})=\mathrm{H}^{q}\bigl{(}\mathscr{D}(\mathbf{R}\pi_{\ast}\mathscr{D}(\Lambda_{C}))\bigr{)}=\mathrm{H}^{q}\bigl{(}\mathscr{D}(\mathbf{R}\pi_{\ast}\mathscr{K}_{C})\bigr{)}\,,$

where
$\mathscr{D}\mathrel{\mathop{:}}=\mathscr{H}om(\makebox[8.00003pt][c]{\raisebox{0.86108pt}{\Tiny\textbullet}},\mathscr{K})$
is the dualizing functor. By (1), we have

$\mathrm{H}^{q}(\mathbf{R}\pi_{\ast}\mathscr{K}_{C})=\mathrm{H}_{-q}(C)=\begin{cases}\Lambda^{\oplus
r}&q=0,\\\ \Lambda(1)^{\oplus n}&q=-2,\\\ 0&\text{otherwise.}\end{cases}$

In other words, all entries of $\mathbf{R}\pi_{\ast}\mathscr{K}_{C}$ are free
$\Lambda$-modules. Therefore we have

$\mathrm{H}^{q}(C,\Lambda)=\begin{cases}\Lambda^{\oplus r}&q=0,\\\
\Lambda(-1)^{\oplus n}&q=2,\\\ 0&\text{otherwise.}\end{cases}$

So (2) is verifies. (c) is also by [8, IX, 4.7]. ∎

######  Remark 2.2.

In fact, every connected component of $(C\otimes_{k}\bar{k})_{\mathrm{red}}$
is a tree of nonsingular rational curves with normal crossings.

##  3\. Calculation of étale homology on surfaces with rational singularities

In this paper, a surface means a two-dimensional integral normal scheme which
satisfies (†). Note that for each surface $X$,
$\delta(x)\mathrel{\mathop{:}}=2-\dim\mathcal{O}_{X,x}$ is a dimension
function on $X$; and for all points $x\in X$, we have
$\delta(x)=\dim\overline{\\{x\\}}$. From now on, all surfaces are tacitly
equipped with this type of dimension function.

We recall the definition of _rational singularity_ (cf. [1] & [11]). A normal
local ring $R$ of dimension $2$ is said to have a _rational singularity_ if
there exists a desingularization $f\colon X\to\operatorname{Spec}(R)$ such
that $\mathrm{H}^{1}(X,\mathcal{O}_{X})=0$. By [11, Proposition (16.5)], for a
normal local ring $R$ of dimension $2$, if one of the four rings $R$,
$\widehat{R}$, $R^{\mathrm{sh}}$ and $\widehat{R^{\mathrm{sh}}}$ has a
rational singularity, so have the other three rings. A surface $X$ is said to
have a rational singularity at a closed point $x$, if the local ring
$\mathcal{O}_{X,x}$ has a rational singularity.

Now we calculate the étale homology on arbitrary surfaces.

######  Proposition 3.1.

Let $X$ be a surface which is the spectrum of a strictly Henselian local ring,
$P$ the closed point of $X$, $U\mathrel{\mathop{:}}=X\setminus P$. Assume that
$U$ is regular, and $X$ admits a desingularization $\pi\colon\widetilde{X}\to
X$ which induces an isomorphism $\pi^{-1}(U)\xrightarrow{\,\sim\,}U$. Let
$E\mathrel{\mathop{:}}=\pi^{-1}(P)$ be the exceptional divisor. Then we have

  1. (1)

The map
$\mathrm{H}^{0}(\mathrm{t}_{X})\colon\Lambda\xrightarrow{\,\sim\,}\mathrm{H}_{4}(X)(-2)$
is an isomorphism.

  2. (2)

$\mathrm{H}_{3}(X)\xrightarrow{\,\sim\,}\mathrm{H}^{1}\bigl{(}E,\Lambda(2)\bigr{)}$.

  3. (3)

There is an exact sequence of $\Lambda$-modules:

$0\to\mathrm{H}_{2}(E)\to\mathrm{H}^{2}\bigl{(}E,\Lambda(2)\bigr{)}\to\mathrm{H}_{2}(X)\to\mathrm{H}_{1}(E)\to
0\,,$

Moreover the kernel of the map $\mathrm{H}_{2}(X)\to\mathrm{H}_{1}(E)$ is a
finite group.

  4. (4)

There is an exact sequence of $\Lambda$-modules:

$0\to\mathrm{H}_{1}(X)\to\mathrm{H}_{0}(E)\xrightarrow{\deg}\Lambda\to 0\,.$

  5. (5)

$\mathrm{H}_{q}(X)=0$ for $q>4$ or $q<1$.

###### Proof.

After applying section 1 to the following Cartesian square,

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{i}}$$\scriptstyle{\tau}$$\textstyle{\square}$$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{X}$
(3.1)

we obtain a long exact sequence

$\cdots\to\mathrm{H}_{n+1}(X)\to\mathrm{H}_{n}(E)\to\mathrm{H}_{n}(P)\oplus\mathrm{H}_{n}(\widetilde{X})\to\mathrm{H}_{n}(X)\to\cdots\,.$
(3.2)

For each $q\in\mathbb{Z}$, there is a canonical isomorphism of
$\Lambda$-modules:

$\mathrm{H}_{q}(\widetilde{X})\xrightarrow{\,\sim\,}\mathrm{H}^{4-q}\bigl{(}\widetilde{X},\Lambda(2)\bigr{)}\xrightarrow{\,\sim\,}\mathrm{H}^{4-q}\bigl{(}E,\Lambda(2)\bigr{)}\,,$

where the first isomorphism is induced by section 1 (since $\widetilde{X}$ is
regular), and the last isomorphism is from the base change theorem via the
square (3.1). Thus $\mathrm{H}_{q}(\widetilde{X})=0$ for $q<2$ and $q>4$. Also
note that $\mathrm{H}_{q}(E)=0$ for $q<0$ and $q>2$; and
$\mathrm{H}_{q}(P)=\begin{cases}0&q\neq 0,\\\ \Lambda&q=0.\end{cases}$ Hence
we may divide (3.2) to three exact sequences

$\displaystyle
0\to\mathrm{H}^{0}\bigl{(}E,\Lambda(2)\bigr{)}\to\mathrm{H}_{4}(X)\to 0,$
(3.3) $\displaystyle
0\to\mathrm{H}^{1}\bigl{(}E,\Lambda(2)\bigr{)}\to\mathrm{H}_{3}(X)\to\mathrm{H}_{2}(E)\to\mathrm{H}^{2}\bigl{(}E,\Lambda(2)\bigr{)}\to\mathrm{H}_{2}(X)\to\mathrm{H}_{1}(E)\to
0,$ (3.4) $\displaystyle
0\to\mathrm{H}_{1}(X)\to\mathrm{H}_{0}(E)\xrightarrow{\deg}\Lambda\to\mathrm{H}_{0}(X)\to
0.$ (3.5)

Now (1) is by the exact sequence (3.3).

From the sequence (3.5), to prove (4) & (5) we have only to prove the map
$\mathrm{H}_{0}(E)\xrightarrow{\deg}\Lambda$ is epimorphic. Since $X$ is
strictly local, the field $k\mathrel{\mathop{:}}=k(P)$ is separably closed.
Let $Q$ be a closed point on $E$. Then
$\deg\mathrm{cl}_{E}(Q)=\bigl{[}k(Q),k\bigr{]}$. Put
$p\mathrel{\mathop{:}}=\operatorname{char}k$. Since $k$ is separably closed,
the number $\bigl{[}k(Q),k\bigr{]}$ is either $1$ or a power of $p$; in both
cases, it is invertible on $\Lambda$ (as $\ell\neq p$). Hence $\deg$ is
surjective.

From the sequence (3.4), to prove (2) & (3) we have only to prove that the
homomorphism
$\mathrm{H}_{2}(E)(-1)\to\mathrm{H}^{2}\bigl{(}E,\Lambda(1)\bigr{)}$ is
injective and its cokernel is a torsion group. Let $E_{1},E_{2},\ldots,E_{n}$
be all irreducible components of $E$. Then $\mathrm{H}_{2}(E)(-1)$ is a free
$\Lambda$-Module with a basis
$\bigl{\\{}\mathrm{cl}_{E}(E_{i})\bigr{\\}}^{n}_{i=1}$. Note that the
composite map

$\mathrm{H}_{2}(E)(-1)\to\mathrm{H}_{2}(\widetilde{X})(-1)\xrightarrow{\,\sim\,}\mathrm{H}^{2}\bigl{(}\widetilde{X},\Lambda(1)\bigr{)}\xrightarrow{\,\sim\,}\mathrm{H}^{2}\bigl{(}E,\Lambda(1)\bigr{)}\xrightarrow[\,\sim\,]{\rho}\Lambda^{\oplus
n}$ (3.6)

sends each $\mathrm{cl}_{E}(E_{i})$ to $\sum\limits^{n}_{j=1}a_{ij}e_{j}$,
where the isomorphism $\rho$ is defined in [8, IX, 4.7],
$e_{1},e_{2},\ldots,e_{n}$ is the standard basis for $\Lambda^{\oplus n}$, and

$a_{ij}=\deg\bigl{(}\mathscr{L}_{\widetilde{X}}(E_{i})|_{E_{j}}\bigr{)}=(E_{i},E_{j})\,,$

where $\mathscr{L}_{\widetilde{X}}(E_{i})$ means the invertible sheaf on
$\widetilde{X}$ associated to the divisor $E_{i}$. In other words, the map
(3.6) is given by the intersection matrix $\bigl{(}(E_{i},E_{j})\bigr{)}$,
which is negative-definite by [11, (14.1)]. Therefore the map (3.6) is
injective with torsion cokernel. ∎

######  Theorem 3.2.

Let the assumptions and the notations be as in section 3. We further assume
that $X$ has a rational singularity at $P$. Then we have

  1. (1)

Let $\mathrm{Cl}(X)$ denote the Weil divisor class group of $X$. Then there is
a canonical isomorphism

$\mathrm{cl}\colon\mathrm{Cl}(X)\otimes_{\mathbb{Z}}\Lambda\xrightarrow{\,\sim\,}\mathrm{H}_{2}(X)(-1)$

which sends each $[Y]\otimes 1$ to $\mathrm{cl}_{X}(Y)$.

  2. (2)

The map
$\mathrm{H}^{0}(\mathrm{t}_{X})\colon\Lambda\xrightarrow{\,\sim\,}\mathrm{H}_{4}(X)(-2)$
is an isomorphism.

  3. (3)

$\mathrm{H}_{q}(X)=0$ for $q\neq 2,4$.

###### Proof.

By [11, (4.1)], any desingularization of a surface with rational singularities
is a product of quadratic transformations. Since $X$ is the spectrum of a
local ring, the exceptional divisor $E$ is connected. Thus the map
$\deg\colon\mathrm{H}_{0}(E)\to\Lambda$ is an isomorphism by section 2 (1a).
Then (2) & (3) are by section 3 and section 2. So we have only to prove (1).

We adopt the assumptions and notation in [11, §14]. Let
$E_{1},E_{2},\ldots,E_{n}$ be all irreducible components of $E$. For each $i$,
let $d_{i}>0$ be the greatest common divisor of all the degrees of invertible
sheaves on $E_{i}$. Put $k\mathrel{\mathop{:}}=k(P)$ and
$p\mathrel{\mathop{:}}=\operatorname{char}k$. For each $i$, let $Q_{i}$ be a
closed point on $\mathrm{Reg}(E_{i})$. Then $Q_{i}$ defines an invertible
sheaf of degree $\bigl{[}k(Q_{i}),k\bigr{]}$ on $E_{i}$. Since the number
$\bigl{[}k(Q_{i}),k\bigr{]}$ is invertible on $\Lambda$, so is $d_{i}$.

Let $\mathbf{E}$ be the additive group of divisors on $\widetilde{X}$
generated by $\\{E_{i}\\}^{n}_{i=1}$. Then $\mathbf{E}$ is a free abelian
group with basis $E_{1},E_{2},\ldots,E_{n}$. Put
$\mathbf{E}^{\ast}\mathrel{\mathop{:}}=\operatorname{Hom}_{\mathbb{Z}}(\mathbf{E},\mathbb{Z})$
and let $\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{n}$ be the dual
basis. We define a homomorphism of groups:

$\vartheta\colon\mathbf{E}\to\mathbf{E}^{\ast}\,,\qquad
E_{i}\mapsto\sum^{n}_{j=1}\tfrac{1}{d_{j}}(E_{i},E_{j})\cdot\varepsilon_{j}\,.$

By [11, (14.3), (14.4) & (17.1)], there is an exact sequence

$0\to\mathbf{E}\xrightarrow{\vartheta}\mathbf{E}^{\ast}\to\mathrm{Cl}(X)\to
0\,.$

Let $e_{1},e_{2},\ldots,e_{n}$ be the basis for the free $\Lambda$-module
$\mathrm{H}^{2}\bigl{(}E,\Lambda(1)\bigr{)}$ defined by the isomorphism in
section 2 (2c). Since each $d_{j}$ is invertible on $\Lambda$, the
homomorphism

$\delta\colon\mathbf{E}^{\ast}\otimes_{\mathbb{Z}}\Lambda\to\mathrm{H}^{2}\bigl{(}E,\Lambda(1)\bigr{)}\,,\qquad\varepsilon_{j}\otimes
1\mapsto d_{j}e_{j}$

is an isomorphism of $\Lambda$-modules. It is easy to check that the following
diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{E}\otimes_{\mathbb{Z}}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta\otimes\mathrm{id}}$(i)$\scriptstyle{\cong}$$\textstyle{\mathbf{E}^{\ast}\otimes_{\mathbb{Z}}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\cong}$$\textstyle{\mathrm{Cl}(X)\otimes_{\mathbb{Z}}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{cl}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{2}(E)(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{2}\bigl{(}E,\Lambda(1)\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{2}(X)(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is commutative with both rows exact, where the isomorphism (i) sends each
$E_{i}\otimes 1$ to $\mathrm{cl}_{E}(E_{i})$. Therefore
$\mathrm{cl}\colon\mathrm{Cl}(X)\otimes_{\mathbb{Z}}\Lambda\xrightarrow{\,\sim\,}\mathrm{H}_{2}(X)(-1)$
is an isomorphism. ∎

By [11, (17.1)], the divisor class group $\mathrm{Cl}(X)$ is a finite group.
So we have:

######  Corollary 3.3.

Let $X$ be a surface having only finitely many singular points, all of which
are rational singularities. Then
$\mathrm{t}_{X}\colon\mathbb{Q}_{\ell}(2)[4]\xrightarrow{\,\sim\,}\mathscr{K}_{X}$
is an isomorphism in
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\mathbb{Q}_{\ell})$.
In particular, $\mathbb{Q}_{\ell}$ is a dualizing complex on
$X_{\textup{\'{e}t}}$ in coefficient $\mathbb{Q}_{\ell}$.

######  Corollary 3.4 (Poincaré Duality).

Let $k$ be a separably closed field on which $\ell$ is invertible. And let $X$
be a surface over $k$. Assume that $X$ has only finitely many singular points,
all of which are rational singularities. Then

  1. (1)

The trace map
$\operatorname{Tr}_{X}\colon\mathrm{H}_{\mathrm{c}}^{4}\bigl{(}X,\mathbb{Q}_{\ell}(2)\bigr{)}\xrightarrow{\,\sim\,}\mathbb{Q}_{\ell}$
is an isomorphism.

  2. (2)

For every $0\leqslant r\leqslant 4$ and every object $\mathscr{F}$ in
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\mathbb{Q}_{\ell})$,
the pairing

$\mathrm{H}_{\mathrm{c}}^{r}(X,\mathscr{F})\times\mathrm{Ext}^{4-r}_{X}\bigl{(}\mathscr{F},\mathbb{Q}_{\ell}(2)\bigr{)}\to\mathrm{H}_{\mathrm{c}}^{4}\bigl{(}X,\mathbb{Q}_{\ell}(2)\bigr{)}\xrightarrow[\,\sim\,]{\operatorname{Tr}_{X}}\mathbb{Q}_{\ell}$

is nondegenerate.

  3. (3)

For every $0\leqslant r\leqslant 4$, the pairing

$\mathrm{H}_{\mathrm{c}}^{r}(X,\mathbb{Q}_{\ell})\times\mathrm{H}^{4-r}\bigl{(}X,\mathbb{Q}_{\ell}(2)\bigr{)}\to\mathrm{H}_{\mathrm{c}}^{4}\bigl{(}X,\mathbb{Q}_{\ell}(2)\bigr{)}\xrightarrow[\,\sim\,]{\operatorname{Tr}_{X}}\mathbb{Q}_{\ell}$

is nondegenerate.

###### Proof.

(3) is a special case of (2). Let $p\colon X\to\operatorname{Spec}k$ be the
structural morphism. By section 3 and section 1, we have isomorphisms

$\mathbb{Q}_{\ell}(2)[4]\xrightarrow[\,\sim\,]{\mathrm{t}_{X}}\mathscr{K}_{X}\xrightarrow{\,\sim\,}\mathbf{R}p^{!}\mathbb{Q}_{\ell}\,.$

Hence we have

$\displaystyle\mathrm{Ext}_{X}^{4-r}\bigl{(}\mathscr{F},\mathbb{Q}_{\ell}(2)\bigr{)}$
$\displaystyle\xrightarrow{\,\sim\,}\operatorname{Hom}_{\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\mathbb{Q}_{\ell})}\bigl{(}\mathscr{F}[r],\mathbb{Q}_{\ell}(2)[4]\bigr{)}$
$\displaystyle\xrightarrow{\,\sim\,}\operatorname{Hom}_{\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\mathbb{Q}_{\ell})}\bigl{(}\mathscr{F}[r],\mathbf{R}p^{!}\mathbb{Q}_{\ell}\bigr{)}$
$\displaystyle\xrightarrow{\,\sim\,}\operatorname{Hom}_{\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(\mathbb{Q}_{\ell})}\bigl{(}\mathbf{R}p_{!}\mathscr{F}[r],\mathbb{Q}_{\ell}\bigr{)}$
$\displaystyle\xrightarrow{\,\sim\,}\mathrm{H}_{\mathrm{c}}^{r}(X,\mathscr{F})^{\vee}\,.$

So (2) is proved. In above isomorphism, letting $r=4$ and
$\mathscr{F}=\mathbb{Q}_{\ell}$, we obtain

$\mathrm{H}_{\mathrm{c}}^{4}(X,\mathbb{Q}_{\ell})^{\vee}\xrightarrow{\,\sim\,}\mathrm{H}^{0}\bigl{(}X,\mathbb{Q}_{\ell}(2)\bigr{)}=\mathbb{Q}_{\ell}(2)\,.$

So (1) is proved. ∎

######  Corollary 3.5.

Let $X$ be a surface having only finitely many singular points, all of which
are rational singularities. Assume further that for any singular point $P$,
$\mathcal{O}_{X,P}^{\mathrm{sh}}$ is factorial. Then
$\mathrm{t}_{X}\colon\mathbb{Z}_{\ell}(2)[4]\xrightarrow{\,\sim\,}\mathscr{K}_{X}$
is an isomorphism in
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\mathbb{Z}_{\ell})$.
In particular, $\mathbb{Z}_{\ell}$ is a dualizing complex on
$X_{\textup{\'{e}t}}$ in coefficient $\mathbb{Z}_{\ell}$.

######  Remark 3.6.

By [11, (17.2)], the ring $\mathcal{O}_{X,P}^{\mathrm{sh}}$ is factorial if
and only if its completion $\widehat{\mathcal{O}}_{X,P}^{\mathrm{sh}}$ is
factorial. See [11, §25] for the complete list of two-dimensional factorial
complete local rings with rational singularities.

##  4\. Applications

In this section, we study perverse sheaves and the weights of smooth sheaves
on surfaces with rational singularities. First we briefly introduce the
t-structures on arbitrary schemes.

Let $X$ be a scheme satisfying (†) equipped with a dimension function
$\delta$. For each point $x$ on $X$, the canonical morphism
$i_{x}\colon\operatorname{Spec}k(x)\to X$ factor as

$\operatorname{Spec}k(x)\xrightarrow{j}\overline{\\{x\\}}\xrightarrow{i}X\,.$

Then we define two functors:

$\displaystyle i_{x}^{\ast}$ $\displaystyle\mathrel{\mathop{:}}=j^{\ast}\circ
i^{\ast}\colon\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})\to\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}\bigl{(}x_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell}\bigr{)}\,,$
$\displaystyle i_{x}^{!}$
$\displaystyle\mathrel{\mathop{:}}=j^{\ast}\circ\mathbf{R}i^{!}\colon\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})\to\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}\bigl{(}x_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell}\bigr{)}\,.$

Now we define a pair of full subcategories
$\bigl{(}\vphantom{\mathbf{D}}^{\mathrm{P}}\mathbf{D}_{\mathrm{c}}^{\mathrm{b},\leqslant
0}(X_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell}),\vphantom{\mathbf{D}}^{\mathrm{P}}\mathbf{D}_{\mathrm{c}}^{\mathrm{b},\geqslant
0}(X_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})\bigr{)}$ of
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$
as following: for every object $\mathscr{F}$ in
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$,

  1. (1)

$\mathscr{F}\in\vphantom{\mathbf{D}}^{\mathrm{P}}\mathbf{D}_{\mathrm{c}}^{\mathrm{b},\leqslant
0}(X_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$ if and only if for any
point $x$ on $X$,
$i_{x}^{\ast}\mathscr{F}\in\mathbf{D}_{\mathrm{c}}^{\leqslant-\delta(x)}(x_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$;

  2. (2)

$\mathscr{F}\in\vphantom{\mathbf{D}}^{\mathrm{P}}\mathbf{D}_{\mathrm{c}}^{\mathrm{b},\geqslant
0}(X_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$ if and only if for any
point $x$ on $X$,
$i_{x}^{!}\mathscr{F}\in\mathbf{D}_{\mathrm{c}}^{\geqslant-\delta(x)}(x_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$.

By [5] or [6], this pair defines a $t$-structure on
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$.
If $f\colon X\to Y$ is a quasi-finite morphism, then the functors
$\mathbf{R}f_{!}$ and $f^{\ast}$ are right t-exact, and the functors
$\mathbf{R}f^{!}$ and $\mathbf{R}f_{\ast}$ are left t-exact. Thus we may
define the functor $f_{!\ast}$.

######  Proposition 4.1.

Let $X$ be a surface having only finitely many singular points, all of which
are rational singularities; $U$ an open dense subset contained in
$X_{\mathrm{reg}}$ and $j\colon U\hookrightarrow X$ the inclusion. Let
$\mathscr{F}$ be a smooth $\overline{\mathbb{Q}}_{\ell}$-sheaf on
$X_{\textup{\'{e}t}}$. Then $\mathscr{F}[2]$ is an object in
$\mathrm{Perv}(X,\overline{\mathbb{Q}}_{\ell})$ and there is a canonical
isomorphism $j_{!\ast}\circ
j^{\ast}\mathscr{F}[2]\xrightarrow{\,\sim\,}\mathscr{F}[2]$.

###### Proof.

Obviously we may assume that $U=X_{\mathrm{reg}}$. Then
$Y\mathrel{\mathop{:}}=X\setminus U$ contains only finitely many singular
points, all of which are rational singularities of $X$. Let $P$ be a point in
$X\setminus U$. Then

$i_{P}^{\ast}\mathscr{F}[2]=\mathscr{F}_{P}[2]\in\vphantom{\mathbf{D}}^{\mathrm{P}}\mathbf{D}_{\mathrm{c}}^{\mathrm{b},\leqslant-2}(X_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})\,.$

Next we calculate $i_{P}^{!}\mathscr{F}[2]$. Put
$\overline{X}\mathrel{\mathop{:}}=\operatorname{Spec}\mathcal{O}_{X,P}^{\mathrm{sh}}$,
and let $\rho\colon\overline{X}\to X$ and $\iota\colon
P\hookrightarrow\overline{X}$ be the canonical morphisms. Since $\overline{X}$
is strictly local and $\mathscr{F}$ is a smooth sheaf,
$\rho^{\ast}\mathscr{F}\xrightarrow{\,\sim\,}\overline{\mathbb{Q}}_{\ell}^{\oplus
r}$ for some $r\in\mathbb{N}$. So we have

$\displaystyle i_{P}^{!}\mathscr{F}[2]$
$\displaystyle=\mathbf{R}\iota^{!}\circ\rho^{\ast}\mathscr{F}[2]\xrightarrow{\,\sim\,}\mathbf{R}\iota^{!}\overline{\mathbb{Q}}_{\ell}^{\oplus
r}[2]$
$\displaystyle\xrightarrow{\,\sim\,}\bigl{(}\mathbf{R}\iota^{!}\mathscr{K}_{\overline{X}}(-2)[-4]\bigr{)}^{\oplus
r}[2]$ $\displaystyle\qquad(\text{\autoref{M-A21}})$
$\displaystyle=\mathbf{R}\iota^{!}\mathscr{K}_{\overline{X}}^{\oplus
r}(-2)[-2]$ $\displaystyle\xrightarrow{\,\sim\,}\mathscr{K}_{P}^{\oplus
r}(-2)[-2]$ $\displaystyle\qquad(\text{\autoref{P-78E}})$
$\displaystyle\xrightarrow{\,\sim\,}\overline{\mathbb{Q}}_{\ell}^{\oplus
r}(-2)[-2]\,.$

Hence
$i_{P}^{!}\mathscr{F}[2]\in\vphantom{\mathbf{D}}^{\mathrm{P}}\mathbf{D}_{\mathrm{c}}^{\mathrm{b},\geqslant
2}(X_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$. This show that
$\mathscr{F}[2]\in\mathrm{Perv}(X,\overline{\mathbb{Q}}_{\ell})$.

Let $i\colon Y\hookrightarrow X$ and $j\colon U\hookrightarrow X$ be the
inclusion. Then
$i^{\ast}\mathscr{F}[2]\in\vphantom{\mathbf{D}}^{\mathrm{P}}\mathbf{D}_{\mathrm{c}}^{\mathrm{b},\leqslant-2}(Y_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$
and
$\mathbf{R}i^{!}\mathscr{F}[2]\in\vphantom{\mathbf{D}}^{\mathrm{P}}\mathbf{D}_{\mathrm{c}}^{\mathrm{b},\geqslant
2}(Y_{\textup{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$. So both
${}^{\mathrm{P}}i^{\ast}\mathscr{F}[2]$ and
${}^{\mathrm{P}}i^{!}\mathscr{F}[2]$ are zero. By section 1, there are two
long exact sequences in $\mathrm{Perv}(X,\overline{\mathbb{Q}}_{\ell})$:

$\displaystyle\cdots\to{}^{\mathrm{P}}j_{!}\circ
j^{\ast}\mathscr{F}[2]\to\mathscr{F}[2]\to
i_{\ast}\circ{}^{\mathrm{P}}i^{\ast}\mathscr{F}[2]\to\cdots\,,$
$\displaystyle\cdots\to
i_{\ast}\circ{}^{\mathrm{P}}i^{!}\mathscr{F}[2]\to\mathscr{F}[2]\to{}^{\mathrm{P}}j_{\ast}\circ
j^{\ast}\mathscr{F}[2]\to\cdots\,.$

Thus ${}^{\mathrm{P}}j_{!}\circ j^{\ast}\mathscr{F}[2]\to\mathscr{F}[2]$ is an
epimorphism and $\mathscr{F}[2]\to{}^{\mathrm{P}}j_{\ast}\circ
j^{\ast}\mathscr{F}[2]$ is a monomorphism. Therefore $j_{!\ast}\circ
j^{\ast}\mathscr{F}[2]=\mathscr{F}[2]$. ∎

Next we study the $\ell$-adic cohomology of surfaces over finite fields. Let
$p\neq\ell$ be a prime number, $q\mathrel{\mathop{:}}=p^{e}$ and
$k\mathrel{\mathop{:}}=\mathbb{F}_{q}$. If $X$ is an algebraic scheme over
$k$, we write $\overline{X}\mathrel{\mathop{:}}=X\otimes_{k}\bar{k}$; and for
each sheaf $\mathscr{F}$ (or complex of sheaves) on $X_{\textup{\'{e}t}}$, we
use $\overline{\mathscr{F}}$ to denote the inverse image of $\mathscr{F}$ on
$\overline{X}_{\textup{\'{e}t}}$.

######  Definition 4.2.

Let $w\in\mathbb{Z}$. An element
$\alpha\in\overline{\mathbb{Q}}_{\ell}^{\ast}$ is said to be _pure of weight
$w$_ if for any $\mathbb{Q}$-embedding
$\iota\colon\overline{\mathbb{Q}}_{\ell}\hookrightarrow\mathbb{C}$, we have
$|\iota(\alpha)|=q^{r/2}$.

######  Definition 4.3.

Let $X$ be an algebraic scheme over $k$.

  1. (1)

A constructible $\overline{\mathbb{Q}}_{\ell}$-sheaf $\mathscr{F}$ on $X$ is
said to be

     * •

_punctually pure of weight $n$_ if for every closed point $x$ on $X$, all
eigenvalues of the geometric Frobenius $F_{x}$ acting on
$\mathscr{F}_{\bar{x}}$ are pure of weight $w$;

     * •

_mixed_ (_of weights $\leqslant w$_), if there exists a finite filtration

$0=\mathscr{F}_{r}\subset\mathscr{F}_{r-1}\subset\cdots\subset\mathscr{F}_{1}\subset\mathscr{F}_{0}=\mathscr{F}$

consisting of constructible $\overline{\mathbb{Q}}_{\ell}$-sheaves, such that
each $\mathscr{F}_{i}/\mathscr{F}_{i+1}$ is punctually pure (of weights
$\leqslant n$).

  2. (2)

An object $\mathscr{F}$ in
$\mathbf{D}_{\mathrm{c}}^{\mathrm{b}}(X_{\textup{\'{e}t}},\widetilde{\mathbb{Q}}_{\ell})$
is said to be

     * •

_mixed of weights $\leqslant w$_ if for every $n\in\mathbb{Z}$,
$\mathrm{H}^{n}(\mathscr{F})$ is mixed of weights $\leqslant n+w$;

     * •

_mixed of weights $\geqslant w$_ if
$\mathrm{D}(\mathscr{F})=\mathbf{R}\mathscr{H}\\!om(\mathscr{F},\mathscr{K}_{X})$
is mixed of weights $\leqslant-w$;

     * •

_pure of weight $w$_ if it is mixed of weights $\leqslant w$ and $\geqslant
w$.

######  Proposition 4.4.

Let $X$ be a complete surface over $k$, $\mathscr{F}$ a smooth
$\overline{\mathbb{Q}}_{\ell}$-sheaf on $X_{\textup{\'{e}t}}$. Assume that $X$
has only finitely many singular points, all of which are rational
singularities; and $\mathscr{F}$ is punctually pure of weight $w$. Then for
every $n\in\mathbb{N}$, the $\overline{\mathbb{Q}}_{\ell}$-module
$\mathrm{H}^{n}(\overline{X}_{\textup{\'{e}t}},\overline{\mathscr{F}})$ is
punctually pure of weight $n+w$.

###### Proof.

Put $U\mathrel{\mathop{:}}=X_{\mathrm{reg}}$ and let $j\colon U\hookrightarrow
X$ be the inclusion. Since $U$ is smooth and $j^{\ast}\mathscr{F}$ is a smooth
$\overline{\mathbb{Q}}_{\ell}$-sheaf, $j^{\ast}\mathscr{F}[2]$ is pure of
weight $w+2$. By section 4,
$\mathscr{F}[2]\in\mathrm{Perv}(X,\overline{\mathbb{Q}}_{\ell})$ and
$j_{!\ast}j^{\ast}\mathscr{F}[2]\cong\mathscr{F}[2]$. So after applying [2,
Corollaire 5.4.3], we obtain that
$j_{!\ast}j^{\ast}\mathscr{F}[2]\cong\mathscr{F}[2]$ is pure of weight $w+2$.
Let $\pi\colon X\to\operatorname{Spec}k$ be the structural morphism. By [3,
(3.3.1) & (6.2.3)], the functor $\mathbf{R}\pi_{\ast}=\mathbf{R}\pi_{!}$ sends
perverse sheaves pure of weight $w+2$ on $X$ to perverse sheaves pure of
weight $w+2$ on $\operatorname{Spec}k$. Hence we prove the proposition. ∎

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