# Bridging the Gap between Model Explanations in
Partially Annotated Multi-label Classification

Youngwook Kim1 Jae Myung Kim2 Jieun Jeong1,3
Cordelia Schmid4 Zeynep Akata2,5 Jungwoo Lee1,3

1Seoul National University 2University of Tübingen 3HodooAI Lab
4Inria, Ecole normale supérieure, CNRS, PSL Research University 5MPI for
Intelligent Systems  Corresponding author.

###### Abstract

Due to the expensive costs of collecting labels in multi-label classification
datasets, partially annotated multi-label classification has become an
emerging field in computer vision. One baseline approach to this task is to
assume unobserved labels as negative labels, but this assumption induces label
noise as a form of false negative. To understand the negative impact caused by
false negative labels, we study how these labels affect the model’s
explanation. We observe that the explanation of two models, trained with full
and partial labels each, highlights similar regions but with different
scaling, where the latter tends to have lower attribution scores. Based on
these findings, we propose to boost the attribution scores of the model
trained with partial labels to make its explanation resemble that of the model
trained with full labels. Even with the conceptually simple approach, the
multi-label classification performance improves by a large margin in three
different datasets on a single positive label setting and one on a large-scale
partial label setting. Code is available at
https://github.com/youngwk/BridgeGapExplanationPAMC.

##  1 Introduction

Multi-label image classification is the task of predicting all labels
corresponding to a given image. Since web-crawled images often contain
multiple objects/concepts [44, 35, 3, 32], the importance of this task is
rising. However, it faces a significant issue of huge annotation costs. We
need C binary labels for each training image to provide exhaustive annotation
for a model that classifies images into C categories. It acts as a severe
obstacle to scaling multi-label classification datasets.

For this reason, partially annotated multi-label classification [13, 17, 24,
11, 21, 2] has recently become an actively studied topic. In this setting,
instead of exhaustive annotation, only a few categories are labeled for each
training image. We can effectively reduce the burden of annotation by adopting
partial annotation strategies.

Figure 1: CAM Observation. We compare the class activation map (CAM) output
from two multi-label classification models: one trained with full labels
($\textit{CAM}_{\textit{full}}$) and the other trained with partial labels and
AN assumption ($\textit{CAM}_{\textit{partial}}$). We observe that the overall
structure of $\textit{CAM}_{\textit{partial}}$ is not much affected by the
noisy false negative labels during training. This observation motivates us to
make $\textit{CAM}_{\textit{partial}}$ similar to
$\textit{CAM}_{\textit{full}}$ by boosting its relatively large attribution
scores. Best viewed in color.

One baseline approach for solving a partially annotated multi-label
classification task is assuming unobserved labels as negative labels (Assume
Negative, AN) [36, 4, 6, 40]. It is a reasonable assumption since most labels
are negative labels in the multi-label scenario [33]. However, this assumption
causes label noise in a form of false negatives since the actually positive
but unannotated labels are incorrectly assumed to be negative. Since this
label noise perturbs the learning process of the model [45, 7, 1, 18], recent
studies on a partially annotated multi-label classification focus on
suppressing the influence of label noise by ignoring or correcting the loss of
samples that are likely to be false negatives[21, 2].

Aside from recent research directions, we delve into “how” false negative
labels influence a multi-label classification model. We conduct control
experiments with two models. One is the model trained with partial labels and
AN assumption where false negative labels exist. The other is the model
trained with full annotations and thus trained without false negatives. We
compare the class activation map (CAM) [49] output between the two models to
see the difference in how each model understands the input image and makes a
prediction result.

Figure 1 shows that a model trained with false negatives still highlights
similar regions to one trained with full annotation. However, the attribution
scores in the highlighted areas are much small. This observation leads us to
think that if we scale up the damaged score of the highlighted region in the
model trained with false negatives, the explanation of this model will become
similar to that of the model trained with full annotation.

To this end, we introduce a simple piece-wise linear function, named BoostLU,
that bridges the gap between the explanation of two models trained with false
negatives and with full annotation each. Concretely, we use the modified CNN
model to get CAM during the forward pass directly[47], and the logit in the
modified CNN model is the mean of attribution scores of CAM. The BoostLU
function is applied element-wisely to the CAM output of the modified CNN to
boost the scores of the highlighted regions, thereby compensating for the
decrease of attribution scores in CAM caused by false negatives. It increases
the logit value for positive labels and thus makes a better prediction.
Furthermore, when we combine BoostLU with the recently proposed methods [21]
that explicitly detect and modify false negatives during training, it helps to
detect false negatives better, thus leading to better performance. As a
result, we achieve state-of-the-art performance on PASCAL VOC [14], MS COCO
[28], NUSWIDE [10], and Openimages V3 [23] datasets in a partial label
setting.

We summarize the contributions of this paper as follows.

  1. 1.

We analyze how the false negative labels affect the explanation of the model
in a partially annotated multi-label classification scenario.

  2. 2.

We propose a simple but effective function, named BoostLU, that compensates
for the damage of false negatives in a multi-label classification model with
little extra computational cost.

  3. 3.

When applied during inference, BoostLU boosts the baseline method (AN)’s test
performance without additional training.

  4. 4.

Combined with recent methods of detecting and modifying false negatives during
training, BoostLU boosts the state-of-the-art performance on single positive
and large-scale partial label settings.

##  2 Related Works

Partially annotated multi-label classification. One primary stream to solve
the partially annotated multi-label classification problem is to view
unobserved labels as missing labels. Earlier works tackled this problem by
solving matrix completion [5, 43, 15] or employing the Bayesian model [19,
37]. However, these works require loading all data into memory at once, thus
making it infeasible to train deep neural networks. Curriculum labeling [13]
proposed a bootstrapping strategy using model prediction. IMCL [17], SE [24],
and SST [8] exploited label correlation and image similarity to generate
regularization losses or pseudo-labels for missing labels. SARB [31] performed
a category-wise mixup on feature space between labeled and unlabeled images to
propagate information into missing labels. Zhou et al. [50] proposed entropy
maximization loss that suppresses gradients from missing labels to promote
learning from observed labels.

Since a significant part of labels is negative in a multi-label setting [33],
there is another stream to treat unobserved labels as negatives and try to
lessen the harmful impact of false negatives. In other words, it views
unobserved labels as noisy labels. ROLE [11] proposed to estimate unobserved
labels while simultaneously regularizing the estimation with an average number
of positive labels online. Kim et al. [21] observed the memorization effect
[1] in a noisy multi-label classification setting that the model learns from
clean labels first. Thus false negative labels are likely to show large loss
values during training. Then they suggested three methods, LL-R, LL-Ct, and
LL-Cp, that prevent false negatives from being memorized by rejecting,
temporally correcting, and permanently correcting samples with large losses,
respectively. P-ASL [2] assigned different scaling rates between annotated
negatives and assumed negatives. It also ignored losses from categories with
high prediction scores or label prior values. In this work, we look at false
negatives differently and study their effect on model explanation.

Class activation mapping. Class activation mapping (CAM) [49] provides
information about where the classification model is attending to generate
prediction scores. There are several follow-up works, including Grad-CAM [34],
which generates model-agnostic attention maps, and CALM [20], which
strengthens the interpretability of attention maps.

Since CAM provides localization ability to classification models, it has been
widely used for various vision tasks, such as weakly supervised object
localization [42, 41, 12, 9] and weakly supervised semantic segmentation [42,
29, 26, 46, 25]. Recently, Zhang et al. [48] utilized CAM in facial expression
recognition in the presence of noisy labels. They found that the model trained
with noisy labels highlights only part of the features and suggested a random
masking strategy to prevent memorizing partial features. Although there is a
similarity in that they inspected the CAM output of the model in noisy label
situations, our work is different since we focus on the noisy multi-label
classification setting with another type of noise.

##  3 Preliminary

(a) Similarity of CAM by Spearman correlation

(b) Top-ranking attribution score

(c) Bottom-ranking attribution score

Figure 2: CAM Analysis on COCO test set. (a): Distribution of Spearman
correlation coefficients between $\textit{CAM}_{\textit{full}}$ and
$\textit{CAM}_{\textit{partial}}$ from the same image. Overall positive
correlation implies that $\textit{CAM}_{\textit{partial}}$ has a structure
similar to $\textit{CAM}_{\textit{full}}$. / (b), (c): Boxplot of the average
of top/bottom 5% of attribution scores, respectively. The damage of false
negative labels to the model mainly lowers the upper attribution scores for
positive labels while maintaining its overall structure in CAM.

This section introduces the formal definition of a partially annotated multi-
label classification in §3.1. Next, we briefly summarize the class activation
map (CAM) in §3.2.

###  3.1 Problem Definition

We aim to train a multi-label classification model with dataset
${\mathcal{D}}$ consisting of pairs of input image ${\bm{x}}$ and partially
annotated label ${\bm{y}}$. Each category can have three kinds of labels: 0,
1, and $\phi$. In other words, ${\bm{y}}\in{\mathcal{Y}}=\\{0,1,\phi\\}^{C}$
where $\phi$ indicates the absence of annotation and $C$ is the number of
total categories. Denote the index set of positive labels, negative labels,
and unannotated labels as ${\mathcal{I}}^{\,p}$, ${\mathcal{I}}^{\,n}$, and
${\mathcal{I}}^{\,\phi}$, respectively. We study the setting where labels are
sparsely annotated, i.e.,
$|{\mathcal{I}}^{\,p}|+|{\mathcal{I}}^{\,n}|\ll|{\mathcal{I}}^{\,\phi}|$.

A straightforward approach to train the model given partial labels is to treat
unannotated labels by assuming negative (AN) and use binary cross-entropy as a
loss function:

$\mathcal{L}_{AN}=\frac{1}{C}\left[\,\,\sum_{i\,\in\,{\mathcal{I}}^{\,p}}\,\mathcal{L}_{+}+\sum_{i\,\in\,{\mathcal{I}}^{\,n}\cup\,{\mathcal{I}}^{\,\phi}}\,\mathcal{L}_{-}\,\right]$
(1)

where $\mathcal{L}_{+}=-\log(\sigma(g_{i}))$,
$\mathcal{L}_{-}=-\log(1\\!-\\!\sigma(g_{i}))$ and $g_{i}$ is a logit for
$i$-th category. However, labels whose true label is positive but unannotated
are incorrectly assumed to be negative and become false negatives. Denote the
index set of true negative and false negative labels as ${\mathcal{I}}^{\,tn}$
and ${\mathcal{I}}^{\,fn}$, then
${\mathcal{I}}^{\,n}\,\cup\,{\mathcal{I}}^{\,\phi}={\mathcal{I}}^{\,tn}\,\cup\,{\mathcal{I}}^{\,fn}$.
We set the approach of training the model with Equation (1) as the baseline
method and investigate the influence of false negatives on the multi-label
classification model.

###  3.2 Recap CAM

Most CNN architectures consist of several convolution layers (Convs), followed
by a Global Average Pooling (GAP) layer [27] and a fully connected (FC) layer.
Let the last convolutional feature map be ${\bm{F}}\in\mathbb{R}^{H\times
W\times D}$, and a weight matrix of the FC layer be
${\bm{W}}\in\mathbb{R}^{C\times D}$ where $(H,W)$ and $D$ are the spatial size
and channel size of the feature map, respectively. We can obtain the class
activation map (CAM) [49] for class c (${\bm{M}}_{c}$) by

${\bm{M}}_{c}=\sum_{d=1}^{D}{\bm{W}}_{cd}{\bm{F}}_{d}\,\,,$ (2)

where ${\bm{F}}_{d}$ denotes $d$-th channel of ${\bm{F}}$. ${\bm{M}}_{c}$
explains the model’s prediction by attributing scores on each pixel.

Instead of performing post-processing to get CAM as in Equation (2), we can
directly get CAM during the forward pass by reordering the last two layers
from Convs-GAP-FC to Convs-1x1Conv-GAP where 1x1Conv is the one-by-one
convolutional layer with the weight ${\bm{W}}$ [47]. The output feature maps
of 1x1Conv become the same as ${\bm{M}}$, and the logit $g_{c}$ becomes

$g_{c}=\frac{1}{HW}\sum_{i=1}^{H}\sum_{j=1}^{W}({\bm{M}}_{c})_{ij}\,\,.$ (3)

Thus, we can interpret each element $({\bm{M}}_{c})_{ij}$ as an attribution
score at spatial location $(i,j)$ contributing to the logit for class $c$. For
the following sections, we utilize this modified architecture to facilitate
the application of our method.

##  4 Impact of False Negatives on CAM

It is well known that neural networks can memorize wrong labels due to their
large model capacities [45]. Likewise, if we train a multi-label
classification model with AN loss (Equation (1)) when given partial labels,
the model is damaged by memorizing false negative labels [21]. It results in
poor performance compared to the model trained with full labels, which false
negatives have not influenced.

To better understand why the model trained with partial labels performs less
than that with full labels, we analyze the behavioral difference between these
two models. Concretely, we use a class activation map (CAM) [49] to explain
each model’s prediction and compare the explanation results. We train two
multi-label classification models on a COCO dataset [28] with the same CNN
architecture ResNet-50 [16]: one model with full labels using binary cross
entropy loss and the other with partial labels using AN loss (Equation (1)).
We denote the CAM output from each model as $\textit{CAM}_{\textit{full}}$ and
$\textit{CAM}_{\textit{partial}}$, respectively.

To analyze the explanation of these two models, we first compute the Spearman
correlation between $\textit{CAM}_{\textit{full}}$ and
$\textit{CAM}_{\textit{partial}}$ on positive labels. We show the distribution
of the correlation values on the test set in Figure 2(a). For comparison, we
consider a 2D Gaussian image centered at the midpoint and calculate the
Spearman correlation coefficient between this Gaussian image and
$\textit{CAM}_{\textit{full}}$. We observe that there is mainly a positive
correlation between $\textit{CAM}_{\textit{full}}$ and
$\textit{CAM}_{\textit{partial}}$, while the correlation of the control group
is distributed widely but mostly around zero. It implies that the overall
structure (i.e., the attribution ranking among pixels) of
$\textit{CAM}_{\textit{partial}}$ is preserved despite the influence of false
negative labels, therefore having a high Spearman correlation with
$\textit{CAM}_{\textit{full}}$. We can also visually inspect the similar
structure between $\textit{CAM}_{\textit{partial}}$ and
$\textit{CAM}_{\textit{full}}$ in Figure 1, where both CAMs highlight similar
regions.

Since we know that the overall structure is similar between
$\textit{CAM}_{\textit{full}}$ and $\textit{CAM}_{\textit{partial}}$, we then
compare the range of attribution scores between $\textit{CAM}_{\textit{full}}$
and $\textit{CAM}_{\textit{partial}}$. Concretely, we compute the mean of the
highest 5% of attribution scores and the lowest 5%, respectively, for each CAM
and summarize the distribution of these values on the test set in Figure 2(b)
and 2(c). Note that we take an average of 5% of scores to reduce the effect of
outliers. We observe that top-ranking attribution scores of
$\textit{CAM}_{\textit{partial}}$ from positive labels drop sharply compared
to $\textit{CAM}_{\textit{full}}$, while these scores from negative labels
remain similar. Also, there is little difference in bottom-ranking attribution
scores between $\textit{CAM}_{\textit{full}}$ and
$\textit{CAM}_{\textit{partial}}$, both on positive and negative labels. It
implies that false negatives mainly affect the model’s understanding in
regions with relatively high attribution scores, especially for positive
labels. Consequently, the decrease of attribution scores at specific regions
in CAM leads to a decrease in the logit value (since logit is the average of
attribution scores on CAM as in Equation (3)), making the model predicts a
lower score for the positive category. The change of gradient during training
can explain the occurrence of this phenomenon.

Gradient analysis. In Equation (1), recall that the BCE loss is
$\mathcal{L}_{+}$ with a positive target and $\mathcal{L}_{-}$ with a negative
one. Their gradients with respect to the logit $g$ are

$\frac{\partial\mathcal{L}_{+}}{\partial
g}=\sigma(g)-1,\,\,\frac{\partial\mathcal{L}_{-}}{\partial g}=\sigma(g)\,\,.$
(4)

For a training image ${\bm{x}}$, the gradient difference on the logit $g$
between partial label (with AN assumption) and full label case is given by

$\displaystyle\frac{1}{C}\left[\,\sum_{i\,\in\,{\mathcal{I}}^{\,p}}\frac{\partial\mathcal{L}_{+}}{\partial
g_{i}}+\sum_{i\,\in\,{\mathcal{I}}^{\,fn}}\frac{\partial\mathcal{L}_{-}}{\partial
g_{i}}+\sum_{i\,\in\,{\mathcal{I}}^{\,tn}}\frac{\partial\mathcal{L}_{-}}{\partial
g_{i}}\right]$ (5)
$\displaystyle-\frac{1}{C}\left[\,\sum_{i\,\in\,{\mathcal{I}}^{\,p}}\frac{\partial\mathcal{L}_{+}}{\partial
g_{i}}+\sum_{i\,\in\,{\mathcal{I}}^{\,fn}}\frac{\partial\mathcal{L}_{+}}{\partial
g_{i}}+\sum_{i\,\in\,{\mathcal{I}}^{\,tn}}\frac{\partial\mathcal{L}_{-}}{\partial
g_{i}}\right]$
$\displaystyle=\frac{1}{C}\left[\,\sum_{i\,\in\,{\mathcal{I}}^{\,fn}}(\frac{\partial\mathcal{L}_{-}}{\partial
g_{i}}-\frac{\partial\mathcal{L}_{+}}{\partial
g_{i}})\right]\,=\,\,\frac{|{\mathcal{I}}^{\,fn}|}{C}\,\,.$

Equation (5) shows that the logit receives more gradients proportional to the
number of false negative labels on a partial label setting. Therefore, as
training progresses, the additional gradients from false negatives are
gradually accumulated in the logit, making the logit smaller than the model
trained on full labels. Since the logit is equal to the average of CAM, the
attribution scores of $\textit{CAM}_{\textit{partial}}$ become lower than that
of $\textit{CAM}_{\textit{full}}$.

##  5 Proposed Method

In this section, we propose a conceptually simple but effective method to make
the model trained with partial labels resemble the model trained with full
labels by mimicking the explanation. We propose a function BoostLU devised to
compensate for the damaged attribution score of the explanation due to false
negatives in §5.1. We then introduce three scenarios that utilize our function
through §5.2 $\sim$ §5.4.

###  5.1 BoostLU

From the modified CNN architecture described in §3.2, define convolutional
layers (Convs-1x1Conv) as $\Phi$. Given an input image ${\bm{x}}$, its class
activation map (CAM) is ${\bm{M}}=\Phi({\bm{x}})$. Our goal is to make the
explanation ${\bm{M}}$ of the model trained with partial labels closer to the
explanation of the model trained with full labels, even if we do not have
access to the full labels, thus improving the prediction performance.

In the previous section, we observe that when a multi-label classification
model is trained with AN loss, the way the model understands images is damaged
by false negatives. However, we also find that this damage is mainly focused
on a drop in the relatively high attribution scores while the overall spatial
structure of CAM is preserved. Based on these findings, we conjecture that if
the damaged high attribution scores are scaled up in the model trained with
partial labels, $\textit{CAM}_{\textit{partial}}$ will become similar to
$\textit{CAM}_{\textit{full}}$. To achieve this, we devise a piece-wise linear
function that boosts the attribution scores that are above a certain
threshold:

$f(x)=\begin{cases}\alpha x+(1-\alpha)\beta,&x\geq\beta\\\
x,&x<\beta\,\,,\end{cases}$ (6)

where $\alpha$ is a scaling factor with $\alpha>1$, and $\beta$ is a threshold
determining whether to boost the score. Since top-ranking attribution scores
on CAM tend to have large positive values for positive labels and around zero
for negative labels (as seen in Figure 2(b)), we search for the values of
$\beta$ around zero. Since we empirically observe no significant difference in
model performance for different $\beta$ (these results are reported in the
Appendix), we only consider the simplest case of $\beta=0$. Then we can
rewrite Equation (6) in a ReLU-like form as

$\text{$\text{BoostLU}$}(x)=max(x,\alpha x)\,\,.$ (7)

By applying BoostLU to each element of CAM, as illustrated in Figure 3,
BoostLU boosts positive attribution scores by $\alpha$ times, which are the
main target to be damaged by false negatives, while maintaining the negative
scores unchanged. These selectively boosted attribution scores are aggregated
through the GAP layer to produce a logit value as

Figure 3: Schematic diagram of applying BoostLU. BoostLU is applied to the
model’s CAM output element-wisely to compensate for the attribution scores
damaged by false negative labels.

$g({\bm{x}})=(\text{GAP}\circ\text{BoostLU}\circ\Phi)({\bm{x}})\,\,.$ (8)

From now on, we will consider three different scenarios for applying BoostLU
in multi-label classification.

###  5.2 Usage 1: BoostLU in inference only

Since the idea of BoostLU comes from analyzing the CAM of a model which
finished training with AN loss, we first propose to apply BoostLU only during
the inference phase of that model. Initially, this model produces low logits
for categories whose label is positive. However, applying BoostLU increases
the corrupted attribution scores and produces higher logits. At the same time,
boosting effect is not much for categories whose label is negative; therefore,
its logits remain almost the same. As a result, prediction scores are better
separated between samples with positive and negative labels, improving average
precision.

###  5.3 Usage 2: BoostLU in both training and inference

Next, we consider applying BoostLU during the training phase with AN loss and
the inference phase. The gradient of logit $g$ with respect to the attribution
score on CAM at location $(i,j)$ (i.e., ${\bm{M}}_{ij}$) then becomes

$\frac{\partial
g}{\partial{\bm{M}}_{ij}}=\begin{cases}\alpha/HW,&{\bm{M}}_{ij}\geq 0\\\
1/HW,&{\bm{M}}_{ij}<0\,\,.\end{cases}$ (9)

Compared to the case that does not use BoostLU, where every spatial location
gets a uniform gradient of $1/HW$, the locations with positive attribution
scores receive gradients boosted by $\alpha$ times. Thanks to the boosted
gradients, these locations are encouraged to produce higher attribution scores
during training when the model receives a positive label. Also, when a true
negative label comes in, these locations are encouraged to produce lower
attribution scores.

However, in practice, we observe only marginal improvement in model
performance. It is because the boosted gradients have an adverse effect when
false negatives come in as input. That is, BoostLU also boosts the wrong
direction of gradients from false negatives, which can be easily seen by
combining Equation (4) and (9):

$\frac{\partial{\mathcal{L}_{-}}}{\partial{{\bm{M}}_{ij}}}=\frac{\partial{\mathcal{L}_{-}}}{\partial{g}}\cdot\frac{\partial{g}}{\partial{{\bm{M}}_{ij}}}\,\,$
(10)

Note that $\partial{\mathcal{L}_{-}}/\partial{g}$ has a wrong sign for false
negatives, and it decreases CAM values.

###  5.4 Usage 3: Combination with Large Loss Modification

To alleviate the problem mentioned above, we propose combining our BoostLU
with recent studies [21, 2] that detect and treat suspicious false negatives
while training multi-label classification models. We especially adopt three
methods, i.e., LL-R, LL-Ct, and LL-Cp [21], since they work on several partial
label settings. When these methods are combined with BoostLU, they suppress
the side effects caused by false negatives. As a result, the model can take
full advantage of the boosted gradients from the positive labels during
training. Moreover, because these combined methods consider samples with
relatively high prediction scores among unobserved labels as false negatives,
BoostLU helps the model detect more false negatives by boosting their logit
values.

##  6 Experiments

To validate the efficacy of our proposed method, we report our experimental
results on two partial label settings: single positive label (§6.1) and large-
scale partial label (§6.2). In both sections, we adopt mean Average Precision
(mAP) as an evaluation metric and report the performance on a test set using
the model weight with the highest mAP in the validation set. We fix our
hyperparameters as $\alpha=5,\,\beta=0$. Next, we present analysis results on
§6.3.

###  6.1 Single positive label

Datasets. We target four multi-label classification datasets: PASCAL VOC 2012
[14], MS COCO 2014 [28], NUSWIDE [10], and CUB [38]. Each dataset is annotated
for 20 classes, 80 classes, 81 concepts, and 312 attributes. Since they are
fully annotated, we only keep one positive label and drop the rest of the
labels for every training image to build a single positive label setting
identical to [11].

Hyperparameter settings. For a fair comparison, we set the same search space
as [11, 50]: $\\{8,16\\}$ for batch size and
$\\{10^{-2},10^{-3},10^{-4},10^{-5}\\}$ for learning rate. We train the model
for 10 epochs with Adam optimizer [22]. LL-R, LL-Ct, and LL-Cp [21] have a
hyperparameter $\Delta_{rel}$ that controls the slope of increase in the
modification rate. We set $\Delta_{rel}=0.5$ for LL-R, $0.2$ for LL-Ct, and
$0.1$ for LL-Cp, respectively. We set a 10x learning rate for the last 1x1Conv
layer.

Implementation details. We also follow identical configurations as [11, 21,
50]. Specifically, 20% of the original training set is used for validation.
ResNet-50 [16] CNN backbone pre-trained on ImageNet [44] is used as a feature
extractor. Each image is resized to 448x448 before being fed to CNN, and only
random horizontal flipping is used for data augmentation during training. Note
that some categories do not have positive labels in the CUB dataset on a
generated single positive label setting. In these categories, we do not apply
BoostLU when training as the benefit from the boosted gradient becomes
weakened.

BoostLU | BoostLU | LL-R | Performance
---|---|---|---
in inference | in training | in training | VOC | COCO
|  |  | 86.10 | 64.58
✓ |  |  | 87.31 | 66.27
✓ | ✓ |  | 86.73 | 65.33
|  | ✓ | 88.24 | 70.60
| ✓ | ✓ | 87.18 | 68.45
✓ |  | ✓ | 88.90 | 70.87
✓ | ✓ | ✓ | 89.27 | 72.82
Table 1: Ablation study on BoostLU and LL-R. We test seven combinations of using BoostLU and LL-R [21] on VOC and COCO datasets. Training a model with both LL-R and BoostLU and applying BoostLU during inference shows the best mAP.  Methods | VOC | COCO | NUS | CUB
---|---|---|---|---
Full Label | 89.42 | 76.78 | 52.08 | 30.90
AN | 85.89 | 64.92 | 42.27 | 18.31
LS [30] | 87.90 | 67.15 | 43.77 | 16.26
ASL [33] | 87.76 | 68.78 | 46.93 | 18.81
ROLE [11] | 87.77 | 67.04 | 41.63 | 13.66
ROLE + LI [11] | 88.26 | 69.12 | 45.98 | 14.86
EM [50] | 89.09 | 70.70 | 47.15 | 20.85
EM + APL [50] | 89.19 | 70.87 | 47.59 | 21.84
LL-R [21] | 88.27 | 70.70 | 48.76 | 19.56
\+ BoostLU (Ours) | 89.29 | 72.89 | 49.59 | 19.80
LL-Ct [21] | 87.79 | 70.29 | 48.08 | 19.06
\+ BoostLU (Ours) | 88.61 | 71.78 | 48.37 | 19.25
LL-Cp [21] | 87.44 | 70.27 | 47.92 | 19.21
\+ BoostLU (Ours) | 87.81 | 71.41 | 48.61 | 19.34
Table 2: Experimental results on various datasets with single positive label setting. Each number shows the average of mAP in three experiments. A bold number means the best performance. Results of methods except for LL-R, LL-Ct, and LL-Cp are taken from [50]. We report the reimplemented results for LL-R, LL-Ct, and LL-Cp with the same hyperparameter search space as [50, 11].  Methods | Group 1 | Group 2 | Group 3 | Group 4 | Group 5 | All Classes
---|---|---|---|---|---|---
CNN-RNN [39] | 68.76 | 69.70 | 74.18 | 78.52 | 84.61 | 75.16
Curriculum Labeling [13] | 70.37 | 71.32 | 76.23 | 80.54 | 86.81 | 77.05
IMCL [17] | 70.95 | 72.59 | 77.64 | 81.83 | 87.34 | 78.07
P-ASL [2] | 73.19 | 78.61 | 85.11 | 87.70 | 90.61 | 83.03
LL-R [21] | 77.76 | 79.07 | 81.94 | 84.51 | 89.36 | 82.53
\+ BoostLU (Ours) | 79.28 | 80.81 | 83.32 | 85.63 | 90.27 | 83.86
LL-Ct [21] | 77.76 | 79.18 | 81.97 | 84.46 | 89.51 | 82.58
\+ BoostLU (Ours) | 79.43 | 80.75 | 83.41 | 85.70 | 90.41 | 83.94
LL-Cp [21] | 77.49 | 79.22 | 81.89 | 84.51 | 89.18 | 82.46
\+ BoostLU (Ours) | 79.53 | 81.04 | 83.40 | 85.85 | 90.39 | 84.04
Table 3: Experimental results on a OpenImages V3 dataset. Each group includes
1,000 classes without overlapping. Group 1 has the smallest annotations, and
Group 5 has the most. The number of annotations increases as the group number
increases. LL-R, LL-Ct, and LL-Cp are reimplemented and the other results are
borrowed from [2]. A bold number shows the best performance.

Results of ablation study. We first conduct ablation studies on PASCAL VOC and
COCO datasets. Its results are reported in Table 1. First, we show the
performance of the model trained with AN loss in the first row. In the second
row, it can be seen that when BoostLU is applied during inference of this
model, its test performance is improved even without additional training. It
confirms the property of BoostLU that compensates for the damaged attribution
score. However, if we further apply BoostLU while training (third row), the
performance improvement is lower than when BoostLU is applied only during
inference. We can observe the side effect of BoostLU that the gradient
received by the region with a positive attribution score is boosted even for
false negative labels.

In the fourth row, we show the performance of LL-R, which rejects large losses
during training. We then train the model by applying both LL-R and BoostLU
during training and BoostLU during inference. Its performance is reported in
the final row, and its improvement is much more significant than the case
where LL-R is not applied (+0.63 v.s. +1.03 on PASCAL, and +0.75 v.s. +2.22 on
COCO). Thanks to LL-R filtering out false negatives, the side effect of the
boosted gradient becomes minimized. Moreover, since our BoostLU helps LL-R
detect false negatives, its advantage is further amplified. We also find in
the last three rows that when we combine BoostLU with LL-R, applying BoostLU
either during training or during inference results in a performance drop
compared to applying it during both phases. This shows that BoostLU plays a
vital role both in training and inference, together with large loss
modification methods. In particular, it is crucial to apply BoostLU during
inference to achieve high performance. Additional discussion about this is
described in the Appendix.

From now on, we will only report the experimental results using the
configuration of the last row (BoostLU in inference + BoostLU in training +
LL-R in training).

Comparison with prior arts. We compare our results with recent state-of-the-
art: Label Smoothing (LS) [30], Asymmetric loss (ASL) [33], ROLE (with
LinearInit) [11], and Entropy-maximization loss (EM) with Asymmetric Pseudo-
Labeling (APL) [50]. We train the network three times and report the average
test performance.

The results are shown in Table 2. We find that applying BoostLU in both
training and inference consistently improves the performance of LL-R, LL-Ct,
and LL-Cp in all datasets, only with little additional computational cost. It
achieves +1.02, +0.82, and +0.37 mAP improvement in VOC, as well as +2.19,
+1.49, and +1.14 mAP improvement in COCO, respectively. Especially the
performance of LL-R + BoostLU shows the most significant increase, achieving
state-of-the-art performance and reaching closest to the full label
performance on VOC, COCO, and NUSWIDE. It also surpasses the previous state-
of-the-art method EM+APL which does not use AN assumption on these datasets.
However, the performance improvement is not that large in CUB. Since CUB has
an annotation for attributes, the number of false negative labels is much
higher, and this may increase the side effect of BoostLU when applied during
training.

###  6.2 Large-scale partial label

Dataset. We target a partially annotated OpenImages V3 [23] dataset which
consists of 3.4M training images, 41,620 validation images, and 125,436 test
images with 5,000 trainable classes (having more than 30 human-verified
samples in the training set and 5 in the valid or test sets). We sort these
classes in ascending order by the number of annotations in the training set
and divide them into five groups of equal size 1,000. We report the mAP score
averaged within each group and the entire 5,000 classes.

Implementation details. We use ImageNet [44] pre-trained ResNet-101 [16] as a
feature extractor, the same as prior works. We follow [21] to set the learning
rate as $2\times 10^{-5}$ and batch size as 288. We train the model for 20
epochs and set $\Delta_{rel}=0.005$. We resize every image to 224x224
resolution and perform a random horizontal flip during training. We set a 10x
learning rate for the last 1x1Conv layer.

Results. We compare our results with prior works: CNN-RNN [39], Curriculum
Labeling [13], IMCL [17], and P-ASL [2]. As shown in Table 3, BoostLU also
works well in a real partial label scenario. Combined with LL-R, LL-Ct, and
LL-Cp, it boosts their performance by a large margin: improvement of +1.33,
+1.36, and +1.58 mAP, respectively. All of the combined methods surpass other
previous methods and achieve state-of-the-art performance. In particular, LL-
Cp + BoostLU shows the highest 84.04 mAP.

Figure 4: Qualitative results. Categories and their corresponding prediction
scores are displayed above the images and CAM results. LL-R + BoostLU is the
closest to the explanation and prediction score of the model trained with full
labels.

###  6.3 Analysis

Qualitative results. Figure 4 visualizes the CAM results from four different
methods. The category corresponding to the CAM is shown above the image. The
prediction score, obtained by averaging attribution scores on CAM and applying
sigmoid activation, is shown above each CAM. First, column (c) shows that a
model trained with AN loss gives low prediction scores due to the damage of
false negatives. Although this model highlights similar regions for a given
input image, the attribution scores of the corresponding regions are
considerably shrunk compared to column (b).

When we perform inference by attaching BoostLU to this model, it can be seen
in column (d) that BoostLU successfully recovers the model’s explanation,
yielding high prediction scores. For LL-R + BoostLU in column (e), its model
explanation is further improved due to the role of LL-R and BoostLU during
training which further accelerates the improvement of the attribution score of
the highlighted region. It is the most similar to the explanation of the model
trained with full annotation (column (b)) compared to other methods.

Synergy effect of BoostLU and large loss modification methods during training.
We train LL-R and LL-R + BoostLU on the COCO dataset with the same
$\Delta_{rel}=0.5$ to make both models reject the same number of samples
during training. We then inspect how many of the rejected samples are false
negative labels. Figure 5 shows the number of false negative labels rejected
by each model per epoch. It can be seen that after the warmup phase (first
epoch), LL-R + BoostLU rejects more false negatives than LL-R in every epoch.
It is because BoostLU boosts the logit value of false negative samples, thus
boosting the large loss modification methods’ ability to detect false
negatives. At the same time, it also reduces the number of true negative
samples that the model incorrectly rejects, further contributing to
performance improvement.

Figure 5: Comparison of the number of rejected false negative labels. BoostLU
helps LL-R detect more false negative labels in every epoch.

##  7 Conclusion

In this paper, we studied the effect of false negative labels on model
explanation when assuming unobserved labels as negatives in a partially
annotated multi-label classification situation. We found that the overall
spatial shape of the explanation tends to be preserved, but the scale of
attribution scores is significantly affected. Based on these findings, we
proposed a conceptually simple piece-wise linear function BoostLU that
compensates for the damaged attribution scores. Through several experiments,
we confirmed that BoostLU successfully contributed to bridging the explanation
of the model closer to the explanation of the model trained with full labels.
Furthermore, combined with large loss modification methods, it achieved state-
of-the-art performance on several multi-label datasets.

Acknowledgements. Youngwook Kim thanks Junghyun Lee and Youngmin Ro for their
valuable help. Jae Myung Kim thanks the European Laboratory for Learning and
Intelligent Systems (ELLIS) PhD program and the International Max Planck
Research School for Intelligent Systems (IMPRS-IS) for support. This work is
in part supported by National Research Foundation of Korea (NRF,
2021R1A4A1030898(10%)), Institute of Information & communications Technology
Planning & Evaluation (IITP, 2021-0-00106 (50%), 2021-0-01059 (20%),
2021-0-00180 (20%)) grant funded by the Ministry of Science and ICT (MSIT),
INMAC, and BK21-plus. This work is also supported by DFG project number
276693517, by BMBF FKZ: 01IS18039A, by the ERC (853489 - DEXIM), and by EXC
number 2064/1 – project number 390727645.

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On flat pullbacks for Chow groups

Nitin Nitsure

Retired Professor, School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400 005, India. email<EMAIL_ADDRESS>
###### Abstract

It is a fundamental property of the Chow groups of algebraic schemes that they
are contra-functorial with respect to flat morphisms between schemes. While
the pullback homomorphism is easy to define at the level of algebraic cycles,
the crucial step is to show that the pullback of cycles preserves rational
equivalence, so that it descends to the Chow groups. The purpose of this note
is to give a natural sheaf theoretic proof of the preservation of rational
equivalence under flat pullback on cycles.

Key words: Algebraic cycles, flat morphisms, rational equivalence, Chow
groups.

Math. Subj. Class. MSC2010 : 14C15 (Primary), 14C25, 14A15 (Secondary).

1\. Introduction

If $f:X\to Y$ is a flat morphism of algebraic schemes over a base field $k$,
such that $f$ is of constant relative dimension $d$, then it induces a graded
homomorphism $f^{*}:Z_{*}(Y)\to Z_{*}(X)$ of degree $d$ between the respective
$\mathbb{Z}$-graded groups $Z_{*}(Y)$ and $Z_{*}(X)$ of algebraic cycles,
defined by sending the generator $[V]\in Z_{n}(Y)$ corresponding to an
$n$-dimensional closed subvariety $V\subset Y$ to the fundamental cycle
$[f^{-1}V]\in Z_{n+d}(X)$ of the closed subscheme $f^{-1}V\subset X$, which is
of pure dimension $n+d$ if non-empty. These pullback homomorphisms can be
easily seen to satisfy (i) $\mathop{\rm id}\nolimits_{X}^{*}$ is the identity
on $Z_{*}(X)$, and (ii) if $f:X\to Y$ and $g:Y\to Z$ are flat morphisms of
constant relative dimensions then $(g\circ f)^{*}=f^{*}\circ g^{*}$. This
makes $Z_{*}$ a contravariant functor from the category of algebraic schemes
and flat morphisms of constant relative dimensions to the category of graded
abelian groups and graded homomorphisms. What is more difficult is to show
that if $f:X\to Y$ is a flat morphism of constant relative dimension $d$, then
$f^{*}:Z_{*}(Y)\to Z_{*}(X)$ maps the graded subgroup $Rat_{*}(Y)\subset
Z_{*}(Y)$ formed by algebraic cycles on $Y$ that are rationally equivalent to
$0$, into the corresponding graded subgroup $Rat_{*}(X)\subset Z_{*}(X)$,
thereby inducing a graded homomorphism $f^{*}:A_{*}(Y)\to A_{*}(X)$ of degree
$d$ between the respective graded Chow groups, which are the quotients
$A_{*}(Y)=Z_{*}(Y)/Rat_{*}(Y)$ and $A_{*}(X)=Z_{*}(X)/Rat_{*}(X)$. This is the
flat pullback for Chow groups in the title of this note.

The proof of the preservation of rational equivalence by flat morphisms, given
by Fulton in Chapter 1 of [3], is based on three main ingredients. The first
ingredient is the fundamental theorem of Chevalley [1] that the push-forward
of algebraic cycles by proper morphisms preserves rational equivalence. The
second ingredient is an alternative description of rational equivalence of
algebraic cycles on a variety $V$ by representing a rational function on $V$
via a flat morphism $\tilde{V}\to{\mathbb{P}}^{1}$ on a suitable variety
$\tilde{V}$ that is proper birational over $V$, and then taking the push-
forward on $V$ of the pullback on $\tilde{V}$ of the algebraic cycle
$[P_{0}]-[P_{\infty}]$ on ${\mathbb{P}}^{1}$. The third ingredient is a lemma
on how the push-forwards of algebraic cycles by proper morphisms commute with
the pullbacks of algebraic cycles by flat morphisms of constant relative
dimensions on algebraic cycles, in a cartesian square.

The purpose of this note is to give a more ‘natural’ proof of the preservation
of rational equivalence by flat morphisms, by using the sheaf theoretic
properties of algebraic cycles and of invertible meromorphic functions. The
sheaf of meromorphic functions ${\mathcal{K}}_{X}$ on a scheme $X$ is a
somewhat neglected topic in most textbooks of algebraic geometry, hence we
recall some necessary basics in Section 2 (the experienced reader can skip
this section). In Section 3, we introduce the site ${\mathcal{C}}$ of
algebraic schemes and flat morphisms of constant relative dimensions, equipped
with the big Zariski topology. Invertible meromorphic functions, and algebraic
cycles, form sheaves ${\mathcal{K}}^{\times}$ and ${\mathcal{Z}}_{*}$ on
${\mathcal{C}}$. In Section 4, we introduce the sheaf ${\mathcal{W}}$ of Weil
divisors on the full sub site ${\mathcal{C}}^{pure}$ of ${\mathcal{C}}$
consisting of pure dimensional schemes, and a homomorphism
$\Phi:{\mathcal{K}}^{\times}\to{\mathcal{W}}$ over ${\mathcal{C}}^{pure}$,
using which we complete the proof of the preservation of rational equivalence.

2\. Sheaves of meromorphic functions

For any ring $R$, let $S(R)\subset R$ denote set of all non-zero divisors, and
let $Q(R)$ denote the total quotient ring $R[S(R)^{-1}]$ of $R$. For any
locally ringed space $(X,{\mathcal{O}}_{X})$ there is a subsheaf of sets
${\mathcal{S}}_{X}\subset{\mathcal{O}}_{X}$ whose sections over an open subset
$U\subset X$ are all $f\in{\mathcal{O}}_{X}(U)$ such that for each $x\in U$
the germ $f_{x}$ lies in $S({\mathcal{O}}_{X,x})$ (in particular,
${\mathcal{S}}_{X}(U)\subset S({\mathcal{O}}_{X}(U))$, but the inclusion can
sometimes be proper). The sheaf ${\mathcal{K}}_{X}$ of meromorphic functions
on $X$ is the sheaf of ${\mathcal{O}}_{X}$-algebras which is the localization
${\mathcal{K}}_{X}={\mathcal{O}}_{X}[{\mathcal{S}}_{X}^{-1}]$ of
${\mathcal{O}}_{X}$ obtained by inverting ${\mathcal{S}}_{X}$ (see EGA IV.20
[2]). If $X$ is a locally noetherian scheme, then the sections of
${\mathcal{S}}_{X}$ and ${\mathcal{K}}_{X}$ over any affine open subscheme
$U=\mathop{\rm Spec}\nolimits A\subset X$, and the stalks
${\mathcal{S}}_{X,x}$ and ${\mathcal{K}}_{X,x}$ at any
$x=\mathfrak{p}\in\mathop{\rm Spec}\nolimits A$, have the following direct
algebraic description (see Mumford [5] Chapter 9, Section 1):
${\mathcal{S}}_{X}(U)=S(A)$,
${\mathcal{S}}_{X,x}=S({\mathcal{O}}_{X,x})=S(A_{\mathfrak{p}})$,
${\mathcal{K}}_{X}(U)=Q(A)$ and
${\mathcal{K}}_{X,x}=Q({\mathcal{O}}_{X,x})=Q(A_{\mathfrak{p}})$. If
$\mathop{\rm Spec}\nolimits B\subset\mathop{\rm Spec}\nolimits A$ are affine
open subschemes of $X$, then the corresponding ring homomorphism $A\to B$ is
flat, so it maps $S(A)$ into $S(B)$ and induces a homomorphism $Q(A)\to Q(B)$.
These are the restriction maps from $\mathop{\rm Spec}\nolimits A$ to
$\mathop{\rm Spec}\nolimits B$ for the sections of the sheaves
${\mathcal{S}}_{X}$ and ${\mathcal{K}}_{X}$. Let ${\mathcal{K}}_{X}^{\times}$
denote the sheaf of multiplicative groups formed by the units in
${\mathcal{K}}_{X}$. It satisfies ${\mathcal{K}}_{X}^{\times}(\mathop{\rm
Spec}\nolimits A)=Q(A)^{\times}$ and
$({\mathcal{K}}_{X}^{\times})_{x}={\mathcal{K}}_{X,x}^{\times}=Q({\mathcal{O}}_{X,x})^{\times}=Q(A_{\mathfrak{p}})^{\times}$
for any affine open subscheme $\mathop{\rm Spec}\nolimits A\subset X$ and
point $x=\mathfrak{p}\in\mathop{\rm Spec}\nolimits A$.

If $f:X\to Y$ is a flat morphism of locally noetherian schemes, then as flat
homomorphisms of rings preserve non zero-divisors, the homomorphism
$f^{\sharp}:f^{-1}{\mathcal{O}}_{Y}\to{\mathcal{O}}_{X}$ induces homomorphisms
$f^{\sharp}:f^{-1}{\mathcal{S}}_{Y}\to{\mathcal{S}}_{X}$,
$f^{\sharp}:f^{-1}{\mathcal{K}}_{Y}\to{\mathcal{K}}_{X}$ and
$f^{\sharp}:f^{-1}{\mathcal{K}}_{Y}^{\times}\to{\mathcal{K}}_{X}^{\times}$. If
$f:X\to Y$ and $g:Y\to Z$ are both flat then
$f^{\sharp}\circ(f^{-1}(g^{\sharp}))=(g\circ f)^{\sharp}$ on ${\mathcal{S}}$,
${\mathcal{K}}$ and ${\mathcal{K}}^{\times}$.

The above description of the sections and stalks of ${\mathcal{S}}_{X}$,
${\mathcal{K}}_{X}$ and ${\mathcal{K}}_{X}^{\times}$ does not in general
extend to non locally noetherian schemes (see Kleiman [4]).

Support $|r|$ and restriction map $res_{X,Y}$.  The quotient sheaf
${\mathcal{K}}_{X}^{\times}/{\mathcal{O}}_{X}^{\times}$ is called the sheaf of
Cartier divisors on $X$. The image of any
$r\in\Gamma(X,{\mathcal{K}}_{X}^{\times})$ in
$\Gamma(X,{\mathcal{K}}_{X}^{\times}/{\mathcal{O}}_{X}^{\times})$ is called
the Cartier divisor of $r$, and is denoted by $\mathop{\rm div}\nolimits(r)$.
The set-topological support of $\mathop{\rm div}\nolimits(r)$, which is a
closed subset of the underlying topological space $|X|$ of $X$, will be called
as the support of $r$, and will be denoted by $|r|$ (the support $\mathop{\rm
supp}\nolimits(r)$ of the section $r\in\Gamma(X,{\mathcal{K}}_{X}^{\times})$
in the usual sheaf theoretic sense, which is by definition the set of all
points $x$ for which the germ $r_{x}\neq 1\in{\mathcal{K}}^{\times}_{X,x}$, is
contained in $|r|$ but $\mathop{\rm supp}\nolimits(r)\neq|r|$ in general). If
$U=X-|r|$ is regarded as an open subscheme of $X$, then
$r|_{U}\in\Gamma(U,{\mathcal{O}}_{U}^{\times})$. If $X=\mathop{\rm
Spec}\nolimits A$ where $A$ is noetherian, then we can write $r=s/t$ where
$s,t\in S(A)$, and hence $|r|\subset V(a)\cup V(b)$ where $V(a),V(b)$ are the
closed subsets defined by $a=0$ and $b=0$. If $\mathfrak{p}$ is a minimal
prime of $A$, then $\mathfrak{p}\subset Zerodiv(A)$, and so
$a\not\in\mathfrak{p}$, $b\not\in\mathfrak{p}$, showing that $\mathfrak{p}\in
U=X-|r|$. For any $r\in\Gamma(X,{\mathcal{K}}_{X}^{\times})$ for a locally
noetherian scheme $X$, the above argument shows that $|r|$ does not contain
any height $0$ points of $X$. Hence if $Y\subset X$ is an irreducible
component of $X$ regarded as a closed integral subscheme, then $U\cap
Y\neq\emptyset$, and therefore the regular function
$r|_{U}\in\Gamma(U,{\mathcal{O}}_{U}^{\times})$ further restricts to a section
$r_{Y}\in\Gamma(Y\cap U,{\mathcal{O}}_{Y}^{\times})$. As $Y$ is integral,
$\Gamma(Y\cap U,{\mathcal{O}}_{Y}^{\times})\subset K(Y)^{\times}$ where $K(Y)$
denotes the function field of $Y$, which in turn equals
${\mathcal{K}}_{Y}(Y)$. Hence we have a restriction homomorphism
$res_{X,Y}:{\mathcal{K}}^{\times}_{X}(X)\to{\mathcal{K}}^{\times}_{Y}(Y):r\mapsto
r_{Y}$ for any irreducible component $Y$ of $X$. If $U\subset X$ is an open
subscheme such that $Y\cap U\neq\emptyset$, then $Y\cap U$ is an irreducible
component of $U$, and it follows from the above definition of restrictions
that $r_{Y}|_{Y\cap U}=(r|_{U})_{Y\cap U}\in K_{Y\cap U}(Y\cap U)$, an
equality that will be useful later. We will denote $r_{Y}|_{Y\cap U}$ and
$(r|_{U})_{Y\cap U}$ by the common symbol $r|_{Y\cap U}$.

For completeness, we include the following algebraic lemma which is used in
proving the above statements about sections and stalks of ${\mathcal{S}}_{X}$
and ${\mathcal{K}}_{X}$.

Lemma 1. (1) For any ring $A$, let the subset ${\mathcal{S}}(A)\subset A$
consist of all $a\in A$ such that for every prime ideal $\mathfrak{p}\subset
A$, the image of $a$ under the localization homomorphism $A\to
A_{\mathfrak{p}}$ is a non zero-divisor in $A_{\mathfrak{p}}$. Then
${\mathcal{S}}(A)\subset S(A)$. If the ring $A$ is noetherian, then
${\mathcal{S}}(A)=S(A)$.

(2) If $A$ is noetherian and $I\subset A$ an ideal such that $Ann(I)=0$ then
$S(A)\cap I\neq\emptyset$, where $Ann(I)\subset A$ denotes the annihilator
ideal of $I$.

(3) Let $f_{1},\ldots,f_{n}$ generate the unit ideal in $A$. If $A$ is
noetherian then the diagrams

$S(A)\to{\textstyle\prod}_{i}\,S(A_{f_{i}})\stackrel{{\scriptstyle\to}}{{\scriptstyle\to}}{\textstyle\prod}_{(j,k)}\,S(A_{f_{j}f_{k}})\mbox{
and
}Q(A)\to{\textstyle\prod}_{i}\,Q(A_{f_{i}})\stackrel{{\scriptstyle\to}}{{\scriptstyle\to}}{\textstyle\prod}_{(j,k)}\,Q(A_{f_{j}f_{k}}),$

obtained by applying the functors $S$ and $Q$ to the exact diagram
$A\to{\textstyle\prod}_{i}\,A_{f_{i}}\stackrel{{\scriptstyle\to}}{{\scriptstyle\to}}{\textstyle\prod}_{(j,k)}A_{f_{j}f_{k}}$
of rings and flat homomorphisms, are exact.

Proof. (1) The inclusion ${\mathcal{S}}(A)\subset S(A)$ is obvious. If $A$ is
noetherian, its set of associated prime ideals $Ass(A)$ is finite and the set
$Zerodiv(A)$ of all zero divisors in $A$ equals $\cup\,Ass(A)$. If
$\mathfrak{q}$ is any prime ideal in $A$, then
$Ass(A_{\mathfrak{q}})=\\{\mathfrak{p}A_{\mathfrak{q}}\,|\,\mathfrak{p}\in
Ass(A)\mbox{ and }\mathfrak{p}\subset\mathfrak{q}\\}$. Hence if $a\in
S(A)=A-\cup\,Ass(A)$, then $a/1\in S(A_{\mathfrak{q}})$, showing
$S(A)\subset{\mathcal{S}}(A)$.

(2) Suppose to the contrary that $I\subset Zerodiv(A)=\cup Ass(A)$. As
$Ass(A)$ is a finite set of prime ideals, by prime avoidance there exists
$\mathfrak{p}\in Ass(A)$ such that $I\subset\mathfrak{p}$. Being an associated
prime, $\mathfrak{p}$ equals $Ann(x)$ the annihilator ideal of some nonzero
element $x\in A$. Hence $x\in Ann(I)$ showing that $Ann(I)\neq 0$.

(3) A proof of this, assuming (2), is given on the page 61 of Mumford [5].
$\square$

3\. Sheaves of algebraic cycles

We will follow the notation and terminology used in Fulton’s Intersection
Theory [3], and we will assume that the reader is familiar with the material
covered in the Chapter 1 of that book. From here onwards ‘schemes’ will mean
algebraic schemes over a chosen base field $k$, and all morphisms will be
assumed to be over $k$. The phrase ‘flat morphism’ will mean a flat morphism
$f:X\to Y$ of schemes which is of a constant relative dimension. A ‘variety’
will mean an integral scheme, not assumed to be separated. If $V$ is a
variety, $R(V)$ will denote the field of rational functions over $V$. In terms
of the sheaf of meromorphic functions on $V$, note that
$R(V)={\mathcal{K}}_{V}(V)$, and
$R(V)^{\times}={\mathcal{K}}_{V}^{\times}(V)$.

Let ${\mathcal{C}}_{k,flat}$ denote the category whose objects are all
algebraic $k$-scheme, and morphisms are all flat $k$-morphisms of constant
relative dimensions. We give it the big Zariski topology, in other words, an
open cover of an object $X$ is a Zariski open cover of $X$. This defines a
site ${\mathcal{C}}_{k,flat,Zar}$ which we will simply denote by
${\mathcal{C}}$. We define a sheaf ${\mathcal{K}}^{\times}$ of abelian groups
on this site as follows. For any object $X$ of ${\mathcal{C}}$ we define
${\mathcal{K}}^{\times}(X)={\mathcal{K}}_{X}^{\times}(X)$, and for any
morphism $f:X\to Y$, we define the restriction homomorphism
${\mathcal{K}}^{\times}(Y)\to{\mathcal{K}}^{\times}(X)$ to be the group
homomorphism induced by
$f^{\sharp}:f^{-1}{\mathcal{K}}_{Y}\to{\mathcal{K}}_{X}$ (it matters here that
$f$ is flat, but it does not matter here whether $f$ has a constant relative
dimension or not).

We next define a presheaf ${\mathcal{Z}}_{*}$ of $\mathbb{Z}$-graded abelian
groups and graded homomorphisms on ${\mathcal{C}}$ as follows. For any object
$X$, we define ${\mathcal{Z}}_{*}(X)=Z_{*}(X)$, and for any morphism $f:X\to
Y$ in ${\mathcal{C}}$, which is by definition a flat $k$-morphism of a
constant relative dimension say $d$, we define the restriction homomorphism
${\mathcal{Z}}_{*}(Y)\to{\mathcal{Z}}_{*}(X)$ to be the graded homomorphism of
degree $d$ induced by the flat pullbacks $f^{*}:Z_{n}(Y)\to Z_{n+d}(X)$ under
$f$. If $V\subset X$ is a closed subvariety of dimension $n$, then $f^{-1}(V)$
is a closed subscheme of $X$ of pure dimension $n+d$, and by definition
$f^{*}[V]$ is the fundamental cycle $[f^{-1}(V)]\in Z_{n+d}(X)$. Here, recall
that if $X$ is a scheme and $W\subset X$ is a closed subscheme of pure
dimension $m$, then the fundamental cycle $[W]\in Z_{m}(W)\subset Z_{m}(X)$ is
defined by

$[W]={\textstyle\sum}_{W_{i}}\,\ell({\mathcal{O}}_{W,W_{i}})[W_{i}]$

where the sum is indexed by all irreducible components $W_{i}$ of $W$, and
${\mathcal{O}}_{W,W_{i}}$ denotes the stalk of ${\mathcal{O}}_{W}$ at the
generic point of $W_{i}$, which is an artin local ring, with length denoted by
$\ell({\mathcal{O}}_{W,W_{i}})$. The map $f^{*}$ so defined on the generators
$[V]$ extends uniquely to the free abelian group $Z_{n}(Y)$ to define a
homomorphism $f^{*}:Z_{n}(Y)\to Z_{n+d}(X)$. By [3] Lemma 1.7.1, for any
closed subscheme $W$ of $Y$ of pure dimension $n$, we have
$f^{*}[W]=[f^{-1}W]$. From this, it follows that for any two arrows $f:X\to Y$
and $g:Y\to Z$ in ${\mathcal{C}}$, we have $(g\circ f)^{*}=f^{*}\circ g^{*}$,
and it is clear that $\mathop{\rm id}\nolimits_{X}^{*}=\mathop{\rm
id}\nolimits_{Z^{*}(X)}$. Hence ${\mathcal{Z}}_{*}$ is a presheaf on
${\mathcal{C}}$.

Proposition 2. The presheaf ${\mathcal{Z}}_{*}$ is a sheaf on the site
${\mathcal{C}}$.

Proof. Let $\alpha=\sum_{i}m_{i}[V_{i}]\in Z_{n}(X)$ where $V_{i}$ are closed
subvarieties of $X$ of dimension $n$, with $V_{i}\neq V_{j}$ for $i\neq j$,
and with $m_{i}\neq 0$ for each $i$. Let $(U_{\lambda})$ be a Zariski open
cover of $X$. If $\alpha|_{U_{\lambda}}=0$ for each $\lambda$, then we will
show that $\alpha=0$. If $V_{i}\cap U_{\lambda}\neq\emptyset$, then $V_{i}$ is
uniquely recovered from $V_{i}\cap U_{\lambda}$ as its closure. Hence as the
$V_{i}$ are distinct, if $V_{i}\cap U_{\lambda}=V_{j}\cap U_{\lambda}$ for
$i\neq j$ then we must have $V_{i}\cap U_{\lambda}=V_{j}\cap
U_{\lambda}=\emptyset$. Hence there can be no cancellations amongst the
coefficients $m_{i}$, and so if $\alpha|_{U_{\lambda}}=0$ then each $V_{i}\cap
U_{\lambda}=\emptyset$. If $\alpha|_{U_{\lambda}}=0$ for each $\lambda$, then
as $(U_{\lambda})$ is a cover of $X$, it follows that $\alpha=0$ (the empty
sum). This shows that ${\mathcal{Z}}_{*}$ is a separated presheaf on
${\mathcal{C}}$.

As ${\mathcal{Z}}_{*}$ is a separated presheaf and as any object $X$ of
${\mathcal{C}}$ is quasicompact, to verify the gluing condition for
${\mathcal{Z}}_{*}$, it is enough to consider a union $X=U^{\prime}\cup
U^{\prime\prime}$ of two open subschemes, together with $n$-cycles
$\alpha^{\prime}=\sum_{i=1}^{p}m_{i}^{\prime}[V_{i}^{\prime}]\in
Z_{n}(U^{\prime})$ and
$\alpha^{\prime\prime}=\sum_{j=1}^{q}m_{j}^{\prime\prime}[V_{j}^{\prime\prime}]\in
Z_{n}(U^{\prime\prime})$ where $p,q\geq 0$, $V^{\prime}_{i}$ are distinct
closed subvarieties of $U^{\prime}$, $V^{\prime\prime}_{j}$ are distinct
closed subvarieties of $U^{\prime\prime}$, and each $m^{\prime}_{i}$ and
$m^{\prime\prime}_{j}$ is nonzero, such that
$\alpha^{\prime}|_{U^{\prime}}=\alpha^{\prime\prime}|_{U^{\prime\prime}}$, and
glue $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ together. By re indexing
the $V^{\prime}_{i}$ and the $V^{\prime\prime}_{j}$, we can assume that for
$i\leq r$, $V^{\prime}_{i}\cap U^{\prime}\cap U^{\prime\prime}\neq\emptyset$
and for $i>r$, $V^{\prime}_{i}\cap U^{\prime}\cap U^{\prime\prime}=\emptyset$.
Similarly, for $j\leq s$, $V^{\prime\prime}_{j}\cap U^{\prime}\cap
U^{\prime\prime}\neq\emptyset$ and for $j>s$, $V^{\prime\prime}_{j}\cap
U^{\prime}\cap U^{\prime\prime}=\emptyset$. Note that $V^{\prime}_{i}$ can be
uniquely recovered from $V^{\prime}_{i}\cap U^{\prime}\cap U^{\prime\prime}$
if $i\leq r$, and similarly, $V^{\prime\prime}_{j}$ can be uniquely recovered
from $V^{\prime\prime}_{j}\cap U^{\prime}\cap U^{\prime\prime}$ if $j\leq s$.
It follows that $r=s$, and as closed subvarieties of $X$, and after re-
indexing $V^{\prime}_{1},\ldots,V^{\prime\prime}_{r}$, we have
$\overline{V_{i}^{\prime}}=\overline{V^{\prime\prime}_{i}}$ for $i\leq r$, and
$m^{\prime}_{i}=m^{\prime\prime}_{i}$ for $i\leq r$ (where $\overline{V}$
denotes the closure of $V$ in $X$). On the other hand, if $i>r$ so that
$V^{\prime}_{i}\cap U^{\prime\prime}=\emptyset$, then $V^{\prime}_{i}$ is
already closed in $X$. Similarly, if $j>r$ then $V^{\prime\prime}_{j}$ is
already closed in $X$. Now consider the cycle

$\alpha={\textstyle\sum}_{a=1}^{r}m^{\prime}_{a}[\overline{V^{\prime}_{a}}]+{\textstyle\sum}_{b=i+1}^{r}m^{\prime}_{b}[V^{\prime}_{b}]+{\textstyle\sum}_{c=r+1}^{q}m^{\prime\prime}_{c}[V^{\prime\prime}_{c}]\in
Z_{n}(X).$

It is immediate from its definition that
$\alpha|_{U^{\prime}}=\alpha^{\prime}$ and
$\alpha|_{U^{\prime\prime}}=\alpha^{\prime\prime}$. $\square$

4\. The sheaf homomorphism from ${\mathcal{K}}^{\times}$ to ${\mathcal{W}}$

Let ${\mathcal{C}}^{pure}$ denote the full subcategory of the category
${\mathcal{C}}$ of all algebraic $k$-schemes and flat $k$-morphisms of
constant relative dimensions, for which the objects are all algebraic
$k$-schemes $X$ of pure dimension (which means any two irreducible component
of $X$ have the same dimension). We regard ${\mathcal{C}}^{pure}$ as a site
under the big Zariski topology.

The sheaf ${\mathcal{Z}}_{*}$ of graded abelian groups on
${\mathcal{C}}^{pure}$, which is the restriction of the sheaf
${\mathcal{Z}}_{*}$ on ${\mathcal{C}}$ defined in the previous section, has a
subsheaf ${\mathcal{W}}$, which we call as the sheaf of Weil divisors. This is
defined as follows. For any $X\in{\mathcal{C}}^{pure}$,

${\mathcal{W}}(X)=Z_{\dim(X)-1}(X)$

regarded as a graded abelian group concentrated in the degree $\dim(X)-1$. We
also regard ${\mathcal{W}}$ simply as a sheaf of abelian groups on
${\mathcal{C}}^{pure}$, defined by the above displayed formula.

For any $X\in{\mathcal{C}}^{pure}$ and any $r\in{\mathcal{K}}^{\times}(X)$,
let $[r]_{X}$ denote the Weil divisor of $r$, that is, $[r]_{X}=[\mathop{\rm
div}\nolimits(r)]$ where $\mathop{\rm div}\nolimits(r)$ is the Cartier divisor
of $r$. By definition,

$[r]_{X}=[\mathop{\rm div}\nolimits(r)]={\textstyle\sum}_{i}\,\mathop{\rm
ord}\nolimits_{V_{i}}(r)[V_{i}]$

where the sum (which is actually a finite sum as $\mathop{\rm
ord}\nolimits_{V_{i}}(r)=0$ for all but finitely many $V_{i}$) is over all
closed subvarieties $V_{i}$ of $X$ of codimension $1$, and $\mathop{\rm
ord}\nolimits_{V_{i}}(r)\in\mathbb{Z}$ is the order of zero (or pole if $<0$)
of $r$ at $V_{i}$, defined as follows. The germ
$r_{i}\in{\mathcal{K}}_{X,V_{i}}^{\times}$ of $r$ at the generic point of
$V_{i}$ can be expressed as a ratio $a_{i}/b_{i}$ where $a_{i},b_{i}\in
S({\mathcal{O}}_{X,V_{i}})$ are non zero-divisors, and by definition
$\mathop{\rm
ord}\nolimits_{V_{i}}(r)=\ell({\mathcal{O}}_{X,V_{i}}/(a_{i}))-\ell({\mathcal{O}}_{X,V_{i}}/(b_{i}))$
where $\ell(R)$ denotes the length of any artin local ring $R$. This is well-
defined, independent of the choice of $a_{i},b_{i}$, as $\mathop{\rm
ord}\nolimits_{V_{i}}$ is an additive function on
$S({\mathcal{O}}_{X,V_{i}})$. As the support $|r|$ is closed and of
codimension $\geq 1$ in $X$ (see Section 1), it contains the generic points of
at most finitely many codimension $1$ subvarieties $V_{i}$ of $X$, so the
above sum that defines $[r]_{X}$ is finite.

Proposition 3. Let $X\in{\mathcal{C}}^{pure}$ have pure dimension $n$, and let
$X_{1},\ldots,X_{p}$ be the irreducible components of $X$, each regarded as a
closed subvariety of $X$ of dimension $n$. Then the following statements hold.

(1) For any $s\in{\mathcal{S}}(X)$, we have $[s]_{X}=[X/(s)]\in Z_{n-1}(X)$
where $[X/(s)]$ is the fundamental cycle of the closed subscheme $X/(s)\subset
X$ defined by the ideal sheaf $(s)\subset{\mathcal{O}}_{X}$.

(2) Let $r\in{\mathcal{K}}^{\times}(X)$, and let
$r_{i}\in{\mathcal{K}}^{\times}(X_{i})$ be its restriction to $X_{i}$ (see
Section 1). Then we have
$[r]_{X}={\textstyle\sum}_{i}\,\ell({\mathcal{O}}_{X,X_{i}})\,[r_{i}]_{X_{i}}\in
Rat_{n-1}(X)\subset Z_{n-1}(X)$.

Proof. (1) If $s\in{\mathcal{S}}(X)$ then the support $|s|$ of $s$ is the
underlying closed subset $|X/(s)|$ of the closed subscheme $X/(s)$ of $X$. Let
$V_{j}$ be the irreducible components of $X/(s)$. By definition,
$[s]_{X}={\textstyle\sum}_{V_{j}}\,\mathop{\rm
ord}\nolimits_{{\mathcal{O}}_{X,V_{j}}}(s)\,[V_{j}]={\textstyle\sum}_{V_{j}}\,\ell({\mathcal{O}}_{X,V_{j}}/(s))\,[V_{j}]$.
On the other hand, the fundamental cycle $[X/(s)]$ of the scheme $X/(s)$ is
given by
$[X/(s)]={\textstyle\sum}_{V_{j}}\,\ell({\mathcal{O}}_{X/(s),V_{j}})\,[V_{j}]$.
As ${\mathcal{O}}_{X,V_{j}}/(s)\cong{\mathcal{O}}_{X/(s),V_{j}}$, the
numerical coefficients of each $V_{j}$ on the two sides are equal, hence (1)
holds.

(2) Now let $r\in{\mathcal{K}}^{\times}(X)$. Let $X_{1},\ldots,X_{p}$, where
$p\geq 1$ be the irreducible components of $X$, each regarded as a closed
subvariety of $X$ of dimension $n$. As the support $|r|$ does not contain any
height $0$ point (see Section 1), there are only finitely many closed
subvarieties $V_{1},\ldots,V_{q}$ of dimension $n-1$ of $X$ (where $q\geq 0$)
that are contained in $|r|$. If $V$ is a closed subvariety of dimension $n-1$
whose generic point does not lie in $|r|$ then $\mathop{\rm
ord}\nolimits_{V}(r)=0$. Hence $[r]_{X}=\sum_{j=1}^{q}\,\mathop{\rm
ord}\nolimits_{{\mathcal{O}}_{X,V_{j}}}(r)[V_{j}]$ (however, it is possible
that $\mathop{\rm ord}\nolimits_{{\mathcal{O}}_{X,V_{j}}}(r)=0$ for some
$V_{j}$, even though $V_{j}\subset|r|$, as the local ring
${\mathcal{O}}_{X,V_{j}}$ may not be regular).

The cycle $[r_{i}]_{X_{i}}$ is by definition the sum
$[r_{i}]_{X_{i}}={\textstyle\sum}_{\\{i\,|\,V_{j}\subset
X_{i}\\}}\,\mathop{\rm
ord}\nolimits_{{\mathcal{O}}_{X_{i},V_{j}}}(r_{i})\,[V_{j}]$. Hence
$[r]_{X}={\textstyle\sum}_{i}\,\ell({\mathcal{O}}_{X,X_{i}})\,[r_{i}]_{X_{i}}$
if and only if

${\textstyle\sum}_{j}\,\mathop{\rm
ord}\nolimits_{{\mathcal{O}}_{X,V_{j}}}(r)[V_{j}]={\textstyle\sum}_{\\{(i,j)\,|\,V_{j}\subset
X_{i}\\}}\,\ell({\mathcal{O}}_{X,X_{i}})\,\mathop{\rm
ord}\nolimits_{{\mathcal{O}}_{X_{i},V_{j}}}(r_{i})\,[V_{j}].$

Comparing the numerical coefficients of $[V_{j}]$ on both sides, the above
holds if and only if for each $j$ we have

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathop{\rm
ord}\nolimits_{{\mathcal{O}}_{X,V_{j}}}(r)={\textstyle\sum}_{\\{i\,|\,V_{j}\subset
X_{i}\\}}\,\ell({\mathcal{O}}_{X,X_{i}})\,\mathop{\rm
ord}\nolimits_{{\mathcal{O}}_{X_{i},V_{j}}}(r_{i}).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ldots(3)$

Let $A={\mathcal{O}}_{X,V_{j}}$, which is noetherian local ring of dimension
$1$. Let $\mathfrak{m}\subset A$ be its maximal ideal and let
$\mathfrak{p}_{1},\ldots,\mathfrak{p}_{k}\subset A$ be its minimal prime
ideals (these are defined by the irreducible components $X_{i}$ which contain
$V_{j}$). For any $a\in S(A)$ the rings $A/(a)$, $A_{\mathfrak{p}_{i}}$ and
$A/(\mathfrak{p}_{i}+(a))$ are artinian. By [3] Lemma A.2.7 (applied by taking
$M=A$ in the statement of that lemma), for any $a\in S(A)$ we have

$\ell(A/(a))={\textstyle\sum}_{\mathfrak{p}_{i}}\,\ell(A_{\mathfrak{p}_{i}})\,\ell(A/(\mathfrak{p}_{i}+(a))).$

In other words, for any $V_{j}$ and any $a\in S({\mathcal{O}}_{X,V_{j}})$ we
have

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathop{\rm
ord}\nolimits_{{\mathcal{O}}_{X,V_{j}}}(a)={\textstyle\sum}_{\\{i\,|\,V_{j}\subset
X_{i}\\}}\,\ell({\mathcal{O}}_{X,X_{i}})\,\mathop{\rm
ord}\nolimits_{{\mathcal{O}}_{X_{i},V_{j}}}(a|_{X_{i}}).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ldots(4)$

If $r^{\prime}$ denotes the germ of $r$ in ${\mathcal{K}}_{X,V_{j}}^{\times}$
then $r^{\prime}=s/t$ for some $s,t\in S({\mathcal{O}}_{X,V_{j}})$. Applying
(4) to $s$ and $t$ and subtracting the results, we get the desired equality
(3).

As each $r_{i}$ is a non-zero rational function on the variety $X_{i}$, by
definition $[r_{i}]_{X_{i}}$ belongs to $Rat_{n-1}(X)$, and hence $[r]_{X}\in
Rat_{n-1}(X)$. $\square$

Proposition 4. The maps
$\Phi_{X}:{\mathcal{K}}^{\times}(X)\to{\mathcal{W}}(X)$ defined by
$\Phi_{X}(r)=[r]_{X}$, as $X$ varies over ${\mathcal{C}}^{pure}$, define a
homomorphism of sheaves of abelian groups
$\Phi:{\mathcal{K}}^{\times}\to{\mathcal{W}}$ on ${\mathcal{C}}^{pure}$. The
homomorphism $\Phi$ factors via the sub presheaf $Rat\subset{\mathcal{W}}$
defined on ${\mathcal{C}}^{pure}$ by $X\mapsto Rat_{\dim(X)-1}(X)\subset
Z_{\dim(X)-1}(X)$.

Proof. We just have to show that $\Phi$ commutes with restrictions, that is,
if $f:X\to Y$ is a flat homomorphism of constant relative dimension $d$
between pure dimensional schemes $X$ and $Y$ of dimensions $n+d$ and $n$
respectively, then the following diagram commutes.

$\begin{array}[]{ccc}{\mathcal{K}}^{\times}(Y)&\stackrel{{\scriptstyle
f^{\sharp}}}{{\to}}&{\mathcal{K}}^{\times}(X)\\\
{\scriptstyle\Phi_{Y}}\downarrow{}{}{}&&~{}~{}~{}\downarrow{\scriptstyle\Phi_{X}}\\\
Z_{n-1}(Y)&\stackrel{{\scriptstyle f^{*}}}{{\to}}&Z_{n+d-1}(X)\end{array}$

If $f:X\hookrightarrow Y$ is an open embedding, then the commutativity of the
above diagram is immediate from the definitions of the objects and the arrows
in it.

So now we consider the general case. Let $r\in{\mathcal{K}}^{\times}(Y)$. Then
we have two sections

$f^{*}\Phi_{Y}(r),\,\Phi_{X}(f^{\sharp}r)\in
Z_{n+d-1}(X)=\Gamma(X,{\mathcal{Z}}_{n+d-1})$

and we have to show that these two are equal, that is, the following equality
holds:

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f^{*}([r]_{Y})=[f^{\sharp}r]_{X}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ldots(5)$

As ${\mathcal{Z}}_{n+d-1}$ restricts to a sheaf (in particular, a separated
presheaf) on the small Zariski site of $X$, to check the equality (5), by
applying the special case of open embeddings, it is enough to check that the
restrictions of its two sides are equal affine locally on $X$.

Given any $x\in X$, let $j:V=\mathop{\rm Spec}\nolimits B\hookrightarrow Y$ be
an affine open subscheme with $f(x)\in V$. Let $i:U=\mathop{\rm Spec}\nolimits
A\hookrightarrow X$ be an affine open subscheme with $x\in U$ and $f(U)\subset
V$. Let $\varphi=f|_{U}:U\to V$, and let $\varphi^{\sharp}:B\to A$ be the
corresponding $k$-algebra homomorphism. Note that $U$ and $V$ have pure
dimensions $n+d$ and $n$, and $\varphi$ is flat of constant relative dimension
$d$.

It follows from the description of meromorphic functions on a noetherian
affine scheme given in Section 1, that
${\mathcal{K}}^{\times}(V)=Q(B)^{\times}$ and
${\mathcal{K}}^{\times}(U)=Q(A)^{\times}$, and the restriction map
$\varphi^{\sharp}:{\mathcal{K}}^{\times}(V)\to{\mathcal{K}}^{\times}(U)$ is
the ring homomorphism $\varphi^{\sharp}:Q(A)\to Q(B)$ induced by the flat
homomorphism $\varphi^{\sharp}:B\to A$. The restriction $r|_{V}=j^{\sharp}r$
of $r\in{\mathcal{K}}^{\times}(Y)$ to
${\mathcal{K}}^{\times}(V)=Q(B)^{\times}$ can be expressed as a quotient
$r|_{V}=b_{1}/b_{2}$ where $b_{1},b_{2}\in S(B)$. We want to show that the
restriction of (5) holds on $U$, that is,

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\varphi^{*}([b_{1}/b_{2}]_{V})=[\varphi^{\sharp}(b_{1}/b_{2})]_{U}\in
Z_{n+d-1}(U)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ldots(6)$

Now, $[b_{1}/b_{2}]_{V}=[b_{1}]_{V}-[b_{2}]_{V}$ and so
$\varphi^{*}([b_{1}/b_{2}]_{V})=\varphi^{*}([b_{1}]_{V})-\varphi^{*}([b_{2}]_{V})$.
On the other hand,
$[\varphi^{\sharp}(b_{1}/b_{2})]_{U}=[(\varphi^{\sharp}b_{1})/(\varphi^{\sharp}b_{2})]_{U}=[\varphi^{\sharp}b_{1}]_{U}-[\varphi^{\sharp}b_{2}]_{U}$.
Hence to prove $(6)$, we just have to show that for any $b\in S(B)$ the
following equality holds:

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\varphi^{*}([b]_{V})=[\varphi^{\sharp}b]_{U}\in
Z_{n+d-1}(U)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\ldots(7)$

As $b\in S(B)={\mathcal{S}}(V)$, by Proposition 3.(1) we have
$[b]_{V}=[\mathop{\rm Spec}\nolimits B/(b)]\in Z_{n-1}(V)$, and so
$\varphi^{*}([b]_{V})=\varphi^{*}[\mathop{\rm Spec}\nolimits B/(b)]$. By
definition of flat pullback on cycles, $\varphi^{*}[\mathop{\rm Spec}\nolimits
B/(b)]=[\varphi^{-1}(\mathop{\rm Spec}\nolimits B/(b))]$, which is the
fundamental cycle of the closed subscheme $\varphi^{-1}(\mathop{\rm
Spec}\nolimits B/(b))$ of $U=\mathop{\rm Spec}\nolimits A$. The schematic
inverse image $\varphi^{-1}(\mathop{\rm Spec}\nolimits B/(b))$ is the closed
subscheme $\mathop{\rm Spec}\nolimits A/(\varphi^{\sharp}b)$ of $\mathop{\rm
Spec}\nolimits A$. The element $\varphi^{\sharp}b$ lies in
$S(A)={\mathcal{S}}(U)$ as $\varphi^{\sharp}:B\to A$ is flat. Hence by
Proposition 3.(1) we have $[\varphi^{\sharp}b]_{U}=[\mathop{\rm Spec}\nolimits
A/(\varphi^{\sharp}b)]$. Hence we have the sequence of equalities

$\varphi^{*}([b]_{V})=\varphi^{*}[\mathop{\rm Spec}\nolimits
B/(b)]=[\varphi^{-1}(\mathop{\rm Spec}\nolimits B/(b))]=[\mathop{\rm
Spec}\nolimits A/(\varphi^{\sharp}b)]=[\varphi^{\sharp}b]_{U}$

in $Z_{n+d-1}(U)$. This proves the equation $(7)$, and hence completes the
proof that $\Phi:{\mathcal{K}}^{\times}\to{\mathcal{W}}$ is a homomorphism of
sheaves. The final statement that $\Phi$ factors via the sub presheaf
$Rat\subset{\mathcal{W}}$ follows from the equality
$[r]_{X}={\textstyle\sum}_{i}\,\ell({\mathcal{O}}_{X,X_{i}})\,[r_{i}]_{X_{i}}$
given by Proposition 3.(2), because by its definition each $[r_{i}]_{X_{i}}$
lies in $Rat_{\dim(X_{i})-1}(X_{i})\subset Rat_{\dim(X)-1}(X)$. $\square$

Remark 5. We do not need to explicitly use the morphism $\tilde{b}:\mathop{\rm
Spec}\nolimits B\to{\mathbb{A}}^{1}=\mathop{\rm Spec}\nolimits k[t]$ defined
by the $k$-homomorphism $k[t]\to B:t\mapsto b$. Note that
$\tilde{b}^{-1}(P_{0})=\mathop{\rm Spec}\nolimits B/(b)$ where
$P_{0}\in{\mathbb{A}}^{1}$ is the closed point defined by $t=0$. As $b\in
S(B)$, the local criterion for flatness implies that $\tilde{b}$ is flat in an
open neighbourhood of $\mathop{\rm Spec}\nolimits B/(b)$. If $B$ is a domain
then $\tilde{b}$ is flat if and only if $b$ is transcendental over $k$, as
follows from the nullstellensatz. The morphism $\tilde{b}$, and implicitly the
above flatness condition when $B$ is a domain, are implicitly used in the
proof of the preservation of rational equivalence under flat morphisms in [3].

Theorem 6. Let $f:X\to Y$ be a flat morphism of constant relative dimension
$d$. Then for any $n$, under the homomorphism $f^{*}:Z_{n}(Y)\to Z_{n+d}(X)$,
the image of $Rat_{n}(Y)$ lies in $Rat_{n+d}(X)$.

Proof. By definition, $Rat_{n}(Y)$ is generated by elements of the form
$[r]_{V}=\Phi_{V}(r)$ where $V\subset Y$ is a closed subvariety of dimension
$n+1$ and $r\in{\mathcal{K}}(V)^{\times}$. Hence we just have to show that
$f^{*}\Phi_{V}(r)$ lies in $Rat_{n+d}(X)$. Let $W=f^{-1}(V)\subset X$ be the
schematic inverse image of $V$. Hence $g=f|_{W}:W\to V$ is again flat and of
constant relative dimension $d$. As $V$ is a variety hence of pure dimension
$n$, this implies that $W$ is a scheme of pure dimension $n+d+1$, and so both
$V,\,W\,\in{\mathcal{C}}^{pure}$, and hence $g$ is an arrow in
${\mathcal{C}}^{pure}$, which is a full subcategory of ${\mathcal{C}}$. Hence
by Proposition 4,

$g^{*}\Phi_{V}(r)=\Phi_{W}(g^{\sharp}r)$

where $g^{\sharp}:{\mathcal{K}}^{\times}(V)\to{\mathcal{K}}^{\times}(W)$ is
the restriction homomorphism under $g$ for the sheaf ${\mathcal{K}}^{\times}$
over ${\mathcal{C}}^{pure}$. By Proposition 3.(2) applied to the scheme $W$
and to $g^{\sharp}r\in{\mathcal{K}}^{\times}(W)$, the cycle
$\Phi_{W}(g^{\sharp}r)=[g^{\sharp}r]_{W}$ lies in the subgroup
$Rat_{n+d}(W)\subset Rat_{n+d}(X)$. Hence the theorem. $\square$

Remark 7. The following rough analogy partly inspired our sheaf theoretic
approach. The sub presheaf $Rat\subset{\mathcal{W}}$ on the site
${\mathcal{C}}^{pure}$, defined by $X\mapsto Rat_{\dim(X)-1}(X)$, is not a
sheaf because a locally principal Weil divisor need not be rationally
equivalent to $0$ globally. This means that the presheaf $X\mapsto A_{*}(X)$
on ${\mathcal{C}}$ is not separated, and hence not a sheaf. The situation is
similar to that for differential forms on manifolds. Let ${\mathcal{M}}$ be
the site formed by all $C^{\infty}$ manifolds and $C^{\infty}$ maps, equipped
with the big Euclidean topology. For any integer $n$, differential $n$-forms
form a sheaf $\Omega^{n}$ on the site ${\mathcal{M}}$, and closed differential
$n$-forms form a subsheaf ${\mathcal{Z}}^{n}$ of $\Omega^{n}$. The exact
$n$-forms only form a sub presheaf
${\mathcal{B}}^{n}\subset{\mathcal{Z}}^{n}$, which however is not a sheaf as a
closed form is locally exact but not necessarily globally exact. The de Rham
cohomology $H_{dR}^{*}$ forms a presheaf which is not separated, so not a
sheaf. Just as we can define $C^{\infty}$-pullbacks on $H_{dR}^{*}$ using the
sheaf homomorphism $d:\Omega^{n-1}\to{\mathcal{Z}}^{n}$, one should be able to
define flat pullbacks on $A_{*}$ using the sheaf homomorphism
$\Phi:{\mathcal{K}}^{\times}\to{\mathcal{W}}$ on ${\mathcal{C}}^{pure}$, with
$Rat$ playing the role of ${\mathcal{B}}^{n}$. This appears to work.

Postscript. It was brought to my attention that our sheaf theoretic approach
to Chow groups has certain similarities with the viewpoint of Rost [6], which
can also be used to derive a parallel proof of the preservation of rational
equivalence under flat pullbacks.

References

[1] C. Chevalley, Les classes d’équivalence rationelle, I, II. Séminaire C.
Chevalley, 2e année, Anneaux de Chow et Applications, Secr. Math. Paris, 1958.

[2] A. Grothendieck, A. and J. Dieudonne, Éléments de Géométrie Algébrique \-
IV4, Publ. Math. IHES 32 (1967).

[3] W. Fulton, Intersection Theory, Springer Verlag, 1984.

[4] S. Kleiman, Misconceptions about $K_{X}$, Enseign. Math. 25 (1979)
203-206.

[5] D. Mumford, Lectures on Curves on an Algebraic Surface, Princeton
University Press, 1966.

[6] M. Rost, Chow groups with coefficients, Doc. Math. J. DMV 1 (1996)
319-393.