how to remove shadow below actionbar with AppCompat.Light.NoActionBar?

I’m not an expert but I run into the same problem just a few hours ago. So the idea here is that with AppCompat we have to manage the library attributes rather than Android attributes. In other words, instead of android:elevation try app:elevation:

<android.support.design.widget.AppBarLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    android:id="@+id/appbar"
    android:layout_width="match_parent"
    android:layout_height="?attr/actionBarSize"
    android:theme="@style/ThemeOverlay.AppCompat.Dark.ActionBar"
    android:fitsSystemWindows="true"
    app:elevation="0dp">

    <android.support.v7.widget.Toolbar
        xmlns:app="http://schemas.android.com/apk/res-auto"
        android:id="@+id/toolbar"
        android:layout_width="match_parent"
        android:layout_height="match_parent"
        app:popupTheme="@style/ThemeOverlay.AppCompat.Light" />

</android.support.design.widget.AppBarLayout>

EDIT :

I just tried another option without AppBarLayout. This way works perfectly fine for me, the shadow is completely gone. So I suspect the problem is in your other View. I don’t think it’s your ToolBar drops the shadow.

<android.support.v7.widget.Toolbar
    xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    app:theme="@style/ThemeOverlay.AppCompat.Dark.ActionBar"
    app:popupTheme="@style/ThemeOverlay.AppCompat.Light"
    android:id="@+id/toolbar"
    android:layout_width="match_parent"
    android:layout_height="wrap_content"
    android:minHeight="?attr/actionBarSize"
    app:elevation="0dp" />