Remember Collected Items

Unity Engine
1

I have a number of scenes.
Each scene has a gem.
I want my game to work like this:
if the player collect the gem, then the game should remember that the it has been collected and which it is (for expample Gem no.3 collected)
Thank in advance.

2

I would save all the collected things in a class for each level.

[Serializable]
public class LevelData
{
    public List<string> CollectedObjects = new List<string>();
}

then you need an object which you can access globally and which helps you to access a certain level data:

public class LevelManager : MonoBehaviour
{
    public static LevelManager Instance; // singleton
    Dictionary<string, LevelData> levelData = new Dictionary<string, LevelData>();

    void Awake()
    {
        Instance = this;
        GameObject.DontDestroyOnLoad(this);
    }

    public void NotifyItemCollected(string itemName)
    {
        if(!levelData.ContainsKey(Application.loadedLevelName))
            levelData.Add(Application.loadedLevelName, new LevelData());

        levelData[Application.loadedLevelName].CollectedObjects.Add(itemName);
    }
}

and then you need a script which you can append to your collectables (probably you already have one). Call Collect() when you collect it:

public class Collectable : MonoBehaviour
{
    public void Collect()
    {
        LevelManager.Instance.NotifyItemCollected(this.name);
    }
}

please note: all code untested and could be improved to make more error checks and stuff.

You need a GameObject with LevelManager attached (one in the very first level… it will stay for all levels)
all the collectables in a level need unique names.

There might be better ways to do it, but this is a simple approach which works if you don’t make any naming mistakes.