AlarmManager可以在应用程序被杀死的情况下在后台工作。它可以使用PendingIntent来启动我们的Activity、BroadcastReceiver或Service。但是，使用Service是一个更好的选择，因为它可以在后台持续运行，而且可以更好地控制时间间隔。以下是一个示例代码，它使用AlarmManager和Service在后台执行操作：

MainActivity.java

public class MainActivity extends AppCompatActivity {
    private Intent serviceIntent;
    private PendingIntent pendingIntent;
    private AlarmManager alarmManager;

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);

        // Create Pending Intent to start Service
        serviceIntent = new Intent(MainActivity.this, MyService.class);
        pendingIntent = PendingIntent.getService(MainActivity.this, 0, serviceIntent, PendingIntent.FLAG_UPDATE_CURRENT);

        // Create Alarm Manager with interval of 1 minute
        alarmManager = (AlarmManager) getSystemService(Context.ALARM_SERVICE);
        long interval = 60 * 1000;
        alarmManager.setRepeating(AlarmManager.RTC_WAKEUP, System.currentTimeMillis() + interval,
                interval, pendingIntent);
    }

    @Override
    protected void onDestroy() {
        super.onDestroy();
        // Cancel Alarm Manager
        alarmManager.cancel(pendingIntent);
    }
}

MyService.java

public class MyService extends Service {
    @Override
    public int onStartCommand(Intent intent, int flags, int startId) {
        // Do some background tasks here
        Toast.makeText(this, "Task is running in the background", Toast.LENGTH_LONG).show();
        return START_STICKY;
    }

    @Override
    public IBinder onBind(Intent intent) {
        return null;
    }
}

在这个示例中，我们使用AlarmManager的setRepeating方法来设置启动我们的Service，该Service运行一些后台任务。我们还在MainActivity的onDestroy方法中取消AlarmManager。