Are chilies a common ingredient in Italian cooking?
Disclosure: I am Italian but no professional, so what I will say is based on my own experience and could be inaccurate/wrong.

There is sure plenty of use for chili peppers in italian cousine. It is really traditional in the southern region of Calabria and close ones, but it is used throughout the whole country.

Together with the already mentioned Arrabbiata pasta sauce, in which spicy chili peppers is the main ingredient, there are several other dishes and cured meat product (sausages and such) that contains or, sometimes, heavily rely on that spicyness ( for example Nduja ).

Also, plenty of other plates can be adjusted to become spicy using dried chili flakes, especially for tomato based dishes, if it is of your taste.