Does Gerrit Cole get an error too for not covering first base?

That was one of the most fun innings ever lol 2 errors by Judge and Volpe and then Cole doesn't cover first and it costs them 5 runs in 1 inning. I'm just wondering why 2 of them got an error but it didn't show an error on the play at first even though it's a routine thing and expected for the pitcher to cover first when the first baseman has to make a more difficult play.

I don't know all the rules or how everything is scored but it just struck me as odd.

Does Gerrit Cole get an error too for not covering first base?
Does Gerrit Cole get an error too for not covering first base?
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