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Let ##(X,\mathcal{O}_X)## and ##(Y,\mathcal{O}_Y)## be locally ringed spaces. If I were to take the product of two, what would the corresponding product space look like? I'd imagine that the new underlying topological space would be ##X\times Y##. However, I don't know what to do about the structure sheaves.

Any nudges in the right direction would be helpful. Thank you!