Proof Verification: $\Omega_{X \times Y/S} \cong p_1^* \Omega_{X/S} \oplus p_2^* \Omega_{Y/S} $

solution-verification algebraic-geometry commutative-algebra

Yes, this is correct.

You might be able to gain some slight improvements by simplifying your tensor products a bit (i.e. $(A\otimes_C B)\otimes_B\Omega_{B/C} \cong A\otimes_C \Omega_{B/C}$) but this is pretty much the right argument. This is a pattern in algebraic geometry: frequently one has something to prove and by a series of reductions it suffices to verify it in the affine case, where it turns in to a problem of pushing explicit elements of rings around.

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