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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
1
vote
Is the "addition" of flat morphisms flat?
This shouldn't be the case.
Let $X = Y = \mathbb{P}^1$ with projective coordinates $[x:y]$, $f = id_X$. $F = \mathcal{O}_X$, $G = \mathcal{O}_X \oplus \mathcal{O}_X(1)$. Let $\phi_1 = id \oplus x$ an …