A flat morphism is a type of morphism between schemes that preserves the property of being flat, meaning it reflects the behavior of modules in a way that ensures that the fibers over each point do not 'collapse'. This property is essential when dealing with local rings and allows for the preservation of certain algebraic structures during base change. In essence, flatness assures that the morphism behaves well with respect to the tensor product, particularly in contexts where one wants to maintain dimensions and regularity.
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