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The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth indoi:10.3390/math9030251 fatcat:ija6dnpnazgwhd3aloeosvjoae