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...Artin s criteria for algebraicity revisited jackhallanddavidrydh abstract using notions of homogeneity we give new proofs m functors and groupoids our methods a more general result unifying two theorems clarifying their dierences introduction classically moduli spaces in algebraic geometry are constructed either projective or by forming suitable quotients his reshaping the foun dations half century ago grothendieck shifted focus to functor points central question became whether certain representable early on he developed formal deformation ory with intent these as main tools proving representability proof existence hilbert picard schemes however is based it was not until ten years later that completed vision series landmark papers particular vastly generalized showed pi card exist great generality also clear correct setting stacks eminent gave precise were claried b conrad andj dejongusingn eron popescudesingularization h flenner exal rst author coherent criterion very streamlined eleg...