By using updated stellar low mass stars models, we can systematically investigate the nucleosynthesis processes occurring in AGB stars, when these objects experience recurrent thermal pulses and third dredge-up episodes. In this paper we present the database dedicated to the nucleosynthesis of AGB stars: the FRUITY (FRANEC Repository of Updated Isotopic Tables & Yields) database. An interactive web-based interface allows users to freely download the full (from H to Bi) isotopic composition, as it changes after each third dredge-up episode and the stellar yields the models produce. A first set of AGB models, having masses in the range 1.5 < M/Msun < 3.0 and metallicities 1e-3 < Z < 2e-2, is discussed here. For each model, a detailed description of the physical and the chemical evolution is provided. In particular, we illustrate the details of the s-process and we evaluate the theoretical uncertainties due to the parametrization adopted to model convection and mass loss. The resulting nucleosynthesis scenario is checked by comparing the theoretical [hs/ls] and [Pb/hs] ratios to those obtained from the available abundance analysis of s-enhanced stars. On the average, the variation with the metallicity of these spectroscopic indexes is well reproduced by theoretical models, although the predicted spread at a given metallicity is substantially smaller than the observed one. Possible explanations for such a difference are briefly discussed. An independent check of the third dredge-up efficiency is provided by the C-stars luminosity function. Consequently, theoretical C-stars luminosity functions for the Galactic disk and the Magellanic Clouds have been derived. We generally find a good agreement with observations.
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