Recently, I started reading scheme theory and in particular all this theory which has been built up by Grothendieck and his "relatives". Whilst I was reading, thought that there must be a reason for all this "new" mathematical entry by the former and the mathematicians after him as well. That is, there must be a deep relation between the structure of these "strange" (I mention that word because we define something pure geometrical from pure algebra) geometrical objects and sheaf theory.
So, my question is if you can mention some results for the topological or geometrical nature of a scheme which arise through the Theory of Sheaves (or that, we can prove it with the help of it).
Moreover I would really like to hear what's the aim behind these heavy construction and machinery. In other words, what the mathematicians thought at this time about the sheaves and how do they help the investigation in terms of scheme theory.
Thank you!