Algebraic geometry over a not-necessarily-closed field, or over the ring of rational integers, requires something beyond maximal ideals. (In particular, number theory requires this.) Grothendieck’s approach to this is to consider not just the maximal ideals but all the prime ideals. This leads to the notion of a scheme. A point of a scheme need not be a closed set. ←This is not our intuitive image of a point. There was resistance to Grothendieck's idea. However, it did not take long for scheme theory to be recognised as a convenient and natural theory. Even though a scheme is not necessary as long as algebraic geometry is considered over an algebraically closed field, the scheme theoretic approach makes the arguments more refined.
Kenji Ueno
