Modern Definitions of The Objects of Algebraic Geometry
See also: spectrum of a ringAlexander Grothendieck then gave the decisive definition, bringing to a conclusion a generation of experimental suggestions and partial developments. He defined the spectrum of a commutative ring as the space of prime ideals with Zariski topology, but augments it with a sheaf of rings: to every Zariski-open set he assigns a commutative ring, thought of as the ring of "polynomial functions" defined on that set. These objects are the "affine schemes"; a general scheme is then obtained by "gluing together" several such affine schemes, in analogy to the fact that general varieties can be obtained by gluing together affine varieties.
The generality of the scheme concept was initially criticized: some schemes are removed from having straightforward geometrical interpretation, which made the concept difficult to grasp. However, admitting arbitrary schemes makes the whole category of schemes better-behaved. Moreover, natural considerations regarding, for example, moduli spaces, lead to schemes which are "non-classical". The occurrence of these schemes that are not varieties (nor built up simply from varieties) in problems that could be posed in classical terms made for the gradual acceptance of the new foundations of the subject.
Subsequent work on algebraic spaces and algebraic stacks by Deligne, Mumford, and Michael Artin, originally in the context of moduli problems, has further enhanced the geometric flexibility of modern algebraic geometry. Grothendieck advocated certain types of ringed toposes as generalisations of schemes, and following his proposals relative schemes over ringed toposes were developed by M. Hakim. Recent ideas about higher algebraic stacks and homotopical or derived algebraic geometry have regard to further expanding the algebraic reach of geometric intuition, bringing algebraic geometry closer in spirit to homotopy theory.
Read more about this topic: Scheme (mathematics)
Famous quotes containing the words modern, definitions, objects, algebraic and/or geometry:
“The higher processes are all processes of simplification. The novelist must learn to write, and then he must unlearn it; just as the modern painter learns to draw, and then learns when utterly to disregard his accomplishment, when to subordinate it to a higher and truer effect.”
—Willa Cather (1873–1947)
“What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
—G.C. (Georg Christoph)
“Luckless is the country in which the symbols of procreation are the objects of shame, while the agents of destruction are honored! And yet you call that member your pudendum, or shameful part, as if there were anything more glorious than creating life, or anything more atrocious than taking it away.”
—Savinien Cyrano De Bergerac (1619–1655)
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“... geometry became a symbol for human relations, except that it was better, because in geometry things never go bad. If certain things occur, if certain lines meet, an angle is born. You cannot fail. It’s not going to fail; it is eternal. I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total. Thus, I’m able to avoid or manipulate or process pain.”
—Louise Bourgeois (b. 1911)