By David Eisenbud, Joe Harris
The idea of schemes is the basis for algebraic geometry proposed and elaborated through Alexander Grothendieck and his co-workers. It has allowed significant growth in classical parts of algebraic geometry resembling invariant conception and the moduli of curves. It integrates algebraic quantity idea with algebraic geometry, pleasing the goals of previous generations of quantity theorists. This integration has ended in proofs of a few of the most important conjectures in quantity concept (Deligne's evidence of the Weil Conjectures, Faltings' evidence of the Mordell Conjecture).
This e-book is meant to bridge the chasm among a primary direction in classical algebraic geometry and a technical treatise on schemes. It makes a speciality of examples, and strives to teach "what goes on" at the back of the definitions. there are numerous routines to check and expand the reader's realizing. the necessities are modest: a bit commutative algebra and an acquaintance with algebraic types, approximately on the point of a one-semester direction. The ebook goals to teach schemes when it comes to different geometric rules, similar to the speculation of manifolds. a few familiarity with those rules is beneficial, although now not required.
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Extra resources for The Geometry of Schemes (Graduate Texts in Mathematics, Volume 197)
Exercise II-2. Consider the map of the aﬃne line Spec K[x] to itself induced by the ring homomorphism K[x] → K[x] mapping x to x2 . Show 50 II. Examples that the scheme-theoretic ﬁber over the point 0 is the subscheme of the line deﬁned by the ideal (x2 ). Among all schemes, those associated to aﬃne varieties over algebraically closed ﬁelds may be characterized as spectra of rings R that are – ﬁnitely generated – reduced algebras – over a ﬁeld – that is algebraically closed. To get a sense of what more general schemes look like, and what they are good for, we will in the remainder of this section and the next consider what may happen if we remove these four restrictions.
To begin with, for any scheme S we deﬁne a quasicoherent sheaf of OS algebras. 3 Relative Schemes 41 U = Spec Rf ⊂ U, we have F (U ) = F (U ) ⊗R OS (U ) = F (U ) ⊗R Rf as R = OS (U )-algebras. We then associate to any quasicoherent sheaf F of OS -algebras on a scheme S a scheme X = Spec F, together with a structure morphism X → S, such that in case S = Spec R is aﬃne we get simply X = Spec F (S) together with the structure morphism X → S induced by the R = OS (S)-algebra structure on F (S). There are a couple ways to do this.
If X = Y , it gives the set on which the maps ϕ, ψ are equal, the equalizer of the maps. 36 I. Basic Deﬁnitions Exercise I-45. Check these assertions! Note that X ×S Y comes with natural projection maps to X and Y making the diagram X ×S Y ✲ X ϕ ❄ ✲ S ψ ❄ Y commute. Indeed, the set X ×S Y may be deﬁned by the following universal property: among all sets Z with given maps to X and Y making the diagram ✲ X Z ❄ Y ϕ ❄ ✲ S ψ commute, X ×S Y with its projection maps is the unique “most eﬃcient” choice in the sense that, given the diagram with Z above, there is a unique map Z → X ×S Y making the diagram Z ✲ X ×S Y ✲ ❄ Y ✲ ✲ X ϕ ψ✲ ❄ S commute.