The Geometry of Schemes David Eisenbud Joe Harris Springer [...]... by example, what various kinds of schemes look like We focus on affine schemes because virtually all of the differences between the theory of schemes and the theory of abstract varieties are encountered in the affine case — the general theory is really just the direct product of the theory of abstract varieties a la Serre and the theory of affine schemes We ` begin with the schemes that come from varieties... way that the classic theory could not handle In both these ways it has made possible astonishing solutions of many concrete problems On the number-theoretic side one may cite the proof of the Weil conjectures, Grothendieck’s original goal (Deligne [1974]) and the proof of the Mordell Conjecture (Faltings [1984]) In classical algebraic geometry one has the development of the theory of moduli of curves,... in the case where R is the affine ring of an algebraic variety V over an algebraically closed field, the points of V correspond precisely to the closed points of Spec R, and the closed points contained in the closure of the point [p] are exactly the points of V in the subvariety determined by p Exercise I-4 (a) The points of Spec C[x] are the primes (x−a), for every a ∈ C, and the prime (0) Describe the. .. schemes of this type in the next chapter This notion of the local ring of a scheme at a point is crucial to the whole theory of schemes We give a few illustrations, showing how to define various geometric notions in terms of the local ring Let X be a scheme (1) The dimension of X at a point x ∈ X, written dim(X, x), is the (Krull) dimension of the local ring OX,x — that is, the supremum of lengths of. .. to the Hilbert polynomial In Chapters IV and V we exhibit a number of classical constructions whose geometry is enriched and clarified by the theory of schemes We begin Chapter IV with a discussion of one of the most classical of subjects in algebraic geometry, the flexes of a plane curve We then turn to blow-ups, a tool that recurs throughout algebraic geometry, from resolutions of singularities to the. .. on Z I.2.2 The Local Ring at a Point The Noetherian property is fundamental in the theory of rings, and its extension is equally fundamental in the theory of schemes: we say that a scheme X is Noetherian if it admits a finite cover by open affine subschemes, each the spectrum of a Noetherian ring As usual, one can check that this is independent of the cover chosen I.2 Schemes in General 27 There is a... to be the image of f via the canonical maps R → R/p → κ(x) Exercise I-2 What is the value of the “function” 15 at the point (7) ∈ Spec Z? At the point (5)? Exercise I-3 (a) Consider the ring of polynomials C[x], and let p(x) be a polynomial Show that if α ∈ C is a number, then (x − α) is a prime of C[x], and there is a natural identification of κ((x − α)) with C such that the value of p(x) at the point... of chains of prime ideals in OX,x (The length of a chain is the number of strict inclusions.) The dimension of X, or dim X, itself is the supremum of these local dimensions Exercise I-36 The underlying space of a zero-dimensional Noetherian scheme is finite (2) The Zariski cotangent space to X at x is mX,x /m2 , regarded as X,x a vector space over the residue field κ(x) = OX,x /mX,x The dual of this... that of tangent space or of openness can be extended from schemes to certain functors This extension represents the beginning of the program of enlarging the category of schemes to a more flexible one, which is akin to the idea of adding distributions to the ordinary theory of functions Since we believe in learning by doing we have included a large number of exercises, spread through the text Their... required Returning to the case of arbitrary f , set X = Xf , R = Rf , fa = f fa ; then X = Spec R and Xfa = Xfa , so we can apply the case already proved to the primed data The proposition is still valid, and has essentially the same proof, if we replace Rf and Rfa by Mf and Mfa for any R-module M Exercise I-20 Describe the points and the sheaf of functions of each of the following schemes (a) X1 = Spec