By Kenji Ueno
It is a sturdy e-book on very important principles. however it competes with Hartshorne ALGEBRAIC GEOMETRY and that's a tricky problem. It has approximately an analogous necessities as Hartshorne and covers a lot an identical rules. the 3 volumes jointly are literally a piece longer than Hartshorne. I had was hoping this may be a lighter, extra simply surveyable ebook than Hartshorne's. the topic comprises a massive volume of fabric, an total survey exhibiting how the components healthy jointly may be very necessary, and the IWANAMI sequence has a few remarkable, short, effortless to learn, overviews of such subjects--which provide evidence suggestions yet refer in other places for the main points of a few longer proofs. however it seems that Ueno differs from Hartshorne within the different path: He provides extra particular nuts and bolts of the elemental structures. total it really is more straightforward to get an outline from Hartshorne. Ueno does additionally provide loads of "insider info" on how you can examine issues. it's a reliable booklet. The annotated bibliography is especially attention-grabbing. yet i must say Hartshorne is better.If you get caught on an workout in Hartshorne this ebook may also help. while you are operating via Hartshorne by yourself, you will discover this substitute exposition beneficial as a spouse. it's possible you'll just like the extra wide simple remedy of representable functors, or sheaves, or Abelian categories--but you'll get these from references in Hartshorne as well.Someday a few textbook will supercede Hartshorne. Even Rome fell after sufficient centuries. yet here's my prediction, for what it's worthy: That successor textbook usually are not extra undemanding than Hartshorne. it's going to benefit from growth seeing that Hartshorne wrote (almost 30 years in the past now) to make an analogous fabric swifter and easier. it is going to comprise quantity conception examples and should deal with coherent cohomology as a unique case of etale cohomology---as Hartshorne himself does in brief in his appendices. it is going to be written through a person who has mastered each point of the maths and exposition of Hartshorne's booklet and of Milne's ETALE COHOMOLOGY, and prefer either one of these books it's going to draw seriously on Grothendieck's great, unique, yet thorny parts de Geometrie Algebrique. in fact a few humans have that point of mastery, particularly Deligne, Hartshorne, and Milne who've all written nice exposition. yet they can not do every thing and not anyone has but boiled this right down to a textbook successor to Hartshorne. in the event you write this successor *please* enable me comprehend as i'm demise to learn it.
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Additional info for Algebraic Geometry 2: Sheaves and Cohomology (Translations of Mathematical Monographs) (Vol 2)
Defined by x ~ Xa. for x E lim Ea. are called the natuml maps +-+-of the projective limit. If the Ea. are rings, modules or groups, and fe homomorphisms of these structures, then lim Ea. is a structure of the same type. The reader can find a +-more detailed description of this construction in Atiyah and Macdonald , Chap. 10. Here we should bear in mind that the condition that the partial ordered set I is directed is not essential for the definition of projective limit. Now we are ready for the final definition: O(U) = limO(D(f)), <-- where the projective limit is taken over all D(f) C U relative to the system of homomorphisms p~~~?
Example 4. Example 3 of a ringed space shows that Spec A is a scheme for any ring A. Schemes of this type are called affine schemes. Ring homomorphisms A: A - t Band morphisms of schemes Spec B - t Spec A are in one-to-one correspondence; the correspondence is given by cp = a A. Example 5. We explain how the notion of quasiprojective variety fits into the framework of schemes. We start from the case of an affine variety X over an algebraically closed field k. The scheme Spec(k[X]) defined in Example 4 is not equal to X even as a set: the points of Spec(k[X]) are all the prime ideals of k[X], which correspond in turn to all the irreducible subvarieties of X, not just its points.
A ---4 B is the natural quotient map, then as sets, Spec A = Spec B, and cp = a>. is the identity map, whereas even on U = Spec B the map 1/Ju = >. is not an isomorphism. Thus a morphism of ringed spaces cannot be reduced to the map of the corresponding topological spaces. Remark 2. The notion of ringed space provides a convenient principle for the classification of geometric objects. Consider, for example, differentiable manifolds. They can be defined as ringed spaces, namely, as those for which every point has a neighbourhood U such that the ringed space U, (')IU is isomorphic to U, 0, where U is a domain in n-dimensional Euclidean space, and (') is the sheaf of differentiable functions on it.
Algebraic Geometry 2: Sheaves and Cohomology (Translations of Mathematical Monographs) (Vol 2) by Kenji Ueno