By Kenji Ueno
It is a strong booklet on vital rules. however it competes with Hartshorne ALGEBRAIC GEOMETRY and that's a tricky problem. It has approximately a similar necessities as Hartshorne and covers a lot a similar principles. the 3 volumes jointly are literally a piece longer than Hartshorne. I had was hoping this may be a lighter, extra simply surveyable booklet than Hartshorne's. the topic comprises a major volume of fabric, an total survey exhibiting how the components healthy jointly may be very beneficial, and the IWANAMI sequence has a few fabulous, short, effortless to learn, overviews of such subjects--which supply facts innovations 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 supplies extra particular nuts and bolts of the fundamental structures. total it really is more uncomplicated to get an outline from Hartshorne. Ueno does additionally supply loads of "insider info" on how one can examine issues. it's a reliable ebook. The annotated bibliography is particularly attention-grabbing. yet i must say Hartshorne is better.If you get caught on an workout in Hartshorne this booklet may also help. when you are operating via Hartshorne by yourself, you will discover this replacement exposition worthy as a better half. chances are you'll just like the extra vast straight forward 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 adequate centuries. yet here's my prediction, for what it truly is worthy: That successor textbook aren't extra hassle-free than Hartshorne. it's going to reap the benefits of development considering the fact that Hartshorne wrote (almost 30 years in the past now) to make an identical fabric speedier and less complicated. it is going to contain quantity idea examples and may deal with coherent cohomology as a different case of etale cohomology---as Hartshorne himself does in brief in his appendices. it will likely be written through somebody who has mastered each element 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 impressive, unique, yet thorny parts de Geometrie Algebrique. after all 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 nobody has but boiled this right down to a textbook successor to Hartshorne. in case you write this successor *please* permit me recognize 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)
It follows that depth(E/zE) = depth(E) - 1 (COT. to prop. 6), whence the fact that EfxE is Cohen-Macaulay. Let E be of dimension n If E is a Cohen-Macaulay Theorem 3. module, then for every system of parameters x = (~1,. , z,) of E , we have the following properties: i) e,(E, n) = e(E/xE) , length of E/xE. i i ) gr,(E) = (E/xE)[X1,. , Xn] iii) Hl(x, E ) = 0 iv) &(x,E) = 0 forsll qZ1. Conversely, if a system of parameters of E satisfies any one of these prop erties, it satisfies all of them and E is a Cohen-Maca&’ module.
Properties and characterizations of regular local Let A be a regular local ring, n = glob dim A , m the maximal ideal of A , k = A/m and M a n~nzem finitely generated A-module. The following proposition compares proj dim, M and depth, M : 2, = KS&, O 1. It follows that: Tor~(Mn, k ) = Torz(Z,-2, k ) Proposition = = Tor,(Z,~,k) = Tor,+l(M, k) = 0: hence M, is free and a) is t,rue.
But if n = m/p, we have the exact sequence: 0 - p/p n In= - m/m2 + nJn2 + 0, and since [n/n’ : k] = dim A/p i we have [p/p n m2 : k] = hta p Thus if ~1,. , zp are elements of p whose images in m/m* form a k-basis of p/p n m2 i then the ideal (~1,. ,zp) is prime and of height p = htn p ; whence p = (21:. , zP) , qed. If p is a prime ideal of a regular ring A 1then the Proposition 23. local ring A, is regular. Indeed, it follows from the properties proved in part C that glob dim A, 5 glob dim A < 03 80 IV.
Algebraic Geometry 2: Sheaves and Cohomology (Translations of Mathematical Monographs) (Vol 2) by Kenji Ueno