书架与偏见 Bookshelf & Prejudices

正如我之前所言，任何一个想要掌握现代代数几何基本语言的学生，应当熟读本书的大部分章节，至少是 Chapter 2 的 Section 1-7 和 Chapter 3 的 Section 2-7。否则，我并不认为这样的人可以真的了解代数几何的基本概念。 纵使现在已经有太多 Hartshorne 之外的代数几何教材 如 Vakil 的 FOAG，GTM197，Griffiths & Harris，the Red Book，Lei Fu，刘青…… 但归根结底，对绝大多数人来说，Hartshorne 依旧是最好的参考书，即便未必是入门的第一选择。

As I have stated before, any student who aspires to master the fundamental language of modern algebraic geometry must work through most chapters of this book, at least Sections 1-7 of Chapter 2 and Sections 2-7 of Chapter 3. Otherwise, I doubt one can truly grasp the core concepts of the field. Even with the plethora of alternatives available today Such as Vakil's FOAG, GTM197, Griffiths & Harris, the Red Book, Lei Fu, Liu Qing... , Hartshorne remains the gold standard for reference, even if it isn't necessarily the gentlest introduction.

至于习题，我认为如果初学者能做完其中一半（起码是我提到的那几个章节的一半习题）， 那已经可以算是初步掌握了这套语言 有些人认为应当独立做上 80% 以上甚至全部的习题，我其实并不认为这是效率最高的方式。除了极少数交换代数功底扎实的学生，这对大部分人来说未免过于苛刻。 ，没必要再去别的书里找什么刻意练习。如果读者觉得练习还不够，或是理解仍然不佳，那么就把剩下的也做一做，包括Chapter 1。

Regarding the exercises, if a student can complete half of them (specifically half of those in the aforementioned sections) they can be considered to have achieved a preliminary understanding. Though some insist on tackling 80% or even the entire set, I personally don't see that as the most efficient path; it’s frankly too harsh for anyone without an exceptionally rock-solid foundation in commutative algebra. There is hardly any need to seek extra deliberate practice in other textbooks. If the reader feels that the exercises are still not enough, or if their understanding is still lacking, then they should also do the remaining ones, including Chapter 1.

书里的行文比较流畅，每一个命题的引用都严谨地依照先前的结论。证明过程也合乎自 Grothendieck 以来代数几何教材的经典哲学：足够详尽，以至于读者大部分时间会觉得是在容易的推论，而非面对诡异而缺乏动机的构造。当然这另一方面也意味着部分证明显得过于冗长，而且这种行文风格本身，也可以视作一种习题。

The prose is quite fluid, with every citation of a proposition strictly following previously proven results. The proofs adhere to the classical philosophy of post-Grothendieck textbooks: they are detailed enough that readers often feel they are performing straightforward derivations rather than struggling with daunting constructions. However, this also leads to some lengthy proofs, and the writing style itself becomes a form of exercise in its own right.

关于我的批判主要仍然集中在它的时代局限性上。其成书已经五十年，书中对某些性质的刻画非常古典，甚至生硬。尤其在层和概形的引入及基本构造，直到后来我自己琢磨出了一套更轻松的定义并检查了等价性后，才算是放下了这一段。对于现在的学生来说，引入更多范畴论的语言，无论是在直观程度还是论证力度上，显然都更有优势。

My critique mainly focuses on its dated nature. Since it was written fifty years ago, some characterizations of properties feel overly classical, if not clunky. The construction of sheaves and schemes, in particular, is quite cumbersome, so much so that I eventually had to devise my own definitions and verify them myself. For today's students, adopting the language of category theory would be far more advantageous in terms of both clarity and conceptual power.

如果说在我眼中 GTM 52 是一本顶尖的代数几何教材，那么 A-M 的《交换代数导引》在我看来顶多算是一本给大一学生的科普手册，它最大的优点是薄，而最大的缺点是它本可以更薄。

If GTM 52 is, in my eyes, an exceptional textbook on algebraic geometry, then Atiyah-MacDonald’s "Introduction to Commutative Algebra" is merely a popular science manual for freshmen. Its greatest strength is its brevity, and its greatest weakness is that it could have been even briefer.

许多人都将其视为研读 GTM 52 之前的指定前置教材，但在我看来，这并没有什么实质性的意义。这种说法的最大诱因，无非是 Hartshorne 在书中频繁直接引用 A-M 的命题。 事实上，整本书最有价值的仅是前四章。 甚至前三章足矣，第四章的准素分解在代数几何中出镜率极低，真正用得多的是相伴素理想，而书中对此几乎没展开讨论。 至于后文的整相关性、赋值环，乃至链条件和维数理论，在代数几何中的分量远没有想象中那么重。在 Hartshorne 中，真正大规模用到它们的地方，只有 Chapter 2 Section 6 的除子理论。

Many regard it as the mandatory prerequisite for GTM 52, but I see little real significance in that. The primary reason for this reputation is simply that Hartshorne frequently cites its propositions directly. In fact, the only truly valuable parts are the first four chapters—or even just the first three. Primary decomposition in Chapter 4 is rarely seen in AG compared to associated primes, which the book barely touches upon. The subsequent topics like integral dependence, valuation rings, and even chain conditions or dimension theory, are far less critical in algebraic geometry—appearing substantially only in the divisor theory of Chapter 2, Section 6.

关于习题，我认为前四章理应全部做完，至少也要覆盖 90% 的内容。 这一方面是为了训练高中生水平的代数基本功， 事实上有些题目确实有些难度，但这种难不在于智力或技巧，而在于时间和熟练度。如果你初次尝试时感到寸步难行，其实只需停下来，在通勤时多琢磨定义。下次再看，你会发现不少题目竟然变得显然了。 不至于在国内极度轻视代数的教学体系下沦为瘸腿；另一方面，这些题目确实也算不上困难。

As for the exercises, one should ideally complete all of the first four chapters, or at least 90%. This serves partly to discipline one's algebraic foundation—ensuring one doesn't end up lame under an educational system that neglects algebra; To be fair, these problems can be tough, but the difficulty lies in time and familiarity rather than raw intellect or tricks. If you're stuck, just spend your commute ruminating on the definitions. You'll find many become "obvious" by the next attempt. and partly because these problems aren't intellectually insurmountable.

至于剩下的习题，大一大二的学生可以权当消遣练练手；但对于大三的学生，正文的重要性远大于习题。 即便二者皆是走马观花，短期内也不会遗漏什么要命的东西。 等到真用上的时候，你自然会回头补。我认为后几章唯一必须理解的命题是：在一维诺特局部整环的情况下，Regular、UFD、PID、整闭、DVR 是等价的。除此之外，一切都没那么重要。 而对于大四还没读过 A-M 前四章且不了解环与模基本语言的学生，我建议就不要往代数方向挤了。

For the remaining exercises, freshmen or sophomores might try them for fun, but for juniors, the main text is far more important than the problems. Even a cursory glance at both won't leave you stranded; you can always backtrack when necessary. The only proposition worth memorizing is the equivalence of Regular, UFD, PID, Integrally Closed, and DVR in 1D Noetherian local domains. Beyond that, nothing else is truly vital. For a senior who hasn't touched the first four chapters or mastered the basic language of rings and modules, I wouldn't recommend pursuing algebra.

因此我始终认为，学习代数几何根本不需要先花一整学期去啃交换代数。一个月足矣；而对于在近世代数中已经熟练掌握环与模语言的人来说，甚至一周就够了。 真正需要在进入代数几何前反复强调的，只有张量积、Hom 函子以及局部化等几个重要的概念及其性质。 这些才是构成现代概型语言和层论的基础，其他的代数性质大可等到具体几何问题出现时再按图索骥。 至于其他内容，正如没有做椭圆偏微分方程的人还会指望靠读同济高数去学多变量微积分一样，当你在前沿研究中真正需要用到更深层次的交换代数（比如奇异点消解）时，A-M 这本科普手册早就帮不上忙了，你自然会去翻阅 Matsumura 甚至 EGA。

Therefore, I firmly believe that studying algebraic geometry does not require dedicating a whole prerequisite semester to commutative algebra. A single month is more than enough; for those already fluent in the language of rings and modules from abstract algebra, a week will suffice. The only concepts that demand rigorous emphasis before diving in are tensor products, Hom functors, and localization. These form the absolute backbone of modern scheme theory and sheaves; other algebraic properties can simply be picked up on the fly when specific geometric problems arise. As for the rest, it is exactly like expecting someone working on elliptic PDEs to rely on a freshman calculus textbook to learn multivariable analysis—by the time your research actually requires deeper commutative algebra (such as for the resolution of singularities), Atiyah-Macdonald will have long ceased to be of any help, and you will naturally turn to Matsumura or EGA anyway.

我对这本书的批判同样在于其古老且保守。许多命题在范畴论中已有现成的结论或陈述，但本书依然固执地采用逐个元素的论证方式。当然，这确实受限于成书之时范畴论尚未成为一种通用且成熟的语言。

My critique of this book also stems from its "ancient and conservative" approach. Many propositions have ready-made categorical counterparts, yet the book insists on element-wise proofs—a byproduct of an era when category theory had not yet become the pervasive and refined language it is today.