在这里，我将以翻译的方式和大家一起读这本“the rising sea” 代数几何我是读过的，只不过是不熟练。但我为什么不选择再重读52呢？一方面是因为手头没有，况且我也嫌弃重复52太过繁碎；另一方面是被格洛腾迪克想象海潮的传说吸引，期望能够从中得到咱对数学的享受。是的，如今的咱更觉得能够享受到自己，享受到阅读与思考中的安宁，学习和研究中的乐趣。比追逐更加抽象，更加罕为人至的数学更加重要。 前言摘要 Chapter 1. Category theory is only language, but it is language with an embedded logic. Category theory is much easier once you realize that it is designed to formalize and abstract things you already know. The initial chapter on category theory prepares you to think cleanly. For example, when someone names something a “cokernel” or a “product”, you should want to know why it deserves that name, and what the name really should mean. The conceptual advantages of thinking this way will gradually become apparent over time. Yoneda’s Lemma — and more generally, the idea of understanding an object through the maps to it — will play an important role. 翻译：第 1 章。范畴论只是语言，但它是具有内在逻辑的语言。一旦你意识到范畴论旨在形式化和抽象你已经知道的事物，它就会容易得多。关于范畴论的第一章让你准备好清楚地思考。例如，当有人将某物命名为“cokernel”或“product”时，您应该想知道为什么它配得上这个名字，以及这个名字的真正含义。随着时间的推移，这种思维方式的概念优势将逐渐显现出来。米田引理——更一般地说，通过映射到物体来理解物体的想法——将发挥重要作用。 Chapter 2. The theory of sheaves again abstracts something you already understand well (see the motivating example of §2.1), and what is difficult is understanding how one best packages and works with the information of a sheaf (stalks, sheafification, sheaves on a base, etc.). 翻译：第 2 章。层论再次抽象了你已经很好地理解的东西（参见§2.1的动机示例），困难的是理解如何最好地打包和使用层的信息（茎， 层化、"a base"上的层等）。 Chapters 1 and 2 are a risky gamble, and they attempt a delicate balance. Attempts to explain algebraic geometry often leave such background to the reader, refer to other sources the reader won’t read, or punt it to a telegraphic appendix. Instead, this book attempts to explain everything necessary, but as little as possible, and tries to get across how you should think about (and work with) these fundamental ideas, and why they are more grounded than you might fear. 翻译：第1章和第2章是一场冒险的赌博，他们尝试一个微妙的平衡。尝试解释代数几何通常会留给读者的背景：读者不会阅读的其他参考来源，或者把它放在一个”telegraphic appendix“上。相反，本书试图解释一切必要的事情，但尽可能少的，尝试向你。说明应该如何思考（和处理）这些基本思想。以及为什么他们比你想的的更通俗。 Chapters 3–5. Armed with this background, you will be able to think cleanly about various sorts of “spaces” studied in different parts of geometry (including differentiable real manifolds, topological spaces, and complex manifolds), as ringed spaces that locally are of a certain form. A scheme is just another kind of “geometric space”, and we are then ready to transport lots of intuition from “classical geometry” to this new setting. (This also will set you up to later think about other geometric kinds of spaces in algebraic geometry, such as complex analytic spaces, algebraic spaces, orbifolds, stacks, rigid analytic spaces, and formal schemes.) The ways in which schemes differ from your geometric intuition can be internalized, and your intuition can be expanded to accomodate them. There are many properties you will realize you will want, as well as other properties that will later prove important. These all deserve names. Take your time becoming familiar with them. 翻译：有了这个背景，您将能够清晰地思考在不同几何中研究的各种各样的空间（包括可微实流形、拓扑空间和复流形），如 局部具着特定形式的赋环空间。 一个概型只是另一种“几何空间”，然后我们准备将许多直觉从“经典几何”转移到这个新设定中。（这也将使您以后考虑代数几何中的其他几何空间，例如复解析空间、algebraic spaces、orbifolds、叠、rigid analytic spaces和形式概型。）你将会接受这些几何直觉，并且扩展至适应他。你会意识到你会想要很多属性，以及后来这些性质被证明很重要。这些都配得上名字。花时间熟练他。 Chapters 6–10. Thinking categorically will lead you to ask about morphisms of schemes (and other spaces in geometry). One of Grothendieck’s fundamental lessons is that the morphisms are central. Important geometric properties should really be understood as properties of morphisms. There are many classes of morphisms with special names, and in each case you should think through why that class deserves a name. 翻译：第6-10章。范畴性的思考会引导你询问概型的态射（和其他几何中的空间）。格罗滕迪克的基本lesson之一是”态射是核心“。重要的几何性质实际上应该理解为态射的性质。有许多类具有特殊名称的态射，在每种情况下，您都应该考虑为什么该类态射值得被命名。 Chapters 11–12. You will then be in a good position to think about fundamental geometric properties of schemes: dimension and smoothness. You may be surprised that these are subtle ideas, but you should keep in mind that they are subtle everywhere in mathematics. 翻译：第11-12章。您将能够很好地考虑概型的基本几何性质：维度和平滑性。你可能会惊讶于这些都是微妙的想法，但你应该记住，这些微妙在数学中无处不在。 Chapters 13–21. Vector bundles are ubiquitous tools in geometry, and algebraic geometry is no exception. They lead us to the more general notion of quasicoherent sheaves, much as free modules over a ring lead us to modules more generally. We study their properties next, including cohomology. Chapter 19, applying these ideas ideas to study curves, may help make clear how useful they are. 翻译：第13-21章。向量丛是几何学中无处不在的工具，代数几何也不例外。它们将我们走向更一般的拟凝聚层的概念，就像环上的自由模将我们引向更一般的模一样。接下来我们研究它们的性质，包括上同调。第19章，将这些想法应用于研究曲线，能够有助于明确它们是多么有用。 Chapters 23–30. With this in hand, you are ready to learn more advanced tools widely used in the subject. Many examples of what you can do are given, and the classical story of the 27 lines on a smooth cubic surface (Chapter 27) is a good opportunity to see many ideas come together. 翻译：第23-30章。有了这个，您准备好学习该在该主题下广泛使用的更高级工具了。这里给出了许多你可以做什么的例子：比如一个经典的故事——在光滑的cubic surface上的27条线（第27章），这是一个很好的机会，能够看到许多想法汇集在一起。 [b]第一章：一点范畴论[/b] [i]格洛腾迪克说：数字 0 或群的概念引入是多么平凡的事情，数学或多或少停滞了数千年，却因为没有人采取过如此幼稚的步骤......[/i] 哇，这也太神秘。 1.1 1.2先跳过 1.31.3 万有性质决定对象同构意义下的唯一性