寤寐无为

Main References

AM=Introduction to Commutative Algebra,by M.F.Atiyah and I.G.Macdonald

GM=Methods of Homological Algebra,by Sergei I.Gelfand and Yuri I.Manin

H=Algebraic Geometry,by Robin Hartshorne

It isn’t a book that you should sit down and read.

—Ravi Vakil

MO1=Math 256A Algebraic Geometry Course Note,by Martin Olsson

http://math.berkeley.edu/~aaron/256a.pdf

RV=Foundations of Algebraic Geometry,by Ravi Vakil

http://math.stanford.edu/~vakil/216blog/

These are course notes by an algebraic geometer who is also a master of exposition.

—Bjorn Poonen

S=Basic Algebraic Geometry,by Igor R.Shafarevich

W=An introduction to Homological Algebra,by Charles A.Weibel

1.Introduction

  Algebraic geometry was classically concerned with the geometric study of solutions to polynomial equations in several variables over C, and in the first half of 20 century it was put on a firm foundation by Zariski, Weil, and others, using the (then new) methods of commutative algebra. This allowed one to work over fields other than C, and made it possible to exploit geometric ideas in previously ‘un-geometric' subjects such as number theory.

—Brian Conrad

2.Affine Varieties

   (reading:S Ch I.2,I.3

                   H Ch I.1)

   (doing:H Ch I:1.1(a),(b),1.2,1.6,1.12)

3.Projective Varieties

   (reading:H Ch I.2)

   I have repeatedly realized that ideas developed in Paris in the 1960’s are simpler than I initially believed, once they are suitably digested.

—Ravi Vakil

4.Category theory(part I)

   (reading:W Appendix A )

   (doing:W A.1.1,A.1.2)

5.Sheaves

   (reading:H Ch II.1)

   (doing:H Ch II 1.2)

6.Schemes(ringed spaces,affine schemes,schemes,Proj and projective space)

   (reading:H Ch II.2 pp.69-77,Ch II.5 pp.108-110)

   (doing:H Ch II 2.1,2.2,2.3)

Main References

AM=Introduction to Commutative Algebra,by M.F.Atiyah and I.G.Macdonald

DF=Abstract Algebra,by David Dummit and Richard Foote

FT=Algebraic Number Theory,by A.Frohlich and M.J.Taylor

The book of Frohlich-Taylor wiil be the ”official” course text;it has lots of nice worked examples and develops background in commutative algebra.But as a text it’s theoretical development is sort of unmotivated.

—Brian Conrad

KC1=Algebraic Number Theory Course Note,by Keith Conrad

http://www.math.uconn.edu/~salisbury/notes/AlgNumThy.pdf

KC2=Local Fields Course Note,by Keith Conrad

http://www.math.uconn.edu/~salisbury/courses_taught.php

Lang1=Algebra,by Serge Lang

MB=Algebraic Number Theory Course Note,by Matthew Baker

http://people.math.gatech.edu/~mbaker/pdf/ANTBook.pdf

Neu=Algebraic Number Theory,by Jurgen Neukirch

PS=Algebraic Theory of Numbers,by Pierre Samuel

1.Prerequisite and introduction

Number fields are finite extensions of the field of rational numbers.They have been extensively studied for about 200 years.Special examples can be found in the work of Gauss from the early part of the 19th century,while the first systematic treatment was given by Kummer in the middle of that century. Gauss was motivated by his desire to extend his celebrated law of quadratic reciprocity,but much early impetus also came from the study of Diophantine equations i.e. the search for rational solutions of polynomial equations.Today number fields continue to play a central role in number theory.They are both fascinating in their own right and essential tools for other work.They are rather concrete objects,with which one can make explicit calculations.At the same time they have a beautiful abstract structure.Gauss' quest for higher reciprocity laws was to a large degree realised with the development of class field theory in the first half of the twentieth century.More recently much effort has been put into developping even more general,`non-abelian' reciprocity laws.This is often referred to as the `Langlands program'.Moreover the modern study of Diophantine equations would be unrecognizable without number fields.

—Richard L. Taylor    

(reading:MB pp.1-7

FT pp.1-7,you could skip the details of discussion about the proof of the two-square theorem,see KC1 pp1-8 in term of a soften description and also help you understand the content of FT p.3.)

2.Integrality,integral closure,ring of integers

   (reading:MB pp.7-12)

3.Traces,norms,discriminants,definition of Dedekind domains

   (reading:MB pp.12-16

                   PS 2.6,2.7,2.8)

4.Unique factorization of ideals

   (reading:MB pp.16-20)

5.Fractional ideals,ideal class groups,finiteness of the ideal class group

   (reading:MB pp.20-29)

6.Minkowski theory

7.Dirichlet unit theorem

8.Primes in field extension

9.Primes in Galois extension(decomposition groups,inertia groups and Frobenius elements)

10.Cyclotomic fields

11.Direct limits and inverse limits

     (reading:Neu Ch IV.2)

12.Introduction to valuation

     (reading:KC2 pp.1-15

                     Neu Ch II.1-II.2)

     (doing:Neu Ch II.1 exercise 1,2,3,4(hint:Neu pp.102 examples))

出发的时间是初一晚上十点五十左右，火车卧铺，倒也习惯。我和妈妈(遗憾的是因为爸爸要在家里带狗狗，所以没有去，我可不放心将胖儿子交给其他人照顾，不过更好的办法是爸爸和妈妈去，我留守家里，这大过年的，而我一向孤僻的性格，实在不太愿意出去玩几天，可没办法，老妈要求我必须去)、大舅舅一家、小舅舅一家再加上小舅舅的战友、朋友一共十七人（原谅我记不清具体的数字啦…）。我们和小舅舅一家人晚上九点半从他们家（四点左右就在我们家吃完晚饭，大约是考虑在火车上要大吃很多零食吧）坐地铁到火车北站，一路除了那些一看装束便像是去火车站的人外，倒也没有多少人，车厢显得空荡荡。十点左右到达火车站，弟弟、大舅舅、大舅妈他们早已在车站外等着，天空有些细雨，很凉，温度不太高。然后检票、进站，在候车厅休息了近一个小时。虽然是一个旅行团（只管住宿、当地火车站来回接送，其余全部自由行）可车票出了一些小问题，妹妹他们不在和我们一节车厢，一番折腾换过来，大舅舅自己卤了很多菜，再加上各自带上的一些零食，以及旅途一开始所特有的新鲜感所带来的兴奋，所有人都没有很早睡，喝酒、吃东西、聊天一直到十二点、一点方才慢慢睡去，我和弟弟在一起，各自上铺。不过好像因为一个叔叔的鼾声实在太大—第二天成为整个车厢低声抱怨的主题，我也和大部分人一样，没有怎么睡好。

第二天上午九点左右到达西昌火车站，下车赶紧拍照（是的，就是在火车站西昌二字下面，汗…）然后坐上旅行团派来的汽车，拉到吃早饭的地方，很糟糕的早餐，我什么也没有吃，就在大厅坐着（这个吃早饭的酒店并不是我们入住的酒店）。早饭完毕之后便是今天唯一的目的地-螺髻山。开车大约一个半小时，沿途就是一些曲折的山路，西昌这个城市晴空万里，一年三百六十五天有三百天都是大太阳（我们在西昌城的三天都是如此），只是早晚温差较大而已，所以嘴唇始终很干燥，喝很多水也不见好，每天清晨起来便是觉得口干舌燥，我虽然喜欢太阳，可也实在不喜欢这种感觉。螺髻山虽然离西昌市区不远，一路听导游接受当地的民俗，倒也不觉得很漫长，我和妈妈坐在汽车头排，听得很仔细，大部分人在养神休息。有趣的便是导游说到西昌彝族的人以黑胖为美...到达之后便是另外一种感觉-冷的可怕，而且随着上山温度一直降低。旅游车到达山底，排了一两个小时的队坐上他们的旅行车上了一段非常曲折的山路，又是排一个小时的队才坐上缆车（我们在山上其实也就呆了一个小时，前后排队花的时间大约是四五小时—小舅舅将这定义成“这就是旅游”），山上气温在零下，积雪很厚。下山我问司机，他讲：我们去的这天应该是这些天气温最低的一天。从景点出发，我感觉比西岭雪山好上很多，尤其是那个大湖，水面已经结冰，四遭全是白色的世界，风大的惊人，人站立都很难。我捧了一些雪，手便被冻的不行。下山坐在缆车上更是冷的可怕（上山虽然气温也很低，可有日光反而很暖和，下山快五点，太阳已经落山），只求快点下山，可缆车慢的很，上下各用四十五分钟左右。然后旅行车将我们拉回入住的酒店，洗澡，稍微休息然后去吃烧烤，可惜没有吃到久仰大名的烤茄子—据导游讲他们西昌人每天的晚饭就是烧烤，我觉得很一般，五花肉和一种像锅贴儿的东西还不错，然后吃完烧烤大家都赶紧回房早早入睡，都很疲劳，两人间，我和弟弟一间房，紧挨四楼电梯口，没有电脑…

第二天一早被大舅舅、妈妈反复催下去吃饭（我可不是睡懒觉，只是不想吃酒店里的早饭）不过饭菜比想象中的好很多，很丰富，西瓜吃了不少。人多就事儿多，每个人整理完大约也是十一点，坐酒店对门的是十七路公交车直接坐到邛海的湿地公园很挤很挤，不过最后倒是坐到座位，最后一排。极好的天气，我们沿着邛海漫步、照相，到了坐船的地方，因为买票是所谓豪华游轮，每人六十，普通的大约是二三十（普通和豪华的区别就是速度快很多—所以不能照相，噢，还有一点是豪华意味着坐的不是木凳而是皮沙发，很晕…）等了大半天，所幸天气很好，坐到小渔村找了一个吃饭的地方，然后和妈妈提前逛了逛，吃了一个烤红薯，很不错—估计是饿了。午饭终于有所谓醉虾（其实就是小活虾，所以我没吃）、烤茄子（失望无比）以及很多家常菜（这些倒不错，例如小炒肉），下午便都在这儿坐着喝茶、斗地主，我一边看看带去的一些东西、一边看看他们斗地主、然后晒太阳，也难得如此这般悠闲，倒也不想多去逛逛—只是第二天发现脸上有脱皮，这里的太阳可真厉害…因为五点半游船就已经收工，回去是坐的野的，从邛海的另外一个方向，一路上司机开的非常快—我略微小担心，又不好发作。回到酒店，洗澡、整理，晚上大家去吃酒店下的火锅鱼，我只吃了一些便毫无胃口，就早早回房休息，一直不太喜欢这种东西。晚上妈妈他们都去打牌，到凌晨一点左右。

第三天一早吃完早饭，便和妈妈他们去市区买茶叶、然后慢慢逛回酒店，西昌这个城市感觉很一般，除了天气，行人倒也很少，路面也比较干净，就是因为日光的原因显得干巴巴。不过草莓好便宜，昨天在邛海是六元一斤，今天在市区居然五元、甚至更便宜。看上去非常新鲜、口感也不错。我是极喜欢草莓的孩子，吃了很多很多，还不尽兴。十一点半左右回到酒店，整理、收拾行李、退房，将行李寄存在酒店，便又坐上是十七路去邛海，今天恰恰遇到一个加班车，很空，和昨天形成鲜明的对比—其实我看人多也是被成都人所闹的，街上随处可见川A的车牌，下午坐野的回酒店，旁边的叔叔问司机这儿的房价多少，司机讲：八九千吧.又补充上一句：就是被你们成都人买高的!好幽默...我们今天比昨天多坐了一站，下车对面就是小吃街，吃了很多羊肉串，二十元十二串，味道还行，午饭是在当地一个公安局的地方，很不错的家常菜，然后他们依然去找地方喝茶、晒太阳。我则自己一人跑去登庐山—当地的一个山名。三块钱的门票，再加上烧香所用七块，一共用了十快钱，这山呢说来就是两个词儿—庙子与猴子，沿途上山其实就是隔一段距离一个庙宇，一路的野猴子，丝毫不怕人。这山大约以前是墓地（旁边紧挨着当地牺牲的抗日战士的墓地），沿途也可见好多老式的坟墓，我三点开始爬山，一直到四点半左右，一问，上面倒也没有多少新鲜的东西，而且要爬到顶还需要一个小时，游人倒也不见少，还有陆续上山都人，偶尔一段山路有些清静。我想了想，时间也不早，这山也实在没有意思，便慢慢下山，其实这时脚也累的不行，沿着邛海走了一大段终于找到“大部队"，坐野的回酒店，妈妈他们又去打牌，九点半才集合呢，我和大舅舅又出去逛了逛，吃了一碗排骨刀削面，我自认为是三天以来吃的最美味的一餐，然后回酒店，又和舅舅朋友的妻子和女儿聊了聊天，很快便到集合时间，拿上行李，坐上旅行社派来的车，去火车站—发现火车站与酒店离的好近，十点五十多的火车晚点很多，大家上车之后再也没有来时的兴奋与精神，各自洗漱完毕之后便早早的睡觉，我和来时一样，依然是上铺，一觉睡到第二天一早，火车到达成都火车南站已经近十点，晚点一个半小时左右。

Dear friends:

  Happy holidays!(or by our custom,I should say happy new year,blablabla.)

  But the truth is that I extremely dislike this word ”new” as if it’s absurd unusual comparing to otherdays.Unfortunately,I found my remark is also absurd so that nowhere to run to it even though I always insist everyday is present.

  Forgive me I definitely don’t have any intresting to express meaningless words like a layman.You know,I care about those person even if he or she don’t care about me,naturally,I don’t care about those person even if he or she care about me.

  Haha,look like a zen and now be serious,let me talk about some plans.

  In future,I don’t(or more accurately plus a word:seldom)use qq、micro-blog anymore.So any information through there will be probably ignore unintentionally.Therefore,if you have anything,please tell me in email(or in person?).Additional beside my learning and working,if I have extra energy I will be concentrate all attention on my blog’s learning articles series since I pretty sure it’s good for me.Certainly,I really want to learn further French(that means I can speak thank you or I love you at least)and learn playing the violin if possible.

  Tomorrow,I will be travel five days and thoroughly separate the way that sit quietly in room alone recently.

  At last,I read an intresting sentence by accident and I quote here as the end of this remark and this year:每一个不想恋爱的人,心里都住在一个不可能的人。

Sincerely

Sun

The main motivation of these articles(see following named my learning roadmap for Algebraic Number Theory and my learning roadmap for Algebraic Geometry )come from thinking through my learning recently,I found my knowledge of these topic(obviously they are the most important topics to my research or just called my research interesting)learned form whether professors or myself is not only in term of solid but also lack of systematic.Due to my preparing for the phd qualifying exam currently(believe me,the word “currently” is not absolutely accurate),I start to reconsider(frankly speaking,almost of are “relearn”) these material and learn futher.Therefore,in order to be more effectively or just only keep on pushing myself,I try editing them and adding process anytime(even the existing contents are drafts,which maybe modify completely in some day).In short,this is purely “My” learning process and the material included are simply those which i have come across or coming across in my own learning.Then,if you’d like to waste of your time have a glance it and find an omission or error,I would be grateful if you would email me at xiaosunliang@gmail.com.

Although I also enjoyed thinking through the structure of these subjects,I’m not qualified to any of these topics.Therefore,almost choices made by related course page、private discussion etc.My contribution to thematerial following mainly consist of collect、consider and practice.

In principal,I‘m not assume any prerequisites even the elementary results for some topic such as module over PID、finite Galois Theory etc)even though this is extremely unreasonable.Of course,I know this choice looks weird,however I insist it for some reasons.On the one side,I wrote some articles about book recommended in past,I have to admit they are extremely terrible,then I really want to rewrite some materials instead. On the another side since I usually read some basic books or lectures again and again,hence I pretty sure the prerequisite part is incomparable huge(or copious?) and mingled with more repeated so you could completely neglect this part.

Anyway the results belong to Abstract Algebra、Commutative Algebra and Homological Algebra etc being everywhere.Nevertheless,I promise they are absolutely go along with the essential contens of Number Theory and Algebraic Geometry in parallel.

Remark:I don't have firsthand experience with all texts,but others have recommended them.Therefore,I still put them in the reference.