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I am a beginner trying to learn about sheaves. I am trying to read Masaki Kashiwara's book "Sheaves on Manifolds", but I find it is not easy for me to understand.

What other books should I read first, with little knowledge about abstact algebra and homological algebra?

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Could you say a bit more about your background, like what courses/books have you already been through, and what is the reason you want to read Kashiwara - do you have a specific goal in mind? Then it will be easier to recommend good background reading. –  Andreas Holmstrom Nov 19 '09 at 23:58
Yes, what makes you reading Kashiwara? It's use as introduction to sheaf theory, the first three chapters look nice. –  Thomas Riepe Nov 20 '09 at 0:53
Perhaps you should instead ask for suggestions regarding good abstract algebra and homological algebra texts. If I recall correctly, the appendix to Kashiwara-Schapira has a brief introduction to category theory. I recommend against reading it linearly. –  S. Carnahan Nov 20 '09 at 17:29
Pierre Schapira has some notes on his website where he does an intro to sheaves. They are the lecture notes from an algebraic topology class: people.math.jussieu.fr/~schapira/lectnotes/AlTo.pdf I personally found these notes useful. Also, to echo somebody elses suggestion, if you want to read about them in the Alg Geometric viewpoint, the notes of Ravi Vakil have an easy to read chapter on sheaves. And the book "Algebraic Geometry 2" by Kenji Ueno is all about sheaf theory and is pretty elementary. –  B. Bischof Jul 3 '10 at 17:58
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6 Answers

The Kashiwara's book is quite focused and technical. I won't recommend it as an introduction to sheaves, since the abstract language of sheaves and homological algebra is most useful when you already know a big class of examples.

If you're planning on hitting algebraic geometry one day, it could be a good idea to start with reading about it now. Any technical book, e.g. Hartshorne or others suggested in this MO question will contain such material as sheaves, functors, derived functors, Verdier duality, etc.

There are also better places to learn about D-modules and related stuff; e.g. note Kashiwara's book says:

(p.411) Although perverse sheaves have a short history ...

and, indeed, 30 years later there are quite a few introductions to perverse sheaves that are easier to read.

I don't know about microlocalization, perhaps this topic should be indeed read from Kashiwara.

Now we'll be able to recommend a more specific text if you tell us what exactly you planned on reading Kashiwara for and where you get stuck!

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Sadly I'm not sure there is a better reference for microlocalization. Ginzburg's paper "Characteristic varieties and vanishing cycles" is a bit more comprehensible, if less comprehensive. –  Ben Webster Nov 22 '09 at 21:26
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I personally won't recommend Bredon's book, rather Iversen's "Cohomology of sheaves" (especially if you are interested in the topological aspects/applications of sheaf theory).

There is also Dimca's "Sheaves in topology". However I should say that the epigraph to this (very good) book is "Do not shoot the pianist", and maybe not without a reason.

If you are more into algebraic geometry, then you should read chapter 2 of Hartshorne.

A classical reference is Godement's "Topologie alge\'ebrique et th\'eorie des faisceaux".

Prerequisites for all of these are some algebra (the definitions of a ring and a module, basically, but if you've never seen complexes before, you may find the presentation a bit dense in the beginning; you'll also need some commutative algebra if you are reading Hartshorne), some basic general topology (and also some theory of smooth manifolds, e.g. partitions of unity, in the case of Iversen's book) and some category theory. You could just start reading Hartshorne or Iversen (depending on what the goal is) and then look up categorical notions that are unfamiliar in MacLane's "Categories for the working mathematician" or on Wikipedia.

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I like Dimca's book but I don't know that I'd recommend it to someone who isn't already comfortable with homological algebra. –  Greg Stevenson Nov 22 '09 at 3:37
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I like: Iversen, Cohomology of sheaves


It serves as an introduction to sheaves and their cohomology without requireing much background. Applications to topology and algebraic geometry are explained. Morover it has an appendix on derived categories.

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I think, the first volume of Harder's Lectures on Algebraic Geometry contains a nice and balanced account of sheaf theory and the cohomology of sheaves. Besides the title, it is not really a book about algebraic geometry. Instead there are many examples from algebraic topology and Riemann surfaces. One should although note that the book contains many typos.

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Assuming you already know undergraduate algebra (and maybe a little basic homological algebra already), the book Methods of Homological Algebra by Gelfand and Manin is a good source for the kinds of things in the first chapter or so of Kashiwara-Schapira, i.e. derived categories, derived functors. I can't remember how elementary its sheaf theory is, but a little background in sheaf theory wouldn't hurt either. Swan's book is probably a kinder starting point for sheaf theory then Bredon.

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There is the book by Bredon called "sheaf theory" but I'm afraid it may not be better but have a look. Of course if algebraic geometry is your goal then you can learn it directly from Hartshorne or Ueno.

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