[Weibel Homological Algebra Djvu
Everardo Laboy
I am interested in learning about the derived categories of coherent sheaves, the work of Bondal/Orlov and T. Bridgeland. Can someone suggest a reference for this, very introductory one with least prerequisites. As I was looking through the papers of Bridgeland, I realized that much of the theorems are stated for Projective varieties (not schemes), I've just started learning Scheme theory in my Algebraic Geometry course, my background in schemes is not very good but I am fine with Sheaves. It would be better if you suggest some reference where everything is developed in terms of Projective varieties.
As for Bridgeland's work, I would recommend reading his papers directly, using Huybrecht's book as a reference (as mentioned by Francesco above, "Fourier-Mukai transforms in algebraic geometry"). Specifically for Bridgeland stability conditions, I also have some short notes onmy homepage, but again, you should also read his original papers.
Let me suggest a perhaps longer program to master derived categories of coherent sheaves. In a comment you said "I don't know much Homological Algebra", so first you have to master the basics of abelian categories and ext and tor of modules. The first chapters of Weibel's "Homological Algebra" may be a useful reference.
This said, I agree with Polizzi's suggestion of Huybrechts' book. The only issue I would point out is that the book concentrates on the bounded derived category of coherent sheaves. It is enough for its application to the classification of non-singular varieties. But if you have varieties with singularities in mind then things may get more complicated. First there is a difference between perfect complexes and bounded complexes of coherent sheaves. Second one sometimes need to use infinite methods, therefore one is forced to consider quasi-coherent sheaves and unbounded derived categories. For this beautiful theory, a good introduction is (as I have suggested here before) the first chapters of Lipman's "Notes on Derived Functors and Grothendieck Duality", (in Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics, no. 1960, Springer, 2009).
Scheme theory won't be that important (as you said, most of the time you are concerned with smooth projective varieties over $\mathbb C$). However, you didn't mention about your experience in homological algebra. Huybrechts' book is a pleasure to read, but it can't serve as a textbook in homological algebra. I would recommend e.g. reading first first few chapters of Weibel's book (1, 2, 3, 5, 10). Gelfand-Manin "Methods of Homological Algebra" is also a good reference here.
Which books would you recommend, for self-studying homological algebra, to a beginning graduate (or advanced undergraduate) student who has background in ring theory, modules, basic commutative algebra (some of Atiyah & Macdonald's book) and some (basic) field theory?
Homological algebra, a fundamental branch of mathematics, explores the intricate relationships between algebraic structures through the study of homology and cohomology theories. It serves as a powerful tool in both pure and applied mathematics, illuminating the connections between algebraic topology, algebraic geometry, and number theory. Emphasising the transformation of mathematical problems into algebraic terms, homological algebra has become indispensable for researchers and students alike, enhancing understanding of complex mathematical concepts.
Homological Algebra is a branch of mathematics that studies abstractions of algebraic structures using the concepts of homology and cohomology. It's a tool used to solve problems in various fields including algebraic topology, group theory, and algebraic geometry, by examining the relationships between objects rather than the objects themselves.
Understanding homological algebra starts with getting to grips with some of its fundamental concepts such as chains, boundary operators, and exact sequences. These elements form the building blocks of homological algebra and help mathematicians understand how algebraic structures interact.
At its core, homological algebra involves the study of mathematical objects and their functions. It uses sequences of algebraic objects and maps between them, known as complexes, to understand properties that are invariant under certain types of transformations.
The concepts of exact sequences and chain complexes are crucial for detecting holes and assessing the shape of mathematical objects without directly observing them. These techniques delve into understanding the underlying structure and solving problems that may seem intractable at first glance.
To bring the concepts of homological algebra to life, let's explore a few simple examples. These will illustrate the use of exact sequences and the role of homological algebra in solving problems.
Example 1: Consider a short exact sequence \[0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0\]. In this context, the sequence is exact if the image of each map is exactly the kernel of the next. This property helps in understanding the relationship between the algebraic objects A, B, and C.
Example 2: Looking at homology in the context of a torus, you can use homology groups to detect the presence of holes. A torus has a homology group that differs from a sphere, indicating it has a different number of holes and thus, a different structure.
Through these examples, you can see how homological algebra enables mathematicians to explore the properties of algebraic structures from a new perspective. This branch of mathematics offers powerful tools for abstraction, allowing for a deeper understanding of the relationships between different mathematical entities.
Exploring the depths of Homological Algebra through Rotman's perspective offers a pathway to understand complex algebraic structures and the interrelations between them. This journey through Homological Algebra will illuminate how this branch of mathematics serves as a cornerstone for numerous fields, providing tools for abstract analysis and problem-solving.
Joseph J. Rotman's work in Homological Algebra stands as a beacon for students and professionals alike. His approach breaks down the subject into digestible sections, starting with the most fundamental concepts and gradually moving towards more complex ideas. By introducing a series of meticulously designed examples and exercises, Rotman ensures the reader not only understands but also applies the concepts of Homological Algebra effectively.
One of Rotman's strengths lies in his ability to connect theory with real-world applications, making abstract concepts more tangible and understandable. His emphasis on the historical development of the subject alongside its practical uses adds depth to the learning journey.
Chain Complex: A sequence of abelian groups or modules connected by boundary operators, where the image of one map is the kernel of the following map, symbolically represented as \[ ... \rightarrow A_n+1 \rightarrow A_n \rightarrow A_n-1 \rightarrow ... \].
A deeper look into exact sequences reveals their ubiquity in various mathematical disciplines. Acting as a cornerstone in Homological Algebra, these sequences not only facilitate the investigation of algebraic structures but also bridge the understanding between different branches of mathematics like topology and algebraic geometry. Rotman leverages exact sequences to explain complex concepts, such as torsion and free modules, by methodically illustrating their roles in understanding the intrinsic properties of algebraic objects.
Homological methods in commutative algebra involve sophisticated techniques that help illuminate the structure and properties of algebraic systems. By focusing on homological approaches, you can gain a deeper understanding of how algebraic entities relate to each other within a commutative context.
Homological algebra serves as a powerful tool in understanding the intricacies of algebraic structures, particularly in commutative algebra. A key aspect includes the study of modules over a commutative ring and how these modules interact through exact sequences and homological dimensions. These concepts are not only foundational but also provide a unified framework to tackle complex algebraic problems.
At the heart of homological methods lie the notions of resolutions and Tor and Ext functors. These allow the exploration of the depth and projective dimension of modules, which are crucial in identifying the properties of rings and algebraic systems.
Projective Dimension: A measure of the complexity of a module with respect to projective resolutions. It is an indicator of the minimal number of projections needed to build the module from a projective module.
Imagine a module M over a ring R, where M has a projective resolution of length 2. This means that we can find projective modules P0, P1, and P2 such that there exists an exact sequence \[0 \rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow M \rightarrow 0\]. This sequence effectively explains how M can be constructed from simpler, projective modules.
Implementing homological methods in algebraic structures involves using specific tools like derived functors, spectral sequences, and local cohomology to analyse and solve problems in algebra. These approaches provide deep insights into the characteristics of modules, ideals, and rings, thereby offering a panoramic view of their algebraic properties and interrelations.
The application of derived functors in calculating cohomology groups paves the way to identifying the extent to which a given algebraic structure deviates from being perfect or semi-simple. This is especially useful in the study of sheaves and the cohomological analyses of algebraic varieties.
795a8134c1