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Commutative Algebra commutative algebra Lectures delivered by Jacob Lurie Notes by Akhil Mathew Fall 2010, Harvard Last updated 12/1/2010 Contents Lecture 1 9/1 x1 Unique factorization 6 x2 Basic definitions 6 x3 Rings of holomorphic functions 7 Lecture 2 9/3 x1 R-modules 9 x2 Ideals 11 Lecture 3 9/8 x1 Localization 13 Lecture 4 9/10 x1 SpecR and the Zariski topology 17 Lecture 5 [Section] 9/12 x1 The ideal class group 21 x2 Dedekind domains 21 Lecture 6 9/13 x1 A basis for the Zariski topology 24 x2 Localization is exact 27 Lecture 7 9/15 x1 Hom and the tensor product 28 x2 Exactness 31 x3 Projective modules 32 Lecture 8 9/17 x1 Right-exactness of the tensor product 33 x2 Flatness 34 Lecture 9 [Section] 9/19 x1 Discrete valuation rings 36 1 Lecture 10 9/20 x1 The adjoint property 41 x2 Tensor products of algebras 42 x3 Integrality 43 Lecture 11 9/22 x1 Integrality, continued 44 x2 Integral closure 46 Lecture 12 9/24 x1 Valuation rings 48 x2 General remarks 50 Lecture 13 [Section] 9/26 x1 Nakayama's lemma 52 x2 Complexes 52 x3 Fitting ideals 54 x4 Examples 55 Lecture 14 9/27 x1 Valuation rings, continued 57 x2 Some useful tools 57 x3 Back to the goal 59 Lecture 15 9/29 x1 Noetherian rings and modules 62 x2 The basis theorem 64 Lecture 16 10/1 x1 More on noetherian rings 66 x2 Associated primes 67 x3 The case of one associated prime 70 Lecture 17 10/4 x1 A loose end 71 x2 Primary modules 71 x3 Primary decomposition 73 Lecture 18 10/6 x1 Unique factorization 76 x2 A ring-theoretic criterion 77 x3 Locally factorial domains 78 x4 The Picard group 78 Lecture 19 10/8 x1 Cartier divisors 81 x2 Weil divisors and Cartier divisors 82 Lecture 20 10/13 x1 Recap and a loose end 85 x2 Further remarks on Weil(R) and Cart(R) 86 x3 Discrete valuation rings and Dedekind rings 86 Lecture 21 10/15 x1 Artinian rings 89 x2 Reducedness 91 Lecture 22 10/18 x1 A loose end 94 x2 Total rings of fractions 95 x3 The image of M ! S−1M 96 x4 Serre's criterion 97 Lecture 23 10/20 x1 The Hilbert Nullstellensatz 99 x2 The normalization lemma 100 x3 Back to the Nullstellensatz 102 x4 Another version 103 Lecture 24 10/22 x1 Motivation 104 x2 Definition 104 x3 Properties of completions 105 Lecture 25 10/25 x1 Completions and flatness 109 x2 The Krull intersection theorem 110 x3 Hensel's lemma 111 Lecture 26 10/27 x1 Some definitions 113 x2 Introduction to dimension theory 114 Lecture 27 10/29 x1 Hilbert polynomials 117 x2 Back to dimension theory 119 Lecture 28 11/1 x1 Recap 121 x2 The dimension of an affine ring 122 x3 Dimension in general 123 x4 A topological characterization 123 Lecture 29 11/3 x1 Recap 125 x2 Another notion of dimension 126 x3 Yet another definition 127 Lecture 30 11/5 x1 Consequences of the notion of dimension 129 x2 Further remarks 130 x3 Change of rings 130 Lecture 31 11/8 x1 Regular local rings 133 x2 A bunch of examples 133 x3 Regular local rings look alike 134 x4 Regular local rings are domains 136 Lecture 32 11/10 x1 Regularity and algebraic geometry 136 x2 Derivations and K¨ahler differentials 138 x3 Relative differentials 139 x4 Examples 140 Lecture 33 11/12 x1 Formal properties of K¨ahlerdifferentials 140 x2 K¨ahlerdifferentials for fields 142 Lecture 34 11/15 x1 Continuation of field theory 145 x2 Regularity, smoothness, and K¨ahler differ- entials 146 Lecture 35 11/17 x1 Basic definitions in homological algebra 149 x2 Ext functors 150 x3 Injective modules 152 Lecture 36 11/19 x1 Depth 153 x2 Cohen-Macaulayness 156 Lecture 37 11/22 x1 Reduced rings 157 x2 Serre's criterion again 158 x3 Projective dimension 159 x4 Minimal projective resolutions 160 Lecture 38 11/24 x1 The Auslander-Buchsbaum formula 162 Lecture 39 11/29 x1 The projective dimension for noetherian local rings 165 x2 Global dimension and regularity 166 Lecture 40 12/1 x1 Applications of Serre's criterion 168 x2 Factoriality 169 Introduction Jacob Lurie taught a course (Math 221) on commutative algebra at Harvard in Fall 2010. These are my \live-TEXed" notes from the course. Conventions are as follows: Each lecture gets its own \chapter," and appears in the table of contents with the date. Some lectures are marked \section," which means that they were taken at a recitation session. The recitation sessions were taught by Jerry Wang. These notes were typeset using LATEX 2.0. I used vim to take the notes. I ran the Perl script latexmk in the background to keep the PDF output automatically updated throughout class. The article class was used for the notes as a whole. The LATEX package xymatrix was used to generate diagrams. Of course, these notes are not a faithful representation of the course, either in the mathematics itself or in the quotes, jokes, and philosophical musings; in particular, the errors are my fault. By the same token, any virtues in the notes are to be credited to the lecturer and not the scribe. Please email corrections to [email protected]. Lecture 1 Notes on commutative algebra Lecture 1 9/1 x1 Unique factorization Fermat's last theorem states that the equation xn + yn = zn has no nontrivial solutions in the integers. There is a long history, and there are many fake proofs. Factor this expression for n odd. Let ζ be a primitive nth root of unity; then we find (x + y)(x + ζy)(x + ζ2y) ::: (x + ζn−1y) = zn: We are tempted to ask how the product decomposition interacts with the power de- composition. The caveat is that though z is an integer, and x + y 2 Z, the others actually live in Z[ζ]. The problem is the ring Z[ζ]. While this is a legitmate ring, and we can talk about primes and things like that, and try to factor into primes, things go wrong. Over Z, factorization is unique up to permuting the factors. Over Z[ζ], you can still get a decomposition into irreducible factors, but in gneeral it is not unique. The ring does not always have unique factorization. In order to think about the failure of unique factorization, Dedekind introduced the theory of ideal numbers, nowp called ideals. Let us look at a failurep of unique factorization. Consider Z[ −5], i.e. complex numbers that look like a + −5b; a; b 2 Z. We can write p p 6 = 2 × 3 = (1 + −5)(1 − −5); you can convince yourself that these are two fundamentally different factorizations, even though all these are irreducible. So 6 admits a nonunique factorization into irreducibles. Dedekind was trying to get to the bottom of what was going on. Let us look at the problem. Ideally, if you have a prime factor of 6, and a decomposition 6 = ab, then that prime factor must divide a or b. This is not true, however.p There is a new idea. Imaginep an ideal number x such that x divides 2, 1 + −5. It doesn't live in the ring Z[ −5]; it can't, because theyp have no common factor. Yet we can talk aboutp divisibility. We know that 2 and 1 + −5 are divisible by x. For instance, 2 + 1 + −5 should be divisible by x. We're going to identify x with the set of all numbers divisible by x. This led to the introduction of an ideal. x2 Basic definitions 1.1 Definition. A commutative ring is a set R with an addition map R × R ! R and a multiplication map R × R ! R that satisfy all the usual identities. For instance, it should form a group under addition. E.g. x + y = y + x, and there are zeros and additive inverses. Actually, I will not write out all the properties. We do not review the definition of a subring, though Lurie did in the class. 6 Lecture 1 Notes on commutative algebra 1.2 Definition. Let R be a ring. An ideal in R is a subset I ⊂ R (\the set of all elements divisible by something, not necessarily in R") satisfying 1. 0 2 I 2. x; y 2 I implies x + y 2 I 3. x 2 I; y 2 R, then xy 2 I. 1.3 Example. If R is a ring, x 2 R, then the set of things divisible by x (i.e. xR) is an ideal. This is denoted (x). It is in fact the smallest ideal containing x. You can do some of the same things with ideals as you can do with numbers. 1.4 Definition. If I;J are ideals in a ring R, the product IJ is defined as the smallest ideal containing xy for all x 2 I; y 2 J. More explicitly, this is the set of all expressions X xiyi for xi 2 I; yi 2 J. 1.5 Example. We have (x)(y) = (xy). p p 1.6 Example. Let R = Z[ −5]. Is there a common factor of 2 and 1 + −5? No, we said. But the \common factor" is the ideal p (2; 1 + −5) generated by both of them. This is not trivial (1); this is easy to check (exercise?). However, it is also not principal.
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