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00 Introduction: Homological and Commutative Algebra 1. Some (October 16, 2009) 00 Introduction: homological and commutative algebra Paul Garrett [email protected] http://www.math.umn.edu/˜garrett/ 1. Some recurring themes 2. Example: what is an indeterminate? 3. Foundational dangers: set theory and universes 4. Resolutions, complexes, snake lemma 5. Pictorial interlude, tensors 6. Adjoint pairs of functors 7. Counting holes: algebraic topology [iou] 8. Appendix: set theory axioms A fundamental idea emphasized by category theory is characterization of objects by their interactions with other objects, rather than constructions and internal details. [1] More generally: Sufficiently many good examples should make formal definitions nearly unnecessary, and should make most basic results obvious. A good methodological and technical viewpoint should enable not merely verification of interesting results, but should assist discovery. Phenomena are more interesting than symbols. 1. Some recurring themes There are many threads here, touched at various levels of sophistication, interacting in many ways. The following are some of the recurring ideas: Naive category theory has the immediate feature of characterizing or describing objects by their mappings to other objects, rather than by their internal structure or construction. The modifier naive indicates that we do not aggressively axiomatize, and do not necessarily worry about reconciling categorical ideas with set theory. Naive set theory is the typical background or foundation for constructing more complicated structures, in a century-old tradition. The modifier naive denotes an informal and non-axiomatic treatment, in contrast to versions of set theory specifically aimed at preventing paradoxes by limiting constructions. Part of the set theory tradition demands that any mathematical discussion be compatible with the explicit or implicit prohibitions of set theory. Linear algebra in the broadest sense takes vector spaces over fields as prototypes for modules over general rings. Decomposition into eigenvectors or more complicated atomic pieces appears everywhere. Collections of modules over rings are the prototypes for abelian categories, including important examples that are not literal categories of modules, such as categories of sheaves or complexes of sheaves. Functors arise immediately in category theory, being maps from one category to another, converting one type of object to another, and converting corresponding maps-of-objects from one type to the other. Homology [1] The occasional definition of new objects as symbols behaving a certain way is a pale shadow of the idea of characterization by interactions, but made clumsy and opaque by entangling half-hearted constructions with under- specification. 1 Paul Garrett: 00 Introduction: homological and commutative algebra (October 16, 2009) and homotopy groups of topological spaces are dramatic first examples. Once we realize that most systematic procedures describe functors, we see that functors are everywhere. Adjoint functors and adjunction relations: many important functors occur in adjoint pairs, satisfying an adjunction relation, in a sense a sort of inversion, but significantly different. Many forgetful or lossy functors, while boring themselves, have interesting adjoints. Yoneda’s small lemma yields left exactness of right adjoints, and right exactness of left adjoints, once and for all. Natural transformations are maps of functors, and allow formulation of the notion of equivalence of functors, the best version of isomorphism of functors. Homology and cohomology arose as counting holes in geometric objects. Geometric interpretations are still important, but the underlying algebra is universally useful. Extensions of rings appear in algebraic number theory and in algebraic geometry, with arithmetic or geometric motivations. Many important features are best described in categorical and homological terms. Complexes eventually appear as more appropriate fundamental objects than individual modules. The structure theorem for finitely-generated modules over principal ideal domains is a prototype. Projective and injective resolutions are the fundamental acyclic resolutions. These and other themes are profitably illustrated with examples, as follow. 2. Example: what is an indeterminate? What is an indeterminatex? We are not asking about a merely unknown literal number, as in the earliest use of algebra, but about the x that appears in polynomials and other functions in calculus. The x is sometimes alleged to be a variable number, or an arbitrary number. It’s not clear what variable means, but the vague idea of motion is generally compatible with the motivating issues of calculus. These are not bad pseudo-definitions to give calculus students. However, the idea that x is merely a variable number does not match our use of it. For example, in the Cayley-Hamilton theorem [2] the indeterminate x is replaced by a linear transformation. This suggests that x is something that can be substituted for. Elementary mathematics has no way to say this with any precision, but a categorical viewpoint succeeds, as follows. Let R be a commutative ring, probably [3] with 1. An R-algebra A is a ring A that is also an R-module, and so that the R-module structure is compatible with the multiplication in A: (r a)b = r (ab) (for r R, a,b A) · · ∈ ∈ For example, R actually sitting inside the center of A gives A an R-algebra structure. But we do also want to allow some collapsing of R in the way that it acts on A. For example, Z/n is a Z-algebra in the obvious fashion. [2] Recall the assertion of the Cayley-Hamilton theorem: let T be an endomorphism of a finite-dimensional vector space V , let χT (x) be the characteristic polynomial det(x · I − T ), with I the identity map on V . Then χT (T ) = 0. The non-proof of this by (falsely) claiming that χT (T ) = det(T − T ) = det 0 = 0 is of interest, if only to provoke us to see why it’s not a proof. [3] Rings with 1 are more intuitive, and we hope that rings have 1, but occasionally there are good reasons to consider rings without units. But at the moment there’s insufficient motivation to get embroiled with issues about lack of units. 2 Paul Garrett: 00 Introduction: homological and commutative algebra (October 16, 2009) An R-algebra homomorphism or map f : A B is a ring homomorphism f : A B that respects the R-algebra structure, in the reasonable sense that→ → f(r a) = r f(a) (for r R and a A) · · ∈ ∈ Given the idea of R-algebra, we can say what x is, albeit indirectly. A free R-algebra on x is an R-algebra F (thinking that F = R[x], but don’t want to prejudice ourselves by notation!) with distinguished element x F , having the universal property that, for any R-algebra A and choice of element ao A, there exists a unique∈ R-algebra map f : F A such that fx = a . ∈ → o This universal property characterizes the free algebra R[x] not by telling what x is, but by telling how it interacts with other R-algebras. The main idea is that we have substituted ao for x, and that the rest of F accompanies x suitably. We do not exclude the possibility that something happens to the ring R, as in quotient maps Z Z/n. → Grant for a moment that the more elementary notion of polynomial ring R[x] really is a free R-algebra on the generator x. Then for a field k we can certainly map x to an endomorphism T of a finite-dimensional k-vectorspace V , mapping the free k-algebra k[x] to the k-algebra k[T ] of k-linear endomorphisms of V [4] generated by T . The more mundane case of replacing x by an element ro of the ring R gives an R-algebra homomorphism R[x] R mapping x ro, called an evaluation homomorphism. A slightly subtler example is Q[x] Q(√→2) by mapping x→ √2. → → We might worry that the mapping characterization of the free algebra F = R[x] accidently under-specified it. After all, we’ve not directly said anything about the internal structure of it. However, mildly amazingly, this seemingly vague and indirect characterization completely determines F = R[x]. Further, the style of the proof uses almost nothing about the specifics of the situation. That is, suppose we had two free R-algebras on single generators: suppose A a free R-algebra on ao, and B a free R-algebra on b . Then we claim that there is a unique isomorphism j : A B of R-algebras such o → that jao = bo. Indeed, using the characterization of A, a as free R-algebra, there is a unique α : A B with αa = b . o → o o On the other hand, using the characterization of B,bo as free R-algebra, there is a unique β : B A with βb = a . Then α β is an R-algebra map of B to itself sending b to itself, and β α is an R-algebra→ map of o o ◦ o ◦ A to itself sending ao to itself. Obviously we imagine that both these composites are the respective identity maps, which would prove the claim. But one small point is missing, as follows. Universal objects have no proper automorphisms: we claim that the only R-algebra map of A to itself sending a to itself is the identity map on A. This and the corresponding fact for B and b would prove that α β o o ◦ and β α are the respective identities, and, thus, that A, ao and B,bo are isomorphic. Using the mapping characterization◦ of A and a in a seemingly silly way, there is a unique R-algebra map A A sending a o → o to ao. On the other hand, the identity map on A certainly has this property. The uniqueness part of the characterization then says that there is no other such map than the identity.
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