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Additive, Abelian, and Exact Categories

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Additive, Abelian, and Exact Categories U.U.D.M. Project Report 2016:48 Additive, abelian, and exact categories Samuel Pettersson Examensarbete i matematik, 15 hp Handledare: Martin Herschend Examinator: Jörgen Östensson December 2016 Department of Mathematics Uppsala University Additive, abelian, and exact categories Samuel Pettersson December 23, 2016 Contents 1 Modules 2 1.1 Basic constructions . 2 1.2 R-module morphisms . 4 1.3 Induced morphisms . 9 1.4 Direct products and sums of modules . 10 2 Categories 16 2.1 Basic constructions . 16 2.2 Duality . 22 2.3 Kernels and cokernels . 25 2.4 Pullbacks and pushouts . 28 2.5 Functors and subcategories . 37 3 Additive categories 43 3.1 Preadditive categories . 43 3.2 Products and coproducts . 46 3.3 Biproducts . 53 3.4 Additive functors . 59 3.5 Matrix notation . 63 3.6 Pullbacks and pushouts via matrices . 72 4 Abelian categories 76 4.1 Basics . 76 4.2 The first isomorphism theorem . 82 4.3 Exact sequences . 87 5 Exact categories 89 5.1 Basics . 89 5.2 Extension-closed subcategories . 98 1 1 Modules In this preliminary section, we will define modules over rings and explore the bare necessities for the sections to come. Most (if not all) definitions and results in this section can be found in any exposition of modules over a ring, e.g., [3] or [4]. 1.1 Basic constructions Definition 1.1. For any ring (with unity) R, a (left) R-module (or module over R) is an abelian group (M; +) equipped with a compatible (left) R-action ·: R × M ! M by R. Spelled out, an R-module should satisfy the following eight axioms, the first four of which are for the abelian group and the last four of which are for the ring action: 1. Addition is associative, i.e., (a + b) + c = a + (b + c) for any a; b; c 2 M 2. Addition is commutative, i.e., a + b = b + a for any a; b 2 M 3. There is an additive identity, denoted by 0, such that 0 + a = a = a + 0 for any a 2 M 4. Any a 2 M has an additive inverse, denoted by −a, such that a + (−a) = 0 = (−a) + a 5. The ring identity 1R acts trivially, i.e., 1R · m = m for any m 2 M 6. A sum r + s of ring elements acts as (r + s) · m = rm + sm, i.e., in a way compatible with the group addition, for any r; s 2 R and m 2 M 7. A product rs of ring elements acts as (rs) · m = r · (s · m), i.e., in a way compatible with a composition of sorts, for any r; s 2 R and m 2 M 8. Any ring element acts additively, i.e., r · (m + n) = r · m + r · n for any r 2 R and m; n 2 M Remark 1.2. Readers familiar with linear algebra will note that the above definition with \ring R" replaced with “field K" would be a definition of a vector space; modules over fields are precisely vector spaces. Accordingly, one may think of modules over rings as being generalized vector spaces with many, but not all, of the properties of vector spaces (see Example 1.32). Remark 1.3. We will concern ourselves only with left R-modules, but the reader should know that there are notions of a right R-module and even of so-called bimodules, which are equipped with both a left ring action and a right ring action (possibly by different rings!). Example 1.4. For any fixed ring R, there is a trivial example of an R-module, namely when the underlying group is trivial (i.e., the group is a singleton f0g), in which case the R-action maps everything to 0. We will refer to this module as the zero module (though to be strict, it might be better to speak of it as a zero module) and denote it by f0g or sometimes just 0. More interestingly, we may turn any ring R into an R-module in a natural manner. Example 1.5. Consider a ring R. It is by definition an abelian group with the additional structure of a multiplication ·: R × R ! R. This multiplication may be viewed as an R-action, seeing as the axioms for the identity, associativity, and compatibility (with the ring addition) for the ring multiplication guarantee that the necessary module axioms hold. We call this module the regular (left) R-module and denote it by RR or sometimes just R. Example 1.6. As we noted, any vector space is a module. For instance, the familiar Euclidean space R3 is an R-module. 2 Similar to how K-modules (for a field K) are the familiar vector spaces over K, the following remark expresses that Z-modules are essentially (this \essentially" is formalized in Definition 2.60) just the underlying abelian groups. Remark 1.7. For any abelian group M, there is a unique ring action by Z endowed with which M is a Z-module, namely the one defined for any n 2 Z and x 2 M by 8 n terms >z }| { >x + ··· + x if n > 0 <> n · x := 0M if n = 0 > >−(x + ··· + x) if n < 0 :> | {z } −n terms It follows fairly readily that this is the only possible candidate for a compatible Z-action on M: suppose that we have a Z-action on M. Then write any n > 0 as the sum 1 + ··· + 1 and use the module axioms repeatedly to obtain n · x = (1 + ··· + 1) · x = (1 · x) + ··· + (1 · x) = x + ··· + x It is a straightforward task to show (using the module axioms) that the ring action of any module behaves as expected for the additive identity and all additive inverses of the ring, i.e., in this case that 0Z · x = 0M and (−n) · x = −(n · x). That the above definition satisfies the module axioms can be shown one axiom at a time through a painstaking and painful consideration of cases depending on the sign of the integers acting on the group. Definition 1.8. Given an R-module M, an R-submodule (or just submodule) of M is a non-empty subset U ⊆ M that is closed under both addition and the action of R, i.e., U is a subgroup of M (viewed as an abelian group) that is closed under the R-action. This makes U with structure inherited from M into a module in its own right. Explicitly, the requirement is that (1) U 6= ?, (2) If m; n 2 U, then (m + n) 2 U, and (3) For any r 2 R, if m 2 U, then (r · m) 2 U. Example 1.9. Any R-module M has the submodules f0g and M (which may coincide). Example 1.10. The submodules of a regular left module RR are (as sets) precisely the left ideals of the ring R. Both submodules and ideals may be viewed as additive subgroups with some additional condition satisfied. Upon closer scrutiny, the additional condition is in both cases closure under ring multiplication by an element from the left. Much like for vector spaces (in fact, the definition is the very same), we may construct from a module M and a submodule U the quotient module M=U. Definition 1.11. Let R be a ring, M be an R-module, and U ⊆ M be a submodule of M. Consider the equivalence relation ∼ on M under which two elements are equivalent if and only if they differ by an element of U: x ∼ y () (x − y) 2 U The quotient module M=U of M by U is the set M=∼ of all equivalence classes (where [x] denotes the equivalence class of x) of this relation M=∼ = f[x] j x 2 Mg 3 endowed with the addition and R-action given by [m] + [n] := [m + n] r · [m] := [rm] In other words, the quotient module is the corresponding quotient group equipped with the above R-action. It is not immediately clear that the quotient module is a well-defined R-module by the above. We verify that it is in the proof of the following proposition. Proposition 1.12. The quotient module M=U is a well-defined R-module. Proof. Verify first that ∼ is an equivalence relation. For reflexivity, x ∼ x () (x − x) = 0 2 U and every submodule contains the zero element. For symmetry, use the fact that U is closed under the R-action and that the ring action in any module respects negation in the ring (a fact that follows readily from the module axioms and the observation that the ring zero acts by sending everything to zero): x ∼ y () (x − y) 2 U =) (−1R) · (x − y) = −(1R · (x − y)) = −(x − y) = (y − x) 2 U () y ∼ x For transitivity, use the fact that U is closed under addition: x ∼ y and y ∼ z () (x − y) 2 U and (y − z) 2 U =) (x − y) + (y − z) = (x − z) 2 U () x ∼ z Thus, the underlying set M=∼ of the quotient module is well-defined. Next, verify that the operations are well-defined. For addition, [x] = [x0] and [y] = [y0] () (x − x0); (y − y0) 2 U =) (x − x0) + (y − y0) = (x + y) − (x0 + y0) 2 U () [x + y] = [x0 + y0] which shows that the right-hand side in the definition of the addition does not depend on the representatives in the left-hand side. For the R-action, we similarly get [x] = [x0] () x − x0 2 U =) r · (x − x0) = (r · x − r · x0) 2 U () [r · x] = [r · x0] Thus, the quotient module at least consists of a well-defined underlying set and well-defined operations, and it only remains to show that the module axioms are satisfied.
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