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Homological Algebra Homological Algebra Irena Swanson Graz, Fall 2018 The goal of these lectures is to introduce homological algebra to the students whose commutative algebra background consists mostly of the material in Atiyah-MacDonald [1]. Homological algebra is a rich area and can be studied quite generally; in the first few lectures I tried to be quite general, using groups or left modules over not necessarily commutative rings, but in these notes and also in most of the lectures, the subject matter was mostly modules over commutative rings. Much in these notes is from the course I took from Craig Huneke in 1989, and I added much other material. All rings are commutative with identity. This is work in progress. I am still adding, subtracting, modifying, correcting errors. Any comments and corrections are welcome. The first version of these notes was used at University Roma Tre between March and May 2010, where my stay was funded by INDAM through Professor Stefania Gabelli. A much corrected version is used in Graz, Austria in the fall of 2018, where I am spending a semester on the Fulbright–NAWI Fellowship. I thank Amr Al Maktry, Lukas Andritsch, Charles Beil, Victor Fadinger, Sophie Frisch, Ana Garcia Elsener, Jordan McMahon, Sarah Nakato, Arnur Nigmetov and Daniel Windisch for all the comments that helped improve these notes. Table of contents: Section 1: Overview, background, and definitions 2 Section 2: Complexes and functors 12 Section 3: General manipulations of complexes 14 Section 4: Koszul complexes 18 Section 5: Projective modules 24 Section 6: Projective, flat, free resolutions 29 Section 7: Manipulating projective resolutions 35 Section 8: Tor 39 Section 9: Regular rings, part I 44 Section 10: Review of Krull dimension 46 Section 11: Primary decomposition of modules 50 Section 12: Regular sequences 52 Section 13: Regular sequences and Tor 57 Section 14: Regular rings, part II 59 1 Section 15: Cohen–Macaulay rings and modules 64 Section 16: Injective modules 67 Section 17: Divisible modules 70 Section 18: Injective resolutions 72 Section 19: A definition of Ext using injective resolutions 76 Section 20: A definition of Ext using projective resolutions 77 Section21:ThetwodefinitionsofExtareisomorphic 78 Section 22: Ext and extensions 80 Section 23: Essential extensions 83 Section 24: Structure of injective modules 87 Section 25: Duality and injective hulls 89 Section 26: More on injective resolutions 94 Section 27: Gorenstein rings 101 Section 28: Bass numbers 104 Section 29: Criteria for exactness 105 Appendix A: Tensor products 111 Bibliography 115 Index 117 1 Overview, background, and definitions 1. What is a complex? A complex is a collection of groups (or left modules) and ho- momorphisms, usually written in the following way: di+1 di Mi+1 Mi Mi 1 , ···→ −→ −→ − →··· where the Mi are groups (or left modules), the di are group (or left module) homomor- phisms, and for all i, d d = 0. Rather than write out the whole complex, we will i ◦ i+1 typically abbreviate it as C , M or (M ,d ), et cetera, where the dot differentiates the • • • • complex from a module, and d denotes the collection of all the di. This d is called the • • differential of the complex. A complex is bounded below if Mi = 0 for all sufficiently small (negative) i; a complex is bounded above if Mi = 0 for all sufficiently large (positive) i; a complex is bounded if M = 0 for all sufficiently large i . For complexes bounded below we i | | abbreviate “0 0 0 ” to one zero module, and similarly for complexes bounded → → → ··· above. A complex is exact at the ith place if ker(di) = im(di+1). A complex is exact if it is exact at all places. 2 A complex is free (resp. flat, projective, injective) if all the Mi are free (resp. flat, projective, injective). (Flat modules are defined in Definition 2.3, projective modules in Section 5 and injective modules in Section 24.) An exact sequence or a long exact sequence is another name for an exact complex. An exact sequence is a short exact sequence if it is of the form i p 0 M ′ M M ′′ 0. → −→ −→ → For any R-modules M and N we have a short exact sequence 0 M M N N 0, → → ⊕ → → with the maps being m (m, 0) and (m, n) n. Such a sequence is called a split 7→ 7→ exact sequence (it splits in a trivial way). But under what conditions does a short exact sequence 0 M K N 0 split? When can we conclude that K = M N (and → → → → 1 ∼ ⊕ more)? We will prove that the sequence splits if N is projective or if ExtR(M,N) = 0. See also Exercise 1.8. Remark 1.1 Every long exact sequence di+1 di Mi+1 Mi Mi 1 ···→ −→ −→ − →··· decomposes into short exact sequences 0 ker di 1 Mi 1 im di 1 0 → − → − → − → 0 ker d M im||d 0 → i → i → i → 0 ker d M im d|| 0, → i+1 → i+1 → i+1 → et cetera. We will often write only parts of complexes, such as for example M M M 0, 3 → 2 → 1 → and we will say that such a (fragment of a) complex is exact if there is exactness at a module that has both an incoming and an outgoing map. 2. Homology of a complex C . The nth homology group (or module) is • ker dn Hn(C ) = . • im dn+1 3. Cocomplexes. A complex might be naturally numbered in the opposite order: i−1 i i 1 d i d i+1 C• : M − M M , ···→ −→ −→ →··· in which case we index the groups (or modules) and the homomorphisms with superscripts rather than the subscripts, and we call it a co-complex. The nth cohomology module of such a co-complex C• is n n ker d H (C•) = n 1 . im d − 3 If (C•,d•) is a co-complex, then −2 −1 0 1 2 2 d 1 d 0 d 1 d 2 d 3 C− C− C C C C ···→ −→ −→ −→ −→ −→ →··· n n After renaming Dn = C− and en = d− we convert the co-complex into the complex: e2 e1 e0 e−1 e−2 D2 D1 D0 D 1 D 2 D 3 ···→ −→ −→ −→ − −→ − −→ − →··· n+2 n+2 The naming can be even shifted: Fn = C− and gn = d− converts the co-complex into the following complex: 0 0 e2(=d ) 1 e1 e0 e−1 e−2 D2(= E ) D1(= E ) D0 D 1 D 2 D 3 ···→ −→ −→ −→ − −→ − −→ − →··· Similarly, any complex can be converted into a co-complex, possibly with shifting. There are reasons for using both complexes and co-complexes (keep reading). 4. Free and projective resolutions. Definition 1.2 A (left) R-module F is free if it is a direct sum of copies of R. If F = i I Rai and Rai = R for all i, then we call the set ai : i I a basis of F . ⊕ ∈ ∼ { ∈ } Facts 1.3 (1) If X is a basis of F and M is a (left) R-module, then for any function f : X M → there exists a unique R-module homomorphism f˜ : F M that extends f. → (2) Every R-module is a homomorphic image of a free module. Definition 1.4 Let M be an R-module. A free (resp. projective) resolution of M is a complex di+1 di d1 Fi+1 Fi Fi 1 F1 F0 0, ···→ −→ −→ − →···→ −→ → where the Fi are free (resp. projective) modules over R (definition of projective modules is in Section 5) and where di+1 di d1 Fi+1 Fi Fi 1 F1 F0 M 0 ···→ −→ −→ − → · · · −→ −→ → → is exact. Free, projective, flat resolutions are not uniquely determined, in the sense that the free modules and the homomorphisms are not uniquely defined, not even up to isomorphisms. Sometimes, mostly in order to save writing time, Fi+1 Fi Fi 1 ···→ → → − →···→ F F M 0 is also called a free (resp. projective) resolution of M. 1 → 0 → → 5. Construction of free resolutions. We use the fact that every module is a homomor- phic image of a free module. Thus we may take F0 to be a free module that maps onto M, and we have non-isomorphic choices there; we then take F1 to be a free module that maps onto the kernel of F M, giving an exact complex F F M 0; after which we 0 → 1 → 0 → → may take F to be a free module that maps onto the kernel of F F , et cetera. 2 1 → 0 4 Example 1.5 Let R be the polynomial ring k[x,y,z] in variables x,y,z over a field k. Let I be the ideal (x3,y3,xyz) in R. We will construct a free resolution of I in a slow way. One point of this example is to see that it is possible to construct resolutions methodically, at least for monomial ideals, and another point is that the explicit construction of resolu- tions takes effort. (One of the goals of the course is to get properties of free resolutions theoretically without necessarily constructing them, but seeing at least one construction is good as well.) By definition we see that the free R-module R3 maps onto I via the map d : R3 R defined by the 1 3 matrix [x3 y3 xyz]. The following elements are in the 0 → × kernel of d0: y3 yz 0 x3 , 0 , xz . −0 x2 y2 − − T We next prove that the kernel of d0 is generated by these three vectors. So let [f g h] be an arbitrary element of the kernel.
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