University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Math Department: Class Notes and Learning Materials Mathematics, Department of 2010 Class Notes for Math 915: Homological Algebra, Instructor Tom Marley Laura Lynch University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mathclass Part of the Science and Mathematics Education Commons Lynch, Laura, "Class Notes for Math 915: Homological Algebra, Instructor Tom Marley" (2010). Math Department: Class Notes and Learning Materials. 8. https://digitalcommons.unl.edu/mathclass/8 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Math Department: Class Notes and Learning Materials by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Class Notes for Math 915: Homological Algebra, Instructor Tom Marley Topics covered are: Complexes, homology, direct and inverse limits, Tor, Ext, and homological dimensions. Also, Koszul homology and cohomology. Prepared by Laura Lynch, University of Nebraska-Lincoln August 2010 Homological Algebra gained popularity in Commutative Algebra in the 1950s when the following open problems were solved: De¯nition Let (R; m) be a Noetherian commutative local ring. Then R is regular if m = (x1; :::; xd) where d = dim R: Solved Open Problems 1. If R is regular, is Rp regular for all prime ideals p of R? (proved by Serre-Auslander-Buchsbaum, '57) 2. If R is regular, is R a UFD? (proved by Auslander-Buchsbaum, '59) 1 Direct Limits De¯nition 1.1. Let C be a category and I a poset. A direct system in C indexed by I is a family of objects fAigi2I in C such that i 1. For i · j; there is a morphism Áj : Ai ! Aj in C: i 2. For all i 2 I; we have Ái = 1Ai : 3. For all i · j · k; the diagram below commutes. Ái j / Ai Aj }} i }} Ák } } Áj ² }~ } k Ak i With this notation, we say fAi;Ájgi;j2I is a direct system. Examples. 1. Let I be any set and give I the trivial order, that is i · j for i; j 2 I if and only if i = j: Then any family of objects of C is automatically a direct system over I: 2. Suppose I = N; fAigi2N is a set of objects, and Ái : Ai ! Ai+1 are morphisms. This de¯nes a direct system where i Ái Ái+1 Áj¡1 Áj : Ai ¡! Ai+1 ¡¡¡!¢ ¢ ¢ ¡¡¡! Aj: i x x x Special Case. Let R be a ring, M an R¡module, x 2 Z(R): Let Mi = M for all i: Then Áj : Mi ¡! Mi+1 ¡!¢ ¢ ¢ ¡! Mj (multiplication by xj¡i) yields a direct system. 3. Let C =<< R ¡ mod >>; M a left R¡module, I 6= ; a set of R¡submodules of M: Note that I is a poset where the A A order is containment. For A; B 2 I with A ⊆ B; let ÁB : A ! B be the inclusion map. Then fA; ÁBgA;B2I is a direct system in C: Special Case. One way to deal with a non-¯nitely generated module, is to consider the above situation where I is the set of all ¯nitely generated submodules of M: Since every element of M is contained in some ¯nitely generated submodule, the direct limit would have to be M: i De¯nition 1.2. Let fAi;Ájg be a direct system in C: A direct limit of the system is an object X of C together with morphisms ®i : Ai ! X such that for all i · j; the following diagram commutes ® A i / X i ~> ~~ Ái ~ j ~~® ² ~~ j Aj i and with the following universal property: If there exists ¯i : Ai ! Y for all i 2 I such that ¯jÁj = ¯i; then there exists a unique morphism γ : X ! Y such that γ®i = ¯i; that is, the following diagram commutes. ¯i / Ai > Y ~~ ® ~~ i ~~ γ ² ~~ X With the above notation, we say X =¡! limAi: i2I Exercise. If the direct limit exists, then it is unique up to isomorphism. i Proof. Let fAi;Ájg be a direct system and suppose (X; ®i); (Y; ¯i) are direct limits. Then we have the following commutative diagram: ® ¯ o j A j / X `@ O j ? Y @@ ~~ @@ i ~~ @ Áj ~ ®i @ ~ @ ~~ ¯i Ai By de¯nition, this gives us maps γ : X ! Y and σ : Y ! X such that ¯i = γ®i and ®i = σ¯i: Then ®i = σγ®i and ¯i = γσ¯i: By the uniqueness of the direct limit maps, since 1X : X ! X and σγ : X ! X with ®i = 1X ®i and ®i = σγ®i; » we see σγ = 1X and similarly γσ = 1Y : Thus X = Y: De¯nition 1.3. A poset I is directed if for all i; j 2 I there exists k 2 I such that i · k; j · k: Proposition 1.4. Let M be a left R¡module, I a directed set, ordered by containment, of submodules (that is, given A; B 2 I; there exists C 2 I such that A [ B ⊆ C). Then ¡!limA = [A2I A: A2I Proof. First notice that [A2I A is an R¡submodule of M as I is directed. De¯ne ®A : A ![A2I A to be inclusion. Then A A ®AÁB = ®B for all A ⊆ B where ÁB : A ! B is also the inclusion map. So it is just left to show the universal property of A direct limits holds. Suppose there exists ¯A : A ! Y for all A 2 I such that ¯AÁB = ¯B for all A ⊆ B: De¯ne γ : [A2I A ! Y as follows: Let x 2 [A: Then x 2 A for some A 2 I and so de¯ne γ(x) = ¯A(x): It is easy to see this is well de¯ned and an R¡module homomorphism (by the directed property of I). Thus lim¡!A = [A2I A: Corollary 1.5. For any R¡module M; M = lim¡! N where I = fN ⊆ MjN is ¯nitely generatedg: N2I Example. Let R ⊆ S be commutative domains, R the integral closure of R in S: In general, R is not a ¯nitely generated R¡module (and thus we can not say R is Noetherian when R is). To get around this, notice that R = [T 2I T = lim¡!T; where T 2I I = fT jR ⊆ T ⊆ S; T is a ring;T is a ¯nitely generated R¡moduleg: Here, T is Noetherian when R is. i Exercise. Let R be a ring, M an R¡module, x 2 Z(R): Recall the direct system given by Mi = M and Áj : Mi ! Mj j¡i » de¯ned by multiplication by x for i · j: Then lim¡!M = Mx: i m Proof. First note that Áj is an R¡module homomorphism as x 2 Z(R): De¯ne ®i : Mi ! Mx by m 7! xi : This is an i i xj¡im m R¡module homomorphism and clearly ®jÁj = ®i as ®jÁj(m) = xj = xi = ®i(m): To show the universal property holds, i m suppose there exists Y and ¯i : Mi ! Y such that for i · j we have ¯jÁj = ¯i: De¯ne γ : Mx ! Y by xi 7! ¯i(m): Then a b k j i k+j k+i ² γ is well-de¯ned: Suppose xi = xj in Mx: Then there exists k such that x (x a¡x b) = 0 which implies x a = x b: a i j+k k+i b Then γ( xi ) = ¯i(a) = ¯j+k+iÁj+k+i(a) = ¯j+k+i(x a) = ¯j+k+i(x b) = ¯j(b) = γ( xj ): ² γ is clearly an R¡module homomorphism. a ² γ®i = ¯i as for a 2 Mi; we see γ(®i(a)) = γ( xi ) = ¯i(a): m m ² γ is unique: Suppose there exists ¸ : Mx ! Y such that ¸®i = ¯i: For xi 2 Mx; we see ¸( xi ) = ¸(®i(m)) = ¯i(m) = m γ(®i(m)) = γ( xi ): Thus ¸ = γ: i i Remark. Let C be a category, I a poset. A morphism F : fAi;Ájg ! fBi;Ãjg of direct systems in C (with index I) is a set of morphisms Fi : Ai ! Bi for all i 2 I such that for all i · j the following diagram commutes: F A j / B O j O j i i Áj Ãj / Ai Bi Fi One easily checks that this makes the direct systems in C over I a category, denoted by DirC(I): \De¯nition" 1.6. A category C is called abelian if ² the Hom sets of any two objects are abelian groups, ² there exists a zero object, denoted 0 (i.e., an object that is initial and terminal - for all objects C; there exist unique morphisms 0 ! C and C ! 0), ² every morphism in C has a kernel and cokernel in C; ² the concept of \exact" makes sense, and ² ¯nite products exist (i.e., if A; B 2 ObjC; then A £ B is). [For a more precise de¯nition, see Wiebel's Appendix] Examples. << R ¡ mod >> and Dir<<R¡mod>>(I) are abelian categories. Most of our examples are concrete categories where the objects are sets and morphisms are de¯ned pointwise. In that situation, we can use our notions of kernel, cokernel, and exact for the above de¯nition. In general, it is more complicated and technical. Theorem 1.7. Let C be an abelian category such that arbitrary sums exist. Then any direct system in C has a direct limit. i Proof. We will prove the theorem in the case that C =<< R ¡ mod >> : Let fAi;Ájgi;j2I be a direct system in C: Let i F = ©i2I Ai: De¯ne ¸i : Ai ! F to be the canonical injection.