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Derived Functors for Hom and Tensor Product: the Wrong Way to Do It

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Derived Functors for Hom and Tensor Product: the Wrong Way to Do It Derived Functors for Hom and Tensor Product: The Wrong Way to do It UROP+ Final Paper, Summer 2018 Kevin Beuchot Mentor: Gurbir Dhillon Problem Proposed by: Gurbir Dhillon August 31, 2018 Abstract In this paper we study the properties of the wrong derived functors LHom and R ⊗. We will prove identities that relate these functors to the classical Ext and Tor. R With these results we will also prove that the functors LHom and ⊗ form an adjoint pair. Finally we will give some explicit examples of these functors using spectral sequences that relate them to Ext and Tor, and also show some vanishing theorems over some rings. 1 1 Introduction In this paper we will discuss derived functors. Derived functors have been used in homo- logical algebra as a tool to understand the lack of exactness of some important functors; two important examples are the derived functors of the functors Hom and Tensor Prod- uct (⊗). Their well known derived functors, whose cohomology groups are Ext and Tor, are their right and left derived functors respectively. In this paper we will work in the category R-mod of a commutative ring R (although most results are also true for non-commutative rings). In this category there are differ- ent ways to think of these derived functors. We will mainly focus in two interpretations. First, there is a way to concretely construct the groups that make a derived functor as a (co)homology. To do this we need to work in a category that has enough injectives or projectives, R-mod has both. The second way is a categorical construction that defines the derived functors as left or right Kan Extension for homotopy categories. To see that these definitions agree see [1]. Now we present some of the reasons why people are mostly interested on the left (right) derived functor LF (RF ) of a right (left) exact functor F ; there is a result that shows the equality of functors 0 L0F = F (R F = F ) this result is equivalent to saying that the left (right) derived functor will give informa- tion related to the exactness of the functor F . However, in this paper we focus on giving an insight on the properties that the left (right) derived functor LF (RF ) of a left (right) exact functor F has. In this paper we will study derived functors that are not as natural, we study the ex- R plicit examples of the functors LHom, ∗LHom and ⊗, the left and right derived functors of the functors Hom(A; −), Hom(−;A) and − ⊗ A respectively. To study these functors L we prove some identities that will relate them to the traditional RHom and ⊗. One of R these results, surprisingly, shows that LHom(M; −) a −⊗M is an adjoint pair whenever M is a finitely generated module. To summarize some of the similarities and differences between the classical derived functors of Hom and ⊗ and the derived functors studied in this paper we present the following table: L R RHom LHom ∗LHom ⊗ ⊗ Type of Resolution Projective Projective Injective Flat Injective or Injective Type of Kan Extension Left Right Right Right Left 2 Statement of Results The main results in this paper are the following three theorems. These relate the wrong R derived functors we study in this paper, LHom and ⊗ to the functors Tor and Ext, when evaluated at some dual modules. In particular, we will look at the dual mod- ule M _ = Hom(M; R), and also consider for an injective cogenerator E the modules M + = Hom(M; E) and M e = Hom(E; M). With this notation, we have the following results: Theorem 2.1. For a finitely presented module M and any module N, we have the following isomorphism in the derived category of R-mod: L LHom(M; N) ' M _ ⊗ N: Theorem 2.2. For a finitely presented module M and any module N, we have the following isomorphism in the derived category of R-mod: R M⊗N ' RHom(M _;N): From Theorem 2.1(6.1) and Theorem 2.2(6.2) we get the following corollary: Corollary 2.1. For a finitely generated module M, the functors R LHom(M; −) a −⊗M; form an adjoint pair. We do not know of any reference where these functors have been studied explicitly. We only know of general results for derived functors and these theorems give more ex- R plicit properties for the functors LHom and ⊗ that resemble properties of the functors L RHom and ⊗; the most surprising of these results being that the wrong derived functors form an adjoint pair. These results may suggest that there may be other interesting properties for the wrong derived functors of other well-known functors, particularly that they might form an adjoint pair. In the case of ⊗ and Hom it was necessary that the module was finitely presented{which can be related to some conditions of commutativity with certain colim- its. If there is a way to extend these results to general adjoint pairs it cannot be done following the proofs of the theorems above as they are dependent on properties of Hom and ⊗. In the next section we show a set of three spectral sequences, one for each of LHom, R ∗LHom and ⊗, that converges to the modules Ext(M; N), Ext(M; N) and M ⊗N respec- tively. We get this result by one of the possible filtrations; while in the other filtration R ∗ we get a different E2 page which contains the functors LHom, LHom and ⊗. We give as an illustrative example the E2 page for LHom. Proposition 2.1. Let M; N be two R modules. Then the spectral sequence that comes from a sign change of the double complex Hom(PM ;QN ) where PM := ··· P2 ! P1 ! P0 ! 0 and QN := ··· Q2 ! Q1 ! Q0 ! 0 are projective resolutions of M and N 1 p respectively, converges to Ep;0 = Ext (M; N) under one of the filtrations and under the other filtration we get the following E2 page: 0 1 2 LHom (M; N) L0Ext (M; N) L0Ext (M; N) ··· 1 1 2 LHom (M; N) L1Ext (M; N) L1Ext (M; N) ··· 2 1 2 LHom (M; N) L2Ext (M; N) L2Ext (M; N) ··· . .. Furthermore, if M is finitely generated and R is noetherian we can use that Hom(M; N) ' _ M ⊗ N and get the E2 page: 0 1 2 Tor0(Ext (M; R);N) Tor0(Ext (M; R);N) Tor0(Ext (M; R);N) ::: 0 1 2 Tor1(Ext (M; R);N) Tor1(Ext (M; R);N) Tor1(Ext (M; R);N) ::: 0 1 2 Tor2(Ext (M; R);N) Tor2(Ext (M; R);N) Tor2(Ext (M; R);N) ::: . .. This result shows that the wrong derived functors are related to the classical derived functors and that we do not need any assumption as finite generation as it is needed for theorems 2.1 and 2.2. The second part of the result is related to theorems 2.1 and 2.2 as we consider the case of M finitely generated and use the same lemmas to get another result. The next section is motivated by the idea that we would wish that the derived functor R ∗LHom had similar properties as those we have proved for LHom and ⊗. However, this functor is more related to the injective cogenerator instead of the projective generator (as R); unfortunately, the injective cogenerator E does not behave that well in this context and for our results we are forced to work with modules E that satisfy special conditions. Theorem 2.3. For any module M and N finitely copresented by E, an injective cogen- erator with endomorphism ring End(E) = S, we have the following isomorphism in the derived category of R-mod: L ∗LHom(M; N) ' M + ⊗ N e: S Finally, in sections 9 and 10 we work to understand how these functors behave in some particular rings. In section 9 we show that the higher (co)homology groups of the wrong derived functors vanish for rings of projective (or injective) dimension 2 or lower. In contrast, in section 10 we prove that we can find examples where the (co)homology groups do not vanish. We prove this fact with an explicit construction using regular local rings of dimension 3 or higher. Theorem 2.4. Let R be a regular local ring of dimension d ≥ 3 and maximal ideal m, let k := R=m be its residue field. Then for any integer 3 ≤ ` ≤ d−1 there exists a module N such that: R N⊗R ' k[−`] in the derived category. We see that the wrong derived functors also have some interesting properties and can be, in some conditions, related to the derived functions that are already studied. The results proved here cannot be straightforwardly extended to general derived functors, but we hope it can give a little more understanding of this subject. 3 Acknowledgements I would like to thank the MIT Mathematics Department for holding the UROP+ Pro- gram under which this work was done. I would also like to thank my mentor Gurbir Dhillon, who suggested this problem and for his guidance while working on it. Finally, I would also like to thank Olga Medrano who helped me review an early version of this paper. 4 Notation and Conventions In this paper we will use the following notation and conventions. The ring of interest will be a commutative ring R and all functors we consider will be functors from R-mod to R-mod unless specified otherwise. We will omit the subindex R in functors HomR(A; B) or ⊗R and so we will only write Hom(A; B) or ⊗ respectively.
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