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Appendix 1 Introduction to Local Cohomology Appendix 1 Introduction to Local Cohomology This appendix is an introduction to local cohomology, including the results used in the text and the connection with the cohomology of sheaves on projective space. For another version, see [Hartshorne/Grothendieck 1967]; for more results, and a very detailed and careful treatment, see [Brodmann and Sharp 1998]. A partial idea of recent work in the subject can be had from the survey [Lyubeznik 2002]. We will work over a Noetherian ring, with a few comments along the way about the differences in the non-Noetherian case. (I am grateful to Arthur Ogus and Daniel Schepler for straightening out my ideas about this.) A1A Definitions and Tools First of all, the definition: If R is a Noetherian ring, Q ⊂ R is an ideal, and M is an R-module, then the zeroth local cohomology module of M is 0 { ∈ | d } HQ(M):= m M Q m = 0 for some d . 0 → HQ is a functor in an obvious way: if ϕ : M N is a map, the induced map 0 0 HQ(ϕ) is the restriction of ϕ to HQ(M). One sees immediately from this that 0 the functor HQ is left exact, so it is natural to study its derived functors, which i we call HQ. For example, suppose that R is a local ring and Q is its maximal ideal. If 0 M is a finitely generated R-module then we claim that HQ(M) is the (unique) largest submodule of M with finite length. On one hand, Nakayama’s Lemma 188 Appendix 1. Introduction to Local Cohomology shows that any finite length submodule is annihilated by some power of Q,and 0 thus is contained in HQ(M). On the other hand, since R is Noetherian and M is 0 finitely generated, HQ(M) is generated by finitely many elements. Some power d 0 of Q annihilates each of them, and thus HQ(M) is a finitely generated module over the ring R/Qn, which has finite length, completing the argument. The same proof works in the case where R is a graded algebra, generated over a field K by elements of positive degree, the ideal Q is the homogeneous maximal ideal, and M is a finitely generated graded R-module. Local Cohomology and Ext We can relate the local cohomology to the more familiar derived functor Ext. Proposition A1.1. There is a canonical isomorphism i ∼ i d HQ(M) = −lim→ ExtR(R/Q ,M), i d → i e where the limit is taken over the maps ExtR(R/Q ,M) ExtR(R/Q ,M) in- duced by the natural epimorphisms R/Qe -- R/Qd for e ≥ d. Proof. There is a natural injection 0 d d - ExtR(R/Q ,M)=Hom(R/Q ,M) M φ - φ(1) { ∈ | d } 0 d whose image is m M Q m =0 . Thus the direct limit lim−→ ExtR(R/Q ,M)= d −lim→ Hom(R/Q ,M) may be identified with the union { ∈ | d } 0 d m M Q m =0 =HQ(M). i d − d − The functor ExtR(R/Q , )isthei-th derived functor of HomR(R/Q , ). Tak- ing filtered direct limits commutes with taking derived functors because of the ex- actness of the filtered direct limit functor [Eisenbud 1995, Proposition A6.4]. i Corollary A1.2. Any element of HQ(M) is annihilated by some power of Q. i d Proof. Any element is in the image of some ExtR(R/Q ,M), which is itself annihilated by Qd. Local Cohomology and Cechˇ Cohomology Another useful expression for the local cohomology is obtained from a Cechˇ complex: Suppose that Q is generated by elements (x1,...,xt). We write [t]= {1,...,t } for the set of integers from 1 to t, and for any subset J ⊂ [t]welet −1 xJ = j∈J xj. We denote by M[xJ ] the localization of M by inverting xJ . A1A Definitions and Tools 189 Theorem A1.3. Suppose that R is a Noetherian ring and Q =(x1,...,xt). i For any R-module M the local cohomology HQ(M) is the i-th cohomology of the complex t - d- −1 d- ··· C(x1,...,xt; M): 0 M M[xi ] 1 - −1 d- ··· - −1 - M[xJ ] M[x{1,...,t}] 0, #J=s whose differential takes an element ∈ −1 ⊂ −1 mJ M[xJ ] M[xJ ] #J=s to the element oJ (k) d(mJ )= (−1) mJ∪{k}, k/∈J where oJ (k) denotes the number of elements of J less than k, and mJ∪{k} denotes −1 −1 −1 the image of mJ in the further localization M (xJ∪{k}) = M[xJ ][xk ]. ˇ Here the terms of the Cech complex are numbered from left to right, counting s −1 M as the 0-th term, and we write C (M)= #J=s M[xJ ] for the term of coho- mological degree s.IfR is non-Noetherian, then the Cechˇ complex as defined here does not always compute the derived functors in the category of R-modules of 0 HI () as defined above, even for finitely generated I; see Exercise A1.7. Rather, it computes the derived functors in the category of (not necessarily quasi-coherent) sheaves of OSpec R modules. For this and other reasons, the general definition of the local cohomology modules should probably be made in this larger cate- gory. As we have no use for this refinement, we will not pursue it further. See [Hartshorne/Grothendieck 1967] for a treatment in this setting. ∈ → −1 Proof. An element m M goes to zero under d : M j M[xj ]ifandonly if m is annihilated by some power of each of the xi. This is true if and only if 0 0 m is annihilated by a sufficiently big power of Q,soH(C(M)) = HQ(M)as required. The complex C(x1,...,xt; M) is obviously functorial in M. Since localiza- tion is exact, a short exact sequence of modules gives rise to a short exact se- quence of complexes, and thus to a long exact sequence in the homology functors i i i H (C(M)). To prove that H (C(M)) = HQ(M) we must show it is the derived 0 0 i functor of HQ(M)=H (C(M)). For this it is enough to show that H (C(M)) = 0 when M is an injective module and i>0 (see for example [Eisenbud 1995, Propo- sition A3.17 and Exercise A3.15]). We need two properties of injective modules over Noetherian rings: Lemma A1.4. Suppose that R is a Noetherian ring, and M is an injective R-module. 190 Appendix 1. Introduction to Local Cohomology ⊂ 0 1. For any ideal Q R, the submodule HQ(M) is also an injective module. 2. For any x ∈ R, the localization map M → M[x−1] is surjective. ⊂ → 0 Proof. 1. We must show that if I R is an ideal and φ : I HQ(M)isa → 0 map, then φ extends to a map R HQ(M). We first extend φ to an ideal containing a power of Q:SinceI is finitely generated, and each generator goes to an element annihilated by a power of Q, we see that for sufficiently large d the ideal QdI is in the kernel of φ. By the Artin–Rees Lemma [Eisenbud 1995, Lemma 5.1], the ideal QdI contains an ideal of the form Qe∩I. It follows that the ⊕ e → 0 e ⊂ map (φ, 0) : I Q HQ(M) factors through the ideal I + Q R. Changing notation, we may assume that I ⊃ Qe from the outset. By the injectivity of M we may extend φ to a map φ : R → M.Since e e ⊂ 0 e φ (Q )=φ(Q ) HQ(M), it follows that some power of Q annihilates Q φ (1), ∈ 0 and thus some power of Q annihilates φ (1); that is, φ (1) HQ(M), so φ is the desired extension. 2. Given m ∈ M and natural number d, we want to show that m/xd is in the image of M in M[x−1]. Since R is Noetherian, the annihilator of xe in R is equal to the annihilator of xd+e in R when e is large enough. Thus the annihilator of xd+e is contained in the annihilator of xem. It follows that there is a map from the principal ideal (xd+e)toM sending xd+e to xem.SinceM is injective, this map extends to a map R → M;writem ∈ M for the image of 1, so that xe+dm = xem.Sincexe(xdm − m)=0,theelementm goes, under the localization map, to m/xd ∈ M[x−1], as required. i To complete the proof of Theorem A1.3 we prove that H (C(x1,...,xt; M)) = 0 for all i>0, when M is an injective module. We apply induction on t,the case t = 0 being obvious. For the case t = 1 we must show that, for any injective R-module M and any x ∈ R, the localization map M → M[x−1] is surjective, and this is the content of part 2 of Lemma A1.4. When t>1, we observe that the submodules −1 ⊂ s M[xJ ] C (x1,...,xt; M) t∈J #J=s −1 for a subcomplex isomorphic to C(x1,...,xt; M)[xJ ][1], where the [1] indicates that the cohomological degree is shifted by 1. Since the quotient involves no terms where xt is inverted, we get a short exact sequence of complexes - −1 - - 0 C(x1,...,xt−1; M)[xt ][1] C(x1,...,xt; M) - - C(x1,...,xt−1; M) 0.
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