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Notes on Koszul Complexes and Commutative Algebra Notes on Koszul complexes and commutative algebra Akhil Mathew January 2009 Abstract I wrote this short set of notes to cover some of the basic ideas of the homolog- ical theory of local rings. I didn’t get that far, only up to the syzygy theorem. The main source was Eisenbud’s book on commutative algebra. In these notes, all rings are commutative, Noetherian, and have an identity element. 1 Krull Dimension 1.1 Krull’s Theorem Let R be a commutative Noetherian ring. Then we define the dimension of R, i.e. dim R, to be the supremum of lengths r of prime chains p0 ⊂ · · · ⊂ pr; where all the inclusions are strict and the pi are prime ideals of R. We define the codimension or height of a prime ideal p (denoted by htp) as the dimension of the local ring Rp; it is the maximum length of a prime ideal descending from p. This notion of dimension is fundamental, and will occupy this section. One of the most important results is: Theorem 1.1 (Krull’s Hauptidealsatz). Let I ⊂ R denote an ideal with g generators. Then any minimal prime p ⊃ I has codimension at most g. To prove it, we introduce the idea of symbolic powers. Let q be prime; then the n-th symbolic power is defined as fs : 9t2 = q s:t: ts 2 qng; it is the inverse image in R of the n-th power of the localized ideal qq ⊂ Rq. Proof. The case g = 0 is trivial. Consider the case g = 1, i.e. with I principal. Then we fix a minimal prime p over I. Localizing at p if necessary, we assume p maximal as well. Suppose there exists a prime ideal q ⊂ p distinct from p. We show that q is minimal, proving the result for g = 1. Now the only prime ideal in the ring R=(x) is the image of p since p is minimal over (x) and maximal in R. Hence, R=(x) is 1 Artinian; therefore any descending sequence of ideals of R containing (x) stabilizes. As an example, consider q(n) + (x). There thus exists N very large such that q(N) ⊃ (x) + q(N+1): In other words, for each w 2 q(n), we can write w = xr + s for s 2q 2 Projective Modules and Global Dimension We will work over a Noetherian ring R. Projective modules are necessary to the de- velopment of homological dimension. We develop some of the properties of projective modules below. A basic familiarity with the definitions is assumed; cf. [?]. 2.1 Projective Modules over Local Rings We want to show that a finitely generated projective module over a local ring R is free. First we need Lemma 2.1 (Nakayama). Let M be a finitely generated module over the local ring R, whose maximal ideal is m. Suppose mM = M. Then M = 0. Proof. Let x1; : : : ; xn be a set of generators for M with n as small as possible. If the set is empty, then M = 0 and we are done. If not, then xn 2 mM and we can write xn = m1x1 + ··· + mnxn; for m1; : : : ; mn 2 m. Hence (1 − mn)xn can be written as a linear combination of x1; : : : ; xn−1. Thus so can xn, 1 − mn being a unit. We could therefore have deleted xn from the list of generators, which was assumed minimal, contradiction. That result holds more generally if R is an arbitrary (not even necessarily Noethe- rian) ring and m denotes the Jacobson radical, i.e. the intersection of all maximal ideals. The proof is the same. Corollary 2.2. With the same hypotheses as above, suppose the images of x1; : : : ; xn 2 M generate the R=m-vector space M=mM. Then they generate M. Proof. Let N denote the submodule of M generated by x1; : : : ; xn. Consider M=N. We have m(M=N) = (mM + N)=N = M=N, since every element of M is congruent mod mM to an element of N. Thus we can apply Nakayama to show that M=N = 0, concluding the proof. We may now apply these results to projective modules: Theorem 2.3. Let M be a finitely generated projective module over the local ring R, with maximal ideal m. Then M is free. 2 Proof. Let x1; : : : ; xn 2 M be such that their images form a basis for the vector space M=mM. Then x1; : : : ; xn generate M by the corollary. Take a free module F of rank n, with generators e1; : : : ; en. We have a surjective homomorphism: F ! M ! 0 mapping the ej ! xj. Since M is projective, the map splits. Therefore we can write (up to isomorphism) F = F1 ⊕ M. Here F1 is the kernel of the map to M. Now, any P element f of F1 is a linear combination of the ej, say f = ajmj for aj 2 R. The P aj are actually in m since the image of f, which is ajxj, is 0 in M. Thus f 2 mF . Moreover, f 2 F1, so f 2 m(F1 ⊕ F2) \ F1 = mF1. Hence F1 = mF1, so F1 = 0. Thus F = M and M is free. Finally, we generalize this result to projective modules in general: Theorem 2.4. Let R be a Noetherian ring and M a finitely generated R-module. Then M is projective if and only if for all maximal ideals m, the localized module Mm is free over Rm. Proof. If M is projective over R, then the localized modules are projective over the localized rings. Indeed, M is a direct summand of a free R-module, and localization preserves direct sums. So one implication is obvious. The other implication follows be- cause projectiveness can be tested for with the Ext functor, the Ext functor commutes with localization on Noetherian rings, and a module whose localizations at maximal ideals vanish vanishes itself. In more detail, M is projective over R if and only if Mm is projective over Rm for all m maximal. Then the last theorem completes the proof. 2.2 Homological Dimension Let R be a Noetherian ring and M a module. The projective dimension of M, denoted by pdM is the minimum natural number n such that a resolution 0 ! Pn ! Pn−1 !···! P0 ! M ! 0 exists with all the Pj projective. If no such chain exists, then we write pdM = 1. We define the global dimension of R, denoted by glR, to be supM pdM, the sup being taken over all R-modules. It is an important theorem of commutative algebra that the notion of global dimen- sion coincides with the usual Krull dimension (cf. [?] or [?]) over regular local rings. We prove this later. Lemma 2.5. We have pdM ≤ n if and only if Exti(M; N) = 0 for all modules N and i > n. Proof. Suppose we have a resolution 0 ! Pn ! Pn−1 !···! P0 ! M ! 0 3 with the Pj projective (with some possibly zero). Then by dimension-shifting we see i i−n that Ext (M; N) = Ext (Pn;N) = 0 (assuming i > n). This gives one implica- tion. We can essentially reverse the argument for the other. Pick a projective (e.g. free) resolution Pn−1 ! Pn−2 !···! P0 ! M ! 0: This can be constructed by an elementary argument, of course. The new part here is the zero at the left. Let Pn be the kernel at the far left, so we have an exact sequence 0 ! Pn ! Pn−1 ! Pn−2 !···! P0 ! M ! 0: Now using dimension-shifting one can easily see that Pn is projective, whence the lemma. We can define the notion of the iojective dimension of a module N, denoted by 2 N, in the same way, considering resolutions 0 ! N ! I0 ! I1 !···! In ! 0 for injective moduels Ij. The same technique as above proves: Lemma 2.6. We have 2 N ≤ n if and only if Exti(M; N) = 0 for all modules M and i > n. Theorem 2.7 (Auslander). The global dimension of R is equal to supN 2 N, and is the smallest n such that all the Exti with i > n vanish for all modules. Moreover, n is the smallest integer such that Extn(R=I; N) = 0 for all ideals I and modules N. Finally, we can restrict our attention to finitely generated modules in the above supremums. Proof. The first part follows immediately from the above lemmas and discussion. Only the last part remains. So fix a module N and an ideal I, and consider an injective resolution 0 ! N ! I0 !···! In−1: Let In be the final cokernel, and so we get an exact sequence 0 ! N ! I0 !···! In−1 ! In ! 0: i i+n We see that Ext (R=I; In) = Ext (R=I; N) = 0 for all i > 0. By a criterion of 1 Baer, this will show that In is injective. All we really need is that Ext (R=I; In) = 0 for all ideals I. We have an exact sequence 0 ! I ! R ! R=I ! 0; which yields to a long exact sequences; chopping off parts of it gives Hom(R; In) ! Hom(I;In) ! 0: This means any homomorphism I ! In extends to R. Using a criterion of Baer (an application of Zorn’s lemma; see [?]), we see that In is actually injective. The last part follows since any module is the direct limit of its finitely generated submodules, and the Ext functors commute with direct limits. Indeed, direct limits are exact and commute with Hom.
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