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Extended Modules EXTENDED MODULES Wolfgang Hassler and Roger Wiegand August 29, 2008 Suppose R and S are local rings and (R; m) ! (S; n) is a flat local homomorphism. Given a ¯nitely generated S-module N, we say N is extended (from R) provided there is an R-module M such that S ­R M is isomorphic to N as an S-module. If such a module M exists, it is unique up to isomorphism (cf. [EGA, (2.5.8)], and it is necessarily ¯nitely generated. The m-adic completion R ! Rb and the Henselization R ! Rh are particularly important examples. One reason is that the Krull-Remak-Schmidt uniqueness theorem holds for direct- sum decompositions of ¯nitely generated modules over a Henselian local ring. Indeed, failure of uniqueness for general local rings stems directly from the fact that some modules over the Henselization (or completion) are not extended. Understanding which Rh-modules are extended is the key to unraveling the direct-sum behavior of R-modules. Throughout, we assume that (R; m) and (S; n) are Noetherian local rings and that R ! S is a flat local homomorphism. Many of our results generalize easily to a mildly non- commutative setting. Moreover, it is not always necessary to assume that our rings are local. Thus, we assume that A is a commutative ring, that B is a faithfully flat commutative A-algebra, and that ¤ is a module-¯nite A-algebra. Given a ¯nitely generated left B ­A ¤-module N, we say that N is extended (from ¤) provided there is a ¯nitely generated left ¤-module M such that B ­A M is isomorphic to N as a B ­A ¤-module. In Sections 1 and 2 of the paper, we examine how the extended modules sit inside the family of all ¯nitely generated modules. In Sections 3 { 4 we consider rings of dimension 2 and 1, respectively, ¯nd criteria for a module to be extended , and show how the extendedness problem for one-dimensional rings reduces to the Artinian case. In Section 5, we ¯nd situations where every ¯nitely generated B-module is a direct summand of an extended module, and in Section 6 we make a few observations about the Artinian case. 1. Two out of three: direct sums Our goal in this section is to prove the following theorem, which generalizes Proposition 3.1 of [FSW]: The research of W. Hassler was supported by the Fonds zur FÄorderung der Wissenschaftlichen Forschung, project number P20120-N18. Wiegand's research was partially supported by NSA Grant H98230-05-1-0243. 1 1.1. Theorem. Let A ! B be a faithfully flat homomorphism of commutative, Noether- ian, semilocal rings. Let ¤ be a module-¯nite A-algebra, and let N; N 0 and N 00 be ¯nitely » 0 00 0 00 generated left B ­A ¤-modules, with N = N © N . If two of the modules N; N ;N are extended, so is the third. The ¯rst step in the proof is to observe that ¯nitely generated modules over ¡ := B ­A ¤ satisfy direct-sum cancellation. This is essentially contained in E. G. Evans's paper [E], but we need a little argument to deal with the non-commutative setting. 1.2. Lemma. Let ¡ be a module-¯nite algebra over a commutative Noetherian semilocal ring B, and let U; V and W be ¯nitely generated left ¡-modules. If U © W =» V © W , then U =» V . Proof. Let E be the endomorphism ring End¡(W ), and let J(¡) denote the Jacobson radical. Then E= J(E) is a module-¯nite B= J(B)-algebra and therefore is Artinian. (Thus E is \semilocal" in the non-commutative sense.) Therefore E has 1 in the stable range (cf. [F1, Theorem 4.4]), and by [E, Theorem 2] we have U =» V . ¤ If, in Theorem 1.1, N 0 and N 00 are extended, then N is obviously extended. The other two implications are symmetric, and so we want to prove that if N and N 0 are extended, then so is N 00. It is convenient to use the notation \X j¤ Y ", for ¤-modules X and Y , to mean that there is a ¤-module Z such that X © Z =» Y . When the ring ¤ is understood, we will write \X j Y ". » 0 » 0 0 Assume, now, that N = B­A M and N = B­A M , where M and M are left ¤-modules. 0 00 » 0 Then (B­A M )©N = B­A M, so B­A M jB­A¤ B­A M. Suppose we can conclude that 0 0 00 » 0 00 » M j¤ M, say, M © M = M. Then we will have (B ­A M ) © (B ­A M ) = B ­A M, and, 00 » 00 by Lemma 1.2, N = B ­A M . Therefore Theorem 1.1 is a consequence of the following result (where we no longer need to assume that B is semilocal): 1.3. Theorem. Let A ! B be a faithfully flat homomorphism of commutative Noetherian rings, with A semilocal. Let ¤ be a module-¯nite A-algebra, and let M and M 0 be ¯nitely 0 0 generated left ¤-modules. If B ­A M jB­A¤ B ­A M, then M j¤ M. Note that this includes the well-known result on faithfully flat descent of isomorphism 0 » 0 0 [EGA, (2.5.8)]: If B ­A M =B­A¤ B ­A M, Theorem 1.3 implies that M j M and M j M , and it follows easily that M 0 =» M. Our proof of the theorem is based on the following beautiful result due to R. Guralnick: 1.4. Theorem [G, Theorem A]. Let (R; m; k) be a local ring and ¤ a module-¯nite R- algebra. Given ¯nitely generated left ¤-modules U and V , there is an integer e = e(U; V ), depending only on U and V , with the following property: For each positive integer f and e+f e+f each ¤-homomorphism » : U=m U ! V=m V , there exists σ 2 Hom¤(U; V ) such that σ and » induce the same homomorphism U=mf U ! V=mf V . (Thus the outer and bottom 2 squares in the diagram below both commute, though the top square may not.) U ¡¡¡¡!σ V ? ? ? ? y y » U=me+f U ¡¡¡¡! V=me+f V ? ? ? ? y y »=σ U=mf U ¡¡¡¡! V=mf V Proof. Choose exact sequences of left ¤-modules ¤(n) ¡!® ¤(m) ! U ! 0; ¯ ¤(n) ¡! ¤(m) ! V ! 0: (m) (n) (n) (m) De¯ne a ¤-homomorphism T : End¤(¤ )£End¤(¤ ) ! Hom¤(¤ ; ¤ ) by T (¹; º) = (n) (m) ¹® ¡ ¯º. Applying the Artin-Rees Lemma to the submodule Im(T ) of Hom¤(¤ ; ¤ ), we get a positive integer e = e(U; V ) such that e+f (n) (m) f (1) Im(T ) \ m Hom¤(¤ ; ¤ ) ⊆ m Im(T ) for each f > 0: e+f e+f Suppose now that » : U=m U ! V=m V . We can lift » to homomorphisms ¹0 and º0 making the following diagram commute: (¤=me+f ¤)(n) ¡¡¡¡!® (¤=me+f ¤)(m) ¡¡¡¡! U=me+f U ¡¡¡¡! 0 ? ? ? ? ? ? (2) º0y ¹0y »y ¯ (¤=me+f ¤)(n) ¡¡¡¡! (¤=me+f ¤)(m) ¡¡¡¡! V=me+f V ¡¡¡¡! 0 (m) (n) Lifting ¹0 and º0 to maps ¹0 2 End¤(¤ ) and º0 2 End¤(¤ ), we see, by commutativity e+f (n) (m) f of (2), that T (¹0; º0) 2 m Hom¤(¤ ; ¤ ). By (1), T (¹0; º0) 2 m Im(T ). Therefore f (m) (n) there is a pair (¹1; º1) 2 m (End¤(¤ )£End¤(¤ )) such that T (¹1; º1) = T (¹0; º0). Set (¹; º) := (¹0; º0)¡(¹1; º1). Then T (¹; º) = 0, so ¹ induces an ¤-homomorphism σ : U ! V . f f f Since ¹ and ¹0 agree modulo m , σ and » induce the same map U=m U ! V=m V . ¤ 1.5. Corollary [G, Corollary 2]. Let (R; m; k) be a local ring and ¤ a module-¯nite R- algebra. Given ¯nitely generated left ¤-modules U and V , put ` := maxfe(U; V ); e(V; U)g, where e(¡; ¡) is as in Theorem 1.4. If U=m`+1U j V=m`+1V , then U j V . Proof. Choose ¤-module homomorphisms » : U=m`+1U ! V=m`+1V and ´ : V=m`+1V ! `+1 U=m U such that ´» = 1U=m`+1U . By Theorem 1.4 there exist ¤-homomorphisms σ : U ! V and ¿ : V ! U such that σ agrees with » modulo m and ¿ agrees with ´ modulo m. By Nakayama's lemma, ¿σ : U ! U is surjective and therefore an automorphism. It follows that U j V . ¤ We need one more preliminary result before we can prove Theorem 1.3. 3 1.6. Lemma [W1, 1.2]. Let A ! B be a faithfully flat homomorphism of commutative rings, let ¤ be a module-¯nite A-algebra, and let U and V be ¯nitely presented left ¤- (r) modules. If B ­A U jB­A¤ B ­A V , then there is a positive integer r such that U j¤ V . ® ¯ Proof. Choose B ­A ¤-homomorphisms B ­A U ¡! B ­A V and B ­A V ¡! B ­A U such that ¯® = 1B­AU . Since ¤V is ¯nitely presented, the natural map B ­A Hom¤(V; U) ! HomB­A¤(B ­A V; B ­A U) is an isomorphism. Therefore we can write ¯ = b1 ­ σ1 + ::: + (r) br ­ σr, with bi 2 B and σi 2 Hom¤(V; U) for each i. Put σ := [σ1 ¢ ¢ ¢ σr]: V ! U. Then " # b1 . (1B ­ σ) . ® = 1B­AU ; br (r) so 1B ­ σ : B ­A V ! B ­A U is a split surjection. It follows that 1B ­ σ induces a (r) surjective map HomB­A¤(B ­A U; B ­A V ) ! HomB­A¤(B ­A U; B ­A U).
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