課程名稱︰代數二 課程性質︰數學系選修，可抵必修代數導論二 課程教師︰林惠雯 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2017/6/19 考試時限（分鐘）：180分鐘 試題 : 1.(50%) (a) Let A be the ring of integers in Q(√(d)). Show that A is a Euclidean d 5 domain but A is not. －5 (b) Show that if M is a finitely generated R-module (R：commutative with 1) and IM = M (I：an ideal of R contained in the Jacobson radical of R), then M = 0. (c) Let R be a local ring and M be a finitely generated projective R-module. Show that M is free. (d) State and show the Going-up theorem. (e) Let R be a Dedekind domain and I≠0 an ideal in R. Show that every ideal in R/I is principal. Deduce that every ideal in R can be generated by at most two elements. (You can use the properties of Dedekind domains directly.) 2.(20%) Justify your answers. n (a) Let N be a Z/12Z-module. Compute Ext (Z/3Z, N), for all n≧0. Z/12Z (b) Compute Tor (Z/12Z, Z/15Z). 1 3.(25%) Let x ,..., x be a regular sequence in a commutative ring R (with 1) 1 n and I be the ideal of R generated by x ,..., x . 1 n (a) Construct a free resolution of R/I. (Justify your answer.) (b) Show that R/I does not have any projective resolution of length shorter than n. n ～ (c) Let M be an R-module. Show that Ext (R/I, M) ＝ M/IM. 4.(25%) Let A, B be abelian categories, F：A → B be an additive left exact functor and R be the class in A adapted to F. (a) Show that the class S of quasi-isomorphisms in A is localizing and so A is S . B ＋ －1 ＋ (b) Show that a triangle in K (R)[S ] which is distinguished in D (A) is R ＋ －1 also distinguished in K (R)[S ]. (Hint： don't forget a map in R ＋ －1 K (R)[S ] is represented by a left roof.) R ＋ (c) Show that the derived functor RF defined in D (A) is exact. --

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