Basic Homological Algebra by M. Scott Osborne

B P P )0 with Hom(P, p)(g) = f, that is, pg = f.

Suppose A E MR. Show that A is flat if and only if A ® I AI is one-to-one for every finitely generated left ideal I. 12. Suppose R is a PID. Hence,showQis a ,flat 7L-module. Note: Q is not projective, as will be established in the next chapter. It is somewhere between amusing and exasperating to attempt this now. ) 13. Suppose R and S are rings, A E MR, B E RMS, and C E SM. Then A OR B E Ms and B ®s C E RM. Show that A®R (B ®s C) (A OR B) ®s C. Note: The "obvious" approach, defining a ® (b ® c) -+ (a ® b) ® c, has the usual difficulty: Why is this well-defined?

Tor(A, B) will simultaneously measure unflatness of A (in MR) and B (in RM). 2 shows why we never applied Hom(C, ) to a projective resolution. The analog of property (b) would not hold, since Hom(C, ) is not right exact. This failure will be more significant in the next section, where we relate Extn to Extn+l. The fact that Hom(C, ) is left exact suggests that something can be done. The thoroughly remarkable result of this (to be discussed in Section 3) will be a cloning of the same Ext we got here.

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