First and foremost, his injury shouldn't be too big of a deal. It's his ankle but it had nothing to do with pitching, he hurt it running the bases. Obviously, that won't be too big of a problem in the AL.
Ok, so I've heard many people voice some concerns about Peavy, and I'd like to address some of them.
Never does well under pressure/never succeeds in big games: Well, according to Baseball Reference, this is blasphemy. In high leverage situations throughout his career, 837 PA to be exact, he has a 2.94 K/BB ratio, a .215 BAA, and hitters only muster a smoking .599 OPS. He's struck out 203 of those batters and gotten 42 DP.
Second, I've heard some people voice concerns over Peavy coming to the AL and how that would affect his numbers. Well, in inter-league play throughout his career, he's got a 3.29 inter-league ERA in 20 starts which is exactly the same as his career average. Against the AL, his numbers are:
.235 BAA
.291 OBP
.663 OPS
120.1 IP
113 K
33 BB
3.42 K/BB
1.17 WHIP
8.5 K/9
He strikes out a lot of hitters and doesn't walk many, and now he's going to be coached by Don Cooper. He keeps the ball down and he's an ace.
Oh yeah, don't forget he's 28 years old and under our control for at least 3 more seasons.