Homework 3 - Topics in Geometry - Spring 2009 | MATH 6490, Assignments of Geometry

Download Homework 3 - Topics in Geometry - Spring 2009 | MATH 6490 and more Assignments Geometry in PDF only on Docsity! TOPICS IN GEOMETRY: SHEAF THEORY MATH 6490, SPRING 2009 HOMEWORK 3 For this homework set F will denote an additive functor from R-MOD to AB. Suppose F is covariant; we say F is left exact if for any short exact sequence 0 // A // B // C // 0 of R-modules, the sequence 0 // F (A) // F (B) // F (C) is exact. In the case when F is contravariant, we would ask for 0 // F (C) // F (B) // F (A) to be ex- act. (You can remember this by noting that after applying F , the 0 is only on the left.) Right exactness is defined similarly: When F is covariant, we say F is right exact if for any short exact sequence 0 // A // B // C // 0 of R-modules, the sequence F (A) // F (B) // F (C) // 0 is exact. In the case when F is contravariant, we would ask for F (C) // F (B) // F (A) // 0 to be exact. (You can remember right exactness by noting that after applying F , the 0 is only on the right.) Exercise 1. If F is left exact then show that R0F is naturally equivalent to F . (You have to separately argue the two cases depending on when F is covariant or contravariant.) Exercise 2. If F is right exact then show that L0F is naturally equivalent to F . (You have to separately argue the two cases depending on when F is covariant or contravariant.) Exercise 3. Let M be a projective module, and F an additive covariant functor. Then show that LnF (M) ' { F (M) , if n = 0 0 , if n ≥ 1. Exercise 4. Let I be an injective module, and F an additive covariant functor. Then show that RnF (I) ' { F (I) , if n = 0 0 , if n ≥ 1. ∗ ∗ ∗ ∗ ∗ ∗ ∗