Homework 2 - Topics in Geometry - Spring 2009 | MATH 6490, Assignments of Geometry

Download Homework 2 - 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 2 Exercise 1. Let F be a field, and (V•, d•) be a complex of finite-dimensional F -vector spaces. Assume that it is a finite complex, i.e., Vn 6= 0 for only finitely many n. Show that∑ i (−1)i dimF (Vi) = ∑ i (−1)i dimF (Hi(V )). Exercise 2. State and prove the snake lemma. Exercise 3. State and prove the five lemma. Exercise 4. Consider two short exact sequences S : 0 // N // A // M // 0 and T : 0 // M // B // K // 0. Note that T starts where S ends. Describe how we might splice these two sequences to get an exact sequence S ◦ T : 0 // N // A // B // K // 0 This operation is called the Yoneda composition of two short exact sequences to get a longer sequence. Philosophically, this is the same as wedging two differential one-forms to get a differential two-form. Which is why this composition is sometimes called the wedge product of two short exact sequences. (For more on Yoneda composition, see MacLane’s classic ‘Homology’.) ∗ ∗ ∗ ∗ ∗ ∗ ∗