AG-Week Four

From UW-Math Wiki
Revision as of 11:07, 24 September 2010 by Hittson (talk | contribs) (→‎Questions)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

Week Four

This is the page with specific information for Week 4 of our Algebraic Geometry Graduate Reading Course

Discussion Leader: Amanda

Schedule

  • For 9/22: Be prepared to discuss 3.4
    • Hand in 6 written up problems
  • For 9/24: Read & be prepared to discuss 3.5
  • For 9/27: Read & be prepared to discuss 3.5, 3.6, 3.7.
    • Meeting with faculty: Bring questions from all of Chapter 3.

Homework

  • 6 problems due 9/22

Questions

  • Why does sheafification being left-adjoint to the forgetful functor "explain" why you don't need to sheafify when taking kernel, and why you need to sheafify when taking cokernel (see 3.4.K)?
    • If $f:A \to B$ is a morphism of sheaves, then we really have a morphism of presheaves $F(f): F(A) \to F(B)$, where $F$ denotes the forgetful functor. Since $F$ is right-adjoint to sheafification, it commutes with kernels, so we "should" have, $ker_{pre} F(f) = F(ker_{sheaf} f)$. Since $F$ is just the forgetful functor, it doesn't really do anything. So taking the kernel in presheaves "should" actually be a kernel in sheaves. So (sh, F) being adjoint explains why we don't need to sheafify with kernels.

Comments

Typos