# Hartshorne Problems



Recommended problems from Hartshorne (c/o Elham Izadi):

**Chapter 1**

- [x] Section 1 ✅ 2022-10-28
    - [x] 1,2,4,5,6 ✅ 2022-10-27
    - [x] 7,8,9,10 ✅ 2022-10-28
- [x] Section 2: ✅ 2022-10-29
    - [x] 2,6,7,8,10 ✅ 2022-10-28
    - [x] Attempt 9 ✅ 2022-10-29

**Chapter 2**

-   [ ] Section 1
    - [x] 1,2,3,4,5 ✅ 2022-10-30
    -   [ ] 6,7,8,17,22
    -   [ ] 9
    -   [ ] 18
-   [ ] Section 2
    -   [ ] 1,2,3,5
    -   [ ] 6,7,8,9,12
    -   [ ] 14,16,17,18,19
    -   [ ] 11 if you want to try something a little harder (See 32.14 of the Stacks project, on universally closed morphisms)
-   [ ] Section 3
    -   [ ] 1,2,3,4,5
    -   [ ] 6,7,8,9,10
-   [ ] Section 4
    -   [ ] 1,2,3,4,6
-   [ ] Section 5
    -   [ ] 1,3,5
    -   [ ] 2,4,7,8,9
    -   [ ] 10,11,12,13
    -   [ ] 14,15,16,17,18
    -   [ ] Note: these problems are especially hard but also especially important and interesting, do as much of them as you can, only please make sure you read them carefully and understand the statements, in many textbooks these are done in the text.
-   [ ] Section 6
    -   [ ] 1,8
    -   [ ] Also prove the following: If $X$ is integral, then any nonzero morphism of invertible sheaves is injective, any generically injective morphism of locally free sheaves is injective *Hint: first prove that a locally free sheaf has no torsion subsheaf, where by a torsion sheaf we mean a sheaf whose support has codimension $>0$.*
    -   [ ] 5,10,
-   [ ] Section 7
    -   [ ] 1,2,3
    -   [ ] 4,5,7,8,9
-   [ ] Section 8
    -   [ ] 1,2,3
    -   [ ] 4, 5