See below for the homework assignments for Math 552.
The Syllabus for course is available here.
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Textbook: Ravi Vakil’s notes, The Rising Sea: Foundations Of Algebraic Geometry, version of Nov. 18, 2017, available here, or along with earlier versions, here.
HW1: Please do the following problems contained in the textbook. The assignment is due Sept. 9.
Section 1.3: A, B, C and read the following paragraph, E, F, G, H, K, L, M, N, S, U, Y; Do one of the following: D or J; Do one of the following: O or T
Section 1.4: B, D, F
Section 1.5: B, C, E
Section 1.6: F, G, J
For 1.3.F it is important to recall the difference between the direct product and direct sum of modules. The direct product is a sequence of elements in each component. The direct sum is a sequence of elements in each component with all but finitely many of these elements being 0.
HW2: Please do the following problems contained in the textbook. Due Sept. 16.
Section 2.2: E, G(a), H
Section 2.3: C, I, J
Section 2.4: C, F, L, M, P
Either 2.2.I or 2.3.A
Choose one of 2.4.I, 2.4.J, or 2.4.K
Either 2.4.N or 2.4.O
Either 2.4.D or 2.4.E
HW3: Please do the following problems from the textbook. Due Sept. 23.
Section 2.5: C,D
Section 2.6: C, D, J
Section 2.7: A, B, F
Section 3.2: C, G, P, Q, R, T
Either 2.6.A or 2.6.B
Either 2.6.F or 2.6.G
HW4: Please do the following problems from the textbook. Due Sept. 30.
Section 3.4: 3.4.D, 3.4.E, 3.4.H
Section 3.5: 3.5.B, 3.5.C, 3.5.E (read 3.5.F as well but you don't need to do it)
Section 3.6: 3.6.F(a). Read all of these problems but only turn in solutions to FIVE of the following problems: 3.6.A, 3.6.B, 3.6.C, 3.6.J, 3.6.K, 3.6.L, 3.6.R, 3.6.S, 3.6.V
Section 3.7: 3.7.B
Section 4.1: 4.1.A, 4.1.B,4.1.D
HW5: Please do the following problems from the textbook. Due Oct. 7.
Section 4.3: 4.3.C, 4.3.F, 4.3.G
Section 4.4: 4.4.A, 4.4.B or C, 4.4.E, 4.4.F
Section 4.5: 4.5.C, 4.5.D, 4.5.E, 4.5.F, 4.5.H, 4.5.I, 4.5.J, 4.5.K
HW6: Please do the following problems from the textbook. Due Oct 16.
Section 6.2/6.3: 6.2.A or 6.3.A, 6.3.E, 6.3.F, 6.3.I
Section 6.4: 6.4.A, 6.4.B, 6.4.D, 6.4.E, 6.4.F, 6.4.G
Section 5.1: 5.1.E, 5.1.H
Section 5.2: 5.2.A or B, 5.2.D, 5.2.H, 5.2.I
Section 5.3: 5.3.C, 5.3.F
Section 5.4: 5.4.A, 5.4.B
HW7: Please do the following problems from the textbook. Due Oct. 21.
Section 5.4: 5.4.E, 5.4.H, 5.4.I, 5.4.J, 5.4.K or L, 5.4.N
Section 5.5: 5.5.A, 5.5.C, 5.5.H, 5.5.I, 5.5.J, 5.5.K, 5.5.L, 5.5.M, 5.5.N
Section 6.3: 6.3.J, 6.3.M
HW8: Please do the following problems from the textbook. Due Oct. 28
Section 6.5: 6.5.A, 6.5.B, 6.5.C, 6.5.E, 6.5.F, 6.5.G
Section 7.1: 7.1.B, 7.1.D
Section 7.2: 7.2.A, 7.2.B, 7.2.E, 7.2.F, 7.2.H
Section 7.3: 7.3.C, 7.3.F, 7.3.K, 7.3.M, 7.3.P, 7.3.R or 7.3.S
Here are the notes on the full proof of Grothendieck's Generic Freeness Lemma and Chevalley's Theorem.
HW9: Please do the following problems from the textbook. Due Nov. 4.
Section 7.4: 7.4.B, 7.4.E
Section 8.1: 8.1.G, 8.1.H, 8.1.J, 8.1.M
Section 8.2: 8.2.B, 8.2.C, 8.2.E, 8.2.G, 8.2.H, 8.2.J, 8.2.P
Section 8.3: 8.3.A, 8.3.C
HW10: Please do the following problems from the textbook. Due Nov. 11.
Section 9.2: 9.2.A, 9.2.B, 9.2.C, 9.2.D, 9.2.I, 9.2.J, 9.2.K
Section 9.3: 9.3.B, 9.3.D or E, 9.3.G
Section 9.4: 9.4.B, 9.4.C, 9.4.D, 9.4.E, 9.4.F
HW11: Please do the following problems from the textbook. Due Nov. 18.
Section 9.6: 9.6.B, 9.6.D
Section 9.7: 9.7.D, 9.7.G, 9.7.H, 9.7.P
Section 10.1: 10.1.A, 10.1.B, 10.1.E, 10.1.J, 10.1.K
Section 10.2: 10.2.B, 10.2.D, 10.2.E, 10.2.G
HW12: Please do the following problems from the textbook. Due Nov. 25.
Section 10.3: 10.3.B
Section 11.1: 11.1.B, 11.1.F, 11.1.G, 11.1.H, 11.1.I, 11.1.J, 11.1.K
Section 11.2: 11.2.A, 11.2.B, 11.2.C, 11.2.D, 11.2.E
Extra HW13: Please do the following problems from the textbook. Due Dec. 2.
Section 11.2: 11.2.I, 11.2.J, 11.2.K
Section 11.3: 11.3.B, 11.3.E, 11.3.F, 11.3.H, 11.3.J
Section 11.4: 11.4.C, 11.4.F-H
In order to do 11.2.K and its sequel 11.3.J, you should read Section 6.7 in the textbook.
You don't need to turn the exercises there in, but they may help you understand the construction.