Math 552 Fall 2019

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

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.  

Solutions to HW1.

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

Solutions to HW2.

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

Solutions to HW3.

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

Solutions to HW4.

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

Solutions to HW5.

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

Solutions to HW6.

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

Solutions to HW7.

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

Solutions to HW8.

Here are the notes on the full proof of Grothendieck's Generic Freeness Lemma and Chevalley's Theorem.

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

Solutions to HW9.

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

Solutions to HW10.

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

Solutions to HW11.

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

Solution to HW12.

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.