MAT 487 Course Webpage Independent Study. Wednesday 9:30AM, Math Tower 4-108 Spring 2018

Course Announcements Announcements about the course will be posted here. Please check the site regularly for announcements (which will also be given in lecture).

Course Description The description in the undergraduate bulletin. A reading course for juniors and seniors. The topics may be chosen by the student with the approval of a supervising member of the faculty, who also takes responsibility for evaluation. A topic that is covered in a course regularly offered by the department is not appropriate for independent study. May be repeated. This independent study course will be focused on commutative ring theory and foundations of algebraic geometry.

Prerequisites Permission of the undergraduate director.

Text The recommended textbooks are the following.

• M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Westview Press, 1969.
• Robin Hartshorne, Algebraic Geometry, Springer, 1977.

Course Learning Objectives

• Acclimate to New Mathematics. Gain familiarity with a new mathematical idea (be it a definition, a result, an algorithm, etc.) through examples, through basic results that involve that idea, and through deeper results that reflect the significance of the idea. Example. Modules over commutative rings. Prime and primary ideals in a commutative ring. Affine algebraic varieties, projective algebraic varieties, and abstract algebraic varieties.
• Apply and Model. Understand how an abstract notion or result can lead to an algorithm or computation arising in a context different from the original notion or result. Understand the necessary hypotheses and limitations of that model. Example. The Hilbert Basis Theorem gives an abstract proof that rings of invariants for finite group actions on a polynomial ring are finitely generated, but Noether's method gives an explicit algorithm for finding a finite set of generators for this ring.
• Specialize. Pass from general theorems, definitions, and methods to specific examples. Be able to compute with those examples. Example. The general Primary Decomposition Theorem of Noether and Lasker for ideals in an arbitrary Noetherian ring specializes to the Chinese Remainder Theorem for ideals in the ring of integers.
• Generalize. Understand examples of ideas, constructions and arguments originally developed in one context yet that extend to another context. Example. The Chinese Remainder Theorem and the structure theorem for finitely generated modules over the ring of integers generalizes to finitely generated modules over every Principal Ideal Domain.
• Prove. For a conjectured result, often expected from examples, heuristics and other indirect evidence, rigorously prove the result using techniques such as proof by induction, proof by contradiction, proof by cases, and more advanced proof techniques. Example. Prove Krull's Principal Ideal Theorem (which is trivially verified for principal ideals in a polynomial ring) for every Noetherian ring using the Artin-Rees Lemma.

Course Outcomes / Key Skills

• Understand the Abelian category of modules over a specified commutative ring, including tensor product, Hom, Tor, Ext, and flatness.
• Understand operations on commutative rings including quotients by ideals, tensor products of two or more ring homomorphisms, passing to fraction rings, and localizations with respect to prime ideals.
• Understand the Primary Decomposition Theorem for finitely generated modules over a Noetherian ring, and how this generalizes the Unique Factorization Theorem for the ring of integers.
• Understand the Going Up Theorem and the Going Down Theorem, and the interpretation of these theorems via Krull dimension.
• Understand the structure theory of Discrete Valuation Rings, Dedekind Domains, and finitely generated modules over these rings.
• Understand completions of rings and modules with respect to an ideal. Understand the Artin-Rees Lemma and its applications (e.g., the completion of a Noetherian ring is also a Noetherian ring).
• Understand the basic results of dimension theory, including Krull's Principal Ideal Theorem.
• Understand the basic definitions of affine and projective algebraic sets, such as the Zariski topology, irreducible algebraic sets, dimension, morphisms, and rational transformations.
• Understand the fundamental theorems of affine and projective varieties, such as Bezout's Theorem.

Lectures The instructor for this course is Jason Starr.

Course meetings are Wednesday mornings, beginning at 9:30AM, in Math Tower 4-108.

Office Hours Office hours for Jason Starr are tentatively scheduled as follows.

• Mondays 11AM — 12NOON, Math Tower P—143 (advising).
• Mondays 9AM — 11AM, Math Tower 4—108.

Grading System The letter grade for the course is based on class participation, weekly problem sets, and a final oral report.

Oral presentation
Every student will give an oral report of approximately 20 minutes in the last few weeks of the semester. The topic of the presentation is to be determined by the student and instructor jointly. It may be on anything related to the course, but not directly covered during course meetings or on problem sets, e.g., a skipped section of the textbook, an interesting sequence of (unassigned) textbook problems, related material from another textbook, etc. The topic should be discussed with the instructor and approved no later than April 15th.

Hand-backs Graded problem sets will be handed back during course meetings. Some fraction of course meetings will be devoted to discussing solutions to assigned problem sets.

Required Syllabi Statements
The University Senate Undergraduate and Graduate Councils have authorized that the following required statements appear in all teaching syllabi (graduate and undergraduate courses) on the Stony Brook Campus.

Americans with Disabilities Act.
If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, Room 128, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential.

Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information go to the following website: http://www.stonybrook.edu/ehs/fire/disabilities.

Academic Integrity Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty is required to report any suspected instances of academic dishonesty to the Academic Judiciary. Faculty in the Health Sciences Center (School of Health Technology Management, Nursing, Social Welfare, Dental Medicine) and School of Medicine are required to follow their school-specific procedures. For more comprehensive information on academic integrity, including categories of academic dishonesty please refer to the academic judiciary website at http://www.stonybrook.edu/commcms/academic_integrity/index.html.

Critical Incident Management Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of University Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Faculty in the HSC Schools and the School of Medicine are required to follow their school-specific procedures.

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