Computational Commutative Algebra
This is an introductory course in computational commutative algebra. Topics in a typical first course in commutative algebra are developed along with computations in Macaulay2. The emphasis will be on concrete computations, more than on giving complete proofs of theorems.
COURSE LAYOUT
Week 1:Introduction: rings and ideals, ring homomorphisms, Hilbert basis theorem, Hilbert Nullstellensatz, introduction to Macaulay2
Week 2:Groebner bases, ideal membership, solving systems of polynomial rings
Week 3:Modules.
Week 4:Associated primes and primary decomposition
Week 5:Associated primes and primary decomposition, ctd.
Week 6:Integral extensions, integral closure, Noether normalization
Week 7:Integral extensions, integral closure, Noether normalization, ctd.
Week 8:Hilbert functions, dimension theory
Week 9:Hilbert functions, dimension theory ctd.
Week 10:Applications to geometry.
Week 11:Homological algebra: depth, Koszul complex
Week 12:Homological algebra: free resolutions, Auslander-Buchsbaum formula.
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