Commutative Algebra: Theorems, Proofs, and Python Implementations (Computational Mathematics Library)

Unlock the algebra that powers modern algebraic geometry, cryptography and computational number theory. This comprehensive reference walks you from first principles to frontier results while placing heavy-traffic topics—Noetherian rings, Gröbner bases, Cohen–Macaulay and Gorenstein conditions—front and center.Every chapter pairs rigorous theorems and proofs with concise, runnable Python snippets so you can verify identities, compute primary decompositions and experiment with Hilbert functions in real time.Key coverage• Foundations: commutative rings with identity, modules, homomorphisms, tensor products• Ideals and quotients, Hilbert’s Basis Theorem, primary decomposition, associated primes• Localization, integral dependence, lying over, going up and going down• Dedekind domains, principal ideal domains, unique factorization domains• Krull dimension, height inequalities and catenary chains• Regular sequences, depth, the Auslander-Buchsbaum formula and Cohen-Macaulay rings• Gorenstein duality, canonical modules, Bass numbers• Flat, projective and injective dimensions, Ext and Tor, spectral sequences• Graded and filtered structures, Rees algebras, Hilbert-Samuel multiplicity• Completion, adic topology, Hensel’s lemma, discrete valuation rings• Normalization algorithms, integral closure of ideals, tight closure in characteristic p• Buchberger’s algorithm and Gröbner bases for efficient computer algebra• Intersection theory via Serre multiplicity, linkage, reduction to complete intersections• Final bridge to geometry: sheaves, affine and projective schemes, morphisms, fibre productsWhether you are refreshing fundamentals or pushing toward research-level topics, this authoritative guide and its executable notebooks will keep your algebra both theoretical and tangible. Read more