This textbook on commutative algebra presents basic results necessary for elementary algebraic geometry and introduces basic homological algebra and homological methods in commutative algebra. Its lecture-notes style will help readers with some mathematical maturity to study it on their own. Motivations are given at a number of places, examples follow every definition, and exercises are given at the end of each section.
N S Gopalakrishnan, after obtaining his masters degree in Mathematics in 1955 from Vivekananda College, Chennai, joined the Tata Institute of Fundamental Research, Mumbai, for research in pure mathematics. He specialised in homological algebra on which he obtained his doctorate from Pune university in 1964. He was a faculty member of Pune University for more than three decades, teaching algebra, topology and related subjects to postgraduate students and also guiding students in their PhDs. He retired from Pune University in 1995.
Preface
1 Modules
1.1 Free modules
1.2 Projective modules
1.3 Tensor products
1.4 Flat modules
2 Localisation
2.1 Ideals
2.2 Local rings
2.3 Localisation
2.4 Applications
3 Noetherian Rings
3.1 Noetherian modules
3.2 Primary decomposition
3.3 Artinian modules
3.4 Length of a module
4 Integral Extensions
4.1 Integral elements
4.2 Integral extensions
4.3 Integrally closed domains
4.4 Finiteness of integral closure
5 Dedekind Domains
5.1 Valuation rings
5.2 Discrete valuation rings
5.3 Dedekind domain
6 Completions
6.1 Filtered rings and modules
6.2 Completion
6.3 I-adic filtration
6.4 Associated graded rings
7 Homology
7.1 Complexes
7.2 Derived functors
7.3 Homological dimension
8 Dimension
8.1 Hilbert Samuel polynomial
8.2 Krull dimension
8.3 Dimension of algebras
8.4 Depth
8.5 Cohen–Macaulay modules
9 Regular Local Rings
9.1 Regular local rings
9.2 Homological characterisation
9.3 Normality conditions
9.4 Complete local rings
10 Some Conjectures
10.1 Big Cohen–Macaulay modules conjecture
10.2 Intersection conjecture
10.3 Zero-divisor conjecture
10.4 Bass’s conjecture