Surveys

Welcome to the section of commalg.org consisting, loosely, of survey articles related to commutative algebra. We say “loosely” because it contains other gems from the internet like books and articles about commutative algebraists.
We’re always happy to hear from you if you’ve got suggestions for other articles you think we should point to — just let us know!
Here is a list from David Eisenbud and Craig Huneke of Accessible papers in Commutative Algebra. It’s a great resource; thanks, David and Craig!

Algebraic Geometry

• Aaron Bertram:
  • A Stab at Some Algebraic Geometry (1996)
  • Algebraic Geometry Notes 2002-03
• Manuel Blickle, Hélène Esnault, Kay Rülling: Characteristic 0 and p analogies, and some motivic cohomology
• Manuel Blickle: A Short Course on Geometric Motivic Integration
• Manuel Blickle and Robert Lazarsfeld: An informal introduction to multiplier ideals
• Jaydeep V. Chipalkatti: Notes on Grassmannians and Schubert Varieties
• David A. Cox:
  • Lectures on Toric Varieties
  • What is a Toric Variety? (notes)
  • What is a Toric Variety? (slides)
  • Introduction to Algebraic Geometry
  • Minicourse on Toric Varieties
  • What is the Multiplicity of a Base Point?
• Cox, Little and O’Shea:
  • Ideals, Varieties, and Algorithms
  • Using Algebraic Geometry
• Cox, Little and Schenck: Toric Varieties
• David Eisenbud:
  • Computation of cohomology
  • An exterior view of modules and sheaves
  • The geometry of syzygies
• David Eisenbud, Mark Green and Joe Harris: Cayley-Bacharach theorems and conjectures
• David Eisenbud and Joe Harris: Progress in the theory of complex algebraic curves
• David Eisenbud and Jee-Heub Koh: Remarks on points in a projective space
• Alexander Grothendieck:
  • EGA I, EGA II, EGA III.1, EGA III.2, EGA IV.1, EGA IV.2, EGA IV.3, EGA IV.4
  • SGA I
• Mark Haiman: Commutative algebra of N points in the plane
• Peter Jørgensen: Non-commutative projective geometry
• Juan Migliore: Experiments in Commutative Algebra and Algebraic Geometry
• Ezra Miller: What is … a toric variety?
• J. S. Milne: Algebraic Geometry Course Notes
• Jürgen Richter-Gebert, Bernd Sturmfels, Thorsten Theobald: First steps in tropical geometry
• Karen E. Smith:
  • Motivic Integration
  • The Hodge Conjecture
  • Rational and non-rational Algebraic Varieties: Lectures of János Kollár
  • Vanishing Theorems, Singularities, and Effective Bounds in Algebraic Geometry Via Prime Characteristic Local Algebra and erratum
• David Speyer and Bernd Sturmfels: Tropical mathematics
• Charles Weibel:
  • A Road Map of Motivic Homotopy and Homology Theory
  • Lectures on motivic cohomology

Lie Algebras

• Joris van Hoboken: Platonic solids, binary polyhedral groups,
  Kleinian singularities and Lie algebras of type A, D, E
• Victor Kac: Lie Algebras Course Notes

Miscellaneous

• Free Online MIT Course Materials
• The Rising Sea: Notes by Daniel Murfet
• Apostolos Beligiannis and Idun Reiten: Homological aspects of torsion theories
• Dave Benson:
  • Commutative algebra and duality in the cohomology of groups
  • Commutative algebra in the cohomology of groups
• Edward Green, Idun Reiten and Øyvind Solberg: Dualities on generalized Koszul algebras
• Dragan Milicic:
  • Lectures on Algebraic Theory of D-Modules
  • Lectures on Derived Categories
• L. J. Ratliff, Jr.: A brief history and survey of the catenary chain conjectures

Homological Conjectures

• Mel Hochster: Some Expository Manuscripts
• Paul Roberts:
  • Recent developments on Serre’s multiplicity conjectures: Gabber’s proof of the nonnegativity conjecture
  • Heitmann’s proof of the direct summand conjecture in dimension 3
  • Cycles and Commutative Algebra

Tight Closure

• Mel Hochster: Some Expository Manuscripts
• Karen E. Smith: An Introduction to Tight Closure
• Irena Swanson: Ten lectures on tight closure

Resolutions and Homological Dimensions

• Lucho Avramov:
  • Homological dimensions and related invariants of modules over local rings
  • Infinite free resolutions
• David Eisenbud:
  • A survey of some results on free resolutions
  • Some directions of recent progress in commutative algebra
  • Introduction to algebras with straightening laws
  • Green’s conjecture: An orientation for algebraists
  • Lectures on the geometry of syzygies. With a chapter by Jessica Sidman
• Vladimir Maşek: Gorenstein dimension of modules
• Claudia Miller: The Frobenius endomorphism and homological dimensions
• Roger Wiegand: What is… a Syzygy?

Connections with Algebraic Topology

• J. P. C. Greenlees:
  • First steps in brave new commutative algebra
  • Spectra for commutative algebraists
• Notes from the Summer School on the Interactions between Homotopy Theory and Algebra, University of Chicago, July 26 to August 6, 2004

Multiplicities and Hilbert Functions

• Irena Peeva and Mike Stillman: Open problems on syzygies and Hilbert functions
• Irena Swanson: Multi-graded Hilbert functions and mixed multiplicities

Ideal Theory

• Irena Swanson: Primary decompositions
• Stefania Gabelli: Characterizing integral domains by semigroups of ideals – Notes for an advanced course in Ideal Theory

Computation and Gröbner Bases

• David A. Cox:
  • Gröbner Bases Tutorial: The Geometry of Elimination
  • Gröbner Bases Tutorial: A Sampler of Recent Developments
• Cox, Little and O’Shea: Ideals, Varieties, and Algorithms
• David Eisenbud: Open problems in computational algebraic geometry
• David Eisenbud, Dan Grayson, Mike Stillman, and Bernd Sturmfels: Computations in algebraic geometry with Macaulay 2
• Juan Migliore: Experiment in Commutative Algebra and Algebraic Geometry
• Bernd Sturmfels: What is a Grobner basis?

Integral Closure

• Swanson and Huneke: Integral Closure of Ideals, Rings, and Modules

Factorization

• Jim Coykendall and Scott Chapman: Half-factorial domains, a survey
• Jim Coykendall: Extensions of half-factorial domains
• Marco Fontana and Muhammad Zafrullah: On v-domains: a survey
• Muhammad Zafrullah:
  • Various Facets of Rings between D[X] and K[X]
  • What v-coprimality can do for you
  • A Question/Answer session on v-domains
• Alfred Geroldinger and Qinghai Zhong: Factorization Theory in Commutative Monoids

K-Theory

• Dan Grayson: A Brief Introduction to Algebraic K-Theory
• Charles Weibel:
  • Algebraic K-theory of rings of integers in local and global fields
  • Algebraic K-Theory

Tributes

• Lucho Avramov: The Work of Jan-Erik roos on the Cohomology of Commutative Rings
• David Eisenbud:
  • Wolfgang Vogel: Reminiscences of a mathematical friendship
  • Preface to Saunders Mac Lane: A mathematical autobiography

Connections with Combinatorics

• Winfried Bruns and Udo Vetter: Determinantal Rings
• David Eisenbud: Introduction to algebras with straightening laws
• Ezra Miller and David Perkinson: Eight Lectures on Monomial Ideals
• Ezra Miller and Vic Reiner: What is geometric combinatorics?—An overview of the graduate summer school

(Co)homology

• Craig Huneke: Lectures on Local Cohomology (Appendix by Amelia Taylor)
• Cristodor Ionescu: Hochschild (Co)homology in Commutative Algebra. A Survey
  

Applications

• N. Eriksson, K. Ranestad, B. Sturmfels and S. Sullivant: Phylogenetic algebraic geometry
• Lior Pachter and Bernd Sturmfels: Tropical geometry of statistical models
• Bernd Sturmfels:
  • Open Problems in Algebraic Statistics
  • Algebraic Recipes for Integer Programming

Interviews

• David Eisenbud, interviewed by Sara Robinson: Mathematics comes from many sources
• Allyn Jackson: Presidential views: Interview with David Eisenbud
• Allyn Jackson: Presidential reflections: Interview with David Eisenbud