Homological conjectures in commutative algebra

Homological conjectures in commutative algebra are proposed statements connecting resolutions, homological dimensions, and intersection behavior of modules and rings, often linking local algebra to geometry.

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Commutative algebra: The study of commutative rings and their modules, including their algebraic and geometric properties. The conjectures arise within this field and use its rings, modules, and localization.
Free resolution: An exact sequence of free modules mapping onto a given module. Conjectures often concern the length, ranks, or eventual behavior of these sequences.
Auslander–Reiten conjecture: The conjecture that a finitely generated module over a Noetherian ring is projective if its self-Ext groups vanish in positive degrees. It asks whether homological vanishing alone can force a module to be projective.
Jean-Pierre Serre: A French mathematician whose work shaped modern algebra, topology, and number theory. His local-algebra work formulated influential intersection and multiplicity conjectures.
Homological conjectures in characteristic p: Homological conjectures studied for rings containing a field of positive characteristic. Positive-characteristic methods proved several central statements, but not the whole family.
Local ring: A commutative ring with exactly one maximal ideal. Many conjectures are formulated for local rings, where dimension and depth can be compared.
Tor functor: A family of derived functors measuring the failure of tensor product to preserve exactness. Vanishing and growth of Tor groups encode intersection and rigidity phenomena.
Direct summand conjecture: The assertion that every module-finite extension of a regular local ring splits as a module over that ring. It links ring extensions and regularity; André proved it in 2016.
Maurice Auslander: An American mathematician known for foundational work in representation theory and homological algebra. His conjectures with Reiten became enduring questions about modules and homological vanishing.
Homological conjectures in mixed characteristic: Homological conjectures for local rings whose residue field and ring have different characteristics. Mixed characteristic posed major obstacles until perfectoid methods resolved several cases.