Math 2.0

Commutative algebra Homological methods Conjecture

Hochster's homological conjectures

Builds on Noetherian ringLocal ringRegular local ringSystem of parametersRegular sequence

Statement

Let (R,𝔪)(R, \mathfrak{m}) be a commutative Noetherian local ring of dimension dd.

  1. Direct summand conjecture: let RR be a regular local ring and R⊆SR \subseteq S a ring extension with SS a finitely generated RR-module. Then RR is a direct summand of SS as an RR-module.
  2. Monomial conjecture: if x1,…,xdx_1, \ldots, x_d is a system of parameters of RR, then (x1x2⋯xd)t∉(x1t+1,…,xdt+1)(x_1 x_2 \cdots x_d)^t \notin (x_1^{t+1}, \ldots, x_d^{t+1}) for every integer t≥0t \ge 0.
  3. Big Cohen–Macaulay module conjecture: there is an RR-module BB such that some system of parameters of RR is a regular sequence on BB. The module BB need not be finitely generated.

Remarks

These are three of the homological conjectures studied and organized by Hochster; there are others, such as the intersection conjectures. Hochster proved all three when RR contains a field, using reduction to positive characteristic and the Frobenius map. The monomial conjecture and the direct summand conjecture are known to be equivalent. In mixed characteristic, André proved the direct summand conjecture and the existence of big Cohen–Macaulay algebras, using perfectoid methods; so the three statements are now theorems. The direct summand conjecture has its own node.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Local ring, concept, no evidence levelLocal ringconceptNoetherian ring, concept, no evidence levelNoetherian ringconceptRegular local ring, concept, no evidence levelRegular local ringconceptRegular sequence, concept, no evidence levelRegular sequenceconceptSystem of parameters, concept, no evidence levelSystem of parametersconceptHochster's homological conjectures, conjecture, no evidence levelHochster's homologica…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Local ring concept
  2. Noetherian ring concept
  3. Regular local ring concept
  4. Regular sequence concept
  5. System of parameters concept
  6. Hochster's homological conjectures conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
About this page
Tag1TNS604ZST
URIm2://1TNS604ZST
Kindconjecture
WritingW0 stub
MSC13D22, 13H05
StatusThis copy of the site has no build data. Run m2 status 1TNS604ZST.
AboutRegular local ring, System of parameters
UsesNoetherian ring, Local ring, Regular local ring, System of parameters, Regular sequence
Changed2 November 2026

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