In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth.
The following list given by Melvin Hochster is considered definitive for this area. In the sequel, , and refer to Noetherian commutative rings; will be a local ring with maximal ideal , and and are finitely generated -modules.
- The Zero Divisor Theorem. If has finite projective dimension and is not a zero divisor on , then is not a zero divisor on .
- Bass's Question. If has a finite injective resolution then is a Cohen–Macaulay ring.
- The Intersection Theorem. If has finite length, then the Krull dimension of N (i.e., the dimension of R modulo the annihilator of N) is at most the projective dimension of M.
- The New Intersection Theorem. Let denote a finite complex of free R-modules such that has finite length but is not 0. Then the (Krull dimension) .
- The Improved New Intersection Conjecture. Let denote a finite complex of free R-modules such that has finite length for and has a minimal generator that is killed by a power of the maximal ideal of R. Then .
- The Direct Summand Conjecture. If is a module-finite ring extension with R regular (here, R need not be local but the problem reduces at once to the local case), then R is a direct summand of S as an R-module. The conjecture was proven by Yves André using a theory of perfectoid spaces.
- The Canonical Element Conjecture. Let be a system of parameters for R, let be a free R-resolution of the residue field of R with , and let denote the Koszul complex of R with respect to . Lift the identity map to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from is not 0.
- Existence of Balanced Big Cohen–Macaulay Modules Conjecture. There exists a (not necessarily finitely generated) R-module W such that mRW ≠ W and every system of parameters for R is a regular sequence on W.
- Cohen-Macaulayness of Direct Summands Conjecture. If R is a direct summand of a regular ring S as an R-module, then R is Cohen–Macaulay (R need not be local, but the result reduces at once to the case where R is local).
- The Vanishing Conjecture for Maps of Tor. Let be homomorphisms where R is not necessarily local (one can reduce to that case however), with A, S regular and R finitely generated as an A-module. Let W be any A-module. Then the map is zero for all .
- The Strong Direct Summand Conjecture. Let be a map of complete local domains, and let Q be a height one prime ideal of S lying over , where R and are both regular. Then is a direct summand of Q considered as R-modules.
- Existence of Weakly Functorial Big Cohen-Macaulay Algebras Conjecture. Let be a local homomorphism of complete local domains. Then there exists an R-algebra BR that is a balanced big Cohen–Macaulay algebra for R, an S-algebra that is a balanced big Cohen-Macaulay algebra for S, and a homomorphism BR → BS such that the natural square given by these maps commutes.
- Serre's Conjecture on Multiplicities. (cf. Serre's multiplicity conjectures.) Suppose that R is regular of dimension d and that has finite length. Then , defined as the alternating sum of the lengths of the modules is 0 if , and is positive if the sum is equal to d. (N.B. Jean-Pierre Serre proved that the sum cannot exceed d.)
- Small Cohen–Macaulay Modules Conjecture. If R is complete, then there exists a finitely-generated R-module such that some (equivalently every) system of parameters for R is a regular sequence on M.
References
- André, Yves (2018). "La conjecture du facteur direct". Publications Mathématiques de l'IHÉS. 127: 71–93. arXiv:1609.00345. doi:10.1007/s10240-017-0097-9. MR 3814651. S2CID 119310771.
- Homological conjectures, old and new, Melvin Hochster, Illinois Journal of Mathematics Volume 51, Number 1 (2007), 151-169.
- On the direct summand conjecture and its derived variant by Bhargav Bhatt.