Homological conjectures in commutative algebra

In mathematics, the 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, A, R, and S refer to Noetherian commutative rings; R will be a local ring with maximal ideal m_R, and M and N are finitely generated R-modules.

1. The Zero Divisor Theorem. If $M\neq 0$ has finite projective dimension (i.e., M has a finite projective (=free when R is local) resolution: the projective dimension is the length of the shortest such) and r ∈ R is not a zero divisor on M, then r is not a zero divisor on R.
2. Bass's Question. If $M\neq 0$ has a finite injective resolution then R is a Cohen–Macaulay ring.
3. The Intersection Theorem. If $M\otimes _{R}N\neq 0$ 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.
4. The New Intersection Theorem. Let 0 → G_n → … → G₀ → 0 denote a finite complex of free R-modules such that ⊕_iH_i(G_•) has finite length but is not 0. Then the (Krull dimension) dim R ≤ n.
5. The Improved New Intersection Conjecture. Let 0 → G_n → … → G₀ → 0 denote a finite complex of free R-modules such that H_i(G_•) has finite length for i > 0 and H₀(G_•) has a minimal generator that is killed by a power of the maximal ideal of R. Then dim R ≤ n.
6. The Direct Summand Conjecture. If R ⊆ S 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.
7. The Canonical Element Conjecture. Let x₁, …, x_d be a system of parameters for R, let F_• be a free R-resolution of the residue field of R with F₀ = R, and let K_• denote the Koszul complex of R with respect to x₁, …, x_d. Lift the identity map R = K₀ → F₀ = R to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from R = K_d → F_d is not 0.
8. Existence of Balanced Big Cohen–Macaulay Modules Conjecture. There exists a (not necessarily finitely generated) R-module W such that m_RW ≠ W and every system of parameters for R is a regular sequence on W.
9. 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).
10. The Vanishing Conjecture for Maps of Tor. Let A ⊆ R → S 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 Tor_i^A(W,R) → Tor_i^A(W,S) is zero for all i ≥ 1.
11. The Strong Direct Summand Conjecture. Let R ⊆ S be a map of complete local domains, and let Q be a height one prime ideal of S lying over xR, where R and R/xR are both regular. Then xR is a direct summand of Q considered as R-modules.
12. Existence of Weakly Functorial Big Cohen-Macaulay Algebras Conjecture. Let R → S be a local homomorphism of complete local domains. Then there exists an R-algebra B_R that is a balanced big Cohen–Macaulay algebra for R, an S-algebra B_S that is a balanced big Cohen-Macaulay algebra for S, and a homomorphism B_R → B_S such that the natural square given by these maps commutes.
13. Serre's Conjecture on Multiplicities. (cf. Serre's multiplicity conjectures.) Suppose that R is regular of dimension d and that M ⊗_R N has finite length. Then χ(M, N), defined as the alternating sum of the lengths of the modules Tor_i^R(M, N) is 0 if dim M + dim N < d, and positive if the sum is equal to d. (N.B. Jean-Pierre Serre proved that the sum cannot exceed d.)
14. Small Cohen–Macaulay Modules Conjecture. If R is complete, then there exists a finitely-generated R-module M ≠ 0 such that some (equivalently every) system of parameters for R is a regular sequence on M.

References

• 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.