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. "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.

# The Zerodivisor Theorem. If "M ≠ 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 zerodivisor on "M", then "r" is not a zerodivisor on "R".
# Bass's Question. If "M ≠ 0" has a finite injective resolution then "R" is a Cohen-Macaulay ring.
# The Intersection Theorem. If "M ⊗_R N ≠ 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".
# 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".
# 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".
# 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 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".
# 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".
# 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 "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".
# 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.
# 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.
# 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. 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 "M ≠ 0" such that some (equivalently every) system of parameters for "R" is a regular sequence on "M".