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, A , R {\displaystyle A,R} , and S {\displaystyle S} refer to Noetherian commutative rings; R {\displaystyle R} will be a local ring with maximal ideal m R {\displaystyle m_{R}} , and M {\displaystyle M} and N {\displaystyle N} are finitely generated R {\displaystyle R} -modules.
The Zero Divisor Theorem. If M ≠ 0 {\displaystyle M\neq 0} has finite projective dimension and r ∈ R {\displaystyle r\in R} is not a zero divisor on M {\displaystyle M} , then r {\displaystyle r} is not a zero divisor on R {\displaystyle R} . Bass's Question. If M ≠ 0 {\displaystyle M\neq 0} has a finite injective resolution, then R {\displaystyle R} is a Cohen–Macaulay ring. The Intersection Theorem. If M ⊗ R N ≠ 0 {\displaystyle 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. The New Intersection Theorem. Let 0 → G n → ⋯ → G 0 → 0 {\displaystyle 0\to G_{n}\to \cdots \to G_{0}\to 0} denote a finite complex of free R-modules such that ⨁ i H i ( G ∙ ) {\displaystyle \bigoplus \nolimits _{i}H_{i}(G_{\bullet })} has finite length but is not 0. Then the (Krull dimension) dim R ≤ n {\displaystyle \dim R\leq n} . The Improved New Intersection Conjecture. Let 0 → G n → ⋯ → G 0 → 0 {\displaystyle 0\to G_{n}\to \cdots \to G_{0}\to 0} denote a finite complex of free R-modules such that H i ( G ∙ ) {\displaystyle H_{i}(G_{\bullet })} has finite length for i > 0 {\displaystyle i>0} and H 0 ( G ∙ ) {\displaystyle H_{0}(G_{\bullet })} has a minimal generator that is killed by a power of the maximal ideal of R. Then dim R ≤ n {\displaystyle \dim R\leq n} . The Direct Summand Conjecture. If R ⊆ S {\displaystyle R\subseteq 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. The Canonical Element Conjecture. Let x 1 , … , x d {\displaystyle x_{1},\ldots ,x_{d}} be a system of parameters for R, let F ∙ {\displaystyle F_{\bullet }} be a free R-resolution of the residue field of R with F 0 = R {\displaystyle F_{0}=R} , and let K ∙ {\displaystyle K_{\bullet }} denote the Koszul complex of R with respect to x 1 , … , x d {\displaystyle x_{1},\ldots ,x_{d}} . Lift the identity map R = K 0 → F 0 = R {\displaystyle R=K_{0}\to F_{0}=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 {\displaystyle R=K_{d}\to 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 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 A ⊆ R → S {\displaystyle A\subseteq R\to 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 ) {\displaystyle \operatorname {Tor} _{i}^{A}(W,R)\to \operatorname {Tor} _{i}^{A}(W,S)} is zero for all i ≥ 1 {\displaystyle i\geq 1} . The Strong Direct Summand Conjecture. Let R ⊆ S {\displaystyle R\subseteq S} be a map of complete local domains, and let Q be a height one prime ideal of S lying over x R {\displaystyle xR} , where R and R / x R {\displaystyle R/xR} are both regular. Then x R {\displaystyle xR} is a direct summand of Q considered as R-modules. Existence of Weakly Functorial Big Cohen-Macaulay Algebras Conjecture. Let R → S {\displaystyle R\to S} 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 B S {\displaystyle B_{S}} 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 M ⊗ R N {\displaystyle M\otimes _{R}N} has finite length. Then χ ( M , N ) {\displaystyle \chi (M,N)} , defined as the alternating sum of the lengths of the modules Tor i R ( M , N ) {\displaystyle \operatorname {Tor} _{i}^{R}(M,N)} is 0 if dim M + dim N < d {\displaystyle \dim M+\dim N<d} , 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 M ≠ 0 {\displaystyle M\neq 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.