Homological conjectures in commutative algebra

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, $$A, R$$, 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 $$M \ne 0$$ has finite projective dimension and $$r \in R$$ is not a zero divisor on $$M$$, then $$r$$ is not a zero divisor on $$R$$.
 * 2) Bass's Question. If $$M \ne 0$$ has a finite injective resolution then $$R$$ is a Cohen–Macaulay ring.
 * 3) The Intersection Theorem. If $$M \otimes_R N \ne 0$$ 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 $$0 \to G_n\to\cdots \to G_0\to 0$$ denote a finite complex of free R-modules such that $$\bigoplus\nolimits_i H_i(G_{\bullet})$$ has finite length but is not 0. Then the (Krull dimension) $$\dim R \le n$$.
 * 5) The Improved New Intersection Conjecture. Let $$0 \to G_n\to\cdots \to G_0\to 0$$ denote a finite complex of free R-modules such that $$H_i(G_{\bullet})$$ has finite length for $$i > 0$$ and $$H_0(G_{\bullet})$$ has a minimal generator that is killed by a power of the maximal ideal of R. Then $$\dim R \le n$$.
 * 6) The Direct Summand Conjecture. If $$R \subseteq S$$ 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.
 * 7) The Canonical Element Conjecture. Let $$x_1, \ldots, x_d$$ be a system of parameters for R, let $$F_\bullet$$ be a free R-resolution of the residue field of R with $$F_0 = R$$, and let $$K_\bullet$$ denote the Koszul complex of R with respect to $$x_1, \ldots, x_d$$. Lift the identity map $$R = K_0 \to F_0 = R$$ to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from $$R = K_d \to F_d$$ is not 0.
 * 8) Existence of Balanced Big Cohen–Macaulay Modules Conjecture. There exists a (not necessarily finitely generated) R-module W such that mRW ≠ 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 $$A \subseteq R \to S$$ 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 $$\operatorname{Tor}_i^A(W,R) \to \operatorname{Tor}_i^A(W,S)$$ is zero for all $$i \ge 1$$.
 * 11) The Strong Direct Summand Conjecture. Let $$R \subseteq S$$ 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 $$R \to S$$ 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 $$B_S$$ 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.
 * 13) Serre's Conjecture on Multiplicities. (cf. Serre's multiplicity conjectures.) Suppose that R is regular of dimension d and that $$M \otimes_R N$$ has finite length.  Then $$\chi(M, N)$$, defined as the alternating sum of the lengths of the modules $$\operatorname{Tor}_i^R(M, N)$$ is 0 if $$\dim M + \dim N < d$$, and is positive if the sum is equal to d. (N.B. Jean-Pierre Serre proved that the sum cannot exceed d.)
 * 14) Small Cohen–Macaulay Modules Conjecture. If R is complete, then there exists a finitely-generated R-module $$M \ne 0$$ such that some (equivalently every) system of parameters for R is a regular sequence on M.