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

1. The Zero Divisor Theorem. If <math>M \ne 0</math> has finite projective dimension and <math>r \in R</math> is not a zero divisor on <math>M</math>, then <math>r</math> is not a zero divisor on <math>R</math>.
2. Bass's Question. If <math>M \ne 0</math> has a finite injective resolution, then <math>R</math> is a Cohen–Macaulay ring.
3. The Intersection Theorem. If <math>M \otimes_R N \ne 0</math> 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 <math>0 \to G_n\to\cdots \to G_0\to 0</math> denote a finite complex of free R-modules such that <math>\bigoplus\nolimits_i H_i(G_{\bullet})</math> has finite length but is not 0. Then the (Krull dimension) <math>\dim R \le n</math>.
5. The Improved New Intersection Conjecture. Let <math>0 \to G_n\to\cdots \to G_0\to 0</math> denote a finite complex of free R-modules such that <math>H_i(G_{\bullet})</math> has finite length for <math>i > 0</math> and <math>H_0(G_{\bullet})</math> has a minimal generator that is killed by a power of the maximal ideal of R. Then <math>\dim R \le n</math>.
6. The Direct Summand Conjecture. If <math>R \subseteq S</math> 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.<ref>Template:Cite journal</ref>
7. The Canonical Element Conjecture. Let <math>x_1, \ldots, x_d</math> be a system of parameters for R, let <math>F_\bullet</math> be a free R-resolution of the residue field of R with <math>F_0 = R</math>, and let <math>K_\bullet</math> denote the Koszul complex of R with respect to <math>x_1, \ldots, x_d</math>. Lift the identity map <math>R = K_0 \to F_0 = R</math> to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from <math>R = K_d \to F_d</math> 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 <math>A \subseteq R \to S</math> 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 <math>\operatorname{Tor}_i^A(W,R) \to \operatorname{Tor}_i^A(W,S)</math> is zero for all <math>i \ge 1</math>.
11. The Strong Direct Summand Conjecture. Let <math>R \subseteq S</math> be a map of complete local domains, and let Q be a height one prime ideal of S lying over <math>xR</math>, where R and <math>R/xR</math> are both regular. Then <math>xR</math> is a direct summand of Q considered as R-modules.
12. Existence of Weakly Functorial Big Cohen-Macaulay Algebras Conjecture. Let <math>R \to S</math> 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 <math>B_S</math> 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 <math>M \otimes_R N</math> has finite length. Then <math>\chi(M, N)</math>, defined as the alternating sum of the lengths of the modules <math>\operatorname{Tor}_i^R(M, N)</math> is 0 if <math>\dim M + \dim N < d</math>, 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 <math>M \ne 0</math> 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.