Homological conjectures in commutative algebra explained

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 .

1. Bass's Question. If

M \ne 0 has a finite injective resolution then R is a Cohen–Macaulay ring.

1. The Intersection Theorem. If

M ⊗ _{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.

1. The New Intersection Theorem. Let

0 \to G _{n\to …} \to G _{0\to} 0 denote a finite complex of free R-modules such that o plus\nolimits _{i} H _{i(G} _{\bullet} ) has finite length but is not 0. Then the (Krull dimension) \dim R \le n .

1. The Improved New Intersection Conjecture. Let

0 \to G _{n\to …} \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 .

1. 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.^[1]

1. 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.

1. 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.
2. 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).
3. 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} ^{A(W,R)} _{i} \to \operatorname{Tor} ^{A(W,S)} _{i} is zero for all i \ge 1 .

1. 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.

1. 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 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.

1. 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 \chi(M, N) , defined as the alternating sum of the lengths of the modules \operatorname{Tor} ^{R(M,} _{i} 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.)

1. 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.

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.

Notes and References

1. La conjecture du facteur direct. Yves. André. Yves André. 1609.00345. 2018. . 127. 71–93. 3814651 . 10.1007/s10240-017-0097-9 . 119310771.