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.

# 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 $R$.
# Bass's Question. If $M \ne 0$ has a finite injective resolution, then $R$ is a Cohen–Macaulay ring.
# 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''.
# 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 $\bigoplus\nolimits_i H_i(G_)$ has finite length but is not 0. Then the (Krull dimension) $\dim R \le n$.
# 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 $H_i(G_)$ has finite length for $i > 0$ and $H_0(G_)$ has a minimal generator that is killed by a power of the maximal ideal of ''R''. Then $\dim R \le n$.
# 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.
# 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.
# 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 $A \subseteq R \to S$ 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 $\operatorname_i^A(W,R) \to \operatorname_i^A(W,S)$ is zero for all $i \ge 1$.
# 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.
# 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.
# 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 $\operatorname_i^R(M, N)$ 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''.)
# 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.

{{DEFAULTSORT:Homological Conjectures In Commutative Algebra
Commutative algebra
Homological algebra
Conjectures
Unsolved problems in mathematics