Homological conjectures in commutative algebra

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, homological conjectures have been a focus of research activity in <a href="/facts/Commutative_algebra/mKtUdPCT">commutative algebra</a> since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various <a href="/facts/Homological_algebra/Jfm8w52l">homological</a> properties of a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> to its internal ring structure, particularly its <a href="/facts/Krull_dimension/KQb2uLtz">Krull dimension</a> and <a href="/facts/Depth_(algebra)/1E19xXdD">depth</a>.
The following list given by <a href="/facts/Melvin_Hochster/rbXPNR0X">Melvin Hochster</a> is considered definitive for this area. In the sequel, 
 
 
 
 A
 ,
 R
 
 
 {\displaystyle A,R}
 
, and 
 
 
 
 S
 
 
 {\displaystyle S}
 
 refer to <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian</a> <a href="/facts/Commutative_ring/QS1r2ovR">commutative rings</a>; 
 
 
 
 R
 
 
 {\displaystyle R}
 
 will be a <a href="/facts/Local_ring/gRTdg5Qx">local ring</a> with maximal ideal 
 
 
 
 
 m
 
 R
 
 
 
 
 {\displaystyle m_{R}}
 
, and 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 are <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> 
 
 
 
 R
 
 
 {\displaystyle R}
 
-modules.

<ol><li>The Zero Divisor Theorem. If 
 
 
 
 M
 ≠
 0
 
 
 {\displaystyle M\neq 0}
 
 has finite <a href="/facts/Projective_module/lHymqhNC">projective dimension</a> and 
 
 
 
 r
 ∈
 R
 
 
 {\displaystyle r\in R}
 
 is not a <a href="/facts/Zero_divisor/5GraAU8o">zero divisor</a> on 
 
 
 
 M
 
 
 {\displaystyle M}
 
, then 
 
 
 
 r
 
 
 {\displaystyle r}
 
 is not a zero divisor on 
 
 
 
 R
 
 
 {\displaystyle R}
 
.</li>
<li>Bass's Question. If 
 
 
 
 M
 ≠
 0
 
 
 {\displaystyle M\neq 0}
 
 has a finite <a href="/facts/Injective_resolution/xEFm83U7">injective resolution</a>, then 
 
 
 
 R
 
 
 {\displaystyle R}
 
 is a <a href="/facts/Cohen%25E2%2580%2593Macaulay_ring/4G8wymtC">Cohen–Macaulay ring</a>.</li>
<li>The Intersection Theorem. If 
 
 
 
 M
 
 ⊗
 
 R
 
 
 N
 ≠
 0
 
 
 {\displaystyle M\otimes _{R}N\neq 0}
 
 has finite length, then the <a href="/facts/Krull_dimension/KQb2uLtz">Krull dimension</a> of N (i.e., the dimension of R modulo the <a href="/facts/Annihilator_(ring_theory)/aTaHR5W7">annihilator</a> of N) is at most the <a href="/facts/Projective_dimension/lHymqhNC">projective dimension</a> of M.</li>
<li>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}
 
.</li>
<li>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}
 
.</li>
<li>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 <a href="/facts/Yves_Andr%25C3%25A9/VpudcUKl">Yves André</a> using a theory of <a href="/facts/Perfectoid_space/xqMFHYEb">perfectoid spaces</a>.</li>
<li>The Canonical Element Conjecture. Let 
 
 
 
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 d
 
 
 
 
 {\displaystyle x_{1},\ldots ,x_{d}}
 
 be a <a href="/facts/System_of_parameters/ALsc1Tc6">system of parameters</a> for R, let 
 
 
 
 
 F
 
 ∙
 
 
 
 
 {\displaystyle F_{\bullet }}
 
 be a free R-resolution of the <a href="/facts/Residue_field/e6nflEmB">residue field</a> of R with 
 
 
 
 
 F
 
 0
 
 
 =
 R
 
 
 {\displaystyle F_{0}=R}
 
, and let 
 
 
 
 
 K
 
 ∙
 
 
 
 
 {\displaystyle K_{\bullet }}
 
 denote the <a href="/facts/Koszul_complex/G8N1ugQK">Koszul complex</a> 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.</li>
<li>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.</li>
<li>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).</li>
<li>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}
 
.</li>
<li>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 <a href="/facts/Direct_summand/x3ljVsP6">direct summand</a> of Q considered as R-modules.</li>
<li>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.</li>
<li>Serre's Conjecture on Multiplicities. (cf. <a href="/facts/Serre%2527s_multiplicity_conjectures/lIA8APPD">Serre's multiplicity conjectures</a>.) 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. <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Jean-Pierre Serre</a> proved that the sum cannot exceed d.)</li>
<li>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 <a href="/facts/Regular_sequence/Nxr2uwh7">regular sequence</a> on M.</li></ol>

<ul><li><a href="http://www.math.lsa.umich.edu/~hochster/homcj.pdf">Homological conjectures, old and new</a>, <a href="/facts/Melvin_Hochster/rbXPNR0X">Melvin Hochster</a>, Illinois Journal of Mathematics Volume 51, Number 1 (2007), 151-169.</li>
<li><a href="https://arxiv.org/abs/1608.08882">On the direct summand conjecture and its derived variant</a> by Bhargav Bhatt.</li></ul>

Homological conjectures in commutative algebra open-in-new

Homological conjectures in commutative algebra