Let R be a regular local ring of Krull dimension d, and let M,N ∈ operatornamemod--R be modules such that M otimes_R N has finite length. Define their intersection multiplicity by chi(M,N)=∑_i=0^d(-1)^i operatornamelength(operatornameTor_i^R(M,N)). Serre's intersection multiplicity conjecture. The following assertions hold: operatornameKdimM+operatornameKdimN≤operatornameKdimR. If operatornameKdimM+operatornameKdimN 0. The vanishing assertion was proved by Roberts, while the full conjecture as stated in the source includes the dimension inequality and positivity assertion.

Serre's intersection multiplicity conjecture

Let $R$ be a regular local ring of Krull dimension $d$ , and let $M,N \in \operatorname{mod}\text{--}R$ be modules such that $M \otimes_R N$ has finite length. Define their intersection multiplicity by

\chi(M,N)=\sum_{i=0}^{d}(-1)^i\operatorname{length}(\operatorname{Tor}_i^R(M,N)).

Serre's intersection multiplicity conjecture. The following assertions hold:

$\operatorname{Kdim}M+\operatorname{Kdim}N\leq\operatorname{Kdim}R$ .
If $\operatorname{Kdim}M+\operatorname{Kdim}N<\operatorname{Kdim}R$ , then $\chi(M,N)=0$ .
If $\operatorname{Kdim}M+\operatorname{Kdim}N=\operatorname{Kdim}R$ , then $\chi(M,N)>0$ . The vanishing assertion was proved by Roberts, while the full conjecture as stated in the source includes the dimension inequality and positivity assertion.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Serre's intersection multiplicity conjecture
Let $(A,\mathfrak{m})$ be a regular local ring. For finitely generated $A$ -modules $M$ and $N$ with $\ell(M\otimes_A N)<\infty$ , define
$\chi^A(M,N)=\sum_{i=0}^{\dim A}(-1)^i\ell(\operatorname{Tor}_i^A(M,N)).$
Serre's intersection multiplicity conjecture. The following statements hold: (a) $\chi^A(M,N)\geq 0$ ; (b) $\dim M+\dim N\leq\dim A$ ; and (c) $\chi^A(M,N)>0$ if and only if $\dim M+\dim N=\dim A$ .

The conjecture concerns the positivity and dimension-theoretic behavior of intersection multiplicities. The paper discusses progress in the unramified case, but the supplied text does not establish the general conjecture's resolution.
source: Chris Skalit, “Intersection Multiplicity of Serre in the Unramified Case”, arXiv:1409.3616 (2014).

Sources & referencesView supporting material

Primary source

Lidia Angeleri Hügel, David Pospisil, Jan Stovicek and Jan Trlifaj, “Tilting, cotilting, and spectra of commutative noetherian rings”, arXiv:1203.0907 (2012).

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Serre's intersection multiplicity conjecture

Equivalent formulations 1

Serre's intersection multiplicity conjecture

Primary source

Progress summary

Solutions 0