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

About 14 years old · traced to 2012

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 1Other wordings

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).

References

Primary source

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

Posted 1 month ago

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

No solutions have been posted yet.