Peskine–Szpiro conjectures on modules of finite projective dimension

Let $(R,\mathfrak m)$ be a Noetherian local ring, and let $M$ and $N$ be finitely generated $R$ -modules such that $\operatorname{pd}_R M<\infty$ and $\ell_R(M\otimes_RN)<\infty$ . The dimension inequality conjecture asserts that $\dim M+\dim N\leq\dim R$ . The supplied sources also identify the strong intersection conjecture and the grade conjecture as further Peskine–Szpiro conjectures in this finite-projective-dimension setting, but do not state their formal hypotheses and conclusions.

References

Primary source

arXiv

Additional references

Counterexamples to Peskine-Szpiro's conjecture on modules of finite projective dimension — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unrefereed manuscript claims to disprove three longstanding conjectures, but the claim has not yet been independently verified.

Peskine and Szpiro introduced these conjectures in 1973; one formulation bounds the combined dimensions of two finitely generated modules over a local ring when their tensor product has finite length. The tracked problem concerns three conjectures that the latest manuscript claims to refute.

Known results

The dimension inequality holds for modules that lift to a regular ring.
Hochster proved it for hypersurfaces; in that case the inequality is an equality.
Peskine and Szpiro’s 1973 paper established the foundational finite-projective-dimension framework.

August 25, 2026 counterexamples claimed

A manuscript titled Counterexamples to Peskine-Szp̀iro's conjecture on modules of finite projective dimension claims counterexamples to the three remaining homological conjectures. If correct, this would overturn the conjectures; the preprint is unrefereed and the retrieved evidence is insufficient for independent assessment.

Current status (as of August 2026): The conjectures remain officially unverified, while an August 2026 preprint claims to disprove all three remaining cases.

Sources

No solutions have been posted yet.