MathDB

Illusie’s SGA6 finiteness conjecture

For every proper pseudo-coherent morphism of schemes f ⁣:X→Yf\colon X\to Y, the derived pushforward preserves pseudo-coherent complexes: Rf∗ ⁣:Dpcoh(X)→Dpcoh(Y)\mathbb{R}f_*\colon D_{\mathrm{pcoh}}(X)\to D_{\mathrm{pcoh}}(Y). Equivalently, for every pseudo-coherent complex EE on XX, Rf∗E\mathbb{R}f_*E is pseudo-coherent on YY.

References

Primary source

arXiv

Additional references

algebraic-geometryderived-algebraic-geometry
Posted · edited

Progress summary

Refreshed
Claimed solved

A new unrefereed preprint claims to settle the conjecture for schemes and extend it to algebraic stacks, but the result has not been independently verified.

Illusie’s conjecture asks whether, for a proper pseudo-coherent morphism f:X→Yf:X\to Y of schemes, Rf∗\mathbb{R}f_* preserves pseudo-coherent complexes. The supplied sources give no original posing date or further historical attribution.

Known results

  • Neeman and Lipman: proper perfect morphisms preserve perfect complexes.
  • A related exposition records the local complete intersection case, proving that proper local complete intersection morphisms preserve perfect complexes.

September 2026 claimed resolution

A preprint by Jack Hall and Oliver Li claims the pseudo-coherence preservation theorem for schemes, extends it to algebraic stacks, and proves additional nontame cases. The September 21, 2026 report identifies the preprint as new and unrefereed.

Current status (as of September 2026): A complete resolution is claimed for schemes and broader classes of algebraic stacks, but the claim remains unverified.

Sources

Solutions 0

No solutions have been posted yet.