Katzman’s seventh-Betti-number question

For every finite flag complex $\Delta$ on $n$ vertices and every pair of coefficient fields $K$ and $L$ , is the seventh total Betti number of its Stanley--Reisner ring independent of the field; that is, does $\beta_7^{K}(K[\Delta])=\beta_7^{L}(L[\Delta])$ always hold, where $\beta_7^{K}(K[\Delta])=\dim_K\operatorname{Tor}^{K[x_1,\ldots,x_n]}_7(K[\Delta],K)$ ? Equivalently, for every finite simple graph $G$ , is the seventh total Betti number in the minimal free resolution of its edge ideal independent of the coefficient field?

References

Primary source

arXiv

Additional references

Field independence of the first seven Betti numbers of flag complexes — arXiv — Omkar Javadekar

combinatorial-commutative-algebra topology

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A new unrefereed preprint claims to settle the question and show that the first seven Betti numbers do not depend on the coefficient field.

Katzman’s question asks whether the seventh Betti number of flag complexes is independent of the coefficient field. Katzman’s 2005 work established independence through the sixth Betti number but left the seventh unresolved.

Known results

Katzman, 2005: the first six Betti numbers are characteristic-independent for degree- $2$ Stanley–Reisner ideals.
Katzman, 2005: all Betti numbers are characteristic-independent with at most $10$ variables.
Katzman, 2005: examples on $11$ vertices show characteristic dependence beginning at the eighth Betti number, leaving the maximal universally independent index at $6$ or $7$ .

September 2026 claimed resolution

Omkar Javadekar’s preprint Field independence of the first seven Betti numbers of flag complexes, reported on September 23, 2026, claims a torsion lower bound for flag-complex homology. It would establish field independence through the seventh Betti number and strengthen the result to all homological degrees. The claim is unrefereed and remains unverified.

Current status (as of September 2026): Katzman’s question has a claimed resolution in an unrefereed preprint, but the result is not independently verified; the earlier sixth-Betti-number bound is established.

Sources

No solutions have been posted yet.