4 problems

Avramov–Buchweitz–Iyengar's total rank conjecture. For every projective flag $D$, the inequality above should hold. The conjecture is false: Iyengar–Walker provided counterexamples…

For a Noetherian commutative ring $R$, let $P$ be a short complex, meaning a non-exact complex of finite-rank projective $R$-modules concentrated in degrees $[0,c]$, where … For a…

Avramov's total rank conjecture. Under the same assumptions as in the Buchsbaum–Eisenbud–Horrocks conjecture,

Let $k$ be an algebraically closed field, let $r$ be a positive integer, let $N=2n$ be an even positive integer, and let $d=(d_1,\ldots,d_N)$ be an $N$-tuple of nonincreasing integ…