3 problems

McCallum–Sharifi conjecture. For all primes $p$, the pairing $\langle\cdot,\cdot\rangle_p$ is nontrivial.

Let $p>2$ be a prime and let $F$ be a field containing a primitive $p$-th root of unity. Write $F(p)$ for the maximal $p$-extension of $F$, let $G=\operatorname{Gal}(F(p)/F)$, and…

Let $X$ be a $2$-dimensional simplicial complex, let ${\mathcal{F}}$ be a sheaf on $X$, and use the notation of the modification process. Set … Assume $M\geq 0$. First-step cup-pro…