A note on cohomological vanishing theorems

We study $cd(M,N):=\sup \{j:H_m^j(M,N)\mathrm{\ne }0\}$, and we prove the following over $AB$-rings: $cd(M,N)<\mathrm{\infty }$ iff $cd(M,N)\le 2dimR$. For locally free over the punctured spectrum, we present the better bound, namely $cd(M,N)<\mathrm{\infty }$ iff $cd(M,N)\le dimR,$ and show this is sharp for maximal Cohen-Macaulay, and prove that this detects freeness of $M$. We present some explicit examples to compute $cd(M,N)$. Now, suppose $R$ is only Cohen-Macaulay and of prime characteristic equipped with the Frobenius map $\phi$. We show for some $n\gg 0$ that $cd{(}^{{\phi }_n}R,M)<\mathrm{\infty }$ iff $i{d}_R(M)<\mathrm{\infty }\mathrm{.}$ This presents some criteria on regularity. Also, some vanishing results on $Ex{t}_R^i{(}^{\phi }R,-)$ are given, where $(-)\in \{R{,}^{\phi }R\}$. We determine conditions under which the vanishing $Ex{t}_R^i{(}^{\phi }R,-)$ of restricted many $i$-th, implies the vanishing of all.