# Failure of SCH at singular cardinals of uncountable cofinality

It is known by work of Woodin and Gitik that that the failure of SCH (singular cardinals hypothesis) at a singular cardinal of cofinality $\omega$ is equiconsistent with the existence of a measurable cardinal $\kappa$ of Mitchell order $\kappa^{++}.$

What can we say about the failure of SCH at a singular cardinal of uncountable cofinality: Let $\kappa$ be a singular cardinals of uncountable cofinality $\lambda.$ Then as far as I know the following are well-known:

(A) If $\lambda > \aleph_1,$ then the following are equiconsistent:

1) SCH fails at $\kappa,$

2) $\kappa$ is a measurable cardinal of of Mitchell order $\kappa^{++}+\lambda$ in some inner model,

(B) If $\lambda=\aleph_1,$ then

1) The failure of SCH at $\kappa$ implies $\kappa$ is a measurable cardinal of of Mitchell order $\kappa^{++}$ in some inner model,

2) If $\kappa$ is a measurable cardinal of of Mitchell order $\kappa^{++}+\lambda$, then in some forcing extension SCH fails at $\kappa$ which is singular of cofinality $\lambda.$

Question 1. Are the above mentioned results true? Are there any more results of the same direction?

Question 2. What can we say if we require $\kappa^{cf(\kappa)} =\kappa^{+\delta}$ for some $\delta>2$?

Would you please also give some references for the above mentioned results, in particular for the consistency results from the optimal hypotheses (from (A2) to (A1) and from (B2) to (B1)).

Remark. An interesting fact, first observed by core model techniques, is that if SCH fails at a singular cardinal $\kappa$ of countable cofinality, then either $\kappa$ is large in some inner model (measurable of Mitchell order $\kappa^{++}$) or $\kappa$ is a limit of an $\omega-$sequence of large cardinals. The results of Gitik show that the second case can happen, and lead him to discover long and short extenders forcings. These techniques were used by him quite recently to give a consistent negative answer to Shelah's PCF conjecture (see his papers "short extenders forcings I" and "short extenders forcings II").

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