I've just stumbled across the following theorem ([here](http://www.math.unt.edu/~moliver/fa04s/JacksonNotes/products.pdf)):


**Theorem** Let $P$ be a partial class order in $M$, a transitive model of ZF (resp. ZFC). Suppose for arbitrarily large cardinals $\kappa$ we have an isomorphism $P \simeq P^- \times P^+$ where $P^-$ is $\kappa^+$-cc and $P^+$ is $\leq\kappa$-closed. For $G$ an $M$-generic for $P$, $M[G]$ satisfies ZF (resp. ZFC).


Now Easton forcing constructs a particular $P\_E$, and at least in Jech and S. Friedman's books GCH is assumed, I believe in order to show that for $\kappa$ any given regular cardinal we have $P\_E = P^{\leq \kappa} \times P^{\gt\kappa}$ where $P^{\leq \kappa}$ is $\kappa^+$-cc. I *think* $\leq\kappa$-closedness of $P^{\gt\kappa}$ comes for free. My first question is this: shouldn't the existence of class-many regular cardinals $\lambda$ such that $\lambda^+ = 2^\lambda$ be enough to satisfy the hypotheses of the above theorem? Could we weaken this to $\lambda^{\lt\lambda} = \lambda$? (or is this equivalent to CH at $\lambda$?)


Secondly (for example in [these notes](http://www.math.unt.edu/~moliver/fa04s/JacksonNotes/continuum.pdf)) we have


**Lemma** The poset $\mathrm{Fn}(\eta,\lambda,\rho)$ of partial functions $\eta \rightharpoonup \lambda$ with domain smaller than $\rho$ is $(\lambda^{\lt\rho})^+$-cc.


My guess is that we need regularity of $\kappa$ so that $P^{\leq \kappa}$, which is a product of posets as in the lemma, is $\kappa^+$-cc. Is this correct? 


I'm actually thinking of *Long* Easton forcing, so I don't want to assume Easton support. Does the above line of reasoning still work? Namely, assume the existence of class-many regular cardinals satisfying CH (or $\lambda^{\lt\lambda} = \lambda$, if that's weaker), and construct the partial class order so that we can fullfil the hypotheses of the theorem above.



Can we weaken GCH in this class forcing?
It's impossible to accurately answer this since you don't say what forcing you have in mind. However, in the "typical" case, the answer is yes.


The "typical" case is when the forcing poset $P$ comes from a "typical" iteration of set forcings. These are "typically" arranged so that for unboundedly many $\kappa$, the partial iteration $P\_\kappa$ up to $\kappa$ consists of well-behaved $\kappa^+$-cc forcings (often enough these are small so that $|P\_\kappa| \leq \kappa$), and the remaining posets in the iteration after $\kappa$ are $\kappa$-closed. The $\Delta$-System Lemma (or even $|P\_\kappa| \leq \kappa$ in nice cases) can "typically" be used to show that $P\_\kappa$ is $\kappa^+$-cc; the "typical" $\Delta$-System Lemma application (or counting argument) here requires $\kappa^{\lt\kappa} = \kappa$. Then a standard variation of the theorem you state shows that $P$ preserves ZFC.


This is the "typical" case but it is only "typical" because it is fairly straightforward to show that such $P$ are tame. Note that the theorem you state deals with a very special case where $P$ splits as a product at $\kappa$. This is not uncommon but in general $P$ will only split as an iteration, but the general idea of the argument "typically" still applies to the more general case.


