In fact, it isn't ambiguous at all. In all three of the extant versions of the Ascension of Isaiah, the Beloved (Christ) comes to earth and is found 'dwelling among men'. So Dr Carrier is wrong if he suggests otherwise. I go into this in detail on the Early Writings forum where I created a thread a couple of years ago on the topic:
http://earlywritings.com/forum/viewtopic.php?f=3&t=4640
Jesus is crucified on earth according to the Ethiopic version, which has Jesus born to Mary and is crucified in Jerusalem. The Slavonic/Latin texts have the Beloved come to earth, to dwell among men. While the location of the crucifixion in the Slavonic/Latin texts is missing, it is clearly below the firmament, since the Beloved descends below the firmament before the story hits the fragmented sections. But the passages earlier in the text implies that the crucifixion takes place on earth, though Ben C Smith -- a man who knows much more than me! -- suggests in the thread that the original author possibly had Hades in mind. But it seems to be either earth or Hades, and not the firmament.
So in the three extant versions of the Ascension of Isaiah, we have:
1. the Beloved descending to earth and dwelling among men in all extant versions
2. No crucifixion in the firmament
That's clear in the extant texts. The text that Carrier refers to in his books and videos is a 'reconstruction', but people reading his book and viewing his videos on the topic come away with the idea that Carrier has actually found a variant text that supports a heavenly crucifixion. However, no such text exists outside of Carrier's imagination.
If anyone doubts this, read the link. I've cited all the pertinent passages so you can check this for yourself.
But, but, Carrier is a genius! Just ask him...