Why was Dominique arrested?

She was accused of Karol's murder, that's why she got jailed.

The same day Karol "dies", Dominique arrives to Poland, something quite "suspicious". Besides the unexpected change of Karol's will makes her the main suspect.

Kieslowski doesn't explain in detail the legal technicisms about her arrest because that's not the point. But in short: the "murder" of Karol is the reason why she was arrested.

I too had a hard time understanding this at first. After soem thought I think he had her set up for his murder and/or fraudulently executing his estate so that she would find herself on the wrong side of the law with all her freedoms taken away in a foreign county. Thereby she would suffer the same fate that he was subjected to in the beginning of the movie so they would have established equality in their relationship and be able to reconcile their marriage.

Seems kinda far fetched to me but, hey, it seemed to work within the confines of the story.

Karol's people altered her passport to make it appear as if she had been in Poland on the day he died. because his entire estate was left to her, it was easy to establish motive. since she wasted no time in executing the will, she looked suspicious

It was never explained in detail but we have to assume that the authorities had more incriminating evidence than merely a doctored passport and a loosely cobbled together motive.

It should have been easy for her to establish that she was in fact in Paris on the day Karol was "murdered". She will have countless witness statements and perhaps material evidence to prove beyond doubt that she was not in Poland and hence that her passport was doctored. This is to say nothing of the records kept by French Passport control establishing beyond doubt the day that she exited France - that this was some time after she was informed of the death and subsequent funeral.
There are too many how, when, and why type questions.
How was she informed that she was a benefactor in Karol's will? Phone call? Letter? And when was she informed? Why was she informed of her inheritance before his funeral? Was it an appeal to her greed, so as to ensure that she would attend the funeral?
How was she supposed to have adjusted Karol's will to make herself the sole benefactor? The draftsman of the will will also be questioned as an accomplice in a premeditated murder conspiracy and he may well explain what really happened.

So many other reasons why she would not be arrested. A flight booked to Hong Kong! Evidence of their sexual liaison the night before she was arrested. An exhumation of the decapitated Russian body etc etc.

Anyway, the way that "dead" Karol was walking the streets in plain view should have rendered all of the above redundant after a short while.

This must be the ultimate pseudo-intellectual movie imho. But hey, switch off your brain and delight in the deeper meaning. Equality man! Whatever.