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You can

ein Buch schreiben

that denotes the whole work, from start to finish.

And if you say

ich habe ein Buch geschrieben

we know you have finished the book.

And you can

an einem Buch schreiben

which denotes being busy working on a book.

And if you say

ich habe an einem Buch geschrieben

then we know you have been working on a book, but whether you have finished it or not is not said (most probably you have not, otherwise you'd been using the first phrase).

The same is obviously valid for any kind of book, including autobiographies.

You can

ein Buch schreiben

that denotes the whole work, from start to finish.

And if you say

ich habe ein Buch geschrieben

we know you have finished the book.

And you can

an einem Buch schreiben

which denotes being busy working on a book.

And if you say

ich habe an einem Buch geschrieben

then we know you have been working on a book, but whether you have finished it or not is not said (most probably you have not, otherwise you'd been using the first phrase).

Also, compare the English I wrote a book vs. I was working on a Book.

The same is obviously valid for any kind of book, including autobiographies.

You can

ein Buch schreiben

that denotes the whole work, from start to finish.

And if you say

ich habe ein Buch geschrieben

we know you have finished the book.

And you can

an einem Buch schreiben

which denotes being busy working on a book.

And if you say

ich habe an einem Buch geschrieben

then we know you have been working on a book, but whether you have finished it or not is not said (most probably you have not, otherwise you'd been using the first phrase).

The same is obviously valid for any kind if book, including autobiographies.

You can

ein Buch schreiben

that denotes the whole work, from start to finish.

And if you say

ich habe ein Buch geschrieben

we know you have finished the book.

And you can

an einem Buch schreiben

which denotes being busy working on a book.

And if you say

ich habe an einem Buch geschrieben

then we know you have been working on a book, but whether you have finished it or not is not said (most probably you have not, otherwise you'd been using the first phrase).

The same is obviously valid for any kind of book, including autobiographies.

You can

ein Buch schreiben

that denotes the whole work, from start to finish.

And if you say

ich habe ein Buch geschreiben

we know you have finished the book.

And you can

an einem Buch schreiben

which denotes being busy working on a book.

And if you say

ich habe an einem Buch geschrieben

then we know you have been working on a book, but whether you have finished it or not is not said (most probably you have not, otherwise you'd been using the first phrase).

The same is obviously valid for any kind if book, including autobiographies.

You can

ein Buch schreiben

that denotes the whole work, from start to finish.

And if you say

ich habe ein Buch geschrieben

we know you have finished the book.

And you can

an einem Buch schreiben

which denotes being busy working on a book.

And if you say

ich habe an einem Buch geschrieben

then we know you have been working on a book, but whether you have finished it or not is not said (most probably you have not, otherwise you'd been using the first phrase).

The same is obviously valid for any kind if book, including autobiographies.