/sci/ - Science & Math

Am I doing this correctly? Suppose I have an associative unital [math]k[/math]-algebra [math]A[/math], augmented by [math]\varepsilon \colon A \to k[/math]. If I want to take the classifying construction, I will first split the algebra into two parts: [math]A \cong I \oplus k[/math], where [math]I[/math] is the augmentation ideal. Then I divide the field [math]k[/math] out and get only that [math]I[/math], which I then move one layer upwards with the suspension [math]s[/math]. Finally, I have [math]B(A) = \sum\limits_{n \ge 0} I^{n \otimes}[/math] as my classifying construction for the algebra.

Now, suppose this [math]B(A)[/math] is simply connected. Then I would have [math]B(A)_1 \cong k[/math], but I also have [math]B(A)_1 = I[/math], so what I would be getting is [math]B(A)_n = I^{n \otimes} \cong k \otimes_k k \otimes_k \cdots \otimes_k k \cong k[/math] for each [math]n \ge 0[/math], and so the whole thing would just be a sum of copies of the field. Can this be correct?