I am a bit confused by the relevant literature I am aware of on the subject (Quillen, Avramov, etc.), so I would like to ask the following question to more expert people than me.

[The question has been edited below using Damian and Unknown's comments, and actually split into two different ones.]

Let $X$ and $Y$ be schemes over a field $k$, and let $f: X\rightarrow Y$ be a morphism locally of finite presentation.

Consider the relative cotangent complex $L_{X/Y}$ of $f$ as a cochain complex in non-positive degrees.

$Question$ $1.$ Suppose we know furthermore that

(a) $\mathbb{L}_{X/Y}$ is perfect,

(b) $\mathbb{L}_{X/Y}$ is of perfect amplitude in $[-n,0]$, for $n>>0$.

Does it follows that $\mathbb{L}_{X/Y}$ is of perfect amplitude in $[-2,0]$ ?

$Question$ $2.$ Suppose we know furthermore (i.e. in addition to (a) and (b)) that

(c) $f$ is of finite Tor-dimension,

Does it follows that $\mathbb{L}_{X/Y}$ is of perfect amplitude in $[-1,0]$ ?

Note that I am not assuming any Noetherian property for $X$ and $Y$.

nothaving the schemes to be noetherian. Are there known counterexamples in this case? – Sereza Nov 24 '12 at 1:21