# Cotangent complex and lci morphisms

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 $Y$.

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I think that you need to assume that $L_{X/Y}$ is a perfect complex and $Y$ noetherian for this to be true. Quillen's conjecture assumes perfectness and noetherianity. –  Damian Rössler Nov 23 '12 at 22:51
To Damian: thanks, you are certainly right about perfection of the cotangent complex (edited). On the other hand, my question was exactly about not having the schemes to be noetherian. Are there known counterexamples in this case? –  Sereza Nov 24 '12 at 1:21
The problem is local so you may assume that $X$ and $Y$ are affine. You may then write the morphism as a limit of morphisms of finite type to spectra of noetherian rings (see for instance EGA IV.3 Prop. 8.9.1). On the hand, the homology of the cotangent complex is compatible with limits in this sense - see Quillen, "On the cohomology..." (4.11), so the result follows. –  Damian Rössler Nov 24 '12 at 12:05
I think the induces are slightly off. For example, the ring map $k[t]/(t^2) \to k$ has a perfect cotangent complex with a non-trivial $H^{-2}$. –  anon Nov 24 '12 at 15:38
@unknown (google): the example is nice, its cotangent complex lives in degrees $[-3,0]$ and is perfect (and you can actaully compute it,using that it is an algebra retract); the morphism is finitely presented but not of finite tor-dimension. If it might help in setting the landscape, there is an example (by Planas-Villanova) of a ring morphism $f:R \rightarrow k$ where $k$ is a field, such that the cotangent complex of $f$ has vanishing $H^{-2}$ and non-vanishing $H^{−3}$. Of course R is non noetherian, and the map is not finitely presented (so it is not a counterexample to Question 2). –  gabriele Nov 28 '12 at 15:41