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I am interested in the concept of dimension of derived and $n$-Artin stacks. Take for example the definition 4.10 of From HAG to DAG: derived moduli stacks. in which they define the dimension of a tangent derived stack as an alternating sum of the of cohomology dimensions of the cotangent complex $\mathbb{R} \Omega_{F}^{1}$:

Definition 4.10 If $x:$ $i$Spec $\mathbb{C} \longrightarrow F$ is a point of a strongly geometric $D$ -stack, then we say that the dimension of $F$ at $x$ is defined if the complex $\mathbb{R} \Omega_{F, x}^{1}$ has bounded and finite dimensional cohomology. If this is the case, the dimension of $F$ at $x$ is defined by $$ \mathbb{R} D i m_{x} F:=\sum(-1)^{i} H^{i}\left(\mathbb{R} \Omega_{F, x}^{1}\right) $$

However I don't understand some issues. Why the the complex $\mathbb{R} \Omega_{F}^{1}$ is concentrated in degrees $(-\infty,1]$? For any derived Artin stack $F$, locally of finite presentation, the cotangent complex is perfect. I expected it to be only concentrated in degrees $[-p,1]$ being $p$ bounded in some way by the highest dimension of the schemes which form the atlas for $F$ (or by the topological space $X$ on you define your $F$). Otherwise what is the utility of those higher terms?


QUESTION 1: Can you have non-zero cohomology for arbitrary high terms in your perfect complex, disregarding the dimension of your scheme or topological space on the perfect complex is defined? This leads to the more general question

QUESTION 2: Can you have examples of derived stacks built over a scheme or topological space $X$ in which the derived degrees of freedom (i.e. its dimension as a derived stack) can grow arbitrarily high independently of the dimensions of $X$? I can easily imagine that freedom for $n$-stacks considering for example the moduli stack of $n$-vector bundles over a projective variety (with $n$ arbitrarily high) but I cannot find such examples in the derived side for derived stacks. Maybe more complicated Artin stacks, like the (higher derived ) stack of perfect complexes on a scheme $X$ can do the job?

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    $\begingroup$ Assuming you're using cohomological gradings, the negative terms in the cotangent complex arise from derived structure or singularities, rather than descent data. A non-LCI ring of finite type will have cotangent complex in degrees $(-\infty,0]$. I think an example for both of your questions is given by CDGAs of the form $A:=k[V[r]]$ for $r>0$; since $H_0A=k$, the underlying space is a point, but the dimension is $(-1)^r(\dim V)$. $\endgroup$ – Jon Pridham Aug 28 '20 at 15:16

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