Classicality of derived Emerton–Gee stack

We construct a derived stack <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> of Laurent <i>F</i>-crystals on , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>O</mml:mi> <mml:mi>K</mml:mi></mml:msub> </mml:math> is the ring of integers of a finite extension <i>K</i> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Q</mml:mi> <mml:mi>p</mml:mi></mml:msub> </mml:math> . We first show that its underlying classical stack <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mmultiscripts><mml:mrow></mml:mrow> <mml:mrow></mml:mrow> <mml:mtext>cl</mml:mtext></mml:mmultiscripts> <mml:mi>X</mml:mi></mml:mrow> </mml:math> coincides with the Emerton-Gee stack <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>X</mml:mi> <mml:mtext>EG</mml:mtext></mml:msub> </mml:math> , i.e. the moduli stack of étale <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>φ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> -modules. Then we prove that the derived stack <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> is classical in the sense that when restricted to truncated animated rings, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> is equivalent to the sheafification of the left Kan extension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>X</mml:mi> <mml:mtext>EG</mml:mtext></mml:msub> </mml:math> along the inclusion from the classical commutative rings to animated rings.