论文标题
同型Skolem-无定理
A homotopical Skolem--Noether theorem
论文作者
论文摘要
The classical Skolem--Noether Theorem [Giraud, 71] shows us (1) how we can assign to an Azumaya algebra $A$ on a scheme $X$ a cohomological Brauer class in $H^2(X,\mathbf G_m)$ and (2) how Azumaya algebras correspond to twisted vector bundles.派生的Skolem-非定理[Lieblich，09]将此结果推广到派生的1类局部准代晶状体中的弱代数，以衍生出派生的内态代数为完美的复合物。我们表明，通常对于共同的$ \ mathscr c^\ otimes $ ot的$ otimes $ therable monoidal quasi类别，并在带有Grothendieck拓扑的准类别上下降，具有上述光纤序列。对于完全支持的完美复杂的$ e $，超过准混合和准分离的方案$ x $，同型组sheaves sheaves sheaves sheaves sheaves sheaves sheaves sheaves sheaves sheaves sheaves sheaves sprepits $π_i（\ mathop {\ mathrm {autrm {aut}}} _ { E，\ Mathrm {id} _e）=π_i（\ Mathop {\ Mathrm {aut}} _ {\ Mathop {\ MathRM {\ Mathrm {alg}} \ Mathop {\ Mathop {\ Mathrm {perf}}}}}}}}}}}} \ Mathop { E}，\ Mathrm {id} _ {\ Mathop {\ Mathbf r \ Mathrm {end} e}}}）$ for $ i \ ge1 $。进一步的应用包括在派生的代数几何形状中，光谱代数几何形状中的模块光谱以及特征0的衍生阿尔盖尔几何形状中的indosoherent滑轮和晶体。
The classical Skolem--Noether Theorem [Giraud, 71] shows us (1) how we can assign to an Azumaya algebra $A$ on a scheme $X$ a cohomological Brauer class in $H^2(X,\mathbf G_m)$ and (2) how Azumaya algebras correspond to twisted vector bundles. The Derived Skolem--Noether Theorem [Lieblich, 09] generalizes this result to weak algebras in the derived 1-category locally quasi-isomorphic to derived endomorphism algebras of perfect complexes. We show that in general for a co-family $\mathscr C^\otimes$ of presentable monoidal quasi-categories with descent over a quasi-category with a Grothendieck topology, there is a fibre sequence giving in particular the above correspondences. For a totally supported perfect complex $E$ over a quasi-compact and quasi-separated scheme $X$, the long exact sequence on homotopy group sheaves splits giving equalities $π_i(\mathop{\mathrm{Aut}}_{\mathop{\mathrm{Perf}}} E,\mathrm{id}_E)=π_i(\mathop{\mathrm{Aut}}_{\mathop{\mathrm{Alg}}\mathop{\mathrm{Perf}}}\mathop{\mathbf R\mathrm{End} E},\mathrm{id}_{\mathop{\mathbf R\mathrm{End} E}})$ for $i\ge1$. Further applications include complexes in Derived Algebraic Geometry, module spectra in Spectral Algebraic Geometry and ind-coherent sheaves and crystals in Derived Algeraic Geometry in characteristic 0.
