Categories of Symmetries and Infinite-Dimensional Groups by Yu A. Neretin

By Yu A. Neretin

There are numerous kinds of infinite-dimensional teams, such a lot of which were studied individually from one another because the Nineteen Fifties. it truly is now attainable to slot those it appears disparate teams into one coherent photo. With the 1st particular development of hidden constructions (mantles and trains), Neretin is ready to express what number infinite-dimensional teams are actually just a small a part of a miles higher item, analogous to the way in which actual numbers are embedded inside of complicated numbers.

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With our definition of Ψ, it seems too difficult to study the conservation conjecture. 4. Let us recall the definition of the functor Φ. 3, we call / A1 the elevation to the n-th power. We let η be the generic en : A1k k point of A1k and s its zero section. We consider the commutative diagrams ηn j (en )η  η j / A1 o k  i s i s. en / A1 o k We then define Φ(A) = Colimn∈N× i∗ j∗ (en )∗η A for every object A of DMQ (η). 13. The following two statements are equivalent: • The functor Ψ : DMct Q (η) / DMct (s) is conservative, Q • The functor Φ : DMct Q (η) / DMct (s) is conservative.

Such a reduction could be interesting. Indeed, the functor φeff is rather explicit and defined on sheaves. Unfortunately, we do not know how to prove that φeff : HItf Q (η) / HIQ (s) is conservative. We should also say that Srinivas gave us a counterexample to the conservation of φeff for fields of positive characteristic. We end by recalling his example. 18. Let e : E curves over a field of positive characteristic k. Fix s ∈ B such that the fiber † Warning: this category is not abelian. Indeed, kernels are not necessarily of finite type.

In this case, we call / C the inclusion. Note also the following commutative diadI : CI grams: Di0 vi / Di O cI,i DI ui / Es > } }} } }} c }} I for i ∈ I. / C for card(I) = 2 form a cover by closed subsets of C. The dI : CI By a variant of the Mayer-Vietoris distinguished triangle for covers by closed subschemes (see [3], chapter II), one proves that any object A ∈ DM(C) is in the triangulated subcategory generated by the set of objects {dI∗ d∗I A | I ⊂ [1, r] and card(I) ≥ 2}. To finish the proof, we will show that for ∅ = I ⊂ [1, r] the object c∗I Ψf I is in the triangulated subcategory generated by the set of objects {(cK,I )∗ Z(m) | I ⊂ K ⊂ [1, r] and m ∈ Z}.

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