Triangulated Categories of Mixed Motives

Witt sheaves and the η-inverted sphere spectrum

“…with S + the unit for SH(k) + and S − the unit for SH(k) − . Combining a theorem of Röndigs-Østvaer [18, Theorem 1.1] with a result of Cisinski-Déglise[5, Theorem 16.2.13] shows that the unit map to the motivic cohomology spectrum HZ induces an isomorphism S + Q → HQ, which in turn defines an equivalence of SH(k) + Q with the homotopy category of HQ-modules. The theorem of Röndigs-Østvaer shows that this category is equivalent to Voevodsky's triangulated category of motives with Q-coefficients, DM (k) Q .In the next section, we will use Theorem 3.4 to give an analogous description of SH(k) − Q (see Theorem 4.2 and Corollary 4.4).…”

The Intersection Motive of the Moduli Stack of Shtukas

“…Moreover, tensoring with P 1 is made invertible. This category is denoted by DAé t (X, Q) in [Ayo14] and by D A,ét (X, Q) in [CD09].…”

“…This leads to the theory of motives. Envisioned by Beilinson, and realised by Voevodsky [Voe00], Levine [Lev98], and Hanamura [Han95] for motives over S = Spec(k), and extended by Ayoub [Ayo07a,Ayo07b,Ayo14] and Cisinski-Déglise [CD09,CD16] to motives over general base schemes S, there now exists a theory of motivic sheaves, i.e., a full six functor formalism for suitable triangulated categories DM(S) = DM(S, Q) of motives with rational coefficients over S. By construction this theory of motives is independent of ℓ, but explicit computations are difficult. One of the main obstacles is the lack of a motivic t-structure on these categories, i.e., the existence of an object h i (X) ∈ DM(Spec k) whose ℓ-adic realization would be H í et (X, Q ℓ ).…”

“…In particular (take τ = Zar), it shows that for all Noetherian schemes of finite Krull dimension X (equipped with a map to S) the category DM(X) is the category of motives as defined in Cisinski-Déglise [CD09], cf. Remark 2.2.2 v).…”

Perfection in motivic homotopy theory

“…After Proposition 2.2.1, it remains to show that unit map id SH(S) → f * f * becomes invertible after inverting the primes in P. By continuity [17,Proposition C.12(4)] and proper base change, we may use a noetherian approximation argument [28,Theorem C.9] to assume that S is noetherian and of finite dimension. Then using [5,Proposition A.3] (and proper base change again), we may assume that S is a henselian local scheme (which is still noetherian and finite-dimensional); we denote its closed point by i : {s} → S and the complement by j : U → S. By the localization theorem, the pair of functors (i * , j * ) is jointly conservative (see [9,Section 2.3]). Since U has dimension strictly lower than that of S, we can argue by induction on the dimension of S to reduce to the case where S = {s}, that is, where S is the spectrum of a field k. Since f is radicial, it is then induced by a purely inseparable field extension k ⊂ K. In characteristic zero (p = 1), we are already done.…”