On the derived category of 1-motives (Q2821320)

In this book, the authors compare the derived category of Deligne's \(1\)-motives (cf. [\textit{P. Deligne}, Publ. Math., Inst. Hautes Étud. Sci. 44, 5--77 (1974; Zbl 0237.14003)]) and Voevodsky's triangulated category of motives (cf. [\textit{V. Voevodsky}, Ann. Math. Stud. 143, 188--238 (2000; Zbl 1019.14009)]). Deligne's \(1\)-motives were the first examples of mixed motives.NEWLINENEWLINELet \({\mathcal M}_{1}(k)\) be the category of \(1\)-motives over a field \(k.\) It is natural to compare \({\mathcal M}_{1}(k)\) with the triangulated category \({\mathrm{DM}}_{\mathrm{gm}}^{\mathrm{eff}}(k)\) as the first category is conjectured to be contained in the heart of the (conjectural) \(t\)-structure of the latter. In fact, when \(k\) is perfect Voevodsky conjectured that the functor: NEWLINE\[NEWLINE {\mathrm{Tot}}^{\mathbb Q} : D^b ({\mathcal M}_{1}(k)\otimes {\mathbb Q} \hookrightarrow {\mathrm{DM}}_{\mathrm{gm}}^{\mathrm{eff}}(k)\otimes {\mathbb Q}NEWLINE\]NEWLINE is fully faithful and its essential image is the thick subcategory generated by motives of smooth curves.NEWLINENEWLINEThe main result of the work is that the functor \({\mathrm{Tot}}^{\mathbb Q}\) has a left adjoint \({\mathrm{LAlb}}^{\mathbb Q}.\) The authors also provide a \({\mathbb Z}[1/p]]\)-integral version of this result. Let \({\mathrm{RPic}}\) be the composition of \({\mathrm{LAlb}}\) with the Cartier duality. \({\mathrm{LAlb}}: {\mathrm{DM}}_{\mathrm{gm}}^{\mathrm{eff}}(k) \rightarrow D^b({\mathcal M}_{1}(k)[1/p]\) is here the \({\mathbb Z}[1/p]\)-integral version of this left adjoint. Applying \({\mathrm{LAlb}}\) to an object \(M\in {\mathrm{DM}}_{\mathrm{gm}}^{\mathrm{eff}}(k)\) and then \(1\)-motivic homology one obtains a series of \(1\)-motives \({\mathrm{L}}_{i}{\mathrm{Alb}}(M), \,i\in {\mathbb Z}.\) In particular taking a motive \(M(X)\) for a variety \(X/k\) one obtains new invariants \({\mathrm{L}}_{i}{\mathrm{Alb}}(X)\) and \({\mathrm{R}}^{i}{\mathrm{Pic}}(X)\) of a variety \(X.\) These bounded complexes are computed by the authors for smooth varieties and partly for singular varieties. Applications include motivic proofs of Roitman type theorems as well as new cases of Deligne's conjectures on \(1\)-motives.NEWLINENEWLINEThe book is written very clearly and is certainly a very good source for readers interested in motives.