Grothendieck first defined the notion of a "motif" as a way of finding a universal cohomology theory for algebraic varieties. Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a perfect field, which has many of the properties expected of the derived category of the conjectural abelian category of motives. The construction of the triangulated category of motives has been extended by Cisinski-D\'{e}glise to a triangulated category of motives over a base-scheme $S$. Recently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme $S$ generated by the motives of smooth projective $S$-schemes, assuming that $S$ is itself smooth over a perfect field. In both constructions, the tensor structure requires $\mathbb{Q}$-coefficients. In my thesis, I show how to provide a tensor structure on the homotopy category mentioned above, when $S$ is semi-local and essentially smooth over a field of characteristic zero. This is done by defining a pseudo-tensor structure on the DG category of motives constructed by Levine, which induces a tensor structure on its homotopy category.
Deep Dive into Tensor Structure on Smooth Motives.
Grothendieck first defined the notion of a “motif” as a way of finding a universal cohomology theory for algebraic varieties. Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a perfect field, which has many of the properties expected of the derived category of the conjectural abelian category of motives. The construction of the triangulated category of motives has been extended by Cisinski-D'{e}glise to a triangulated category of motives over a base-scheme $S$. Recently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky’s category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme $S$ generated by the motives of smooth projective $S$-schemes, assuming that $S$ is itself smooth over a perfect fi
Abstract. Grothendieck first defined the notion of a "motif" as a way of finding a universal cohomology theory for algebraic varieties. Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a perfect field, which has many of the properties expected of the derived category of the conjectural abelian category of motives. The construction of the triangulated category of motives has been extended by Cisinski-Déglise to a triangulated category of motives over a base-scheme S. Recently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme S generated by the motives of smooth projective S-schemes, assuming that S is itself smooth over a perfect field. In both constructions, the tensor structure requires Q-coefficients. In my thesis, I show how to provide a tensor structure on the homotopy category mentioned above, when S is semi-local and essentially smooth over a field of characteristic zero. This is done by defining a pseudo-tensor structure on the DG category of motives constructed by Levine, which induces a tensor structure on its homotopy category.
Dedicated to my beloved grandma Juthika Ganguly who taught me to be patient and perseverant. It is also dedicated to my parents Pranab and Jayati Banerjee whose patient support and constant encouragement has been with me all the way since the beginning of my studies.
Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a perfect field, which has many of the properties expected of the derived category of the conjectural abelian category of motives. The construction of the triangulated category of motives has been extended by ) to a triangulated category of motives over a base-scheme S. Hanamura has also constructed a triangulated category of motives over a field, using the idea of a “higher correspondence”, with morphisms built out of Bloch’s cycle complex. Recently, Bondarko (in [Bon09]) has refined Hanamura’s idea and limited it to smooth projective varieties to construct a DG category of motives. Assuming resolution of singularities, the homotopy category of this DG category is equivalent to Voevodsky’s category of effective geometric motives. Soon after, Levine (in [Lev09]) extended this idea to construct a DG category of “smooth motives” over a base-scheme S generated by the motives of smooth projective S-schemes, where S is itself smooth over a perfect field. Its homotopy category is equivalent to the full subcategory of Cisinski-Déglise category of effective motives over S generated by the smooth projective S-schemes. Both these constructions lack a tensor structure in general. However, passing to Q-coefficients, Levine replaced the cubical construction with alternating cubes, which yields a tensor structure on his DG category.
In [BV08], Beilinson and Vologodsky constructs a “homotopy tensor structure” on the DG catgegory of Voevodsky’s effective geometric motives. That is, they constructed a “pseudo-tensor structure” (see Definition 2.1.2) on the DG category D M (defined in [BV08], § 6.1), such that the corresponding pseudo-tensor structure on its homotopy category is actually a tensor structure.
In the following, we construct a pseudo-tensor structure on a DG category dg e Cor S which induces a tensor structure on the homotopy category of DG complexes, such that, in case S is semi-local and essentially smooth over a field of characteristic zero, it induces a tensor structure on the category of smooth motives over S. Thus, we prove Theorem 1. Suppose S is semi-local and essentially smooth over a field of characteristic zero. Then, there is a tensor structure on the category SmM ot eff gm (S) of smooth effective geometric motives over S making it into a tensor triangulated category.
See Theorem 3.1.1 and corollary 3.1.2 for a more precise statement. The pseudo-tensor structure is constructed in two main steps:
• We show that a pseudo-tensor structure on a DG category C induces a pseudo-tensor structure on the catgeory Pre-Tr(C) (defined in 1.1.2). Then we prove that if the pseudo-tensor structure on C induces a tensor structure on the homotopy category H 0 C, then we have an induced tensor structure on K b (C). • For a tensor category C, with a cubical object with comultiplication, Levine constructs a DG category dgC (see 1.2.5). We produce a pseudo-tensor structure on dg e C that induces a tensor structure on its homotopy category, assuming some additional technical conditions. Under these conditions, the DG categories dgC and dg e C are quasi-equivalent, so the homotopy categories of complexes are equivalent triangulated categories. In Chap
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