@@VisualMath Possibly but I was thinking this was a way to derive the triangulated category or at least the distinguished triangle of a projective variety. This triangulated category would have a stable homotopy theory. This is usually described by a mapping cone with a short exact sequence. I think the short exact sequence of the mapping cone in dimension n is given by the Hopf fibration S^1 -> S^2n+1 -> CP^n with spheres S^n. Projective varieties are subsets of projective spaces. Projective spaces are examples of Fano manifolds. Fano manifolds are Kähler manifolds which have a complex structure. Thus, a n-dimensional projective variety is a subset of the complex projective space CP^n. Fibrations like the Hopf fibration are examples of Puppe sequence. With the long exact sequence of a Puppe sequence I think the axioms 1-3 of a triangulated categories can be proven making it pre-triangulated category. I’m not sure about the octahedral axiom even though most pre-triangulated are also triangulated categories.
@@VisualMath It is known if one assumes these are smooth projective variety(= effective Chow motive). Then, it is a triangulated category following Voevodsky. See Proposition 2.1.4 in “The triangulated categories of motives of a field” (2000) by Voevodsky. Also note 3.5.4 which gives a Gysin distinguished triangle. Voevodsky derives the triangulated category of geometric motives over a perfect k as DM_gm(k) and equates it to smooth projective varieties there. Gysin triangles as they apply to projective morphisms (of smooth schemes) at 2.1.2 is explored in “Around the Gysin triangles I” (2011) by Déglise. My argument was missing some details. These can be found in the following. See pg. 350 of “An introduction to algebraic topology” (1988) by Rotman for the coexact Puppe sequence which I refer to as the long exact sequence and think can be used to prove axoims 1-3 of triangulated categories. Theorem 11.53 in Rotman gives the Hopf fibration S^2n+1 -> CP^n with the fiber S^1. I have corrected this fiber in my previous comment. Note Proposition 4.66 for the Puppe sequence of a fibration and (1) and (2) in “Algebraic topology” (2001) by Hatcher. Also, in Hatcher Theorem 4.58 gives a useful description of the cofibration sequence version of a Puppe sequence using unions. In“Triangulation categories” (2001) by Neeman the three axioms of pre-a triangulated category are given by Definition 1.1.2 and a triangulated category with its additional axiom is given by Definition 1.3.13 with mention of uncertainty of a counter example of a pre-triangulated category that is not a triangulated category. The Eckmann-Hilton duality describes the duality between cofibration sequences of suspensions ΣX and fibration sequences of loop spaces ΩY. This duality gives a way to switch between coexact Puppe sequences induced by cofibrations and exact Puppe sequences induced by fibrations. A useful application of this duality is given by Theorem 11.12 and Lemma 11.38 in Rotman. Projective varieties are Kähler manifolds with its metric induced by the Fubini-Study metric on them. See Example 1.5 and note the last paragraph of Proposition 4.4 of “The Kodaira embedding theorem” (2010) notes by Massarenti. Also, note the Fubini-Study metric can be derived from the Hopf fibration S^1 -> S^2n+1 -> CP^n mentioned earlier.
Projective varieties are cones. As cones don’t they have a Puppe sequence aka cofiber sequence which gives them a stable homotopy theory?
Yes, that should be right. I think this is quite extensively studied for so-called Chow varieties.
@@VisualMath Possibly but I was thinking this was a way to derive the triangulated category or at least the distinguished triangle of a projective variety. This triangulated category would have a stable homotopy theory.
This is usually described by a mapping cone with a short exact sequence. I think the short exact sequence of the mapping cone in dimension n is given by the Hopf fibration S^1 -> S^2n+1 -> CP^n with spheres S^n. Projective varieties are subsets of projective spaces. Projective spaces are examples of Fano manifolds. Fano manifolds are Kähler manifolds which have a complex structure. Thus, a n-dimensional projective variety is a subset of the complex projective space CP^n. Fibrations like the Hopf fibration are examples of Puppe sequence. With the long exact sequence of a Puppe sequence I think the axioms 1-3 of a triangulated categories can be proven making it pre-triangulated category. I’m not sure about the octahedral axiom even though most pre-triangulated are also triangulated categories.
@@Jaylooker Not sure, I have never seen that anywhere. Do you know a reference?
@@VisualMath It is known if one assumes these are smooth projective variety(= effective Chow motive). Then, it is a triangulated category following Voevodsky. See Proposition 2.1.4 in “The triangulated categories of motives of a field” (2000) by Voevodsky. Also note 3.5.4 which gives a Gysin distinguished triangle. Voevodsky derives the triangulated category of geometric motives over a perfect k as DM_gm(k) and equates it to smooth projective varieties there. Gysin triangles as they apply to projective morphisms (of smooth schemes) at 2.1.2 is explored in “Around the Gysin triangles I” (2011) by Déglise.
My argument was missing some details. These can be found in the following.
See pg. 350 of “An introduction to algebraic topology” (1988) by Rotman for the coexact Puppe sequence which I refer to as the long exact sequence and think can be used to prove axoims 1-3 of triangulated categories. Theorem 11.53 in Rotman gives the Hopf fibration S^2n+1 -> CP^n with the fiber S^1. I have corrected this fiber in my previous comment. Note Proposition 4.66 for the Puppe sequence of a fibration and (1) and (2) in “Algebraic topology” (2001) by Hatcher. Also, in Hatcher Theorem 4.58 gives a useful description of the cofibration sequence version of a Puppe sequence using unions. In“Triangulation categories” (2001) by Neeman the three axioms of pre-a triangulated category are given by Definition 1.1.2 and a triangulated category with its additional axiom is given by Definition 1.3.13 with mention of uncertainty of a counter example of a pre-triangulated category that is not a triangulated category. The Eckmann-Hilton duality describes the duality between cofibration sequences of suspensions ΣX and fibration sequences of loop spaces ΩY. This duality gives a way to switch between coexact Puppe sequences induced by cofibrations and exact Puppe sequences induced by fibrations. A useful application of this duality is given by Theorem 11.12 and Lemma 11.38 in Rotman.
Projective varieties are Kähler manifolds with its metric induced by the Fubini-Study metric on them. See Example 1.5 and note the last paragraph of Proposition 4.4 of “The Kodaira embedding theorem” (2010) notes by Massarenti. Also, note the Fubini-Study metric can be derived from the Hopf fibration S^1 -> S^2n+1 -> CP^n mentioned earlier.