In algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

 and algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, A¹ homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

, to algebraic varieties and, more generally, to schemes

Scheme (mathematics)

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...

. The theory is due to Fabien Morel and Vladimir Voevodsky

Vladimir Voevodsky

Vladimir Voevodsky is a Russian American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002.- Biography :...

. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval [0,1], which is not an algebraic variety, with the affine line A¹, which is. The theory requires a substantial amount of technique to set up, but has spectacular applications such as Voevodsky's construction of the derived category

Derived category

In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...

 of mixed motives.

Construction of the A¹ homotopy category

A¹ homotopy theory is founded on a category called the A¹ homotopy category. This is the homotopy category for a certain closed model category whose construction requires several steps.

Most of the construction works for any site

Grothendieck topology

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...

 T. Assume that the site is subcanonical

Grothendieck topology

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...

, and let Shv(T) be the category of sheaves of sets on this site. This category is too restrictive, so we will need to enlarge it. Let Δ be the simplicial category

Simplicial category

In mathematics, the simplex category is the category of finite ordinals and order preserving maps. It is used to define simplicial and cosimplicial objects.-Formal definition:...

, that is, the category whose objects are the sets {0}, {0, 1}, {0, 1, 2}, and so on, and whose morphisms are order-preserving functions. We let Δ^opShv(T) denote the category of functors Δ^op → Shv(T). That is, Δ^opShv(T) is the category of simplicial objects on Shv(T). Such an object is also called a simplicial sheaf on T. The category of all simplicial sheaves on T is a Grothendieck topos.

A point of a site T is a geometric morphism x^* : Shv(T) → Set, where Set is the category of sets. We will define a closed model structure on Δ^opShv(T) in terms of points. Let be a morphism of simplicial sheaves. We say that:

• f is a weak equivalence if, for any point x of T, the morphism of simplicial set
  Simplicial set
  In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space...
  
  s is a weak equivalence.
• f is a cofibration if it is a monomorphism.
• f is a fibration if it has the right lifting property with respect to any cofibration which is a weak equivalence.

The homotopy category of this model structure is denoted .

This model structure will not give the right homotopy category because it does not pay any attention to the unit interval object. Call this object I, and denote the final object of T by pt. We assume that I comes with a map μ : I × I → I and two maps i₀, i₁ : pt → I such that:

• If p is the canonical morphism I → pt, then
  • μ(i₀ × 1_I) = μ(1_I × i₀) = i₀p.
  • μ(i₁ × 1_I) = μ(1_I × i₁) = 1_I.
• The morphism is a monomorphism.

Now we localize the homotopy theory with respect to I. A simplicial sheaf is called I-local if for any simplicial sheaf the map


induced by i₀ : pt → I is a bijection. A morphism is an I-weak equivalence if for any I-local , the induced map


is a bijection. The homotopy theory of the site with interval (T, I) is the localization of Δ^opShv(T) with respect to I-weak equivalences. This category is called .

Finally we may define the A¹ homotopy category. Let S be a finite dimensional Noetherian scheme, and let Sm/S denote the category of smooth schemes over S. Equip Sm/S with the Nisnevich topology

Nisnevich topology

In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives...

to get the site (Sm/S)_Nis. We let the affine line A¹ play the role of the interval. The above construction determines a closed model structure on , and the corresponding homotopy category is called the A¹ homotopy category.