Hill yield criterion

2014-01-25T10:32:35Z

2.189.153.26: /* Hill 1993 yield criterion */

{{Use dmy dates|date=June 2013}}
In [[algebraic geometry]] and [[algebraic topology]], a branch of [[mathematics]], '''A1 homotopy theory''' is a way to apply the techniques of algebraic topology, specifically [[homotopy]], to [[algebraic varieties]] and, more generally, to [[scheme (mathematics)|schemes]]. The theory is due to [[Fabien Morel]] and [[Vladimir Voevodsky]]. 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 '''A'''1, 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]] of [[mixed motive (math)|mixed motive]]s and the proof of the [[Milnor conjecture|Milnor]] and [[Norm residue isomorphism theorem|Bloch-Kato conjecture]]s.

==Construction of the A1 homotopy category==
'''A'''1 homotopy theory is founded on a category called the '''A'''1 homotopy category. This is the homotopy category for a certain [[closed model category]] whose construction requires two steps.

Most of the construction works for any [[Grothendieck topology|site]] ''T''. Assume that the site is [[Grothendieck topology|subcanonical]], 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]], 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 Δop''Shv''(''T'') denote the category of functors Δop → ''Shv''(''T''). That is, Δop''Shv''(''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 Δop''Shv''(''T'') in terms of points. Let <math>f : \mathcal{X} \to \mathcal{Y}</math> 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]]s <math>x^*f : x^*\mathcal{X} \to x^*\mathcal{Y}</math> 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 <math>\mathcal{H}_s(T)</math>.

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(''i''0 × 1''I'') = μ(1''I'' × ''i''0) = ''i''0''p''.
** μ(''i''1 × 1''I'') = μ(1''I'' × ''i''1) = 1''I''.
* The morphism <math>i_0 \amalg i_1 : \text{pt} \amalg \text{pt} \to I</math> is a monomorphism.
Now we localize the homotopy theory with respect to ''I''. A simplicial sheaf <math>\mathcal{X}</math> is called ''I''-local if for any simplicial sheaf <math>\mathcal{Y}</math> the map

:<math>\text{Hom}_{\mathcal{H}_s(T)}(\mathcal{Y} \times I, \mathcal{X}) \to \text{Hom}_{\mathcal{H}_s(T)}(\mathcal{Y}, \mathcal{X})</math>

induced by ''i''0 : ''pt'' → ''I'' is a bijection. A morphism <math>f : \mathcal{X} \to \mathcal{Y}</math> is an ''I''-weak equivalence if for any ''I''-local <math>\mathcal{Z}</math>, the induced map

:<math>\text{Hom}_{\mathcal{H}_s(T)}(\mathcal{Y}, \mathcal{Z}) \to \text{Hom}_{\mathcal{H}_s(T)}(\mathcal{X}, \mathcal{Z})</math>

is a bijection. The homotopy theory of the site with interval (''T'', ''I'') is the localization of Δop''Shv''(''T'') with respect to ''I''-weak equivalences. This category is called <math>\mathcal{H}(T, I)</math>.

Finally we may define the '''A'''1 homotopy category. Let ''S'' be a finite dimensional [[Noetherian scheme]], and let ''Sm''/''S'' denote the category of [[smooth morphism|smooth]] schemes over ''S''. Equip ''Sm''/''S'' with the [[Nisnevich topology]] to get the site (''Sm''/''S'')Nis. We let the affine line '''A'''1 play the role of the interval. The above construction determines a closed model structure on <math>\Delta^\text{op}\text{Shv}_\text{Nis}(\text{Sm}/S)</math>, and the corresponding [[homotopy category]] is called the '''A1 homotopy category'''. By construction, it is such that for any ''X'' in ''Sm''/''S'', there is an isomorphism
:<math>X \times_S \mathbf A^1_S \cong X</math>
in the homotopy category.

==Properties of A¹ homotopy theory==
The setup, especially the [[Nisnevich topology]], is chosen as to make [[algebraic K-theory]] representable by a spectrum, and in some aspects to make a proof of the Bloch-Kato conjecture possible.

After the Morel-Voevodsky construction there have been several different approaches to A¹ homotopy theory by using other model category structures or by using other sheaves than Nisnevich sheaves (for example, Zariski sheaves or just all presheaves). Each of these constructions yield the same homotopy category.

There are two kinds of spheres in the theory: those coming from the multiplicative group playing the role of the 1-sphere in topology, and those coming from the simplicial sphere (considered as constant simplicial sheaf). This leads to a theory of motivic spheres <math>S^{p,q}</math> with two indices. To compute the homotopy groups of motivic spheres would also yield the classical stable homotopy groups of the spheres, so in this respect A¹ homotopy theory is at least as complicated as classical homotopy theory.

==References==
* {{Citation | last1=Morel | first1=Fabien | last2=Voevodsky | first2=Vladimir | author2-link=Vladimir Voevodsky | title='''A'''1-homotopy theory of schemes | url=http://archive.numdam.org/article/PMIHES_1999__90__45_0.pdf | accessdate=9 May 2008 | mr=1813224 | year=1999 | journal=[[Publications Mathématiques de l'IHÉS]] | issue=90 | pages=45–143 | doi=10.1007/BF02698831 | volume=90}}
*{{Citation | last1=Voevodsky | first1=Vladimir | author1-link=Vladimir Voevodsky | title=Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998) | url=http://www.mathunion.org/ICM/ICM1998.1/Main/00/Voevodsky.MAN.ocr.pdf | year=1998 | journal=Documenta Mathematica | issn=1431-0635 | chapter='''A'''1-homotopy theory | pages=579–604 | mr=1648048}}

{{DEFAULTSORT:A Homotopy Theory}}
[[Category:Algebraic geometry]]
[[Category:Homotopy theory]]

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Hill yield criterion