J. F. Jardine. Simplicial objects in a Grothendieck topos. In Applications of algebraic K-theory to algebraic geometry and number theory, Part I, number 55 in Contemporary Mathematics, pages 193--239. American Mathematical Society, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Note that it suffices to consider Z=p coefficients, where p is a prime distinct from the characteristic of k. If k is the algebraic closure of the finite field F , then this conjecture holds for any algebraic group G [1] It also holds for any k for the stable groups GL, Sp, O [10] [3]. In the case k = C , we have an isomorphism [5] p. H ffl et (BG C ; Z=p) H ffl (BG top ; Z=p) where BG top is the classifying space of the Lie group G top . This space classifies principal G bundles; i.e. fibrations G E B: For example, if G = GL n , the space BGL n ....

J. Jardine, Simplicial objects in a Grothendieck topos, Contemp. Math. 55 (1986), 193--239.

....R n (A) Ext n Ab(E) Z; A) D(Z; A[n] ho Ab(E) op (Z; K(A;n) ho E op (1; K(A;n) between the Quillen model categories of N indexed chain complexes of abelian group objects, resp. simplicial abelian group objects resp. simplicial objects in the topos E. See Jardine [9] for more details, and Jardine [11] for the statement on non abelian H 1 . The homotopical approach of the present paper is quite robust, allowing one to experiment with combinatorial expressions for other global homotopy sets (or groups) The argument has a modular structure: each of the last ....

....F ] U; X) is a bijection for arbitrary locally brant U , X. But for such U , X, frac LAFib 1 F E op F (U; X) is of course the same as frac LAFib 1 E op (U; X) 2.2. Remark. Ken Brown established his theorem in his axiomatic setting of a category of brant objects ; see also Jardine [9]. Curiously, this is the only place in the argument where a concept of homotopy is needed other than the one hiding in 0 of a category: either simplicial homotopy of simplicial sheaves or the Quillen Brown notion of right homotopy via path objects. Proof of Proposition 3. We begin with one of ....

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J. F. Jardine. Simplicial objects in a Grothendieck topos. In Applications of algebraic K-theory to algebraic geometry and number theory, Part I, number 55 in Contemporary Mathematics, pages 193-239. American Mathematical Society, 1986.

....map H ffl et (BG k ; Z=p) Gamma H ffl (BG;Z=p) is an isomorphism for all primes p not equal to the characteristic of k. Gabber s rigidity theorem [2] implies that this map is indeed an isomorphism for the stable general linear group GL (this is due to Suslin [6] for k = C and to Jardine [3] for arbitrary k) Similarly, a proof of an unstable version of rigidity would lead to a proof of the unstable Friedlander Milnor Conjecture. In this note we consider unstable rigidity for the group PGL 2 over an elliptic curve E. We assume that E is defined by the equation F (x; y) 0, where F ....

J. Jardine, Simplicial objects in a Grothendieck topos, Contemp. Math. 55 (1986), 193--239.

....equivalence in the right pushout diagram above then w 0 is also a weak equivalence. We define the A 1 fibrations in Spc to be the class F A 1 of all morphisms with the right lifting property relative to trivial cofibrations. The following result is based upon results Joyal [Jo] and Jardine [J]. Its proof requires the use of simplicial sheaves, and will be sketched shortly. Theorem 1.9. The classes C, W A 1 and F A 1 form a proper (left and right) closed model structure on Spc. Definition 1.10. The homotopy category Hot = Hot A 1 (k) of schemes over k is the category obtained from ....

J. F. Jardine, Simplicial objects in a Grothendieck Topos, AMS Contemp. Math., vol. 55, 1986, pp. 193--239.

....= CH n (X; i) where CH (X; are the higher Chow groups introduced by S.Bloch [1] 2 Here we prove that the first conjecture is true. The method of the proof is close to the methods developed previously by A.Suslin [19] 20] O.Gabber (unpublished) H.Gillet and R.Thomason [5] and R. Jardine [10] for the computation of algebraic K theory of algebraically closed fields. In section 3 we prove a rather general version of the rigidity theorem of Suslin, Gabber, Gillet and Thomason (theorem (4.4) below) The crucial role in the application of this rigidity theorem to the problem in question is ....

J.F. Jardine. Simplicial objects in a Grothendieck topos. Contemporary Math., 55(1):193--239, 1986.

....general results about simplicial sheaves on sites which will be later applied to our study of the homotopy category of schemes. In the first part (Section 2.1) we describe the main features of the homotopy theory of simplicial sheaves on a site. Many results of this part can be found in [20] and [17], 18] Surprisingly, there are some nontrivial things 2 to be proven in relation to basic functoriality of the homotopy categories of simplicial sheaves. This is done in Section 2.1.6. In Section 2.2 we prove a general theorem which shows that there is a good way to invert any set of morphisms ....

....f : X Y is called a local fibration (resp. trivial local fibration) if for any point x of T the corresponding morphism of simplicial sets x (X ) x (Y) is a Kan fibration (resp. a Kan fibration and a weak equivalence) The list of most important properties of local fibrations can be found in [17]. We will only recall the following result. For simplicial sheaves X , Y denote by (X ; Y) the quotient of Hom(X ; Y) S 0 (X ; Y) with respect to the equivalence relation generated by simplicial homotopies, i.e. the set of connected components of the simplicial function object S(X ; Y) and call ....

J.F. Jardine. Simplicial objects in a Grothendieck topos. Contemporary Math., 55(1):193--239, 1986.

....showed that for a topological space X the category of locally fibrant sheaves of spectra on X is a category of fibrant objects, which is something a little bit weaker than a closed model structure. This has been extended to simplicial objects in an arbitrary Grothendieck topos by Jardine in [15]. This paper will appear in the Journal of Pure and Applied Algebra. It is also Chapter 1 of the author s PhD thesis. If you fetch this paper, please acknowledge this by sending an email to the address below. The thesis is also available in printed form, requests for this can be sent to the ....

J. F. Jardine. Simplicial objects in a Grothendieck topos. In Applications of algebraic K-theory to algebraic geometry and number theory, part I, volume 55, part I of Contemp. Math., pages 193--239. Amer. Math. Soc., Providence, R.I., 1986.

.... homological algebra, with naturality one of the first things being investigated [26] and has continued with and been strenghtened by the development of topos theory [43] and the use of categorical methods in homotopical [46, 27, 31, 7, 12, 16] and homological algebra [41] including K theory [10, 33, 47], and elsewhere. Recently, the interaction has intensified, with the connection between braids and tangles on the one hand and braided and tortile tensor categories on the other [35, 36] and in the theory of operads [44, 6] The above connection between categories and chain complexes, and between ....

J. F. Jardine, Simplicial objects in a Grothendieck topos, in "Applications of algebraic Ktheory to algebraic geometry and number theory, part I", vol. 55, part I of Contemp. Math., pp. 193--239, Amer. Math. Soc., Providence, R.I., 1986.

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J.F. Jardine, Simplicial objects in a Grothendieck topos, Comtemp. Math. 55 (1986), 193--239.

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J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemp. Math 55 (1986), 193-- 239.

....a finite simplicial set behave like coverings. In particular, from this point of view, every simplicial set is locally a Kan complex (Lemma 31) and the methods for manipulating homotopy types then follow almost by exact analogy with the theory of locally fibrant simplicial sheaves or presheaves [5], 6] In that same language, we can show that every fibration which is a weak equivalence has the local right lifting property with respect to all inclusions of finite simplicial sets (Lemma 33) and then this becomes the main idea leading to the coincidence of fibrations as defined here and ....

J.F. Jardine, Simplicial objects in a Grothendieck topos, Comtemp. Math. 55 (1986), 193--239.

....Proposition 4. Suppose that G is a stack, and that U is an object of C. Then there is an isomorphism [U, BG] # = # 0 G(U) Proof. There is an isomorphism [U, BG] # = lim # V #(V, BG) where the colimit is indexed over hypercovers V # K(U, 0) of the object U . We can assume [6] that these hypercovers are of the form V . # U for some sheaf epimorphism V # U , and then a cofinality argument allows us to presume that the sheaf epi # V i # U arises from some covering family V i # U . The canonical map # 0 G(U) # #(V . BG) is the composite # 0 G(U) # ....

J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemporary Math. 55 (1986), 193--239.

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. J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemporary Math. 55 (1986), 193--239.

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. J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemporary Math. 55 (1986), 193--239.

.... ( isomorphism classes of non degenerate symmetric bilinear forms of rank n over K ) BOn ] where the thing on the right denotes morphisms from the terminal (simplicial) object to the classifying simplicial object BOn in the category of simplicial sheaves on the big etale site (SchjK ) et [5], 7] The group scheme On represents a sheaf of groups on this site (by faithfully flat descent) and BOn is the simplicial sheaf whose n simplices consist of the sheaf On Theta Delta Delta Delta Theta On (n fold product) with faces and degeneracies defined in the usual way: d 0 (g n ; ....

.... by mapping the non trivial element of Z=2 to the element oe w 2 On (Q) defined by reflection in the hyperplane orthogonal to the vector w = 2; 1; 0; 0) But then the image of oe w in Q = Q ) 2 under the spinor norm ffi : On (Q) Q = Q ) 2 is ffi(oe w ) 2 2 1 2 ] [5], so that Sp(ae) can be identified with the cup product [7] Delta [5] 2 H 2 et (K; Z=2) By the Merkurjev Suslin theorem, to show that [7] Delta [5] 6= 0 in H 2 et (Q; Z=2) it is enough to show that the symbol f7; 5g is not a 2 divisible element of K 2 (Q) To see this, we compute the tame ....

[Article contains additional citation context not shown here]

. J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemp. Math. 55(I) (1986), 193--239.

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. J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemporary Math. 55 (1986), 193--239.

....henselization of the local ring at x. The purpose of this note is to show that this theorem may be promoted to a similar statement for Karoubi L theory, away from characteristic 2. A computation of the groups ffl L (k; Z= follows, along the lines of the computation of K (k; Z= given in [5]. The method is to prove the isomorphism conjecture for the groups ffl O(k) This is not a generalization of Suslin s arguments [15] 16] Typeset with L A M S T E X. 1. Complements. Suppose, for the rest of this paper, that the characteristic of the algebraically closed field k is not 2, ....

....w T or(Z= ffl L i Gamma1 (k) F NaN w 0 for i = 1 and 2. Lemmas 3 and 4 imply that is an isomorphism when i = 1. Lemmas 4 and 5 imply that is an isomorphism when i = 2. Theorem 1 is proved, by Lemma 2. Let (Smj k ) et denote the site of smooth k schemes with the etale topology, as in [5], and let the algebraic group ffl O n;n represent a sheaf of groups on this site. The sheaf of groups ffl O is defined by ffl O = lim Gamma n ffl O n;n where the filtered colimit is taken in the sheaf category on (Smj k )et. Consider the canonical map ffl : Gamma ffl O(k) ffl O; where ....

[Article contains additional citation context not shown here]

. J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemporary Math. 55,I (1986), 193--239.

No context found.

J. F. Jardine. Simplicial objects in a Grothendieck topos. In Applications of algebraic K-theory to algebraic geometry and number theory, Part I, number 55 in Contemporary Mathematics, pages 193--239. American Mathematical Society, 1986.

No context found.

J. F. Jardine, Simplicial objects in a Grothendieck Topos, AMS Contemp. Math., vol. 55, 1986, pp. 193--239.

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