Theorem 1.1 if k is the algebraic closure eld of a nite eld, then the ayer-vietoris sequence for the above covering! K i (End (X))! K i (End (X 0 )) K

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1 The K-Theory of Schemes with Endomorphisms Is a Global Theory Dongyuan Yao Abstract We show that when X = Pk 1, the projective line over the eld k, with the open covering fx 0 = Spec(k[x]); X 1 = Spec(k[y])g, the ayer-vietoris sequence of the K-theory of the schemes with endomorphisms for the covering is not exact for all K i when i = 2n + 1 and n 1. This leads to the conclusion that the K-theory of schemes with endomorphisms is a global theory. Key Words: Vector Bundles, Endomorphisms, Scheme, K-theory, ayer-vietoris sequence. x1 Introduction Let X be a scheme. Let End (X) denote the category where objects are all pairs (F; f) with F a vector bundle on X and f an endomorphism of F, and the morphisms in End (X) from (F; f) to (G; g) are those morphisms from F to G which commute with the endomorphisms f and g. End (X) becomes an exact category when we dene (F; f)! (G; g)! (H; h) to be short exact if and only if F! G! H is as vector bundles. By the K-theory of X with endophisms we means the K-thoery of the exact category End (X). In the K-theory of schemes, Thomason and Trobaugh's remarkable work ([Th-Tr]) shows that the K-theory of schemes enjoys very good local and global relation: there are localization exact sequences for open subschemes of the scheme and ayer-vietoris exact sequences for open coverings of the scheme. Particularly the latter tells that the K-theory of schemes is a local theory, it is determined by the local data. When the K-theory of schemes with endomorphisms is under study, it is natural and basic to ask if it also enjoys the same local and global relation as the K-theory of schemes does, in particular if we still have the exact ayer-vietoris sequences for open coverings. The answer is NO. The purpose of this paper is to show through an example that the ayer-vietoris sequence of the K-theory of schemes with endomorphisms for an open covering may not be exact. This indicates that the K-theory of schemes with endomorphisms is a global theory rather than a local one. The example we are going to study is X = P 1 k, the projective line over the eld k, with the open covering f X 0 = Spec(k[x]); X 1 = Spec(k[y])g. We will show 1

2 Theorem 1.1 if k is the algebraic closure eld of a nite eld, then the ayer-vietoris sequence for the above covering! K i (End (X))! K i (End (X 0 )) K i (End (X 1 ))! K i (End (X 0 \ X 1 ))! K i?1 (End (X))! is not exact at K i (End (X 0 \ X 1 )) for all i = 2n + 1; n 1. In [Ya], the author showed that the above ayer-vietoris sequence is not exact at K i (End (X 0 \ X 1 )) for i = 0. Since K 0 sometimes behaves dierently from higher K i 's, it was raised by some readers that further evidences are needed to support the assertion that the K-theory of schemes with endomorphisms is a global theory. The current assay is an eort to respond to this concern. Acknowledgement The author would like dedicate the current paper to his late thesis advisor Robert Thomason for his many years of teaching and inspiration. This paper was prepared while the author was visiting David Webb at Dartmouth College. any thanks go to him for helpful conversations and to Dartmouth College for hospitality. x2. Notations and Recollections Let X be a scheme. The forget map (F; f)! F gives a functor from End (X) to the category of all vector bundles over X. This forget functor is clearly splitting with the splitting injection F! (F; 0). Denote End i (X) = ker(k i (End (X))! K i (X)): When X = Spec(A), we write End i (A) for End i (X). Given a scheme X, let S ~ be the multiplicatively closed set of all polynomials of the form f(t ) = 1 + a 1 T + + a n T n where all a i 2?(O X ; X) are global sections on X, i.e., S ~ = 1 + T?(OX ; X)[T ]?(O X[T ] ; X[T ]). Here X[T ] = X Spec(Z[T ]). We form a new scheme X ~ = S ~?1 X[T ] in the following way: Locally for any ane open subscheme U of X, U = Spec(A), denote SU ~ = the image of S ~ under the restriction map?(ox ; X)[T ]!?(O X ; U)[T ] = A[T ]. Let U ~ = Spec( S ~?1 U A[T ]). Clearly these locally dened ane schemes can glue up and form the scheme X. ~ We have the splitting injective map ' : X! X ~ which is induced locally by the surjective splitting ring map S ~?1 U A[T ]! A by setting T = 0. 2

3 ' Let EK i (X) = ker(k i ( X) ~! K i (X)). When X = Spec(A), we write EK i (A) for EK i (X). In [Ya] we showed Theorem 2.1 If X is a quasi-compact scheme with an ample family of line bundles, then with the above notations, we have End i (X) = EK i+1 (X): Let X 0 and X 1 be two open subschemes of X such that X = X 0 [ X 1. Since ayer-vietoris sequence of the K-theory of schemes for the covering :! K i (X)! K i (X 0 ) K i (X 1 )! K i (X 0 \ X 1 )! K i?1 (X)! is exact, we see that the ayer-vietoris sequence for the K-theory of schemes with endomorphisms for the covering! K i (End (X))! K i (End (X 0 )) K i (End (X 1 ))! K i (End (X 0 \ X 1 ))! K i?1 (End (X))! is exact if and only if the sequence! End i (X)! End i (X 0 )End i (X 1 )! End i (X 0 \X 1 )! End i?1 (X)! is exact. Now let X = P 1 k, the projective line over a eld k, X 0 = Spec(k[x]) and X 1 = Spec(k[y]). Then fx 0 ; X 1 g, glueing up along x! y?1, is a covering of X. We assume k is the algebraic algebraically closed eld over a nite eld. Then Theorem 1.1 is equivalent to the following Theorem 2.2 With the above notations, the sequence! End i (P 1 k )! End i (k[x]) End i (k[y])! End i (k[x; x?1 ])! End i?1 (P 1 i?1) is not exact at End i (k[x; x?1 ]) for i = 2n + 1, n 1. The proof of Theorem 2.2 will occupy the section 3. x3. Proofs 3

4 Let A be one of the rings k; k[x]; k[y] or k[x; x?1 ]. ~ S = 1 + T A[T ]. ~S (A[T ]) denotes the abelian category of all nitely generated ~ S-torsion A[T ]-modules. Lemma 3.1 We have isomorphisms : for all i. End i (A) = K i ( ~S (A[T ])) Proof Since A is regular, we have K i+1 (A) = K i+1 (A[T ]). We see that is splitting injective and K i+1 (A[T ])! K i+1 ( ~ S?1 A[T ]) End i (A) = EK i+1 (A) = coker(k i+1 (A[T ])! K i+1 ( ~ S?1 A[T ])): Applying Quillen's localiztion theorem for the K-theory of abelian categories, we have the long exact sequence:! K i+1 (A[T ])! K i+1 ( ~ S?1 A[T ])! K i ( ~S (A[T ]))! : This long exact sequence splits into short exact sequences 0! K i+1 (A[T ])! K i+1 ( ~ S?1 A[T ])! K i ( ~S (A[T ]))! 0: So we have End i (A) = K i ( ~S (A[T ])). Proposition 3.2 1) End i (P 1 k ) = ( L a2k K i(k)) ( L a2k K i(k)). 2) For A = k[x] (or A = k[y]), we have short exact sequences 0! End i (k[x])! K i (Q(k[x; T ]=(f)))! K i?1 (k)! 0: (a;b)2k k Here Q(k[x; T ]=(f)) denotes the fraction eld of the domain k[x; T ]=(f), and irr: stands for irreducible polynomials in k[x; T ]. 3) For A = k[x; x?1 ], we have the long exact sequence! End i (k[x; x?1 ])! 4 K i (Q(k[x; x?1 ; T ]=(f)))

5 Proof 1) Since! (a;b)2k k K i?1 (k)! : K i ( ~ P 1 k ) = K i(p 1 B) = K i (B) K i (B) where B = (1 + T k[t ])?1 k[t ], we have End i (P 1 k ) = EK i+1 (P 1 k ) = ker(k i+1 ( ~ P 1 k )! K i+1(p 1 k )) = ker(k i+1 (B) K i+1 (B)! K i+1 (k) K i+1 (k)) = EK i+1 (k) EK i+1 (k) = End i (k) End i (k): Since k is algebraically closed, a polynomial f 2 ~ S = 1 + T k[t ] is irreducible if and only if f = 1 + at for some a 2 k = k? f0g. Then the devissage theorem gives End i (k) = K i ( ~S (k[t ])) = K i (k[t ]=(f)) = a2k K i (k): 2) Let ~ S 2 (k[x; T ]) denote the Serre subcategory of ~S (k[x; T ]) of those whose supports are of codimension 2, i.e., if P is a prime ideal in k[x; T ] such that P 6= 0, then height(p ) 2. Applying Quillen's Localiztion theoren for the K-theory of abelian categories, we have the long exact sequence! K i ( ~ S 2 (k[x; T ]))! K i( ~S (k[x; T ]))! K i ( ~S (k[x; T ])= ~ S 2 (k[x; T ]))! K i?1( ~S (k[x; T ]))! : First we see that we have the equivalence for the quotient category ~S (k[x; T ])= ~ S 2 (k[x; T ]) = a P [ n odf(k[x; T ] P =P n P ) where P runs through all prime ideals of height=1 in k[x; T ] such that f 2 P for some f 2 ~ S = 1 + T k[x; T ], and odf(r) denote the abelian category of all nitely generated R-modules. Since k[x; T ] is a UFD, such a P must be (f), the ideal generated by f for some irreducible polynomial f 2 ~ S. 5

6 By the devissage theorem, we have K i ( a P [ n odf(k[x; T ] P =P n P ) = K i (Q(k[x; T ]=(f))): Since dim(k[x; T ]) = 2, a prime ideal of height 2 containing some f 2 ~ S must be a maximal ideal of the form (1 + at; x? b) for a; b 2 k and a 6= 0. We see that ~ S 2 (k[x; T ]) = a P [ n odf(k[x; T ] P =P n P ) where P runs through all the maximal ideals in k[x; T ] which have the form P = (1 + at; x? b) for some a; b 2 k and a 6= 0. So, we have S K i ( ~ 2 (k[x; T ])) = K i (k[x; T ]=(1 + at; x? b)) = K i (k): (a;b)2k k (a;b)2k k What remains is to show the localization long exact sequence splits into short exact sequences, i.e., to show that K i ( ~ S 2 (k[x; T ]))! K i( ~S (k[x; T ])) is a zero map. To that end, let 1 (k[x; T ]=(f)) denote the category of all nitely generated k[x; T ]=(f)-modules whose supports are prime ideals of codim=1. where f is a give nonzero polynomial in k[x; T ]. Clearly S ~ (k[x; T ]) = lim 2 f 2 S ~ 1 (k[x; T ]=(f)): So we need to show that for each f 2 ~ S, the map induced by the inclusion j is a zero map on the K-theory K i (j) : K i ( 1 (k[x; T ]=(f)))! K i ( ~S (k[x; T ])): For any 2 1 (k[x; T ]=(f)), let P 1 = (1 + a 1 T; x? b 1 ); : : : ; P l = (1 + a l T; x? b l ) be all the minimal prime ideals over ann() (the annihilating ideal of ). Then for some integer e, we have (P 1 P l ) e ann(), thus g(t ) = ((1 + a 1 T ) (1 + a l T )) e 2 ann() and h(x) = ((x? b 1 ) (x? b l )) e 2 ann(). 6

7 Since h(x) 2 k[x] k[t ][x] is monic in x, k[x; T ]=(h(x)) is a nitely generated free k[t ]-module. Since is a nitly generated k[x; T ]=(h(x))- module, is a nitely generated k[t ]. So k[t ] k[x; T ] is a nitely generated k[x; T ]-module. Since g(t ) 2 S ~ and g( k[t ] k[x; T ]) = 0, k[t ] k[x; T ] 2 ~S (k[x; T ]). The following short exact sequence is well-known: 0! k[t ] k[x; T ] T??! k[t ] k[x; T ]!! 0 where is the endomorphism on induced by the action of T on. Therefore we have an exact sequence of exact functors from 1 (k[x; T ]=(f)) to ~S (k[x; T ]): 0! k[t ] k[x; T ]! k[t ] k[x; T ]! j! 0: By the additivity theorem, we see that K i (j) is a zero map. 3) Let ~ S 2 (k[x; x?1 ; T ]) be the subcategory of ~S (k[x; x?1 ; T ]) of all those whose supports are maximal ideals. Then ~S (k[x; x?1 ; T ])= ~ S 2 (k[x; x?1 ; T ]) = a P [ n odf(k[x; x?1 ; T ] P =P n P ) where P runs through all prime ideals of height=1 in k[x; x?1 ; T ] such that f 2 P for some f 2 S ~ = 1 + T k[x; x?1 ; T ]. Such a P must be (f), the ideal generated by f for some irreducible polynomial f 2 S. ~ So we have K i ( ~S (k[x; x?1 S ; T ])= ~ 2 (k[x; x?1 ; T ])) = K i (Q(k[x; x?1 ; T ]=(f))): f 2 S;irr: ~ Since any maximal ideal P in k[x; x?1 ; T ] that contains some f 2 ~ S has the form P = (1 + at; x? b) for a; b 2 k, we have K i ( ~ S 2 (k[x; x?1 ; T ])) = (a;b)2k k K i (k): Finally an application of Quillen's localization theorem gives the stated long exact sequence in the Lemma. In the proof of the theorem 2.2, we will need the following fact: 7

8 Lemma 3.3 Suppose we have the following commutative diagram in an abelian category where all rows and columns are exact: 0! A! B! C! 0 # f # g #! N! P # h # Q 0 Then there is a subobject R of Q and a surjective map R! coker(g). Proof Applying the snake lemma to the diagram 0! A! B! C! 0 # a f # g # 0! ker()! N! P; we have a surjection: coker(a f)! coker(g), where a :! ker() is the canonical map. a is surjective since the row is exact. So the composite coker(f)! coker(a f)! coker(g) is surjectve. Since the rst column is exact, take R = coker(f) = Im(h) Q. Proof of Theorem 2.2 E 0 = E 1 = E 01 = Write K i (Q(k[x; T ]=(f))); F 0 = K i (Q(k[y; T ]=(f))); F 1 = K i (Q(k[x; x?1 ; T ]=(f))); F 01 = (a;b)2k k (a;b)2k k K i?1 (k); K i?1 (k); (a;b)2k k K i?1 (k) (notice that the ~ S has dierent meanings for dierent rings). Then we have the following commutative diagram 0! End i (k[x]) End i (k[y])! E 0 E 1! F 0 F 1! 0 # # g # End i (k[x; x?1 ]! E 01! F 01 # # End i?1 (P 1 k ) 0 8

9 The two rows are exact from the Proposition 3.2. The last column is also exact since the maps from F 0 and F 1 to F 01 are just the projections. Suppose the rst column is exact. By the Lemma 3.3, there is a subgroup of End i?1 (P 1 k ) which maps surjectively onto coker(g). When i = 2n + 1 for n 1, End i?1 (P 1 k ) = ( a2k K 2n (k)) ( a2k K 2n (k)) = 0: So we have coker(g) = 0 when i = 2n + 1 for n 1. But clearly K i (Q(k[x; x?1 ; T ]=(1 + (x + x?1 )T ))) is direct summamd of coker(g), so it surces to show that K i (Q(k[x; x?1 ; T ]=(1 + (x + x?1 )T ))) is not zero to lead to contradiction. Since k[x; x?1 ; T ]=(1 + (x + x?1 )T ) = k[x; x?1 ; (x + x?1 )?1 ], we see that Q(k[x; x?1 ; T ]=(1 + (x + x?1 )T )) = k(x). So K i (Q(k[x; x?1 ; T ]=(1 + (x + x?1 )T ))) = K i (k(x)) = K i (k[x]) = K i (k) 6= 0 when i = 2n + 1 for all n 1, where the last inequality is the well know fact from Quillen and Suslin ([Qu 2] or [Su]). References [Gr1] D. Grayson, The K-theory of Endomorphisms, J. of Algebra 48(1977), [Gr2] D. Grayson, Higher algebraic K-theory II (after Quillen), Algebraic K-Theory: Evanston 1976, Springer Lect. Notes ath 551(1976), [Qu 1] D. Quillen, Higher algebraic K-theory I, Higher K-theories, Springer Lect. Notes ath 341(1973), [Qu 2] D. Quillen, On the cohomology and K-theory of generallinear group over nite elds, Ann. of ath. 96(1972), [Su] A. Suslin, On the K-theory of algebraically closed elds, Inventiones ath. 73(1983), [Th-Tr] R. Thomason and T.Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift III, Progress in ath. 88, Birkhauser 1990, [Wa] F. Waldhausen, Algebraic K-theory of spaces, Algebraic and Geometric Topology, Springer Lect. Notes ath. 1126(1985), [Ya] D. Yao, The K-theory od vector bundles with endomorphisms over a scheme, to appear in J. of Alg. 9

10 Department of athematics, Washington University in St. Louis, St. Louis, O yao@math.wustl.edu 10