Publication: El teorema de Riemann-Roch y el morfismo de Gysin en geometría aritmética
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2016-08-16
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Universidad Complutense de Madrid
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Abstract
Le th eor eme de Riemann-Roch originale a rme que pour tout morphisme propre f : Y ! X entre vari et es quasi-projectifs lisses sur un corps, et tout el ement a 2 K0(Y ) du groupe de Grothendieck des br es vectoriels on a ch(f!(a)) = f {u100000}Td(Tf ) ch(a) (cf. [BS58]). Ici ch est le caract ere de Chern, Td(Tf ) est la classe de Todd du br e tangent relative et f et f! sont les images directes de l'anneau de Chow et K0 respectivement. Apr es, Baum, Fulton et MacPherson ont d emontr e en [BFM75] le th eor eme de Riemann-Roch pour des morphismes localement intersection compl ete entre des sch emas alg ebriques (sch emas s epar es et localement de type ni sur un corps) projectifs et singuli eres. En [FG83] Fulton et Gillet ont d emontr e le th eor eme sans hypoth eses projectifs. L'extension a la th eorie K sup erieure pour des sch emas r eguli eres sur une base fut d emontr e par Gillet en [Gil81]. Le th eor eme de Riemann-Roch qu'il prouve est pour des morphismes projectifs entre des sch emas lisses et quasi-projectifs. Donc, dans le cas des sch emas sur un corps, le r esultat de Gillet n'inclus pas le th eor eme de [BFM75]. La plus grande g en eralisation du th eor eme de Riemann-Roch que je connais est [D eg14] et [HS15], o u D eglise et Holmstrom-Scholbach obtiennent ind ependamment le th eor eme de Riemann- Roch pour la K-th eorie sup erieure et les morphismes projectifs lic entre sch emas r eguli eres sur une base noetherienne de dimension nie...
NOTA 520 8 El teorema de Riemann-Roch original de Grothendieck a rma que para todo mor smo propio f : Y ! X, entre variedades irreducibles quasiproyectivas lisas sobre un cuerpo, y todo elemento a 2 K0(Y ) del grupo de Grothendieck de brados vectoriales se satisface la relaci on ch(f!(a)) = f {u100000}Td(Tf ) ch(a) (cf. [BS58]). Recu erdese que ch denota el car acter de Chern, Td(Tf ) la clase de Todd del brado tangente relativo y f y f! las im agenes directas en el anillo de Chow y K0 respectivamente. M as tarde Baum, Fulton MacPherson probaron en [BFM75] el teorema de Riemann-Roch para mor smos localmente intersecci on completa entre esquemas algebraicos (es decir, esquemas separados localmente de tipo nito sobre cuerpo) proyectivos singulares. En [FG83] Fulton y Gillet probaron el teorema sin hip otesis proyectivas. La notable extensi on a la teor a K superior para esquemas regulares sobre una base fue probada por Gillet en [Gil81]. El teorema de Riemann-Roch all probado es para mor smos proyectivos entre esquemas lisos quasiproyectivos. Sin embargo, obs ervese que en el caso de esquemas sobre cuerpo el resultado de Gillet no recupera el teorema de [BFM75]. La mayor generalizaci on del teorema de Riemann-Roch que yo conozco es [D eg14] y [HS15] donde D eglise y Holmstrom-Scholbach obtuvieron independientemente el teorema de Riemann-Roch para teor a K superior y mor smos proyectivos lic entre esquemas regulares sobre una base noetheriana nito dimensional...
The original Grothendieck's Riemann-Roch theorem states that for any proper morphism f : Y ! X, between nonsingular quasiprojective irreducible varieties over a eld, and any element a 2 K0(Y ) of the Grothendieck group of vector bundles the relation ch(f!(a)) = f {u100000}Td(Tf ) ch(a) holds (cf. [BS58]). Recall that ch denotes the Chern character, Td(Tf ) the Todd class of the relative tangent bundle and f and f! the direct image in the Chow ring and K0 respectively. Later Baum, Fulton and MacPherson proved in [BFM75] the Riemann-Roch theorem for locally complete intersection morphisms between singular projective algebraic schemes (i.e., locally of nite type separated schemes over a eld). In [FG83] Fulton and Gillet proved the theorem without projective assumptions on the schemes. The remarkable extension to higher K-theory and schemes over a regular base was proved by Gillet in [Gil81]. The Riemann-Roch theorem proved there is for projective morphisms between smooth quasiprojective schemes. However, note that in the case over a eld Gillet's theorem does not recover the result of [BFM75]. The furthest generalization of the Riemann-Roch theorem I know is [D eg14] and [HS15] where D eglise and Holmstrom-Scholbach independently obtained the Riemann-Roch theorem for higher K-theory and projective lci morphisms between regular schemes over a nite dimensional noetherian base... NOTA 520 8
The original Grothendieck's Riemann-Roch theorem states that for any proper morphism f : Y ! X, between nonsingular quasiprojective irreducible varieties over a eld, and any element a 2 K0(Y ) of the Grothendieck group of vector bundles the relation ch(f!(a)) = f {u100000}Td(Tf ) ch(a) holds (cf. [BS58]). Recall that ch denotes the Chern character, Td(Tf ) the Todd class of the relative tangent bundle and f and f! the direct image in the Chow ring and K0 respectively. Later Baum, Fulton and MacPherson proved in [BFM75] the Riemann-Roch theorem for locally complete intersection morphisms between singular projective algebraic schemes (i.e., locally of nite type separated schemes over a eld). In [FG83] Fulton and Gillet proved the theorem without projective assumptions on the schemes. The remarkable extension to higher K-theory and schemes over a regular base was proved by Gillet in [Gil81]. The Riemann-Roch theorem proved there is for projective morphisms between smooth quasiprojective schemes. However, note that in the case over a eld Gillet's theorem does not recover the result of [BFM75]. The furthest generalization of the Riemann-Roch theorem I know is [D eg14] and [HS15] where D eglise and Holmstrom-Scholbach independently obtained the Riemann-Roch theorem for higher K-theory and projective lci morphisms between regular schemes over a nite dimensional noetherian base... NOTA 520 8
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Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Álgebra, leída el 25-01-2016