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# Geometry over đŊ1

ä¸ å ãŽ å ãããĒã äŊ ã å­ å¨ ããã¨ äģŽ åŽ ããã¨ , ããŽ ä¸ ãŽ äģŖ æ° åšž äŊ å­Ļ ã æ§ į¯ ããã¨ãã ãŽã¯ã¨ãĻã č åŗ æˇą ã ãã­ã¸ ã§ ã¯ã ã§ãã ã åŽ é , æ§ ã ãĒ 𝔽 1 -geometry ãŽ æ æĄ ã ãã ã

äž ãã° , Deitmar ãŽ æ æĄ [ Dei05 ] ã¯ , å¯ æį° ãĢ å¯ž åŋ ããããŽã , å¯ æ ãĸãã¤ã ã¨ãŋãĒ ããã¨ã§ãã ã ããã¯ GL ãŽ éĄ äŧŧ ã å¯ž į§° įž¤ ã§ãããã¨ãã čĒ įļ ãĒ ãĸã¤ããĸ ãĢ æ ãã ã [ Dei08 ] ã§ã¯ , étale morphism ãĒãŠã č ããããĻããã , Thas ãŽ [ Tha16 ] ã§ã¯ , ã ãŽ Deitmer ãŽ 𝔽 1 -scheme ã¨ Thas čĒ čēĢ ãŽ åŽ įžŠ ããããŽãĢã¤ããĻ čŋ° ãšãããĻ ãã ã

æ§ ã ãĒ ãĸãã­ ãŧ ã ãĢã¤ããĻãŽ æĻčĻŗ ã¨ ä¸ģ ãĒ įĩ æ ãĢã¤ããĻã¯ , Lorscheid ãŽ č§Ŗ čĒŦ [ Lor18 ] ãĢãžã¨ããããĻãã ã

æ é å integral 𝔽 1 scheme ã¨ toric variety ã¨ãŽ éĸ äŋ ã čĒŋ ãšãããĻãã ã Manin ã¯ 𝔽 1 ä¸ ãŽ analytic function ã [ Man10 ] ã§ č ããĻããã , ããã§ã¯ Deitmar ããŽ 𝔽 1 scheme ãŽ åŽ įžŠ ãŽãããĢ categorical ãĒ åŽ įžŠ ã ãã§ãĒã , ﬁnite extension 𝔽 1 n ã é čĻ ã§ãããã¨ ã čŋ° ãšãããĻãã ã ãããĻ ããŽ post ãĢ æ¸ ãããĻãããããĢ , ã 𝔽 1 ä¸ ãŽ algebra ã¨ čĻ ãĒã , 𝔽 1 scheme ãã ä¸ , ãããĻ ä¸ ãŽ scheme ã äŊ ããã¨ãã§ ãã ã

• ã¯ 𝔽 1 ä¸ ãŽ algebra ã¨ãŋãĒããšã

𝔽 1 ä¸ ãŽ äģŖ æ° åšž äŊ å­Ļ ãããŗããã¨ Tits ãŽ “geometry” ã¨ãŽ éĸ äŋ ãĢã¤ããĻã¯ , [ CC11b ] ã§ č­° čĢ ãããĻãã ã Chevalley įž¤ ã¨ãŽ éĸ äŋ ãĢã¤ããĻã čŋ° ãšãããĻãã ã Cortiñas ã [ CnHWW15 ] ã¯ , toric variety ãŽ K -theory ã čĒŋ ãšããŽãĢ äŊŋ ãŖ ãĻ ãã ã

𝔽 1 ãŽ algebraic K -theory ã¯ į éĸ ãŽ åŽ åŽ ããĸãã ãŧ įž¤ ã¨ãŋãĒããŽã åĻĨ åŊ ãĒãããĢ æ ã ãã , ãã ä¸ čŦ ãŽ 𝔽 1 -scheme ãŽ algebraic K -theory ãĢã¤ããĻã¯ , Chu ã¨ Lorscheid ã¨ Santhanam ãŽ [ CLS12 ] ã§ č ããããĻãã ã Chu ã¨ Morava ãŽ [ CM ] ã čĻ ãã¨ã ã ã ãããĢããã¨ , Abel įž¤ G ãŽ 𝔽 1 ä¸ ãŽ group algebra 𝔽 1 [ G ] ãĢ å¯ž ããĻã¯ , å å

ã æ ã įĢ ã¤ããã§ãã ã

Durov ã¯ [ Dur ] ã§ , Arakelov geometry ãŽ ä¸ čŦ å ã¨ããĻ 𝔽 1 ã å æŦ ãã æ  įĩ ãŋã æ§ į¯ ãããã¨ããĻãã ã ãããĢããã¨ 𝔽 1 -module ãŽ ããĸã­ã¸ ãŧ äģŖ æ° ã¯ åē įš äģ ã simplicial set ãŽ ããĸãã ãŧ äģŖ æ° , ã¤ãžã å¤ å¸į ãĒ ããĸãã ãŧ čĢ ã¨ č ãããŽã č¯ ãããã§ãã ã ãã ã¯ , ä¸ č¨ ãŽãããĢ algebraic K -theory ã¨ããĻ į éĸ ãŽ åŽ åŽ ããĸãã ãŧ įž¤ ã įž ãããã¨ãã ã åĻĨ åŊ ãĢ æ ãã ã

Connes ã¨ Consani [ CC16 ] ã¯ , Segal ãŽ Γ-space ãŽ discrete į , ã¤ãžã Γ-set ã į¨ ã ããã¨ã æ æĄ ããĻãã ã Γ-space ã¯ inﬁnite loop space , ã¤ãžã connective spectrum ãĢ å¯ž åŋ ããããŽã§ããã , Lydakis [ Lyd99 ] ã¯ Γ-space ãŽ category ãĢ monoidal strucutre ã åŽ įžŠ ã , connective ring spectrum ãŽ category ãŽ model ã æ§ æ ãã ã Connes ã¨ Consani ã¯ , Γ-set ãŽ category ãŽ monoid object ã 𝕊 -algebra ã¨ åŧ ãŗ , ãããžã§ åŊŧ į­ ã æ æĄ ããĻãã hyperring ã semiring ãĢãã ãĸãã­ ãŧ ã ã įĩą å ããããŽã¨ããĻ æ æĄ ããĻ ãã ã

Tropical algebraic geometry ã¨ãŽ éĸ éŖ ãĢã¤ããĻã¯ , Giansiracusa ã¨ Giansiracusa ãŽ [ GG ] ããã ã Connes ã¨ Consani [ CC11a ] ã , tropical ãĒ ä¸ į ã¨ãŽ éĸ äŋ ã æ æ ããĻ ãã ã