Smoothness in Relative Geometry

Smo othness in Relativ e Geometry Florian Mart y fmart ymath.ups-tlse.fr Univ ersité T oulouse I I I - Lab oratoire Emile Piard Abstrat In [TV a℄, Bertrand T o ën and Mi hel V aquié dened a s heme theory for a losed monoidal ategory ( C , ⊗ , 1) . In this artile, w e dene a notion of smo othness in this relativ e (and not neessarily additiv e) on text whi h generalize the notion of smo othness in the ategory of rings. This generalisation onsists pratially in  hanging homologial niteness onditions in to homotopial ones using Dold-Kahn orresp ondene. T o do this, w e pro vide the ategory s C of simpliial ob jets in a monoidal ategory and all the ategories sA − mod , sA − alg ( A ∈ scomm ( C ) ) with ompatible mo del strutures using the w ork of Rezk in [ R℄. W e giv e then a general notions of smo othness in sC omm ( C ) . W e pro v e that this notion is a generalisation of the notion of smo oth morphism in the ategory of rings, is stable under omp ositions and homotopi pushouts and pro vide some examples of smo oth morphisms in N − al g and C omm ( S et ) . Con ten ts Abstrat 1 In tro dution 1 Preliminaries 2 1 General Theory 3 1.1 Simpliial Categories and Simpliial Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Compatly Generated Mo del ategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Categories of Mo dules and Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Finiteness Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 A Denition for Smo othness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Simpliial Preshea v es Cohomology 8 2.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Simpliial preshea v es Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Category of Preshea v es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Obstrution Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Simpliial Mo dules Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Examples 15 3.1 The Category ( Z − mod, ⊗ Z , Z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 The ategory S et . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Some Others examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 In tro dution In [TV a ℄, Bertrand T o ën and Mi hel V aquié dened a s heme theory for a losed monoidal ategory ( C , ⊗ , 1) . In this artile, w e dene a notion of smo othness in this relativ e on text whi h generalize the notion of smo othness in the ategory of rings. The motiv ations for this w ork are that in teresting ob jets in the non additiv e on texts C = E ns or N − mod are exp eted not to b e s hemes but Sta ks. A theorem asserts for C = Z − mod that quotien ts of s hemes b y smo oth group s hemes are in fat algebrai sta ks. A near theorem is exp eted in a relativ e on text. The rst step is 1 to get a denition for smo oth morphism. The follo wing theorem giv e the go o d denition of smo othness that an b e generalised to the relativ e on text: Theorem 0.1 . Assume C = Z − mod A morphism of rings A → B is smo oth if and only if i. The ring B is nitely pr esente d in A − al g . ii. The morphism A → B is at. iii. The ring B is a p erfe t  omplex of B ⊗ A B mo dules. The atness of A → B is in fat equiv alen t to T or dim A B = 0 hene the t w o last onditions are homologial niteness onditions. By the orresp ondene of Dold-Kan, the seond ondition an then b e tradued in an homotopial ondition. Finally , a result from [TV℄ asserts that B is a p erfet omplex in ch ( B ⊗ A B ) if and only if it is homotopially nitely presen table in sB ⊗ A B − mod . W e pro vide then the ategory s C of simpliial ob jets in a monoidal ategory and all the ategories sA − mod , sA − alg with mo del strutures using the w ork of Rezk in [R℄. The lassial funtors b et w een the ategories A − mod, A − al g , A ∈ sC omm ( C ) indue Quillen funtors b et w een the orresp onding simpliial ategories. W e giv e the follo wing general denition for smo othness Denition 0.2. Let A b e in sC omm ( C ) , a morphism B → C in sA − alg is smo oth if and only if i. The simplial algebra C is homotopially nitely presen ted in sB − al g . ii. The simpliial algebra C has T or dimension 0 on B . iii. The morphism C ⊗ h B C → C is homotopially nitely presen ted in sC ⊗ h B C − mod . The rst ondition imply the rst ondition of 0.1 ([TV℄, 2.2.2.4) and there is equiv alene for smo oth morphisms of rings. When A, B , C are just rings, the T or dimension 0 imply that the deriv ed tensor pro dut is w eakly equiv alen t to the tensor pro dut. The equiv alene with previous theorem for rings is then lear. W e pro v e that relativ e smo oth morphisms are stable under omp osition and pushouts of Algebras. W e nally pro vide examples of smo oth morphism in relativ e non-additiv e on texts , for C = N − mod or C = E ns . In partiular the ane lines F 1 → N and N → N [ X ] are smo oth resp etiv ely in sC omm ( S et ) and s N − mod . Preliminaries Let ( C , ⊗ , 1) b e a omplete and o omplete losed symmetri monoidal ategory . In the ategory C , there exists a notion of omm utativ e monoid and for a giv en omm utativ e monoid A , of A -mo dule. Let C omm ( C ) denotes the ategory of omm utativ e monoids (with unit y) in C and for A ∈ C omm ( C ) , A − mod denotes the ategory of A -mo dules. It is w ell kno wn that the ategory A − mod is a losed monoidal tensored and otensored ategory , omplete and o omplete. The ategory C omm ( A − mod ) will b e denoted b y A − alg and is desrib ed b y the equiv alene A/C omm ( C ) ⋍ A − alg . A pushout in A − alg is a tensor pro dut in the sense that for omm utativ e monoids B , C ∈ A − al g B ⊗ A C ⋍ B ` A C . All along this w ork , ( C , ⊗ , 1) is a lo ally nitely presen table monoidal ategory i.e. v erify that the (Y oneda) funtor i : C → P r ( C 0 ) , where C 0 is the full sub ategory of nitely presen ted ob jets, is fully faithful, that C 0 is stable under tensor pro dut ans on tains the unit y . The funtor H om C (1 , − ) , denoted ( − ) 0 : X → X 0 is alled underlying set funtor. F or k ∈ C 0 , the funtors H om C ( k , − ) , denoted ( − ) k : X → X k are alled w eak underlying set funtor. It is kno wn that if C is a lo ally nitely presen table monoidal ategory , so are its ategories of mo dules A − mod , A ∈ sC omm ( C ) Let J denotes the set of isomorphism lasses of the ob jets of C 0 . W e assume that the adjuntion C ( r e sp C o mm ( C )) / / E ns J o o indued is monadi. Th us, on simpliial ategories, the theorem of Rezk 1.2 will pro vide a (go o d) mo del struture. Moreo v er, as the forgetful funtor from A − mod to C preserv es olimits for a monoid A the funtor from A − mod to E ns J is also monadi. This is a onsequene of the  haraterisation of monadi funtors (see ). There are t w o fundamen tal adjuntions: C ( −⊗ A ) / / A − mod i o o C L / / C omm ( C ) i o o where the forgetful funtor i is a righ t adjoin t and the funtor free asso iated monoid L is dened b y L ( X ) := ` n ∈ N X ⊗ n /S n . In these adjuntions, C an b e replaed b y B − mod for B ∈ C omm ( C ) , and L b y L B dened b y L B ( M ) := ` n ∈ N M ⊗ B n /S n . Let ϕ (resp ϕ B ) and ψ (resp ψ B ) denote these adjuntions for the ategory C (resp B − mod ). F or X ∈ C and M ∈ A − mod , ϕ : H om C ( X, M ) → H om A − mod ( X ⊗ A, M ) is easy to desrib e : 2 ϕ : f → µ M ◦ I d A ⊗ f ϕ − 1 : g → g ◦ ( I d X ⊗ i A ) ◦ r − 1 X Let s C denotes the ategory of simpliial ob jets in C . There is a funtor onstan t simpliial ob jet denoted k from C to s C whi h is righ t adjoin t to the funtor π 0 from s C to C dened b y π 0 ( X ) := C ol im ( X [1] / / / / X [0] o o ) . The tensor pro dut of C indues a tensor pro dut on s C , its unit y elemen t is k (1) . F or A in C omm ( C ) , sA − mod and sA − al g will denote resp etiv ely the simpliial ategories sk ( A ) − mod and sk ( A ) − al g . As sC omm ( C ) ⋍ C omm ( s C ) , w e will alw a ys refer to simpliial ategory of omm utativ e monoids in s C as sC omm ( C ) . The funtor indued b y L on simplial ategories will b e denoted sL . The funtor i : C → P r ( C 0 ) indues a funtor si : s C → sP r ( s C 0 ) . W e need nally h yp otheses to endo w s C , sC omm ( C ) , and for A ∈ sC omm ( C ) , sA − mod and sA − al g with ompatible mo del strutures. One solution of this question is to assume that the natural funtors from s C and sC omm ( C ) to sS et J are monadi, where J is the set of isomorphism lasses of J . The  haraterisation of monadi funtors of [B ℄ implies that for an y omm utativ e simpliial monoid A , the indued funtors from sA − mod to sS et J is also monadi. 1 General Theory 1.1 Simpliial Categories and Simpliial Theories Denition 1.1. A simpliial theory is a monad (on sS et J ) omm uting with ltered olimits. Theorem 1.2 . (R ezk) L et T b e a simpliial the ory in sS et J , then T − al g admits a simpliial mo del strutur e. f is a W e ak e quivalen e or a br ation in T − al g if and only if so is its image in sS et J (for the pr oje tive mo del strutur e). Mor e over, this Mo del  ate gory is right pr op er. Prop osition 1.3. Mo del strutur es on the simpliial  ate gories. i. L et A = ( A p ) b e a  ommutative monoid in s C . The monadi adjuntions A p − mod / / S et J o o indu e a monadi adjuntion sA − mod / / sS et J o o i.e. ther e is an e quivalen e sA − mod ⋍ T A − al g = wher e T A is the monad indu e d by adjuntion. In p artiular s C ⋍ T 1 − al g . ii. L et A = ( A p ) b e a  ommutative monoid in s C . The monadi adjuntions A p − al g / / E ns J o o indu e a monadi adjuntion sA − al g / / sS et J o o o o i.e. ther e is an e quivalen e sA − alg ⋍ T c A − alg wher e T c A is the monad indu e d by adjuntion. In p artiular sC omm ( C ) ⋍ T c 1 − al g . Pro of As explained in the preliminaries, this is due to the  haraterisation of monadi funtors ([B℄).  R emark 1.4 . The righ t adjoin ts funtors all omm ute with ltered olimits. So do the monads whi h are then simpliial theories on sS et J . Corollary 1.5. L et A b e a  ommutative monoid in s C . The  ate gories s C and sA − mod and sC omm ( C ) ar e Mo del  ate gories. Mor e over, the funtors ( A ⊗ − ) and sL ar e left Quil len and their adjoints pr eserve by  onstrution we ak e quivalen es and br ations. Theorem 1.6 . The Cate gory s C ( resp sA − mod ) is a monoidal mo del  ate gory Pro of: The pro of for s C and sA − mod are similar, so let us pro v e it for s C . Let I , I ′ b e resp etiv ely the sets of generating obration and generating trivial obration. I and I ′ are the image b y the left adjoin t funtor resp etiv ely of generating obration and generating trivial obration in sS et J . W e juste ha v e to pro v e (f [H℄  hap IV) that I  I is a set of obrations and I  I ′ and I ′  I are sets of trivial obrations. It is true for generating obration and generating trivial obration in sS et J , whi h are all morphisms onen trated in a xed lev el. Moreo v er, it is easy to v erify that the funtor sK omm utes with  of morphisms onen trated in one lev el. So it is true in s C . The seond axiom is learly v eried, as 1 is obran t.  3 1.2 Compatly Generated Mo del ategories Denition 1.7. Let M b e a obran tly generated simpliial mo del ategory and I b e the set of generating obrations. i. An ob jet X ∈ I − cell is stritly nite if and only if there exists a nite sequene ∅ = X 0 / / X 1 / / ... / / X n = X and ∀ i a pushout diagram: X i / /   X i +1   A u i / / B with u i ∈ I . ii. An ob jet X ∈ I − cell is nite if and only if it is w eakly equiv alen t to a stritly nite ob jet. iii. An ob jet X is homotopially nitely presen ted if and only if for an y ltered diagram Y i , the morphism : H ocol im i M ap ( X , Y i ) → M ap ( X , H ocol im i Y i ) is an isomorphism in H o ( sS et ) iv. A mo del ategory M is ompatly generated if it is ellular, obran tly generated and if the domains and o domains of generating obration and generating trivial obration are obran t, ω -ompat and ω -small (Relativ e to M ). Prop osition 1.8. L et M b e a  omp atly gener ate d mo del  ate gory. i. F or any lter e d diagr am X i , the natur al morphism H ocol im i X i → C ol im i X i is an isomorphism in H o ( M ) . ii. Assume that lter e d  olimits ar e exat in M . Then homotopi al ly nitely pr esente d obje ts in M ar e exatly obje ts e quivalent to we ak r etr ats of stritly nite I − cel l obje ts. Prop osition 1.9. i. The simpliial mo del  ate gory sS et J is  omp atly gener ate d. ii. The  ate gories of simpliial algebr as over a simpliial the ory ar e  omp atly gener ate d. Lemma 1.10. L et A b e in sC omm ( C ) . L et u j b e the family of images by the left adjoint funtor in sA − mod (r esp sA − al g ) of elements ∗ j of sS et J dene d by ∗ on level j and ∅ on other levels. A ny  o domains of a gener ating  obr ations of sA − mod (r esp sA − al g ) is we akly e quivalent to an obje t u j . A ny domain of a gener ating  obr ation is we akly e quivalent, for a given element j in J , to an obje t obtaine d fr om the initial obje t (denote d ∅ ) and u j in a nite numb er of homotopi pushouts. Pro of: Generating obrations of sA − mod are images of generating obrations of sS et J b y the left adjoin t funtor. Gen- erating obrations of sS et are morphisms δ ∆ p → ∆ p . Their o domain is on tratible, th us so are the o domains of generating obrations of sS et J for the pro jetiv e mo del struture, and their image b y b y the left adjoin t is w eakly equiv alen t to the unit y 1 . F or the domains, onsider the relation δ ∆ p +1 ⋍ ∆ p +1 ` h δ ∆ p ∆ p +1 ⋍ ∗ ` h δ ∆ p ∗ and δ ∆ 0 = ∅ . Domains of generating obration in sS et J for the pro jetiv e mo del struture are ob jets ( δ ∆ p,j ) p ∈ N , j ∈ V dened in lev el i 6 = j b y ∅ and in lev el j b y δ ∆ p and v erify the relation ( δ ∆ p,j ) ⋍ ∗ j ` h ( δ ∆ p − 1 ,j ) ∗ j Clearly δ ∆ 0 ,j = ∅ and δ ∆ 1 ,j = ∗ j . Let u p,j denote the image of δ ∆ p,j . F or all j , u p,j is obtained in a nite n um b er of pushouts from ∅ and u j .  Corollary 1.11. of pr op osition 1.9 and lemma 1.10 . i. The Simpliial Mo del  ate gories s C , sA − mod ( A ∈ sC omm ( C ) ), sC omm ( C ) and sA − al g ( a ∈ sC omm ( C ) ) ar e  omp atly gener ate d. ii. Homotopi al ly nitely pr esente d obje ts of sA − mod ( r espsA − al g ) ar e exatly obje ts we akly e quivalent to we ak r etr ats of stritly nite I − C e ll obje ts. 4 iii. The sub- ate gory of H o ( ssA − mod ) ( resp H o ( sA − al g )) H o ( sA − mod ) c ( resp H o ( sA − al g ) c ) Consisting of homotopi al ly nitely pr esente d obje ts is the smal lest ful l sub- ate gory of H o ( sA − mod ) (r esp H o ( sA − alg ) )  ontaining the family ( u 1 ,j ) j ∈ V ( resp ( u 1 ,j a ) j ∈ V ) , and stable under r etr ats and homotopi pushouts. Pro of of iii : Let D b e the smallest full sub-ategory of H o ( s C ) on taining ( u j ) j ∈ V ( resp ( u j A ) j ∈ V ) , the initial ob jet ∅ and stable under retrats and homotopi pushouts. Clearly , b y ii , as ( ∅ → u j ) j ∈ V are generating obrations of s C , H o ( s C ) c on tains the family ( u j ) j ∈ V , and is stable under retrats and homotopi pushouts. Th us D ⊂ H o ( s C ) c . Reipro ally , let X b e an ob jet of H o ( s C ) c , b y ii , X is isomorphi to a w eak retrat of a stritly nite I − cel l ob jet. Therefore, there exists n and X 0 ...X n su h that: ∅ = X 0 / / X 1 / / ... / / X n = X and ∀ j ∈ { 0 , .., n − 1 } , ∃ K → L , a generating obration su h that: X j / / X j +1 K / / O O L O O is a pushout diagram. No w, as domains and o domains of generating obrations are in D , X is in D .  1.3 Categories of Mo dules and Algebras Prop osition 1.12. L et A b e in sC omm ( C ) and B b e a simpliial monoid in sA − al g ,  obr ant in sA − mod . i. The for getful funtor fr om sB − mod to sA − mod pr eserves  obr ations ii. The for getful funtor fr om sA − al g to sA − mod pr eserves  obr ations whose domain is  obr ant in sA − mod . In p artiular, it pr eserves  obr ant obje ts. Pro of In ea h ase, w e just ha v e to pro v e it for generating obrations and then generalize it to an y obration b y the small ob jet argumen t. Pro of of i : First, w e  ho ose a generating obration in sB − mod . As generating obration of sB − mod are images of generating obrations of sS et J , w e set L → M , a generating obration in sS et J . Let K A , K A denotes resp etiv ely the left adjoin t funtors (from sS et J ) for sA − mod and sB − mod . The axiom of stabilit y under  implies that the morphism ( ∅ → B )  ( K A ( L ) → K A ( M )) is a obration in sA − mod . This morphism is in fat K B ( L ) = B ⊗ A K A ( L ) → B ⊗ A K A ( M ) = K B ( M ) , hene generating obrations of sB − mod are obrations in sA − mod . Pro of of ii : As for i , let N → M b e a generating obration in sS et J . Let L s denotes the funtor  free asso iated omm utativ e monoid of sS et J . The funtors L (resp sL A in sA − mod ) and K A are dened b y olimits and so omm ute up to isomorphisms. That means that K A ◦ L s ⋍ sL A ◦ K A . So the generating obration of sA − al g orresp onding to N → M is isomorphi to K A ( L ( N )) → K A ( L ( M )) . T o pro v e that it is a obration in sA − mod , w e ha v e then to pro v e that the morphism L ( N ) → L ( M ) is injetiv e lev elwise and this is lear as for an y morphism N ⊗ n /S n → M ⊗ n /S n is injetiv e. Th us an y generating obration of sA − alg is a obration in sA − mod . In fat it is a generating obration of sA − mod . T o use the small ob jet argumen t (of sA − al g ), w e need to v erify that it preserv es obrations in sA − mod . In fat, w e need to  he k that an homotopi pushout in sA − al g of a obration in sA − mod is still a obration in sA − mod . W e let the reader v erify that it is a onsequene of the axiom of stabilit y b y  . Finally , the forgetful funtor preserv es obrations and, as A is obran t in sA − mod , an y obran t ob jet of sA − alg is also obran t in sA − mod .  Lemma 1.13. L et A → B ∈ sC omm ( C ) b e a trivial  obr ation b etwe en  obr ant obje ts. The  ate gories of mo dule ar e e quivalent i.e. H o ( sA − mod ) ⋍ H o ( sB − mod ) . Pro of: W e m ust pro v e that for X obran t in sA − mod and y bran t in sB − mod , ϕ a ( f ) : x ⊗ A B → y is a w eak equiv alene in sB − mod if and only if so is f : X → Y in sA − mod . By previous lemma, A → B is a trivial obration in sA − mod . Th us as X is obran t, using the axiom of stabilit y under  , g : X → B ⊗ A X is a w eak equiv alene in sA − mod . By onstrution of the adjuntion ϕ A , the follo wing diagram is omm utativ e : 5 X / / f 3 3 X ⊗ A B / / ϕ A ( f ) ) ) Y ⊗ A B / / Y Th us f = g ◦ ϕ A ( f ) . Finally , ϕ A ( f ) is a w eak equiv alene in sA − mod if and only if so it is in sB − mod and the two out of thr e e axiom ends the pro of.  Prop osition 1.14. L et f : A → B ∈ sC omm ( C ) b e a we ak e quivalen e b etwe en  obr ant obje ts. The  ate gories of mo dule ar e e quivalent ie H o ( sA − mod ) ⋍ H o ( sB − mod ) . Let r c b e the bran t replaemen t of sC omm ( C ) , then b y previous lemma, the homotopial ategories of mo dules o v er A and r c A (resp B and r c B ) are equiv alen t. Th us A and B an b e tak en bran t and f is an homotop y equiv alene i.e. ∃ g su h that f ◦ g and g ◦ f are homotopi to iden tit y . The follo wing diagrams are omm utativ e: B I d A A A A A A A A i 0   B 1 h / / B B i 1 O O f ◦ g > > } } } } } } } } H o ( B − mod ) H o ( B 1 − mod ) i ∗ 0 O O i ∗ 1   H o ( B − mod ) ( f ◦ g ) ∗ v v m m m m m m m m m m m m m h ∗ o o I d h h Q Q Q Q Q Q Q Q Q Q Q Q Q H o ( B − mod ) where i 0 and i 1 are obrations and ha v e the same righ t in v erse p i.e. su h that p ◦ i 1 = p ◦ i 0 = I d B . the morphism h is a trivial bration th us i 0 is a w eak equiv alene. By previous lemma, i ∗ 0 is an equiv alene of ategories. Th us so is p ∗ . As i ∗ 1 and i ∗ 0 are b oth in v erses of p ∗ , they are isomorphi and i ∗ 1 is also an equiv alene. Finally , h ∗ is an equiv alene and so is ( f ◦ g ) ∗ . The same metho d pro v e that ( g ◦ f ) ∗ is an equiv alene.  1.4 Finiteness Conditions Denition 1.15. Let q c b e a obran t replaemen t in sC omm ( C ) and f : A → B b e a morphism in sC omm ( C ) . ⊲ The morphism f is homotopially nite (denoted hf ) if B is homotopially nitely presen ted in sq c A − mod . ⊲ The morphism f is homotopially nitely presen ted (denoted hf p ) if B is homotopially nitely presen ted in sq c A − al g . R emark 1.16 . The morphism A → B is hf (resp hf p ) if and only if the morphism q c A → q c B is hf (resp hf p ). The morphism q c B → B is alw a ys hf . Lemma 1.17. The hf (r esp hfp) morphisms ar e stable under  omp osition. Pro of The pro ofs for hf morphisms and hf p morphisms are analogous so let us pro v e it for hf morphisms. Let A → B → C b e the omp osition of t w o hf morphisms. There is a diagram q c A / /   q c B / /   q c C   A / / B / / C and forgetful funtors F 1 : sq c C − mod → sq c B − mod and F 2 : sq c B − mod → s q c A − mod . The image F 1 ( q c C ) of q c C is homotopially nitely presen ted in sq c B − mod hene w eakly equiv alen t to a retrat of a nite homotopial olimit of q c B in H o ( sq c B − mod ) . The forgetful funtor F 2 preserv es retrats, equiv alenes, nite olimits, obran t ob jets and obrations whose domain is obran t. Th us it also preserv es nite homotopial olimit and sends q c C to a retrat of a nite homoto ipal olimit of q c B in H o ( sq c A − mod ) . As q c B is homotopially nitely presen ted in sq c A − mod , and as homotopially nitely presen ted ob jets are stable under retrats, equiv alenes and nite homotopial olimit, C is sen t b y F 2 ◦ F 1 in sq c A − mod c . Hene A → C is nite.  6 Lemma 1.18. The hf (r esp hfp) morphims ar e stable under homotopi pushout of simpliial monoids. Pro of: The pro ofs for hf morphisms and hf p morphisms are analogous so let us pro v e it for hf morphisms. Let A → B and A → C b e in sC omm ( C ) su h that the rst is nite. Let q cA b e the obran t replaemen t of q c A − al g , it is w eakly equiv alen t to q c and the ob jet q c A B is homotopially nitely presen ted in sq c A − mod . Let us pro v e that B ⊗ h A C ⋍ q cA B ⊗ q c A q c C (in H o ( q c A − mod ) , Reedy lemma) is homotopially nitely presen ted in q c C − mod . The forgetful funtor sq c C − mod → sq c A − mod preserv es ltered olimits and w eak equiv alenes hene it preserv es homotopial ltered olimits. Th us the deriv ed funtor − ⊗ q c A q c C preserv es homotopially nitely presen ted ob jets. So B ⊗ h A C is homotopially nitely presen ted in H o ( q c C ) .  1.5 A Denition for Smo othness Denition 1.19. A morphism A → B in sC omm ( C ) is formally smo oth if the morphism B ⊗ h A B → B is hf . R emark 1.20 . This denition do es not generalise the denition of formal smo othness in the sense of rings. Ho w ev er, the orresp onding notion of smo othness is a generalisation of the lassial notion of smo othness, as it will b e pro v ed in this artile. Prop osition 1.21. F ormal ly smo oth morphisms ar e stable under  omp osition. Pro of: b y previous remarks, it an b e assumed that A is obran t in sC omm ( C ) , B is obran t in sA − al g and C is obran t in sB − al g . Let A → B → C b e the omp osition of t w o formally smo oth morphisms . The morphisms B ` A B → B and C ` B C → C are hf . The follo wing diagram omm utes and is learly o artesian : C C ` A C O O / / C ` B C f f L L L L L L L L L L L B ` A B O O / / B O O Th us, if it is obran t for the Reedy stuture, it will b e homotopially o artesian. The morphisms B ⋍ B ` A A → B ` A B and B ` A B → C ` A C are images b y the left Quillen funtor col im , of lear Reedy obrations (see [A ℄ for a desriptions of these obrations), th us are obrations. In partiular B ` A B and C ` A C are obran t and the diagram onsidered is Reedy obran t. Finally , the morphism C ` A C → C ` B C is hf as a pushout of hf morphisms and C ` A C → C is hf as a omp osition of hf morphisms.  Prop osition 1.22. F ormal ly smo oth morphism ar e stable under homotopi pushout. Pro of: Let u : A → B b e a formally smo oth morphism and C b e a omm utativ e A -algebra. By previous remarks it an b e assumed that A and c are obran ts in sC omm ( C ) and that B is obran t in sA − alg . Let D denote the homotopi pushout of B ⊗ A C and u ′ denote the morphism from B to D . Clearly: D ⊗ C D ⋍ B ⊗ A C ⊗ C B ⊗ A C ⋍ B ⊗ A D Th us the follo wing diagram omm utes : B ⊗ A B x x r r r r r r r r r r I d ⊗ A f   m B / / B f   D ⊗ C D / / B ⊗ A D m D / / D And is o artesian : B ⊗ B ⊗ A B B ⊗ A B ⊗ A C ⋍ B ⊗ A C ⋍ D 7 Moreo v er it is learly obran t as B ⊗ A − preserv e obrations. Finally b y stabilit y of hf morphism under homotopi pushouts, the morphism C → D is formally smo oth.  Denition 1.23. Let A b e in sC omm ( C ) and M b e in sA − mod . i. The ob jet M is n -trunated if M ap s C ( X, M ) is n -trunated in sS et , ∀ X ∈ s C . ii. The T or-Dimension of M in sA − mod is dened b y T or dim A ( M ) = inf { n st M ⊗ h A X is n + p − tru ncated ∀ X ∈ sA − mod p − truncated } iii. A morphism of monoids A → B has T or dimension n if T or dim A ( B ) = n . Lemma 1.24. T or dimension zer o morphisms ar e stable under  omp osition and homotopi pushout. Pro of: Let A → B → C b e the omp osition of t w o T or dimension zer o morphisms. Let M b e a p trunated A -mo dule, M ⊗ A C ⋍ M ⊗ A B ⊗ B C . As T or dim A ( B ) = 0 , M ⊗ A B is a p trunated B -mo dule. As T or dim B ( C ) = 0 , M ⊗ A C is a p trunated C -mo dule. Th us T or dim A ( C ) = 0 . Let A → B b e a T or dimension zer o morphism and A → C b e a morphism in C omm ( C ) . Let M b e in C -mo d and let D denote the pushout B ⊗ A C . W e ha v e M ⊗ C B ⊗ A C ⋍ M ⊗ A B . Th us, T or dim A ( B ) = 0 implies T or dim C ( D ) = 0 .  Denition 1.25. A morphism A → B in C omm ( C ) is smo oth if it is formally smo oth, hf p and has T or-Dimension zero. A morphism of ane s heme is smo oth if the orresp onding morphism of monoids is smo oth. W e sa y that an ane s heme X is smo oth if the morphism X → S pec ( 1 ) is smo oth. Theorem 1.26 . Smo oth morphisms ar e stable under  omp osition and homotopi pushout. Pro of: This a a orollary of 1.24 , 1.21 , 1.22 , 1.17 and 1.18 .  2 Simpliial Preshea v es Cohomology In the artile [T1 ℄, B. T o ën dene a ohomology for a onneted and p oin ted simpliial presheaf. W e will dene here a ohomology for a general simpliial presheaf. This theory will b e used to n examples of smo oth morphisms of omm utativ e monoids (in sets). The referenes ited in this setion are [T1 ℄, [GJ℄ and [J℄. 2.1 Denitions In this setion, D is a ategory and sP r ( D ) is the ategory of simpliial preshea v es o v er D . Denition 2.1. ([GJ℄ V I . 3 ) Soit F ∈ sP r ( D ) . The to w er of n -trunations of F is a P ostnik o v to w er: ... / / τ ≤ n F / / τ ≤ n − 1 F / / ... / / τ ≤ 1 F / / τ ≤ 0 F . Denition 2.2. Let F b e a simpliial presheaf. ⊲ The funtor π 0 ( F ) : D → E ns is dened b y π 0 ( F ) : X → π 0 ( F ( X )) . ⊲ The ategory ( D /F ) 0 is the full sub ategory of sP r ( D ) /F whose ob jets are in D . ⊲ The funtor π n ( F ) : ( D /F ) 0 → E ns is dened b y π n ( F )( X , u ) = π n ( F ( X ) , u ) . Denition 2.3. Let G b e a simpliial group. 8 ⊲ The bisimpliial set E ( G, 1) is dened b y E ( G, 1) p,q = G q p . ⊲ The lassifying spae of G , denoted K ( G, 1) , is giv en b y the diagonal of the bisimpliial set E ( G, 1) /G . More preisely K ( G, 1) n = G n n /G n . It is ab elian if G is ab elian. ⊲ The endofuntor of ab elien groups K ( G, 1) ◦ n is denoted K ( G, n ) . R emarks 2.4 . As the diagonal of E ( G, 1) is p oin ted (b y iden tit y), the simpliial set K ( G, 1) is also p oin ted. In partiular, π n ( K ( G, 1) , ∗ ) ⋍ π n − 1 ( G, e g ) . This onstrution is funtorial (in G ) and then extends to preshea v es of simpliial groups. 2.2 Simpliial preshea v es Cohomology It is neessary to w ork in the prop er ategory to onstrut a ohomology for a simpliial presheaf F whi h is not onneted or p oin ted. In fat the 1 -trunation of F is the nerv e N G of a group oid G and in the ategory sP r ( D ) / N G , F b eomes onneted and p oin ted. But in this ategory , there is no lear onstrution fro lassifying spaes. The solution of this problem is giv en b y a Quillen equiv alene with the ategory sP r ( D /G ) , for a w ell  hosen ategory D /G . W e  ho ose no w a simpliial presheaf F . The Category of Preshea v es The left adjoin t funtor g ( − ) Denition 2.5. The ategory D /G is the ategory whose ob jets are ouples ( X, x ) , x : X → N G , and whose morphisms from ( X, x ) to ( Y , y ) are ouples ( f , u ) where f : X → Y and u : y ◦ f ⋍ x in G ( X ) ⋍ π 1 F ( X ) . Next step is to onstrut a funtor g ( − ) : D /G → sP r ( D / N G ) . Denition 2.6. Let ( X, x, ) b e in D /G . Dene a presheaf of group oïds G X,x on D . Ths image of S ∈ D is the group oid desrib ed as follo w - The ob jets are triples ( u, y , h ) , u : S → X , y ∈ G ( S ) , and h : x ◦ u ⋍ y ∈ G ( S ) . - A morphism from ( u, y , h ) to ( u ′ , y ′ , h ′ ) is an endomorphism k of S su h that k ∗ ( h : x ◦ u → y ) = h ′ : x ′ ◦ u ′ → y ′ . Let ˘ X denote the nerv e of this group oid. R emark 2.7 . There is a omm utativ e diagram of preshea v es of group oids X x / / j " " D D D D D D D D G G X,x l = = z z z z z z z z where l is the pro jetion on G and j is giv en for S ∈ D b y j ( S ) : u ∈ H om D ( S, X ) → ( u, x ◦ u, I d ) ∈ G X,x . Applying the funtor nerv e, one get a morphism ˘ x := N l : ˘ X → G . It denes a funtor ˘ ( − ) : D /G → sP r ( D ) / N G ( X, x ) → ( ˘ X , ˘ x ) Denition 2.8. The funtor g ( − ) : D /G → sP r ( D ) / N G is dened b y g ( − ) : ( X , x ) → ( e X , e x ) := Q ( ˘ X , ˘ x ) where Q is a obran t replaemen t in sP r ( D ) / N G . R emarks 2.9 . This funtor has a k an extension to sP r ( D /G ) , still denoted g ( − ) : sP r ( D /G ) → S pr ( D ) / N G . In fats, the ategory sP r ( D /G ) is equiv alen t to the ategory sP r ( D ) N G dened in [J ℄ and the equiv alene of ategory w e are onstruting is onstruted in a dieren t w a y and a more general situation in [ J℄. 9 The righ t adjoin t funtor ( − ) 1 W e onstrut no w the (righ t) adjoin t of g ( − ) , denoted ( − ) 1 . Denition 2.10. The funtor ( − ) 1 : sP r ( D ) / N G → sP r ( D /G ) is dened b y ( − ) 1 : ( H , u ) → H 1 := ( X , x ) → H om ∆ sP r ( D ) /N G (( e X , e x ) , ( H , u )) where H om ∆ is the simpliial H om . As ( e X , e x ) is onstruted obran t, the funtor ( − ) 1 is righ t Quillen and its adjoin t is then left Quillen. W e pro v e no w that R ( − ) 1 omm ute with homotop y olimits. W e need to reall rst some prop erties. Denition 2.11. Let ( H, h ) b e in sP r ( D ) / N G and ( X, x ) b e in D /G . Dene an ob jet ( H X , h x ) b y the homotop y pullba k diagram H X / /   H h   X x / / N G Lemma 2.12. L et ( H, f ) b e an homotopy  olimit, H ⋍ H ocoli m ( H i ) , in sP r ( D ) / N G and let ( X, x ) b e in D /G . ⋄ Ther e is an isomorphism H X ⋍ H ocoli m ( H i ) X in H o ( sP r ( D ) / N G ) . ⋄ Ther e is an isomorphism RH 1 ( X ) ⋍ M ap sP r ( D ) /X (( X, I d ) , ( H X , h x )) . Corollary 2.13. The funtor R ( − ) 1  ommute with homotopy  olimits. Pro of: Let H b e isomorphi to H ocol im ( H i ) and ( X, x ) b e in D /G . RH 1 ( X ) ⋍ M ap sP r ( D ) /X (( X, I d ) , ([ H ocoli m ( H i )] X , [ H ocol im ( h i )] x )) ⋍ M ap sP r ( D ) /X (( X, I d ) , ( H ocoli m [( H i ) X ] , H ocol im [( h i ) x ])) ⋍ H ocol im ( R ( H i ) 1 ( X ))  The Equiv alene Prop osition 2.14. The Quil len funtors g ( − ) and ( − ) 1 dene a Quil len e quivalen e. Pro of: The funtor g ( − ) omm utes with homotop y olimits and as an y ob jet in sP r ( D /G ) is an homotop y olimit of represen table ob jets H ⋍ H ocol im ( X i ) , its image an b e omputed in terms of represen table ob jets, i.e. e H ⋍ H ocol im ( f X i ) . The short exat sequene H 1 → H → τ ≤ 1 H pro v es that ( − ) 1 preserv es w eak equiv alenes. Then ( e H ) 1 ⋍ ( H ocolim ( f X i )) 1 ⋍ H ocol im ( X i ) ⋍ H . If H is obran t in sP r ( D /G ) and H ′ is bran t in sP r ( D ) / N G , w e onsider a morphism b et w een short exat sequenes H / /   e H   / / N G I d   H ′ 1 / / H ′ / / N G Applying the funtors π i , it is lear that if e H → H ′ is an equiv alene, so is H → H ′ 1 and reipro ally , if H → H ′ 1 , the homotopi b ers of e H → H ′ up on N G are equiv alenes th us so is e H → H ′ .  10 The Cohomology Denition 2.15. Let F b e in sP r ( D ) , a lo al system on F is a presheaf of ab elian groups on D /G , where G v eries N G ⋍ τ ≤ 1 F . A Morphism of lo al system is a morphism of preshea v es of ab elian groups. The ategory of lo al systems on F will b e denoted sy sloc ( F ) . The n-th lassifying spae of M is denoted K ( M , n ) and its image b y L g ( − ) is denoted L ˜ K ( M , n ) . R emark 2.16 . The ob jet L ˜ K ( M , n ) is  haraterised up to equiv alene b y the fat that π n ( L ˜ K ( M , n )) ⋍ M , π 1 ( L ˜ K ( M , n )) ⋍ π 1 ( F ) , π 0 ( L ˜ K ( M , n )) ⋍ π 0 ( F ) and that its other homotop y preshea v es of groups are trivial. Denition 2.17. Let F b e in sP r ( D ) and M b e a lo al system on F . The n-th ohomology group of F with o eien t in M is H n ( F, M ) := π 0 M ap sP r ( D ) /N G ( F, L ˜ K ( M , n )) The standard example of lo al system is π n . Indeed, it has b een dened on ( D /F ) 0 but it learly lifts to D /G . The imp ortan t theorem is here. Theorem 2.18 . L et G b e a gr ouoid. F or al l m , the funtor H m ( N G, − ) : S y sl oc ( N G ) → Ab M → H m ( N G, M ) is isomorphi to the n-th derive d funtor of the funtor H 0 ( N G, − ) . Pro of: There is an equiv alene b et w een the ategory of simpliial ab elian group preshea v es, denoted sAb ( D /G ) , on D /G and the ategory of omplexes of ab elian group preshea v es with negativ e or zero degree, denoted C − ( D /G, Ab ) . This is a generalisation of Dold-Kan orresp ondene. There is a orresp ondene b et w een quasi-isomorphism of omplexes and w eak equiv alenes of simpliial preshea v es, and then an indued equiv alene b et w een the homotopial ategories : Γ : D − ( D /G, Ab ) ⋍ H o ( sAb ( D ) /G ) The deriv ed funtors of H 0 are then giv en b y H m der ( D /G, M ) ⋍ H om D − ( D /G,Ab ) ( Z , M [ m ]) Where Z is regarded as a omplex onen trated in degree zero and M [ m ] is onen trated in degree − m , with v alue M . As Γ( Z ) is the onstan t presheaf with b er Z , still denoted Z , and as Γ( M [ m ]) is equiv alen t to K ( M , m ) , Γ indues an isomorphism : H om D − ( D /G,Ab ) ( Z , M [ m ]) ⋍ H om H o ( sAb ( D /G )) ( Z , K ( M , m )) Finally , the adjuntion b et w een the ab elianisation funtor,denoted Z ( − ) from sP r ( D /G ) to sAb ( D /G ) and the forgetful funtor giv es H om D − ( D /G,Ab ) ( Z , M [ m ]) ⋍ H om H o ( sP r ( D /G )) ( ∗ , K ( M , m )) ⋍ H m ( N G, M ) .  Obstrution Theory There is an homotopi pullba k diagram in sP r ( D /G ) : τ ≤ n F 1 / /   *   τ ≤ n − 1 F 1 / / K ( π n ( F ) , n + 1) As F 1 is 1 -onnex, this pullba k diagram is a (funtorial) generalisation to presheaf of the diagram giv en b y the prop osition 5.1 of [GJ℄. By the quillen equiv alene (( − ) 1 , g ( − )) , there is an homotopi pullba k diagram: 11 τ ≤ n F / /   N G   τ ≤ n − 1 F / / L e K ( π n ( F ) , n + 1) If H → τ ≤ n − 1 F is a morphism in H o ( sP r ( D ) / N G ) , it has a lift to τ ≤ n F if and only if it is send to a zero elemen t in the group π 0 M ap sP r ( D ) /N G ( H, L e K ( π n ( F ) , n + 1)) This group an b e desrib ed in terms of ohomology . Indeed, if G ′ is a group oid su h that N G ′ ⋍ τ ≤ 1 H . Let u denote the morphism u : N G ′ → N G . T o simplify the notations, w e still write H for what w e sould all u ∗ H . There is a Quillen adjuntion: sP r ( D ) / N G ′ u ∗ / / D / N G −× N G N G ′ o o whi h indues an isomorphism M ap sP r ( D ) /N G ( H, L e K ( π n ( F ) , n + 1)) ⋍ M ap sP r ( D ) /N G ′ ( H, L e K ( π n ( F ) , n + 1) × N G N G ′ ) . There is a lear w eak equiv alene L e K ( π n ( F ) , n + 1) × N G N G ′ ⋍ L e K ( π n ( F ) ◦ u ∗ , n + 1) , th us : π 0 M ap sP r ( D ) /N G ( H, L e K ( π n ( F ) , n + 1)) ⋍ H n +1 ( H, π n ( F ) ◦ u ∗ ) 2.3 Simpliial Mo dules Cohomology It is w ell kno wn that for a omm utativ e monoid B in ( S et, × , F 1 ) , there is an equiv alene sP r ( B B ) ⋍ sB − mod where B B is the ategory with one ob jet with a set of endomorphisms isomorphi to B . W e will iden tify these t w o ategories in this part. Let no w A b e a omm utativ e monoid in sets and B → A b e a morphism of omm utativ e monoids. W e are in a partiular ase of previous setion, the ategory D is B B and the presheaf of group oids G is just A . Let M b e a lo al system on B B , there is an isomorphism H n ( A, M ) ⋍ π 0 M ap sB − mod/ A ( A, L e K ( M , n + 1)) . Let Z denote the ab elianization funtor from B − mod/ A to the ategory of ab elian group ob jets in B − mod/ A , denoted Ab ( B − mod/ A ) . There is an equiv alene b et w een Ab ( B − mod/ A ) and the ategory of A graduated Z ( B ) -mo dules, denoted Z ( B ) − mod A − gr ad . The follo wing funtor realizes this equiv alene, its in v erse is the forgetful funtor. Θ : ( M f / / A ) ∈ Ab ( sB − mod/ A ) → ⊕ m ∈ A f − 1 ( m ) ∈ Z ( B ) − mod A − gr ad This equiv alene lifts to simpliial ategories and it is easy to see that H n +1 ( A, M ) ⋍ π 0 M ap Z ( B ) − mod A − gr ad ( Z ( A ) , L e K ( M , n + 1)) (1) Here is the prop osition that in terrests us. Prop osition 2.19. L et B → A b e a morphism of  ommutative monoids in sets. The morphism B → A is hf if and only if ⋄ Z ( A ) is homotopi al ly nitely pr esente d in Z ( B ) − mod A − gr ad . ⋄ A is homotopi al ly nitely pr esente d for the 1 -trun ate d mo del strutur e i.e. in the  ate gory B − Gpd . Pro of Let us pro v e rst the easiest part. Let A b e an homotopially nitely presen ted ob jet in sB − mod . Let sB − mod ≤ 1 denotes the ategory sB − mod endo w ed with its 1 -trunated mo del struture. In the adjuntions 12 sB − mod I d / / sB − mod ≤ 1 I d o o sB − mod Z / / sZ ( B ) − mod/ A i o o sB − mod/ AZ Z / / sZ ( B ) − mod A − gr ad i o o the left adjoin t funtors preserv e w eak equiv alenes and obrations th us the righ t adjoin ts preserv e homotopially nitely presen table ob jets. Let us no w pro v e the hardest part. W e start with this lemma: Lemma 2.20. Ther e exists m 0 ∈ N suh that for any lo  al system M and al l n ≥ m 0 H n ( A, M ) ⋍ ∗ . Pro of The isomorphism 1 pro v es that the ohomology of A is isomorphi to the Ext funtors of Z ( A ) in sZ ( B ) − mod A − gr ad . Moreo v er, there is an equiv alene of ab elian ategories sZ ( B ) − mod A − gr ad ⋍ C − ( B B / A, Ab ) whi h indues b y 2.18 an equiv alene with the deriv ed funtors of H 0 . In partiular as Z ( A ) is homotopially nitely presen ted, the deriv ed funtors of H 0 v anished after a set rank denoted m 0 .  R emark 2.21 . T w o orollaries omes no w. They are a onsequene of this lemma and the follo wing short exat sequene, C ∈ A/sB − mod M ap sB − mod/τ ≤ n − 1 C ( A, τ ≤ n C ) / / M ap sB − mod ( A, τ ≤ n C )   M ap sB − mod/ N G ( A, L e K ( π n ( C ) , n + 1)) M ap sB − mod ( A, τ ≤ n − 1 C ) o o Corollary 2.22. L et A v / / C b e in A/sB − mod . F or al l i ≥ 1 , for al l n ≥ n i = n 0 + i + 1 π 0 M ap sB − mod ( A, τ ≤ n − 1 C ) ⋍ π 0 M ap sB − mod ( A, τ ≤ n C ) π i ( M ap sB − mod ( A, τ ≤ n − 1 C ) , v ) ⋍ π i ( M ap sB − mod ( A, τ ≤ n C ) , v ) Pro of W e rst pro v e that the simpliial set M ap sB − mod/τ ≤ n − 1 C ( A, τ ≤ n C ) is not empt y . There are pushout squares : A × h L e K ( π n ( C ) ,n +1) N G / /   τ ≤ n C / /   N G s   A / / τ τ ≤ n − 1 C / / L e K ( π n ( C ) , n + 1) p [ [ where p ◦ s = I d . There are then equiv alenes M ap sB − mod/τ ≤ n − 1 F ( A, τ ≤ n C ) ⋍ M ap sB − mod/L e K ( π n ( C ) ,n +1) ( A, N G ) ⋍ M ap sB − mod/ A ( A, A × h L e K ( π n ( C ) ,n +1) N G ) . Let f b e the morphism from A to L e K ( π n ( C ) , n + 1) . There is a morphism p ◦ f : A → N G . As the ohomology of A v anished for n ≥ n 0 , the elemen ts s ◦ p ◦ f and f of the ohomology group are equals and th us p ◦ f ∈ π 0 M ap sB − mod/L e K ( π n ( C ) ,n +1) ( A, N G ) . Then, for i = 0 , the orollary is a lear onsequene of lemma 2.20 and the short exat sequene of remark 2.21 . No w, Let us study the ase i > 0 . As N G × h L e K ( π n ( C ) ,n +1) N G ⋍ L e K ( π n ( C ) , n + 1) , w e obtain A × h L e K ( π n ( C ) ,n +1) N G ⋍ L e K ( π n ( C ) ◦ v ∗ , n ) 13 Th us π i ( M ap sB − mod/τ ≤ n − 1 C ( A, τ ≤ n C ) , v ) ⋍ π i (( M ap sB − mod/ A ( A, L e K ( π n ( C ) ◦ v ∗ , n )) , q ) ⋍ H n − i ( A, π n ( C )) where q is the natural morphism from A to L e K ( π n ( C ) ◦ v ∗ , n ) . W e dedue then the result from lemma 2.20 and the short exat sequene of remark 2.21 .  Corollary 2.23. L et A v / / C b e in A/sB − mod . The p ointe d tower of br ations ( M ap sB − mod ( A, τ ≤ n C ) , v )  onver ges  ompletly in the sense of [GJ℄. Pro of It an b e  he k ed with the orollary 2.21 of the omplete on v ergene lemma of [ GJ℄.  Corollary 2.24. F or al l i ≥ 0 , al l n ≥ n i and al l A v / / C in A/sB − mod , ther e ar e isomorphisms π i ( M ap sB − mod ( A, C ) , v ) ⋍ l im n ∈ N π i ( M ap sB − mod ( A, τ ≤ n C ) , v ) ⋍ π i ( M ap sB − mod ( A, τ ≤ n i C ) , v ) Pro of The rst isomorphism is a onsequene of Milnor exat sequene ([GJ℄, 2.15) and the v anishing of the l im 1 indued b y the omplete on v ergene. The seond isomorphism is a onsequene of orollary 2.22  Let us no w reall a w ell kno wn lemma with whi h w e will pro v e the last te hnial lemma neessary for the pro of of 2.19 . Lemma 2.25. L et X f / /   Y   Z g / / T b e a  ommutative squar e in sSet wher e g is a we ak e quivalen e. The morphism f is a we ak e quivalen e if and only if for al l z ∈ Z , the homotopi b ers X z and Y g ( z ) ar e simultane ously empty and e quivalent when not empty. Here is the last te hnial lemma: Lemma 2.26. L et C ⋍ H ocol im α ∈ Θ ( C α ) b e an homotopi al lter e d  olimit. Ther e is a we ak e quivalen e in sS et M ap sB − mod ( A, C ) ⋍ H ocol imM ap sB − mod ( A, C α ) . Pro of By indution on the trunation lev el n of C . This is an h yp othesis of 2.19 for n = 1 . Let us assume that is is true for n − 1 . Let C b e an n -trunated ob jet in sB − mod and ¯ u b e in H ocol imM ap sB − mod ( A, τ n − 1 C α ) , represen ted b y u ∈ M a p sB − mod ( A, τ n − 1 C α 0 ) . Let ˜ u denote its image in M ap sB − mod ( A, C ) . The ltered ho olimit along Θ is w eak equiv alen t to the ho olimit along α 0 /θ . W e will use previous lemma, omputing the b ers along u as in the follo wing diagram: H ocol im α 0 / Θ M ap sB − mod/τ n − 1 C α (( A, u α ) , C α ) / /   M ap sB − mod/τ n − 1 C (( A, ˜ u n − 1 ) , C )   H ocol im α 0 / Θ M ap sB − mod ( A, C α ) / /   M ap sB − mod ( A, C )   H ocol im α 0 / Θ M ap sB − mod ( A, τ n − 1 C α ) / / M ap sB − mod ( A, τ n − 1 C ) Where u α : A u / / C α 0 / / C α and ˜ u n − 1 : A ˜ u / / C / / τ n − 1 C . Let us sho w rst that the b ers are sim ultaneously empt y . The naturel morphism 14 H ocol im α 0 / Θ M ap sB − mod ( A, τ n − 1 C α ) → M ap sB − mod ( A, τ n − 1 C ) ¯ u → ˜ u indues the naturel morphism on ohomology groups H ocol im α 0 / Θ H n +1 ( A, π n C α ) → H n +1 ( A, π n C ) whi h is a w eak equiv alene. Indeed, the H n are isomorphi to E xt funtors in sZ ( B ) − mod A − gr ad whi h omm ute with ltered ho olimits b y the rst h yp othesis of 2.19 . The images of ¯ u and ˜ u in the ohomology groups v anish then sim ultaneously , and the b ers are sim ultaneously empt y . Let us assume no w that the b ers are unempt y and pro v e that they are equiv alen t. The funtors π i omm ute with homotopial ltered olimits, applying them on the b ers, w e get the follo wing natural morphism col im α 0 / Θ π i M ap sB − mod/τ n − 1 C α (( A, u α ) , C α ) → π i M ap sB − mod/τ n − 1 C (( A, ˜ u n − 1 ) , C ) As these π i are in fat isomorphi to H n − 1 , these morphisms are isomorphisms. By 2.25 , this ends the pro of of the lemma.  Let us no w pro v e 2.19 . Let v : A → C b e in A/sB − mod su h that C ⋍ H ocl olim ( C α ) . Let us pro v e that the morphism H ocol im ( M ap sB − mod ( A, C α )) → M ap sB − mod ( A, C ) is a w eak equiv alene. Let i b e a p ositiv e in teger, to  he k if the image of this morphism b y π i is an isomorphism, w e an just onsider the ase C n -trunated b y 2.24 . As the trunation omm uta with homotopial ltered olimits, this is a onsequene of 2.26 . This ends the pro of of 2.19 .  3 Examples 3.1 The Category ( Z − mod, ⊗ Z , Z ) In lassial algebrai geometry , the notion of (pro jetiv e) resolution is obtained using  hain omplex of mo dules or rings. In fats, onsidering the orresp ondene of Dold-Kan this metho d is equiv alen t to taking obran t resolution in the simplial ategory (f [Q ℄). Theorem 3.1 . (Dold-Kahn  orr esp ondan e) L et A b e a ring. Ther e is an e quivalen e of  ate gories: sA − mod ⋍ C h ( A − mod ) ≥ 0 and ∀ i π i ( M ap ( Z , X )) ⋍ H i ( X ) . In p artiular, it indu es a  orr esp onden e b etwe en we ak e quivalen es and quasi-isomorphisms. R emark 3.2 . Let A b e a ring. Generating obrations of C h ( A − mod ) ≥ 0 are lev elwise equal to { 0 } → A or I d A . Denition 3.3. Let A b e a rings, M , N b e t w o A -mo dules. i. Dene T or A ∗ ( M , N ) := H ∗ ( M ⊗ L A N ) . ii. Dene E xt ∗ A ( M , N ) := H ∗ ( RH om A − mod ( M , N )) . iii. Dene the pro jetiv e dimension of M b y: P r oj Dim A ( M ) := inf { n st E xt n +1 A ( M , − ) = { 0 } } . iv. Dene the T or-dimension of M b y: T or Dim A ( M ) := inf { n st ∀ X p − tru ncated T or A i ( M , X ) = { 0 } ∀ i > n + p } R emark 3.4 . The funtor of Dold-Kan orresp ondene is a strong monoidal funtor, as a onsequene the T or dimension an b e omputed with π i instead of H i . Lemma 3.5. L et X b e in H o ( sS et ) and M b e in s Z − mod (r esp sA − mod , for A a ring) 15 ⋄ The obje t X is n -trun ate d if and only if M ap ( ∗ , X ) ⋍ M ap ( S i , X ) ∀ i > n in H o ( sS et ) . ⋄ The obje t M is n -trun ate d if and only if M ap s Z − mod ( Z , M ) (r esp M ap sA − mod ( A, Z ) ) is n -trun ate d in H o ( sS et ) . Pro of F or the rst statemen t, b y2.25 , w e an onsider equiv alen tly the homotopi b ers of this morphism up on M ap ( ∗ , X ) . The b er of M ap ( ∗ , X ) is a p oin t and the b er of M ap ( S i , X ) is M ap sS et/ ∗ ( S i , X ) . As π j M ap sS et/ ∗ ( S i , X ) ⋍ π i + j ( X ) , the equiv alene is lear. F or the seond statemen t, an y ob jet in s Z − mod is an homotopial olimit of free ob jets, i.e. ∀ N ∈ s Z − mod there exists a family of sets ( λ i ) i ∈ I su h that q N ⋍ hocol im I ` λ i Z in H o ( s Z − mod ) . Assume that M ap s Z − mod ( Z , M ) is n -trunated. M ap s Z − mod ( N , M ) ⋍ hol im I Q λ i ( M ap s Z − mod ( Z , M )) , hene is an homotopial limit of n -trunated ob jets. b y i , n -trunated ob jets in sS et are learly stable under homotopial limits.  Lemma 3.6. (f [TV℄) L et u : A → B b e in s Z − mod . The morphism u is at if and only if i. The natur al morphism π ∗ ( A ) ⊗ π 0 ( A ) π 0 ( B ) → π 0 ( B ) is an isomorphism. ii. The morphism π 0 ( u ) is at. In p artiular, if A is  obr ant and n -trun ate d, u at implies B n -trun ate d. R emark 3.7 . [TV ℄ Let A → B b e in Z − al g . The morphism A → B is at if and only if T or Dim A ( B ) = 0 . W e giv e no w the lemmas neessary to the theorem of omparison of the notions of smo othness in rings and relativ e smo othness. Lemma 3.8. L et A → B b e a smo oth morphism of rings. Ther e exists a pushout squar e A ′ / /   B ′   A / / B suh that A ′ → B ′ is a smo oth morphism of no etherian rings. Pro of: This is the ane ase in the orollary 17 . 7 . 9( b ) of [ EGAIV ℄.  Lemma 3.9. L et A → B and A → C b e two morphisms in Z − al g . If B is a p erfe t  omplex of B ⊗ A B mo dules then D := B ⊗ A C is a p erfe t  omplex of D ⊗ C D mo dules. Pro of: P erfet omplexes are learly stable under base  hange. As D ⊗ C D ⋍ B ⊗ A D , the natural morphism D ⊗ C D → D is a pushout of B ⊗ A B → B hene D is a p erfet omplex.  Lemma 3.10. L et A b e a no etherian ring. Every at A -mo dule of nite typ e is pr oje tive. Lemma 3.11. Assume that A is a no etherian ring and  onsider A → B ∈ Z − al g , B of nite typ e. Ther e is an e quivalen e b etwe en i. The ring B is of nite T or-dimension on A . ii. The ring B is of nite pr oje tive dimension on A . The part ii ⇒ i is lear, if B has a nite pro jetiv e resolution 0 → P n → ... → B , then for i ≥ n , T or i +1 ( M , − ) ⋍ T or i − n ( P n +1 , − ) and P n +1 = 0 . Reipro ally , if T or Dim A b < + ∞ , let ... → P n → ... → B b e a free resolution of B . The mo dule P n /im ( P n +1 ) has T or dimension 0 b y previous form ula hene is at b y 3.7 . As A is no etherian and B is of nite t yp e, it is pro jetiv e and w e ha v e a lear nite pro jetiv e resolution.  16 Lemma 3.12. L et u : A → B b e in rings. Assume that A is an algebr ai al ly lose d eld, then ther e is an e quivalen e ⋄ The morphism u is formal ly smo oth in the sense of rings. A ny morphism x : B → A in rings pr ovides A with a strutur e of B -mo dule of nite pr oje tive dimension over B . Lemma 3.13. L et u : A → B b e a nitely pr esente d at morphism in rings. The morphism u is smo oth if and only if for al l algebr ai al ly lose d eld K under A , K → K ⊗ A B is smo oth. Theorem 3.14 . A morphism A → B in Z − al g is smo oth in the sense of rings if and only if i. The ring B is nitely pr esente d in A − al g . ii. The morphism A → B is at. iii. The ring B is a p erfe t  omplex of B ⊗ A B -mo dules. Pro of: Let us no w pro v e the rst part of the theorem. Assume that A → B is smo oth. i and ii are lear. Let us pro v e iii . By 3.8 , as iii is stable under pushout, w e just ha v e to pro v e it for A and B no etherian. Let us pro v e rst that B ⊗ A B → B is of nite T or dimension (hene of nite pro jetiv e dimension b y 3.11 ). Let L b e an algebraially losed eld in A − alg . Set B L := B ⊗ A L . Clearly B ⊗ B ⊗ A B L ⋍ B L ⊗ B L ⊗ L B L L hene omputing the T or dimension of B o v er B ⊗ A B is equiv alen t to ompute the T or dimension of B L o v er B L ⊗ L B L . The morphism L → B L → B L ⊗ L B L is smo oth, b y omp osition of smo oth morphisms, o v er an algebraially losed eld. The ring B L ⊗ L B L is then smo oth on aeld, hene regular. No w, B L is a mo dule of nite t yp e on this regular ring th us it is a p erfet omplex on it. In partiular, it is of nite pro jetiv e dimension hene of nite T or dimension. Finally , B is of nite T or dimension hene of nite pro jetiv e dimension o v er B ⊗ A B . As previously , B of nite t yp e o v er B ⊗ A B . As these rings are no etherian, B is a p erfet omplex. Indeed, B has a nite pro jetiv e resolution b y ( P i ) . Ea h P i is of nite T or dimension hene of nite pro jetiv e dimension. Let us pro v e the seond part of the theorem. Let A → B b e a morphism of rings v erifying i , ii and iii . Let K b e an algebraially losed eld under A . W e will use 3.13 and 3.12 . Let x : B → K b e in Z − al g . The follo wing omm utativ e diagram is an homotopi pushout: B ⊗ K B I d ⊗ K x / /   B ⊗ K K ⋍ B x   B x / / K Th us K has nite pro jetiv e dimension in B − mod . Finally , b y 3.13 , K → B is smo oth in the sense of rings. As it is true for an y K , b y 3.12 , A → B is smo oth in the sense of rings.  Here is no w the omparison theorem. Theorem 3.15 . L et A → B b e a morphism of rings. It is smo oth if and only if it is smo oth in the sense of rings. Pro of The t w o follo wing lemmas, and remark 3.7 pro v e the theorem. Lemma 3.16. [TV ℄ L et A → B b e a morphism in Z − al g . i. if A → B is hf p , then it is nitely pr esente d in Z − al g . ii. if A → B is smo oth and nitely pr esente d, then it is hf p . Lemma 3.17. r eftv L et A → B b e a morphism of rings. The ring B is a p erfe t  omplex of B -mo dules if and only if A → B is hf .  17 3.2 The ategory S et The most diult problem onsists in nding examples of formally smo oth morphisms. The Lemma 2.19 giv es us a  haraterisation of these morphisms in the relativ e on text C = S et . The funtor nerv e and the funtor "fundamen tal group oid" dene a Quillen equiv alene b et w een the ategory sB − m od endo w ed with its 1 -trunated mo del struture and the ategory B − Gpd . Moreo v er, this last ategory is ompatly generated and th us its ltered H ocol im an b e omputer as ltered olimits. Here is the form ula to do this Lemma 3.18. L et I b e a lter e d diagr am and F : I → Gpd . The  olimit of F  onsists of ⋄ On obje ts ( C ol imF ) 0 := C ol im ( f g ◦ F ) wher e f g is the for getful funtor fr om Gpd to S et . ⋄ On morphisms, for ¯ x, ¯ y ∈ C oli m ( f g ◦ F ) r epr esente d by x ∈ F ( i ) and y ∈ F ( i ′ ) . Ther e exists k under i and i ′ suh that H om H ocolim ( F ) ( ¯ x, ¯ y ) := C ol im k/ I ( H om F ( j ) (( l i,j ) ∗ )( x ) , ( l i ′ ,j ) ∗ )( y )) wher e l i,j : i → j and l i ′ ,j : i ′ → j . W e also need to desrib e the deriv ed enri hed Homs. Lemma 3.19. L et B b e a monoid in S et . Ther e is an e quivalen e of  ate gories b etwe en H o ( B − Gpd ) and the  ate gory [ B − Gpd ] whose obje ts ar e B -gr oup oids and morphisms ar e isomorphism lasses of funtors. In p artiular, for two B -gr oup oids G and G ′ , RH om ∆ ≤ 1 B − gpd ( G, G ′ ) ⋍ H om ∆ ≤ 1 [ B − gpd ] ( G, G ′ ) in H o ( Gpd ) , wher e the exp onent ∆ ≤ 1 me ans that the Homs ar e enrihe d on gr oup oids. Lemma 3.20. The  ommutative monoid N is homotopi al ly nitely pr esente d for the 1 -trun ate d mo del strutur e i.e. in the  ate gory ( N × N ) − Gpd . Let N 2 denotes N × N . Let F : J → Gpd b e a funntor from a ltered diagram I to Gpd . W e ha v e to pro v e H ocol im ( H om ∆ ≤ 1 [ N 2 − Gpd ] ( N , F ( − ))) ⋍ H om ∆ ≤ 1 [ N − Gpd ] ( N , H ocol im ( F )) W e let the reader v erify that the follo wing funtor denoted ϕ dene an equiv alene of group oids. Let ¯ H b e in H ocol im ( H om ∆ ≤ 1 [ N 2 − Gpd ] ( N , F ( − ))) represen ted b y H ∈ H om [ N 2 − gpd ] ( N , F ( j )) . W e dene ϕ on ob jets b y ϕ : ¯ H → ˆ H := n → ¯ H ( n ) No w, b y onstrution, an y morphism ¯ η in H ocol im ( H om ∆ ≤ 1 [ N 2 − Gpd ] ( N , F ( − ))) has a represen tan t η : G → G ′ ∈ H om H om [ N 2 − gpd ] ( N ,F ( j )) ( G, G ′ ) . W e dene ϕ on morphisms b y ϕ : ¯ η → ˆ η := n → ¯ η n  Lemma 3.21. The  ommutative gr oup Z is homotopi al ly nitely pr esente d for the 1 -trun ate d mo del strutur e i.e. in the  ate gory ( Z × Z ) − Gpd . Pro of This is the same pro of as previous lemma, replaing N b y Z .  Corollary 3.22. The morphisms F 1 → N and F 1 → Z ar e smo oth. In p artiular, the ane sheme Gl 1 , F 1 ⋍ S pe c ( Z ) , also denote d G m, F 1 in [TV a℄, is smo oth. Pro of They are learly hf p and of T or dimension zero. Their diagonal is hf for the 1 -trunated mo del struture, th us, w e just ha v e to  he k that the diagonal of their ab elianisation is hf in the simpliial graduated ategory giv en in 2.19 . The ab elianisation of N is Z [ X ] and the ab elianisation of Z is Z ( X ) , and the morphisms Z [ X ] ⊗ Z Z [ X ] → Z [ X ] and Z ( X ) ⊗ Z Z ( X ) → Z ( X ) are hf resp etiv ely in s ( Z [ X ] ⊗ Z Z [ X ]) − M od N − gr ad and s ( Z ( X ) ⊗ Z Z ( X )) − M od Z − g r ad .  18 Corollary 3.23. Pour tout n , le shéma Gl n, F 1 est lisse. Pro of This s heme is isomorphi to S pec ( Q E n ` E n Z ) ([TV a℄), where E n is the set of in tegers from 1 to n , th us as opro duts in C omm ( S et ) are pro duts in S et , it is isomorphi to S pec ( Z n 2 ) . The pro dut in set is the tensor pro dut, th us as a nite tensor pro dut of nite olimits of homotopially nitely presen table ob jet, this monoid is homotopially nitely presen table, i.e. F 1 → Z n 2 is a morphism hf p . F or the same reason, the T or dimension is still zero. W e need then to pro v e that a nite tensorisation of the formally smo oth morphism ∗ → Z (in the relativ e sense 1.19 ) b y itself is still formally smo oth. The pushout diagram Z 2 / /   Z   Z k / / Z k − 1 pro v es that Z k → Z k − 1 is hf for an y in teger k and b y omp osition Z 2 k → Z k is hf for an y in teger k . Finaly for ev ery n , F 1 → Z n 2 is smo oth, hene Gl n , F 1 is smo oth.  3.3 Some Others examples If ( C , ⊗ , 1 ) is a symmetri monoidal ategory as desrib ed in the preliminaries, its asso iated ategory of simplial ob jets has simpliial Homs,denoted H om ∆ , and there is an adjuntion s C H om ∆ (1 , − ) / / sS et sK 0 o o where sK 0 (( X n ) n ∈ N ) = ( ` X n 1) n ∈ N . One v eries easily that as 1 is obran t, nitely presen table, and as H om ∆ (1 , − ) preserv es w eak equiv alenes (b y onstrution of the mo del struture on C ), the funtor sK 0 preserv e homotopially nitely presen table ob jets. In partiular, sK 0 preserv es hf morphisms and formally smo oth morphisms. Restriting the adjuntion to the ategories of algebra, where w eak equiv alenes and homotopial ltered olimits are obtained with the forgetful funtor, it is also lear that sK 0 ( u ) preserv es hf p morphisms. W e write then the follo wing prop osition. Prop osition 3.24. L et u : A → B b e a smo oth morphism in C omm ( S et ) , then sK 0 ( u ) is smo oth if and only if sK 0 ( B ) is of nite T or dimension over sK 0 ( A ) . This giv es partiular examples. Indeed, in ev ery on text the ane line orresp ond to the morphism 1 → 1[ X ] := ` N 1 and the s heme G m to the morphism 1 → 1( X ) := ` Z 1 . W e write then the follo wing theorem. Theorem 3.25 . The ane line and the sheme G m ar e smo oth in any  ontext wher e, r esp e tively, 1[ X ] and 1( X ) ar e of nite T or dimension over 1 . This theorem an b e applied in partiular to the on text N − mod . The follo wing lemma pro vides us, in this on text, examples of morphisms of T or-dimension 0 . Lemma 3.26. L et A → B b e in C omm ( N − mod ) suh that B is fr e e over A . The monoid B has T or-dimension 0 over A . Pro of : Let M ∈ A − mod b e a n -trunated mo dule. There exists a set λ su h that B ⋍ ` λ A . Th us B ⊗ L A M ′ ⋍ C oprod λ QM in Q c A − mod where Q, Q c are obran t replaemen t resp etiv ely in Q c A − mod and C omm ( N − mod ) . Th us as this opro dut is a pro dut in set, w e get B ⊗ L A M ′ ⋍ C ol im λ ′ f ini ⊂ λ Q λ ′ QM As funtors π i omm ute with pro duts in sets and ltered olimits, the T or dimenson of B o v er A is zero. Theorem 3.27 . Examples in N − mod . ⋄ The ane line in N − mod , A 1 N , is smo oth. ⋄ The sheme G m, N r elative to N − mod is smo oth. 19 W e onlude with a last theorem Theorem 3.28 . L et C b e a r elative  ontext in the sense of [ M ℄ and A → B b e a Zariski op en immersion in C omm ( C ) , with A  obr ant in C omm ( C ) and B  obr ant in A − al g . The morphism A → B is smo oth. Pro of A Zariski op en immersion is alw a ys formally smo oth, its diagonal is ev en an isomorphism. Th us w e will need to pro v e that it is hf p and of T or dimension zero. First, if there exists f ∈ A 0 ,an ob jet of the underlying set of A , su h that B ⋍ A f , the result is lear. Indeed, A f is giv en b y a ltered olimit of A th us is of T or dimension zero. Let us pro v e that it is hf p . It is lear that A → A [ X ] is homotopially nitely presen ted, then as ev erything is obran t, w e an write A f as a nite olimit of A [ X ] ([M ℄) whi h is in fats a nite homotopial olimit and th us nally A → A f is hf p . No w if B dene a Zariski op en ob jet of A , w e an write B it as a ok ernel of pro duts of A f . As funtors π i omm ute with pro duts, the pro duts preserv e w eak equiv alenes and it is then lear that A → B is hf p . F or the T or dimension, reall that there is a nite family of funtor reeting isomorphisms B − mod → A f − mod . Let M b e a p -trunated A -mo dule. This family sends M ⊗ L A B and its n -trunations, n > p to the same mo dule QM f (Q is the obran t replaemen t of A − mod ) th us learly M ⊗ L A B is p trunated and T or Dim A ( B ) = 0 .  Referenes [A℄ V. Angeltv eit - Enri hed Reedy Categories - Pr o  e e dings of the A meri an Mathemati al So iety , V ol. 136, n um 7, july 2008, pp 2323-2332. [B℄ K. S. Bro wn - Cohomology of groups - Graduate texts in mathematis, 87 - Springer-V erlag, New Y ork-Berlin , 1982. x+308 pp. [B℄ F. Boreux - Handb o ok of Categorial Algebra I I - Cambridge University Pr ess 1994 - 443 pp. [EGAIV℄ A. Grothendie k - Elémen ts de géométrie algébrique IV - étude lo ale des s hémas et des morphismes de s hémas, partie IV - Inst. Hautes études si. Publ. Math. , n um 32, 1967, 361pp. [H℄ M. Ho v ey - Mo del Categories - Mathematial Surv eys and Monographs, 63 - A meri an Mathemati al So iety, Pr oviden e, RI, 1999. xii+209pp. [J℄ J.F. Jardine - Diagrams and T orsors - K-the ory 37 (2006) n um 3 pp 291-309 [GJ℄ P . Go erss and J.F. Jardine - Simpliial Homotopy The ory. Progr. Math. 174, birk auser, 1999. [M℄ F. Mart y - Relativ e zariski Op en Morphisms - Pré-publiation math/ [ML℄ S. Ma Lane - Categories for the w orking mathematiian - Graduate text in mathematis, 5 - Springer-V erlag, New Y ork-Berlin , 1971. ix+262pp. [Q℄ D. Quillen - On the (Co)-homology of Comm utativ e Rings - Appli ations of  ate gori al A lgebr a, Pr o . of the Symp osium in Pur e Mathematis , 1968, New Y ork - AMS, 1970. [R℄ C. Rezk - Ev ery Homotop y Theory of Simpliial Algebras A dmits a Prop er Mo del - prepubliation math/0003065 . [T1℄ B. T o ën - Champs Anes - Sele ta mathemati a, New Series , 12, 2006, pp 39-135 [T2℄ B. T o ën - Deriv ed algebrai geometry . [TV a℄ B.T o ën, M. V aquié - Under Sp e(Z) - pré-publiation math/0509684 . [TV℄ B. T o ën, G. V ezzozi - Homotopial Algebrai Geometry I I: Geometri Sta ks and Appliations - Memoirs of the A meri an Mathemati al So iety - V ol 193, 2008, 230pp. 20