LIMITS OF ALGEBRAIC STACKS

Transcription

1 LIMITS OF ALGEBRAIC STACKS 0CMM Contents 1. Introducton 1 2. Conventons 1 3. Morphsms of fnte presentaton 1 4. Descendng propertes 6 5. Descendng relatve objects 6 6. Fnte type closed n fnte presentaton 7 7. Other chapters 10 References 11 0CMN 0CMP 1. Introducton In ths chapter we put materal related to lmts of algebrac stacks. Many results on lmts of algebrac stacks and algebrac spaces have been obtaned by Davd Rydh n [Ryd08]. 2. Conventons We contnue to use the conventons and the abuse of language ntroduced n Propertes of Stacks, Secton Morphsms of fnte presentaton 0CMQ Ths secton s the analogue of Lmts of Spaces, Secton 3. There we defned what t means for a transformaton of functors on Sch to be lmt preservng (we suggest lookng at the characterzaton n Lmts of Spaces, Lemma 3.2). In Crtera for Representablty, Secton 5 we defned the noton lmt preservng on objects. Recall that n Artn s Axoms, Secton 11 we have defned what t means for a category fbred n groupods over Sch to be lmt preservng. Combnng these we get the followng noton. 0CMR Defnton 3.1. Let S be a scheme. Let f : X Y be a 1-morphsm of categores fbred n groupods over (Sch/S) fppf. We say f s lmt preservng f for every drected lmt U = lm U of affne schemes over S the dagram colm X U X U of fbre categores s 2-cartesan. f colm Y U f Y U Ths s a chapter of the Stacks Project, verson 4decab0f, compled on Oct 15,

2 LIMITS OF ALGEBRAIC STACKS 2 0CMS 0CMT Lemma 3.2. Let S be a scheme. Let f : X Y be a 1-morphsm of categores fbred n groupods over (Sch/S) fppf. If f s lmt preservng (Defnton 3.1), then f s lmt preservng on objects (Crtera for Representablty, Secton 5). Proof. If for every drected lmt U = lm U of affne schemes over U, the functor colm X U (colm Y U ) YU X U s essentally surjectve, then f s lmt preservng on objects. Lemma 3.3. Let p : X Y and q : Z Y be 1-morphsms of categores fbred n groupods over (Sch/S) fppf. If p : X Y s lmt preservng, then so s the base change p : X Y Z Z of p by q. Proof. Ths s formal. Let U = lm I U be the drected lmt of affne schemes U over S. For each we have (X Y Z) U = X U YU Z U Fltered colmts commute wth 2-fbre products of categores (detals omtted) hence f p s lmt preservng we get colm(x Y Z) U = colm X U colm YU colm Z U = X U YU colm Y U colm YU colm Z U = X U YU colm Z U = X U YU Z U ZU colm Z U 0CMU as desred. = (X Y Z) U ZU colm Z U Lemma 3.4. Let p : X Y and q : Y Z be 1-morphsms of categores fbred n groupods over (Sch/S) fppf. If p and q are lmt preservng, then so s the composton q p. Proof. Ths s formal. Let U = lm I U be the drected lmt of affne schemes U over S. If p and q are lmt preservng we get colm X U = X U YU colm Y U = X U YU Y U ZU colm Z U 0CMV as desred. = X U ZU colm Z U Lemma 3.5. Let p : X Y be a 1-morphsm of categores fbred n groupods over (Sch/S) fppf. If p s representable by algebrac spaces, then the followng are equvalent: (1) p s lmt preservng, (2) p s lmt preservng on objects, and (3) p s locally of fnte presentaton (see Algebrac Stacks, Defnton 10.1). Proof. In Crtera for Representablty, Lemma 5.3 we have seen that (2) and (3) are equvalent. Thus t suffces to show that (1) and (2) are equvalent. One drecton we saw n Lemma 3.2. For the other drecton, let U = lm I U be the drected lmt of affne schemes U over S. We have to show that colm X U X U YU colm Y U

3 LIMITS OF ALGEBRAIC STACKS 3 s an equvalence. Snce we are assumng (2) we know that t s essentally surjectve. Hence we need to prove t s fully fathful. Snce p s fathful on fbre categores (Algebrac Stacks, Lemma 9.2) we see that the functor s fathful. Let x and x be objects n the fbre category of X over U. The functor above sends x to (x U, p(x ), can) where can s the canoncal somorphsm p(x U ) p(x ) U. Thus we assume gven a morphsm (α, β ) : (x U, p(x ), can) (x U, p(x ), can) n the category of the rght hand sde of the frst dsplayed arrow of ths proof. Our task s to produce an and a morphsm x U x U whch maps to (α, β U ). Set y = p(x ) and y = p(x ). By (Algebrac Stacks, Lemma 9.2) the functor X y : (Sch/U ) opp Sets, V/U {(x, φ) x Ob(X V ), φ : f(x) y V }/ = s an algebrac space over U and the same s true for the analogously defned functor X y. Snce (2) s equvalent to (3) we see that X y s locally of fnte presentaton over U. Observe that (x, d) and (x, d) defne U -valued ponts of X y and X y. There s a transformaton of functors β : X y X y, (x/v, φ) (x/v, β V φ) n other words, ths s a morphsm of algebrac spaces over U. We clam that U U (x,d) U (x,d) X y β X y commutes. Namely, ths s equvalent to the condton that the pars (x U, β U ) and (x U, d) as n the defnton of the functor X y are somorphc. And the morphsm α : x U x U exactly produces such an somorphsm. Argung backwards the reader sees that f we can fnd an such that the dagram U U (x,d) U (x,d) X y β X y commutes, then we obtan an somorphsm x U x U whch s a soluton to the problem posed n the precedng paragraph. However, the dagonal morphsm : X y X y U X y s locally of fnte presentaton (Morphsms of Spaces, Lemma 28.10) hence the fact that U U equalzes the two morphsms to X y, means that for some the morphsm U U equalzes the two morphsms, see Lmts of Spaces, Proposton CMW Lemma 3.6. Let p : X Y be a 1-morphsm of categores fbred n groupods over (Sch/S) fppf. The followng are equvalent (1) the dagonal : X X Y X s lmt preservng, and

4 LIMITS OF ALGEBRAIC STACKS 4 (2) for every drected lmt U = lm U of affne schemes over S the functor s fully fathful. colm X U X U YU colm Y U In partcular, f p s lmt preservng, then s too. Proof. Let U = lm U be a drected lmt of affne schemes over S. We clam that the functor colm X U X U YU colm Y U s fully fathful f and only f the functor colm X U X U (X Y X ) U colm(x Y X ) U s an equvalence. Ths wll prove the lemma. Snce (X Y X ) U = X U YU X U and (X Y X ) U = X U YU X U ths s a purely category theoretc asserton whch we dscuss n the next paragraph. Let I be a fltered ndex category. Let (C ) and (D ) be systems of groupods over I. Let p : (C ) (D ) be a map of systems of groupods over I. Suppose we have a functor p : C D of groupods and functors f : colm C C and g : colm D D fttng nto a commutatve dagram Then we clam that colm C p colm D s fully fathful f and only f the functor f g C D A : colm C C D colm D B : colm C C,C D C,f gf colm(c D C ) s an equvalence. Set C = colm C and D = colm D. Snce 2-fbre products commute wth fltered colmts we see that A and B become the functors A : C C D D and B : C C,C D C,f gf (C D C ) Thus t suffces to prove that f C p D f g s a commutatve dagram of groupods, then A s fully fathful f and only f B s an equvalence. Ths follows from Categores, Lemma 34.9 (wth trval,.e., punctual, base category) because C D p p 0CMX Ths fnshes the proof. C,C D C,f gf (C D C ) = C A,C D D,A C Lemma 3.7. Let S be a scheme. Let X be an algebrac stack over S. If X S s locally of fnte presentaton, then X s lmt preservng n the sense of Artn s Axoms, Defnton 11.1 (equvalently: the morphsm X S s lmt preservng).

5 LIMITS OF ALGEBRAIC STACKS 5 0CMY Proof. Choose a surjectve smooth morphsm U X for some scheme U. Then U S s locally of fnte presentaton, see Morphsms of Stacks, Secton 26. We can wrte X = [U/R] for some smooth groupod n algebrac spaces (U, R, s, t, c), see Algebrac Stacks, Lemma Snce U s locally of fnte presentaton over S t follows that the algebrac space R s locally of fnte presentaton over S. Recall that [U/R] s the stack n groupods over (Sch/S) fppf obtaned by stackyfyng the category fbred n groupods whose fbre category over T s the groupod (U(T ), R(T ), s, t, c). Snce U and R are lmt preservng as functors (Lmts of Spaces, Proposton 3.8) ths category fbred n groupods s lmt preservng. Thus t suffces to show that fppf stackyfcaton preserves the property of beng lmt preservng. Ths s true (hnt: use Topologes, Lemma 13.2). However, we gve a drect proof below usng that n ths case we know what the stackyfcaton amounts to. Let T = lm T λ be a drected lmt of affne schemes over S. We have to show that the functor colm[u/r] Tλ [U/R] T s an equvalence of categores. Let us show ths functor s essentally surjectve. Let x Ob([U/R] T ). In Groupods n Spaces, Lemma 23.1 the reader fnds a descrpton of the category [U/R] T. In partcular x corresponds to an fppf coverng {T T } I and a [U/R]-descent datum (u, r j ) relatve to ths coverng. After refnng ths coverng we may assume t s a standard fppf coverng of the affne scheme T. By Topologes, Lemma 13.2 we may choose a λ and a standard fppf coverng {T λ, T λ } I whose base change to T s equal to {T T } I. For each, after ncreasng λ, we can fnd a u λ, : T λ, U whose composton wth T T λ, s the gven morphsm u (ths s where we use that U s lmt preservng). Smlarly, for each, j, after ncreasng λ, we can fnd a r λ,j : T λ, Tλ T λ,j R whose composton wth T j T λ,j s the gven morphsm r j (ths s where we use that R s lmt preservng). After ncreasng λ we can further assume that s r λ,j = u λ, pr 0 and t r λ,j = u λ,j pr 1, and c (r λ,jk pr 12, r λ,j pr 01 ) = r λ,k pr 02. In other words, we may assume that (u λ,, r λ,j ) s a [U/R]-descent datum relatve to the coverng {T λ, T λ } I. Then we obtan a correspondng object of [U/R] over T λ whose pullback to T s somorphc to x as desred. The proof of fully fathfulness works n exactly the same way usng the descrpton of morphsms n the fbre categores of [U/T ] gven n Groupods n Spaces, Lemma Proposton 3.8. Let f : X Y be a morphsm of algebrac stacks. The followng are equvalent (1) f s lmt preservng, (2) f s lmt preservng on objects, and (3) f s locally of fnte presentaton. Proof. Assume (3). Let T = lm T be a drected lmt of affne schemes. Consder the functor colm X T X T YT colm Y T Let (x, y, β) be an object on the rght hand sde,.e., x Ob(X T ), y Ob(Y T ), and β : f(x) y T n Y T. Then we can consder (x, y, β) as an object of the algebrac stack X y = X Y,y T over T. Snce X y T s locally of fnte presentaton Ths s a specal case of [EG15, Lemma ]

6 LIMITS OF ALGEBRAIC STACKS 6 (as a base change of f) we see that t s lmt preservng by Lemma 3.7. Ths means that (x, y, β) comes from an object over T for some and unwndng the defntons we fnd that (x, y, β) s n the essental mage of the dsplayed functor. In other words, the dsplayed functor s essentally surjectve. Another formulaton s that ths means f s lmt preservng on objects. Now we apply ths to the dagonal of f. Namely, by Morphsms of Stacks, Lemma 26.7 the morphsm s locally of fnte presentaton. Thus the argument above shows that s lmt preservng on objects. By Lemma 3.5 ths mples that s lmt preservng. By Lemma 3.6 we conclude that the dsplayed functor above s fully fathful. Thus t s an equvalence (as we already proved essental surjectvty) and we conclude that (1) holds. The mplcaton (1) (2) s trval. Assume (2). Choose a scheme V and a surjectve smooth morphsm V Y. By Crtera for Representablty, Lemma 5.1 the base change X Y V V s lmt preservng on objects. Choose a scheme U and a surjectve smooth morphsm U X Y V. Snce a smooth morphsm s locally of fnte presentaton, we see that U X Y V s lmt preservng (frst part of the proof). By Crtera for Representablty, Lemma 5.2 we fnd that the composton U V s lmt preservng on objects. We conclude that U V s locally of fnte presentaton, see Crtera for Representablty, Lemma 5.3. Ths s exactly the condton that f s locally of fnte presentaton, see Morphsms of Stacks, Defnton Descendng propertes 0CPX Ths secton s the analogue of Lmts, Secton 4. 0CPY 0CPZ Stuaton 4.1. Let Y = lm I Y be a lmt of a drected system of algebrac spaces wth affne transton morphsms. We assume that X s quas-compact and quas-separated for all I. We also choose an element 0 I. Lemma 4.2. In Stuaton 4.1 assume that X 0 Y 0 s a morphsm from algebrac stack to Y 0. Assume X 0 s quas-compact and quas-separated. If Y Y0 X 0 Y s separated, then Y Y0 X 0 Y s separated for all suffcently large I. Proof. Wrte X = Y Y0 X 0 and X = Y Y0 X 0. Choose an affne scheme U 0 and a surjectve smooth morphsm U 0 X 0. Set U = Y Y0 U 0 and U = Y Y0 U 0. Then U and U are affne and U X and U X are smooth and surjectve. Set R 0 = U 0 X0 U 0. Set R = Y Y0 R 0 and R = Y Y0 R 0. Then R = U X U and R = U X U. Wth ths notaton note that X Y s separated mples that R U Y U s proper as the base change of X X Y X by U Y U X Y X. Conversely, we see that X Y s separated f R U Y U s proper because U Y U X Y X s surjectve and smooth, see Propertes of Stacks, Lemma 3.3. Observe that R 0 U 0 Y0 U 0 s locally of fnte type and that R 0 s quas-compact and quas-separated. By Lmts of Spaces, Lemma 6.13 we see that R U Y U s proper for large enough whch fnshes the proof. 5. Descendng relatve objects 0CN3 Ths secton s the analogue of Lmts of Spaces, Secton 7. 0CN4 Lemma 5.1. Let I be a drected set. Let (X, f ) be an nverse system of algebrac spaces over I. Assume

7 LIMITS OF ALGEBRAIC STACKS 7 (1) the morphsms f : X X are affne, (2) the spaces X are quas-compact and quas-separated. Let X = lm X. If X s an algebrac stack of fnte presentaton over X, then there exsts an I and an algebrac stack X of fnte presentaton over X wth X = X X X as algebrac stacks over X. Proof. By Morphsms of Stacks, Defnton 26.1 the morphsm X X s quascompact, locally of fnte presentaton, and quas-separated. Snce X s quascompact and X X s quas-compact, we see that X s quas-compact (Morphsms of Stacks, Defnton 7.2). Hence we can fnd an affne scheme U and a surjectve smooth morphsm U X (Propertes of Stacks, Lemma 6.2). Set R = U X U. We obtan a smooth groupod n algebrac spaces (U, R, s, t, c) over X such that X = [U/R], see Algebrac Stacks, Lemma Snce X X s quas-separated and X s quas-separated we see that X s quas-separated (Morphsms of Stacks, Lemma 4.10). Thus R U U s quas-compact and quas-separated (Morphsms of Stacks, Lemma 4.7) and hence R s a quas-separated and quas-compact algebrac space. On the other hand U X s locally of fnte presentaton and hence also R X s locally of fnte presentaton (because s : R U s smooth hence locally of fnte presentaton). Thus (U, R, s, t, c) s a groupod object n the category of algebrac spaces whch are of fnte presentaton over X. By Lmts of Spaces, Lemma 7.1 there exsts an and a groupod n algebrac spaces (U, R, s, t, c ) over X whose pullback to X s somorphc to (U, R, s, t, c). After ncreasng we may assume that s and t are smooth, see Lmts of Spaces, Lemma 6.3. The quotent stack X = [U /R ] s an algebrac stack (Algebrac Stacks, Theorem 17.3). There s a morphsm [U/R] [U /R ], see Groupods n Spaces, Lemma We clam that combned wth the morphsms [U/R] X and [U /R ] X (Groupods n Spaces, Lemma 19.2) we obtan an somorphsm (.e., equvalence) The correspondng map [U/R] [U /R ] X X [U/ p R] [U / p R ] X X on the level of presheaves of groupods as n Groupods n Spaces, Equaton (19.0.1) s an somorphsm. Thus the clam follows from the fact that stackfcaton commutes wth fbre products, see Stacks, Lemma Fnte type closed n fnte presentaton 0CQ0 Ths secton s the analogue of Lmts of Spaces, Secton 11. 0CQ1 Lemma 6.1. Let f : X Y be a morphsm from an algebrac stack to an algebrac space. Assume: (1) f s of fnte type and quas-separated, (2) Y s quas-compact and quas-separated. Then there exsts a morphsm of fnte presentaton f : X Y and a closed mmerson X X of algebrac stacks over Y. Proof. Wrte Y = lm I Y as a lmt of algebrac spaces over a drected set I wth affne transton morphsms and wth Y Noetheran, see Lmts of Spaces, Proposton 8.1. We wll use the materal from Lmts of Spaces, Secton 22.

8 LIMITS OF ALGEBRAIC STACKS 8 Choose a presentaton X = [U/R]. Denote (U, R, s, t, c, e, ) the correspondng groupod n algebrac spaces over Y. We may and do assume U s affne. Then U, R, R s,u,t R are quas-separated algebrac spaces of fnte type over Y. We have two morpsms s, t : R U, three morphsms c : R s,u,t R R, pr 1 : R s,u,t R R, pr 2 : R s,u,t R R, a morphsm e : U R, and fnally a morphsm : R R. These morphsms satsfy a lst of axoms whch are detaled n Groupods, Secton 13. Accordng to Lmts of Spaces, Remark 22.5 we can fnd an 0 I and nverse systems (1) (U ) 0, (2) (R ) 0, (3) (T ) 0 over (Y ) 0 such that U = lm 0 U, R = lm 0 R, and R s,u,t R = lm 0 T and such that there exst morphsms of systems (1) (s ) 0 : (R ) 0 (U ) 0, (2) (t ) 0 : (R ) 0 (U ) 0, (3) (c ) 0 : (T ) 0 (R ) 0, (4) (p ) 0 : (T ) 0 (R ) 0, (5) (q ) 0 : (T ) 0 (R ) 0, (6) (e ) 0 : (U ) 0 (R ) 0, (7) ( ) 0 : (R ) 0 (R ) 0 wth s = lm 0 s, t = lm 0 t, c = lm 0 c, pr 1 = lm 0 p, pr 2 = lm 0 q, e = lm 0 e, and = lm 0. By Lmts of Spaces, Lemma 22.7 we see that we may assume that s and t are smooth (ths may requre ncreasng 0 ). By Lmts of Spaces, Lemma 22.6 we may assume that the maps R U U,s R gven by s and R R and R U U,t R gven by t and R R are somorphsms for all 0. By Lmts of Spaces, Lemma 22.9 we see that we may assume that the dagrams T q R p t R s U are cartesan. The unqueness of Lmts of Spaces, Lemma 22.4 then guarantees that for a suffcently large the relatons between the morphsms s, t, c, e, mentoned above are satsfed by s, t, c, e,. Fx such an. It follows that (U, R, s, t, c, e, ) s a smooth groupod n algebrac spaces over Y. Hence X = [U /R ] s an algebrac stack (Algebrac Stacks, Theorem 17.3). The morphsm of groupods (U, R, s, t, c, e, ) (U, R, s, t, c, e, ) over Y Y determnes a commutatve dagram X X Y Y

9 LIMITS OF ALGEBRAIC STACKS 9 (Groupods n Spaces, Lemma 20.1). We clam that the morphsm X Y Y X s a closed mmerson. The clam fnshes the proof because the algebrac stack X Y s of fnte presentaton by constructon. To prove the clam, note that the left dagram U U U Y Y U 0CQ2 X X X Y Y X s cartesan by Groupods n Spaces, Lemma 24.3 and the results mentoned above. Hence the rght commutatve dagram s cartesan too. Then the desred result follows from the fact that U Y Y U s a closed mmerson by constructon of the nverse system (U ) n Lmts of Spaces, Lemma 22.3, the fact that Y Y U Y Y X s smooth and surjectve, and Propertes of Stacks, Lemma 9.4. There s a verson for separated algebrac stacks. Lemma 6.2. Let f : X Y be a morphsm from an algebrac stack to an algebrac space. Assume: (1) f s of fnte type and separated, (2) Y s quas-compact and quas-separated. Then there exsts a separated morphsm of fnte presentaton f : X Y and a closed mmerson X X of algebrac stacks over Y. Proof. Frst we use exactly the same procedure as n the proof of Lemma 6.1 (and we borrow ts notaton) to construct the embeddng X X as a morphsm X X = Y Y X wth X = [U /R ]. Thus t s enough to show that X Y s separated for suffcently large. In other words, t s enough to show that X X Y X s proper for suffcently large. Snce the morphsm U Y U X Y X s surjectve and smooth and snce R = X X Y X U Y U t s enough to show that the morphsm (s, t ) : R U Y U s proper for suffcently large, see Propertes of Stacks, Lemma 3.3. We prove ths n the next paragraph. Observe that U Y U Y s quas-separated and of fnte type. Hence we can use the constructon of Lmts of Spaces, Remark 22.5 to fnd an 1 I and an nverse system (V ) 1 wth U Y U = lm 1 V. By Lmts of Spaces, Lemma 22.9 for suffcently large the functoralty of the constructon appled to the projectons U Y U U gves closed mmersons V U Y U (There s a small msmatch here because n truth we should replace Y by the scheme theoretc mage of Y Y, but clearly ths does not change the fbre product.) On the other hand, by Lmts of Spaces, Lemma 22.8 the functoralty appled to the proper morphsm (s, t) : R U Y U (here we use that X s separated) leads to morphsms R V whch are proper for large enough. Composng these morphsms we obtan a proper morphsms R U Y U for all large enough. The functoralty of the constructon of Lmts of Spaces, Remark 22.5 shows that ths s the morphsm s the same as (s, t ) for large enough and the proof s complete.

10 LIMITS OF ALGEBRAIC STACKS Other chapters Prelmnares (1) Introducton (2) Conventons (3) Set Theory (4) Categores (5) Topology (6) Sheaves on Spaces (7) Stes and Sheaves (8) Stacks (9) Felds (10) Commutatve Algebra (11) Brauer Groups (12) Homologcal Algebra (13) Derved Categores (14) Smplcal Methods (15) More on Algebra (16) Smoothng Rng Maps (17) Sheaves of Modules (18) Modules on Stes (19) Injectves (20) Cohomology of Sheaves (21) Cohomology on Stes (22) Dfferental Graded Algebra (23) Dvded Power Algebra (24) Hypercoverngs Schemes (25) Schemes (26) Constructons of Schemes (27) Propertes of Schemes (28) Morphsms of Schemes (29) Cohomology of Schemes (30) Dvsors (31) Lmts of Schemes (32) Varetes (33) Topologes on Schemes (34) Descent (35) Derved Categores of Schemes (36) More on Morphsms (37) More on Flatness (38) Groupod Schemes (39) More on Groupod Schemes (40) Étale Morphsms of Schemes Topcs n Scheme Theory (41) Chow Homology (42) Intersecton Theory (43) Pcard Schemes of Curves (44) Adequate Modules (45) Dualzng Complexes (46) Dualty for Schemes (47) Dscrmnants and Dfferents (48) Local Cohomology (49) Algebrac and Formal Geometry (50) Algebrac Curves (51) Resoluton of Surfaces (52) Semstable Reducton (53) Fundamental Groups of Schemes (54) Étale Cohomology (55) Crystallne Cohomology (56) Pro-étale Cohomology Algebrac Spaces (57) Algebrac Spaces (58) Propertes of Algebrac Spaces (59) Morphsms of Algebrac Spaces (60) Decent Algebrac Spaces (61) Cohomology of Algebrac Spaces (62) Lmts of Algebrac Spaces (63) Dvsors on Algebrac Spaces (64) Algebrac Spaces over Felds (65) Topologes on Algebrac Spaces (66) Descent and Algebrac Spaces (67) Derved Categores of Spaces (68) More on Morphsms of Spaces (69) Flatness on Algebrac Spaces (70) Groupods n Algebrac Spaces (71) More on Groupods n Spaces (72) Bootstrap (73) Pushouts of Algebrac Spaces Topcs n Geometry (74) Chow Groups of Spaces (75) Quotents of Groupods (76) More on Cohomology of Spaces (77) Smplcal Spaces (78) Dualty for Spaces (79) Formal Algebrac Spaces (80) Restrcted Power Seres (81) Resoluton of Surfaces Revsted Deformaton Theory (82) Formal Deformaton Theory (83) Deformaton Theory (84) The Cotangent Complex

11 LIMITS OF ALGEBRAIC STACKS 11 (85) Deformaton Problems Algebrac Stacks (86) Algebrac Stacks (87) Examples of Stacks (88) Sheaves on Algebrac Stacks (89) Crtera for Representablty (90) Artn s Axoms (91) Quot and Hlbert Spaces (92) Propertes of Algebrac Stacks (93) Morphsms of Algebrac Stacks (94) Lmts of Algebrac Stacks (95) Cohomology of Algebrac Stacks (96) Derved Categores of Stacks (97) Introducng Algebrac Stacks (98) More on Morphsms of Stacks (99) The Geometry of Stacks Topcs n Modul Theory (100) Modul Stacks (101) Modul of Curves Mscellany (102) Examples (103) Exercses (104) Gude to Lterature (105) Desrables (106) Codng Style (107) Obsolete (108) GNU Free Documentaton Lcense (109) Auto Generated Index References [EG15] Matthew Emerton and Toby Gee, scheme-theoretc mages of morphsms of stacks. [Ryd08] Davd Rydh, Noetheran approxmaton of algebrac spaces and stacks, math.ag/ (2008).