LIMITS OF ALGEBRAIC STACKS
|
|
- Denis Lloyd
- 5 years ago
- Views:
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).
Lecture 7: Gluing prevarieties; products
Lecture 7: Glung prevaretes; products 1 The category of algebrac prevaretes Proposton 1. Let (f,ϕ) : (X,O X ) (Y,O Y ) be a morphsm of algebrac prevaretes. If U X and V Y are affne open subvaretes wth
More informationPOL VAN HOFTEN (NOTES BY JAMES NEWTON)
INTEGRAL P -ADIC HODGE THEORY, TALK 2 (PERFECTOID RINGS, A nf AND THE PRO-ÉTALE SITE) POL VAN HOFTEN (NOTES BY JAMES NEWTON) 1. Wtt vectors, A nf and ntegral perfectod rngs The frst part of the talk wll
More informationDescent is a technique which allows construction of a global object from local data.
Descent Étale topology Descent s a technque whch allows constructon of a global object from local data. Example 1. Take X = S 1 and Y = S 1. Consder the two-sheeted coverng map φ: X Y z z 2. Ths wraps
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationwhere a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets
5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory
More informationDIFFERENTIAL FORMS BRIAN OSSERMAN
DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationwhere a is any ideal of R. Lemma Let R be a ring. Then X = Spec R is a topological space. Moreover the open sets
11. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationThe Pseudoblocks of Endomorphism Algebras
Internatonal Mathematcal Forum, 4, 009, no. 48, 363-368 The Pseudoblocks of Endomorphsm Algebras Ahmed A. Khammash Department of Mathematcal Scences, Umm Al-Qura Unversty P.O.Box 796, Makkah, Saud Araba
More informationDIFFERENTIAL SCHEMES
DIFFERENTIAL SCHEMES RAYMOND T. HOOBLER Dedcated to the memory o Jerry Kovacc 1. schemes All rngs contan Q and are commutatve. We x a d erental rng A throughout ths secton. 1.1. The topologcal space. Let
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More information= s j Ui U j. i, j, then s F(U) with s Ui F(U) G(U) F(V ) G(V )
1 Lecture 2 Recap Last tme we talked about presheaves and sheaves. Preshea: F on a topologcal space X, wth groups (resp. rngs, sets, etc.) F(U) or each open set U X, wth restrcton homs ρ UV : F(U) F(V
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More information20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.
20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationALGEBRA SCHEMES AND THEIR REPRESENTATIONS
ALGEBRA SCHEMES AND THEIR REPRESENTATIONS AMELIA ÁLVAREZ, CARLOS SANCHO, AND PEDRO SANCHO Introducton The equvalence (Carter dualty) between the category of topologcally flat formal k-groups and the category
More informationALGEBRA SCHEMES AND THEIR REPRESENTATIONS
ALGEBRA SCHEMES AND THEIR REPRESENTATIONS AMELIA ÁLVAREZ, CARLOS SANCHO, AND PEDRO SANCHO Introducton The equvalence (Carter dualty) between the category of topologcally flat formal k-groups and the category
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationINTERSECTION THEORY CLASS 13
INTERSECTION THEORY CLASS 13 RAVI VAKIL CONTENTS 1. Where we are: Segre classes of vector bundles, and Segre classes of cones 1 2. The normal cone, and the Segre class of a subvarety 3 3. Segre classes
More informationREAL ANALYSIS I HOMEWORK 1
REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More informationCOMPLETING PERFECT COMPLEXES
COMPLETING PERFECT COMPLEXES HENNING KRAUSE, WITH APPENDICES BY TOBIAS BARTHEL AND BERNHARD KELLER Dedcated to the memory of Ragnar-Olaf Buchwetz. Abstract. Ths note proposes a new method to complete a
More informationFINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There
More informationMATH CLASS 27. Contents
MATH 6280 - CLASS 27 Contents 1. Reduced and relatve homology and cohomology 1 2. Elenberg-Steenrod Axoms 2 2.1. Axoms for unreduced homology 2 2.2. Axoms for reduced homology 4 2.3. Axoms for cohomology
More informationarxiv: v4 [math.ag] 3 Sep 2016
BLUE SCHEMES, SEMIRIG SCHEMES, AD RELATIVE SCHEMES AFTER TOË AD VAQUIÉ OLIVER LORSCHEID arxv:1212.3261v4 [math.ag] 3 Sep 2016 ABSTRACT. It s a classcal nsght that the Yoneda embeddng defnes an equvalence
More informationOn functors between module categories for associative algebras and for N-graded vertex algebras
On functors between module categores for assocatve algebras and for N-graded vertex algebras Y-Zh Huang and Jnwe Yang Abstract We prove that the weak assocatvty for modules for vertex algebras are equvalent
More informationÉTALE COHOMOLOGY. Contents
ÉTALE COHOMOLOGY GEUNHO GIM Abstract. Ths note s based on the 3-hour presentaton gven n the student semnar on Wnter 2014. We wll bascally follow [MlEC, Chapter I,II,III,V] and [MlLEC, Sectons 1 14]. Contents
More informationTHE CARTIER ISOMORPHISM. These are the detailed notes for a talk I gave at the Kleine AG 1 in April Frobenius
THE CARTIER ISOMORPHISM LARS KINDLER Tese are te detaled notes for a talk I gave at te Klene AG 1 n Aprl 2010. 1. Frobenus Defnton 1.1. Let f : S be a morpsm of scemes and p a prme. We say tat S s of caracterstc
More informationRestricted Lie Algebras. Jared Warner
Restrcted Le Algebras Jared Warner 1. Defntons and Examples Defnton 1.1. Let k be a feld of characterstc p. A restrcted Le algebra (g, ( ) [p] ) s a Le algebra g over k and a map ( ) [p] : g g called
More informationLECTURE V. 1. More on the Chinese Remainder Theorem We begin by recalling this theorem, proven in the preceeding lecture.
LECTURE V EDWIN SPARK 1. More on the Chnese Remander Theorem We begn by recallng ths theorem, proven n the preceedng lecture. Theorem 1.1 (Chnese Remander Theorem). Let R be a rng wth deals I 1, I 2,...,
More information( 1) i [ d i ]. The claim is that this defines a chain complex. The signs have been inserted into the definition to make this work out.
Mon, Apr. 2 We wsh to specfy a homomorphsm @ n : C n ()! C n (). Snce C n () s a free abelan group, the homomorphsm @ n s completely specfed by ts value on each generator, namely each n-smplex. There are
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES. 1. Introduction
ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES TAKASHI ITOH AND MASARU NAGISA Abstract We descrbe the Haagerup tensor product l h l and the extended Haagerup tensor product l eh l n terms of
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationOn the smoothness and the totally strong properties for nearness frames
Int. Sc. Technol. J. Namba Vol 1, Issue 1, 2013 On the smoothness and the totally strong propertes for nearness frames Martn. M. Mugoch Department of Mathematcs, Unversty of Namba 340 Mandume Ndemufayo
More informationA Brown representability theorem via coherent functors
Topology 41 (2002) 853 861 www.elsever.com/locate/top A Brown representablty theorem va coherent functors Hennng Krause Fakultat fur Mathematk, Unverstat Belefeld, Postfach 100131, 33501 Belefeld, Germany
More informationDOLD THEOREMS IN SHAPE THEORY
Volume 9, 1984 Pages 359 365 http://topology.auburn.edu/tp/ DOLD THEOREMS IN SHAPE THEORY by Harold M. Hastngs and Mahendra Jan Topology Proceedngs Web: http://topology.auburn.edu/tp/ Mal: Topology Proceedngs
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationEQUIVALENCES OF DERIVED CATEGORIES AND K3 SURFACES DMITRI ORLOV
EQUIVALENCES OF DERIVED CATEGORIES AND K3 SURFACES DMITRI ORLOV Abstract. We consder derved categores of coherent sheaves on smooth projectve varetes. We prove that any equvalence between them can be represented
More informationCrystalline Cohomology
FREIE UNIVERSITÄT BERLIN VORLESUNG WS 2017-2018 Crystallne Cohomology Le Zhang February 2, 2018 INTRODUCTION The purpose of ths course s to provde an ntroducton to the basc theory of crystals and crystallne
More informationVariations on the Bloch-Ogus Theorem
Documenta Math. 51 Varatons on the Bloch-Ogus Theorem Ivan Pann, Krll Zanoullne Receved: March 24, 2003 Communcated by Ulf Rehmann Abstract. Let R be a sem-local regular rng of geometrc type over a feld
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationA Note on \Modules, Comodules, and Cotensor Products over Frobenius Algebras"
Chn. Ann. Math. 27B(4), 2006, 419{424 DOI: 10.1007/s11401-005-0025-z Chnese Annals of Mathematcs, Seres B c The Edtoral Oce of CAM and Sprnger-Verlag Berln Hedelberg 2006 A Note on \Modules, Comodules,
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationSUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION
talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 71457-44776
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More informationDISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization
DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1.
More informationON THE GRAYSON SPECTRAL SEQUENCE. Andrei Suslin
ON THE GRAYSON SPECTRAL SEQUENCE Andre Susln Introducton The man purpose of these notes s to show that Grayson s motvc cohomology concdes wth the usual defnton of motvc cohomology - see [V2, S-V] for example
More informationR n α. . The funny symbol indicates DISJOINT union. Define an equivalence relation on this disjoint union by declaring v α R n α, and v β R n β
Readng. Ch. 3 of Lee. Warner. M s an abstract manfold. We have defned the tangent space to M va curves. We are gong to gve two other defntons. All three are used n the subject and one freely swtches back
More informationp-adic Galois representations of G E with Char(E) = p > 0 and the ring R
p-adc Galos representatons of G E wth Char(E) = p > 0 and the rng R Gebhard Böckle December 11, 2008 1 A short revew Let E be a feld of characterstc p > 0 and denote by σ : E E the absolute Frobenus endomorphsm
More informationINVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS
INVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS HIROAKI ISHIDA Abstract We show that any (C ) n -nvarant stably complex structure on a topologcal torc manfold of dmenson 2n s ntegrable
More informationFixed points of IA-endomorphisms of a free metabelian Lie algebra
Proc. Indan Acad. Sc. (Math. Sc.) Vol. 121, No. 4, November 2011, pp. 405 416. c Indan Academy of Scences Fxed ponts of IA-endomorphsms of a free metabelan Le algebra NAIME EKICI 1 and DEMET PARLAK SÖNMEZ
More informationA CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS
Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs
More informationInternational Journal of Algebra, Vol. 8, 2014, no. 5, HIKARI Ltd,
Internatonal Journal of Algebra, Vol. 8, 2014, no. 5, 229-238 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.12988/ja.2014.4212 On P-Duo odules Inaam ohammed Al Had Department of athematcs College of Educaton
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More informationLecture 5 Decoding Binary BCH Codes
Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationDERIVED CATEGORIES OF STACKS. Contents 1. Introduction 1 2. Conventions, notation, and abuse of language The lisse-étale and the flat-fppf sites
DERIVED CATEGORIES OF STACKS Contents 1. Introduction 1 2. Conventions, notation, and abuse of language 1 3. The lisse-étale and the flat-fppf sites 1 4. Derived categories of quasi-coherent modules 5
More informationTopics in Geometry: Mirror Symmetry
MIT OpenCourseWare http://ocw.mt.edu 18.969 Topcs n Geometry: Mrror Symmetry Sprng 2009 For normaton about ctng these materals or our Terms o Use, vst: http://ocw.mt.edu/terms. MIRROR SYMMETRY: LECTURE
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationA Smashing Subcategory of the Homotopy Category of Gorenstein Projective Modules
Appl Categor Struct (2015) 23: 87 91 DOI 10.1007/s10485-013-9325-8 of Gorensten Projectve Modules Nan Gao eceved: 26 October 2012 / Accepted: 8 January 2013 / Publshed onlne: 26 July 2013 The Author(s)
More informationAppendix B. Criterion of Riemann-Stieltjes Integrability
Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for
More informationNICHOLAS SWITALA. Week 1: motivation from local cohomology; Weyl algebras
BASIC THEORY OF ALGEBRAIC D-MODULES NICHOLAS SWITALA I taught a 12-week mn-course on algebrac D-modules at UIC durng the autumn of 2016. After each week, I posted lecture notes. What follows s smply a
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationExercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
More informationMath 101 Fall 2013 Homework #7 Due Friday, November 15, 2013
Math 101 Fall 2013 Homework #7 Due Frday, November 15, 2013 1. Let R be a untal subrng of E. Show that E R R s somorphc to E. ANS: The map (s,r) sr s a R-balanced map of E R to E. Hence there s a group
More informationn-strongly Ding Projective, Injective and Flat Modules
Internatonal Mathematcal Forum, Vol. 7, 2012, no. 42, 2093-2098 n-strongly Dng Projectve, Injectve and Flat Modules Janmn Xng College o Mathematc and Physcs Qngdao Unversty o Scence and Technology Qngdao
More informationReview of metric spaces. 1. Metric spaces, continuous maps, completeness
(March 14, 2014) Revew of metrc spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [Ths document s http://www.math.umn.edu/ garrett/m/mfms/notes 2013-14/12a metrc spaces.pdf] We
More informationTHE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION
THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION THOMAS MARLEY AND MARCUS WEBB Abstract. Let R be a commutatve Noetheran local rng of prme characterstc p and f : R R the Frobenus
More informationErrata to Invariant Theory with Applications January 28, 2017
Invarant Theory wth Applcatons Jan Drasma and Don Gjswjt http: //www.wn.tue.nl/~jdrasma/teachng/nvtheory0910/lecturenotes12.pdf verson of 7 December 2009 Errata and addenda by Darj Grnberg The followng
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationBallot Paths Avoiding Depth Zero Patterns
Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,
More informationSites, Sheaves, and the Nisnevich topology
Stes, Sheaves, and the Nsnevch toology Bran Wllams Pretalbot 2014 1 Bundles and schemes The noton of a sheaf on a toologcal sace X s a famlar one. Gven a vector bundle E X one can consder, for each oen
More informationJournal of Computer and System Sciences
JID:YJCSS AID:2848 /FLA [m3g; v1.143-dev; Prn:12/12/2014; 14:21] P.1 1-21) Journal of Computer and System Scences ) Contents lsts avalable at ScenceDrect Journal of Computer and System Scences www.elsever.com/locate/jcss
More informationarxiv: v1 [math.ag] 7 Nov 2008
A Smpson correspondence n postve characterstc Mchel Gros, Bernard Le Stum & Adolfo Qurós November 10, 2008 arv:0811.1168v1 [math.ag] 7 Nov 2008 Contents 1 Usual dvded powers 2 2 Hgher dvded powers 4 3
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationMAT 578 Functional Analysis
MAT 578 Functonal Analyss John Qugg Fall 2008 Locally convex spaces revsed September 6, 2008 Ths secton establshes the fundamental propertes of locally convex spaces. Acknowledgment: although I wrote these
More informationSubset Topological Spaces and Kakutani s Theorem
MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered
More informationarxiv: v3 [math.ct] 22 Jul 2015
HIGHER GALOIS THEORY MARC HOYOIS arxv:1506.07155v3 [math.ct] 22 Jul 2015 Abstract. We generalze toposc Galos theory to hgher topo. We show that locally constant sheaves n a locally (n 1)-connected n-topos
More information2 More examples with details
Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and
More informationAli Omer Alattass Department of Mathematics, Faculty of Science, Hadramout University of science and Technology, P. O. Box 50663, Mukalla, Yemen
Journal of athematcs and Statstcs 7 (): 4448, 0 ISSN 5493644 00 Scence Publcatons odules n σ[] wth Chan Condtons on Small Submodules Al Omer Alattass Department of athematcs, Faculty of Scence, Hadramout
More information17. Coordinate-Free Projective Geometry for Computer Vision
17. Coordnate-Free Projectve Geometry for Computer Vson Hongbo L and Gerald Sommer Insttute of Computer Scence and Appled Mathematcs, Chrstan-Albrechts-Unversty of Kel 17.1 Introducton How to represent
More informationDeriving the X-Z Identity from Auxiliary Space Method
Dervng the X-Z Identty from Auxlary Space Method Long Chen Department of Mathematcs, Unversty of Calforna at Irvne, Irvne, CA 92697 chenlong@math.uc.edu 1 Iteratve Methods In ths paper we dscuss teratve
More informationHomotopy Type Theory Lecture Notes
15-819 Homotopy Type Theory Lecture Notes Evan Cavallo and Stefan Muller November 18 and 20, 2013 1 Reconsder Nat n smple types s a warmup to dscussng nductve types, we frst revew several equvalent presentatons
More informationa b a In case b 0, a being divisible by b is the same as to say that
Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :
More informationStates and exceptions are dual effects
States and exceptons are dual effects Jean-Gullaume Dumas, Domnque Duval, Laurent Fousse, Jean-Claude Reynaud LJK, Unversty of Grenoble August 29, 2010 Workshop on Categorcal Logc n Brno (ths s a long
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationarxiv: v2 [math.ct] 1 Dec 2017
FUNCTORIAL CHARACTERIZATIONS OF FLAT MODULES arxv:1710.04153v2 [math.ct] 1 Dec 2017 Abstract. Let R be an assocatve rng wth unt. We consder R-modules as module functors n the followng way: f M s a (left)
More informationSOME MULTILINEAR ALGEBRA OVER FIELDS WHICH I UNDERSTAND
SOME MULTILINEAR ALGEBRA OER FIELDS WHICH I UNDERSTAND Most of what s dscussed n ths handout extends verbatm to all felds wth the excepton of the descrpton of the Exteror and Symmetrc Algebras, whch requres
More informationGraph Reconstruction by Permutations
Graph Reconstructon by Permutatons Perre Ille and Wllam Kocay* Insttut de Mathémathques de Lumny CNRS UMR 6206 163 avenue de Lumny, Case 907 13288 Marselle Cedex 9, France e-mal: lle@ml.unv-mrs.fr Computer
More informationPolynomials. 1 More properties of polynomials
Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a
More informationarxiv: v4 [math.ac] 20 Sep 2013
arxv:1207.2850v4 [math.ac] 20 Sep 2013 A SURVEY OF SOME RESULTS FOR MIXED MULTIPLICITIES Le Van Dnh and Nguyen Ten Manh Truong Th Hong Thanh Department of Mathematcs, Hano Natonal Unversty of Educaton
More information42. Mon, Dec. 8 Last time, we were discussing CW complexes, and we considered two di erent CW structures on S n. We continue with more examples.
42. Mon, Dec. 8 Last tme, we were dscussng CW complexes, and we consdered two d erent CW structures on S n. We contnue wth more examples. (2) RP n. Let s start wth RP 2. Recall that one model for ths space
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More information