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 f (U) V, then (f U,ϕ U) : (U,O X U) (V,O Y V) s a morphsm of affne varetes. Proof. We clam that (f U,ϕ U) s the morphsm of rnged spaces assocated wth ψ : O Y (V) O X (U). We must show the followng: () for each p U, f (p) V corresponds to the maxmal deal ψ 1 (m p ); and () for each g O Y (V), ϕ Vg : O Y (V g ) O X (U f 1 V g ) s the canoncal morphsm O Y (V) g O X (U) ψ(g) nduced by ψ : O Y (V) O X (U). By Defnton 6.5, there exst affne open covers {V } of Y, and {U j },j of each f 1 V such that each restrcton (f j,ψ j ) : (U j,o X U j ) (V,O Y V ) s a morphsm of affne varetes. By Lemma 6.8, we can cover each V V by opens V k whch are smultaneously dstngushed as subsets of V and V. We can also cover each U U j f 1 V k by opens U jkl smultaneously dstngushed n U and U j. By Proposton 6.2, the U jkl and V k are affne varetes. As an exercse, check that the restrcted morphsms (U jkl,o X U jkl ) (V k,o Y V j ) nhert the property of beng morphsms of affne varetes. Replacng {U j },j wth {U jkl },j,k,l, and {V } wth {V k },k, we may assume that the U j U and the V V are dstngushed opens. Consder (). Choose and j such that p U j, and let V V and U j U the dstngushed open correspondng to a O Y (V) and b j O X (U), respectvely. By Proposton 4.14, we have a commutatve dagram O Y (V) O Y (V) a O Y (V ) ψ λ ψ µ j ψ j,v O X (U) O X (U) bj O X (U j ) n whch λ and µ j are the localzaton maps, and ψ s nduced by the unversal property of the localzaton. By Exercse 4.6(), m p = µ 1 j (m p) for some maxmal deal m p O X (U) bj. Hence, ψ 1 (m p ) = λ 1 (ψ 1 (m p )). By hypothess, ψ 1 (m p ) s the maxmal deal correspondng to f (p) = f j (p) V. Clam () follows from the sheaf condton. Corollary 2. Let (f,ϕ) : (X,O X ) (Y,O Y ) and (g,ψ) : (Y,O Y ) (Z,O Z ) be morphsms of algebrac prevaretes. Then (g,ψ) (f,ϕ) s also a morphsm of algebrac prevaretes. In partcular, algebrac prevaretes over k form a category, whch we denote by PreVar k. 1
Proof. Ths composte s gven by (gf,ψ g ϕ). As g f O X = (gf ) O X, ths s a morphsm of rnged spaces. Choose fnte affne open covers {W } of Z, {V j },j of g 1 W, and {U jk },j,k of f 1 V j. By Proposton 1, (U jk,o X U jk ) (V j,o Y V j ) and (V j,o Y V j ) (W,O Z W ) are morphsms of affne varetes. Snce compostes of morphsms of affne varetes are stable under composton, (gf,ψ g ϕ) s a morphsm of algebrac prevaretes. Exercse 3. The functor X (X,O X ) from the category of algebrac sets to that algebrac prevaretes s fully fathful. Notaton: We wll frequently suppress structure sheaves from the notaton when referrng to algebrac prevaretes. If X s an algebrac prevarety, O X wll denote ts structure sheaf, unless otherwse specfed. We wll denote the morphsm (f,ϕ) of algebrac prevaretes by the symbols for the contnuous map f of underlyng topologcal spaces. 2 Glung Defnton 4. An open mmerson s a morphsm of prevaretes (j,ϕ) : (X,O X ) (Y,O Y ) that factors as (X,O X ) (j,ϕ ) (U,O Y U) (j,d OY U) (Y,O Y ), where j : U Y s the ncluson of an open subprevarety, and (j,ϕ ) s an somorphsm. Proposton 5. Let X 1,..., X n be algebrac prevaretes. For each 1,j n, let U j X be an open subset, and f j : U j U j an somorphsm of rnged spaces such that: () for each,j, f j = f 1 j ; () for each,j,k, f j (U j U k ) = U j U jk ; () for each,j,k, f k U j U k = (f jk U j U jk )(f j U j U k ). There exsts an algebrac prevarety X, unque up to unque somorphsm, and open mmersons ϕ : X X such that, for each famly of morphsms {ψ : X Y} such that ψ U j = (ψ j U j )f j for each,j, there exsts a unque morphsm ψ : X Y such that ψϕ = ψ for each. We say that X s obtaned by glung the X along the U j. Proof. The topologcal space underlyng X s the quotent of X by the equvalence relaton generated by x f j (x) for each x U. The contnuous map underlyng ϕ s nduced by the canoncal map X X. The structure sheaf O X s defned by O X (U) {(g ) O X (ϕ 1 U) g U j ϕ 1 U = g j f j U j ϕ 1 Exercse 6. Complete the proof of Proposton 5 by showng that (X,O X ) s a prevarety wth the requred propertes. Example 7. Every algebrac prevarety s obtaned by teratng ths process, startng from fntely many affne varetes. Here are a couple mportant examples of non-affne varetes. U}. 2
() Let X 1 = X 2 = A 1 k, U 1 = U 2 = A 1 k {0}. Glung X 1 and X 2 along the dentty x x : U 1 U 2, we obtan the affne lne wth doubled orgn. () Let X 1 = X 2 = A 1 k, U 1 = U 2 = A 1 k {0}. Glung X 1 and X 2 along the somorphsm x 1/x : U 1 U 2, we obtan the projectve lne P 1 k. Remark 8. If X s obtaned by glung the prevaretes X along the U j as n Proposton 5 and Y s prevarety, then a morphsm of rnged spaces f : Y X s a morphsm of prevaretes f and only f f : f 1 X X s a morphsm of prevaretes for each. Proposton 9. If X and Y are algebrac prevaretes over k, and Y s affne, then the map (f,ϕ) ϕ Y : mor PreVark (X, Y) mor CAlg(k) (O Y (Y), O X (X)) s bjectve,.e., specfyng a morphsm from any prevarety nto any affne varety s equvalent to specfyng the nduced morphsm on global sectons of structure sheaves. Proof. Choose an affne open cover X = U. By Lemma 6.8, we can cover each U U j by affne opens U jk whch are dstngushed n both U and U j. By Proposton 5, gvng a morphsm f : X Y s equvalent to gvng morphsms f : U Y such that f U U j = f j U U j for each,j, or, equvalently, such that f U jk = f j U jk for each,j,k. Snce Y, U, U jk are affne, ths s equvalent to gvng maps of k-algebras O Y (Y) O X (U ) such that the squares O Y (Y) O X (U ) O X (U j ) O X (U jk ) commute. As O X s a sheaf, ths s equvalent to gvng a map O Y (Y) O X (X). 3 Products Defnton 10. If C s a category and X, Y C, then ther product, f t exsts, s an object X Y C equpped wth morphsms π 1 : X Y X and π 2 : X Y Y such that, for each Z C and each par of morphsms f : Z X and g : Z Y, there exsts a unque morphsm (f,g) : Z X Y such that f = π 1 (f,g) and g = π 2 (f,g): f Y Z (f,g) X Y π 2 g π 1 X If the product X Y exsts, then t s unque up to unque somorphsm. If the product X Y exsts for each X, Y C and C admts a fnal object, we say that C admts fnte products. 3
Lemma 11. The tensor product A k B of two affne k-algebras A and B s another such. Proof. Suppose α A k B and α r = 0 for some r Z >0. Wrte α = n =1 a b wth a A, b B. We may assume the b are lnearly ndependent over k: f b 1 = n =2 c b for some c k, for example, then n a b = a 1 ( n ) n c b + a b = =1 =2 =2 n (c a 1 + a ) b and we may thus elmnate any lnear dependences n the b. For each maxmal deal m A, the quotent A A/m k nduces a map A k B k k B B. Ths map sends α to a nlpotent element of B, hence to 0. Snce the b are lnearly ndependent over k, we must have a m for each. Ths s true for each m, so each a belongs to the ntersecton of all maxmal deals of A. By Hlbert s Nullstellensatz, ths ntersecton s the radcal of the zero deal, so each a s nlpotent, hence 0. Thus, α = 0. Lemma 12. The category of affne k-varetes admts fnte products. Proof. Snce the category of affne k-varetes s ant-equvalent to the category of affne k-algebras, t suffces to show that the category of affne k-algebras admts fnte coproducts, where coproducts are defned by reversng the drectons of the arrows n the defnton of products. Frst, note that k s an ntal object n the category of affne k-algebras,.e., every affne k-algebra A admts a unque k-algera map k A. Suppose A and B are two affne k-algebras. Ther tensor product A k B s reduced by Lemma 11. By defnton, gvng a k-algebra map A k B C s equvalent to gvng a par of k-algebra maps A C and B C, so A k B s a coproduct n the category of affne k-algebras. Lemma 13. If X and Y are affne k-varetes, then ther product X k Y n the category of affne k-varetes s also a product n PreVar k. Proof. Gvng a par of morphsms Z X and Z Y s equvalent to gvng a par of maps O X (X) O Z (Z) and O Y (Y) O Z (Z), whch s equvalent to gvng a map O X k Y(X k Y) = O X (X) k O Y (Y) O Z (Z), whch s equvalent to gvng a morphsm Z X k Y. Exercse 14. () If X and Y are prevaretes over k and ther product X k Y exsts, show that ponts of X k Y are n natural bjecton wth ordered pars (p,q) wth p X and q Y. () By consderng A 1 k k A 1 k = A2 k, show as an exercse that the Zarsk topoloy on a product of affne varetes s strctly fner than the product topology n general. Proposton 15. The category PreVar k admts fnte products. =2 4
Proof. The affne k-algebra k corresponds to the algebrac prevarety whose underlyng topologcal space s a pont, and whose structure sheaf s gven by 0 and k. There s clearly a unque map from each prevarety to ths one, so t s a fnal object. Let X and Y be prevaretes, wth fnte affne open covers X = U and Y = j V j. The products U k V j exst by Lemmata 12 and 13. Glue the U k V j along the (U V j ) (U k V l ) = (U U k ) (V j V l ), and let X k Y denote the resultng prevarety. The projecton morphsms X k Y X and X k Y Y are nduced by the unversal property of glung and Remark 8. We clam that X k Y s a product n the category of prevaretes over k. Gvng a par of morphsms Z X and Z Y s equvalent to gvng compatble famles of morphsms f 1 U U and g 1 V j V j, whch s equvalent to gvng a compatble famly of morphsms f 1 U g 1 V j U k V j by Lemma 13. The latter s equvalent to gvng a morphsm Z X k Y, snce the f 1 U g 1 V j cover Z. 5