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 an affne scheme s motvated by the correspondence between affne varetes and fntely generated algebras over a feld, wthout nlpotents. The dea s that we should be able to assocate to any rng R, a topologcal space X, and a set of contnuous functons on X, whch s equal to R. In practce ths s too much to expect and we need to work wth a slghtly more general object than a contnuous functon. Now f X s an affne varety, the ponts of X are n correspondence wth the maxmal deals of the coordnate rng A = A(X). Unfortunately f we have two arbtrary rngs R and S, then the nverse mage of a maxmal deal won t be maxmal. However t s easy to see that the nverse mage of a prme deal s a prme deal. Defnton 5.1. Let R be a rng. X = Spec R denotes the set of prme deals of R. X s called the spectrum of R. Note that gven an element of R, we may thnk of t as a functon on X, by consderng t value n the quotent. Example 5.2. It s nterestng to see what these functons look lke n specfc cases. Suppose that we take X = Spec k[x, y]. Now any element f = f(x, y) k[x, y] defnes a secton of O X (X). Suppose that we consder a maxmal deal of the form p = x a, x b. Then the value of f at p s equal to the class of f nsde the quotent k[x, y] R/p = x a, x b. If we dentfy the quotent wth k, under the obvous dentfcaton, then ths s the same as evaluatng f at (a, b). Now consder Z. Suppose that we choose an element n Z. Then the value of n at the prme deal p = p s equal to the value of n modulo p. For example, consder n = 60. Then the value of ths functon at the pont 7 s equal to 60 mod 7 = 4 mod 7. Moroever 60 has zeroes at 2, 3 and 5, where both 3 and 5 are ordnary zeroes, but 2 s a double zero. Suppose that we take the rng R = Z[x]/ x 2. Then the spectrum contans only one element, the prme deal x. Consder the element x R. Then x s zero on the unque element of the spectrum, but t s not the zero element of the rng. Now we wsh to defne a topology on the spectrum of a rng. We want to make the functons above contnuous. So gven an element 1
f R, we want the set { p f(p) = 0 } = { p f p }, to be closed. Gven that any deal a s the unon of all the prncpal deals contaned n t, so that the set of prme deals whch contan a s equal to the ntersecton of prme deals whch contan every prncpal deal contaned n a and gven that the ntersecton of closed sets s closed, we have an obvous canddate for the closed sets: Defnton 5.3. The Zarsk topology on X s gven by takng the closed sets to be V (a) = {p R a p }, where a s any deal of R. Lemma 5.4. Let R be a rng. Then X = Spec R s a topologcal space Moreover the open sets form a base for the topology. Proof. Easy check. U f = { p R f / p }, By what we sad above, the Zarsk topology s the weakest topology so that the zero sets of f R are closed. Example 5.5. Let k be a feld. Then Spec k conssts of a sngle pont. Now consder Spec k[x]. If k s an algebracally closed feld, then by the Nullstellensatz, the maxmal deals are n correspondence wth the ponts of k. However, snce k[x] s an ntegral doman, the zero deal s a prme deal. Snce k[x] s a PID, the proper closed sets of X consst of fnte unons of maxmal deals. The closure of the pont ξ = 0 s then the whole of X. In partcular, not only s the Zarsk topology, for schemes, not Hausdorff or T 2, t s not even T 1. Now consder k[x, y], where k s an algebracally closed feld. Prme deals come n three types. The maxmal deals correspond to ponts of k 2. The zero deal, whose closure conssts of the whole of X. But there are also the prme deals whch correspond to prme elements f k[x, y]. The zero locus of f s then an rreducble curve C, and n fact the closure of the pont ξ = f s then the curve C. The proper closed sets thus consst of a fnte unon of maxmal deals, unon nfnte sets of maxmal deal whch consst of all ponts belongng to an affne curve C, together wth the deal of each such curve. Now suppose that k s not algebracally closed. For example, consder Spec R[x]. As before the closure of the zero deal conssts of the whole of X. The maxmal deals come n two flavours. Frst there are the 2
deals x a, where a R. But n addton there are also the deals x 2 +a, where a s a postve real number. The set V (x 2 +y 2 = 1) does not contan any deals of the frst knd, but t contans many deals of the second knd. Now suppose that we take Z. In ths case the maxmal deals correspond to the prme numbers, and n addton there s one pont whose closure s the whole spectrum. In ths respect Spec Z s very smlar to Spec k[t]. Lemma 5.6. Let X be the spectrum of the rng R and let f R. If U f = U g then f n = b g, where b 1, b 2,..., b k R. In partcular U f s compact. Proof. Suppose that we are gven an open cover of U f. Snce sets of the form U g form a base for the Zarsk topology, we may assume that U f = U g. Takng complements, we see that V ( f ) = V ( g ) = V ( g ). Now V (a) conssts of all prme deals that contan a, and the radcal of a s the ntersecton of all the prme deals that contan a. Thus f = g. But then, n partcular, f n s a fnte lnear combnaton of the g and the correspondng open sets cover U f. As ponted out above, we need a slghtly more general noton of a functon than the one gven above: Defnton 5.7. Let R be a rng. We defne a sheaf of rngs O X on the spectrum of R as follows. Let U be any open set of X. A secton σ of O X (U) s by defnton any functon s: U p U R p, where s(p) R p, whch s locally represented by a quotent. More precsely, gven a pont q U, there s an element f R such that U f U and such that the secton σ U s represented by a/f n, for some a R. An affne scheme s then any locally rnged space somorphc to the spectrum of a rng wth ts assocated sheaf. A scheme s a locally 3
rnged space, whch s locally somorphc, as locally rnged space, to an affne scheme. It s not hard to see that O X (U) s a rng (sums and products are defned n the obvous way) and that we do n fact have a sheaf rather than just a presheaf. The key result s the followng: Lemma 5.8. Let X be an affne scheme, somorphc to the the spectrum of R and let f R. (1) For any p X, the stalk O X,p s somorphc to the local rng R p. (2) The rng O X (U f ) s somorphc to R f. In partcular O X (X) R. Proof. We frst prove (1). There s an obvous rng homomorphsm O X,p R p, whch just sends a germ (g, U) to ts value g(p) at p. On the other hand, there s an obvous rng homomorphsm, R O X,p, whch sends an element r R to the par (r, X). Suppose that f / p. Then (1/f, U f ) defnes an element of O X,p, and ths element s an nverse of (f, X). It follows, by the unversal property of the localsaton, that there s a rng homomorphsm, R p O X,p, whch s the nverse map. Hence (1). Now we turn to the proof of (2). As before there s an obvous rng homomorphsm, R O X (U f ), whch nduces a rng homomorphsm R f O X (U f ). We have to show that ths map s an somorphsm. We frst consder njectvty. Suppose that a/f n R f s sent to zero. Then the mage of a/f n s equal to zero n R p, for every f / p. But then there s an element h / p such that ha = 0 n R. Let a be the annhlator of a n R. Then h a and h / p, so that a s not a subset of p. Snce ths holds for every p U f, t follows that V (a) U f =. But then f a so that f l a, for some l. It follows that f l a = 0, so that a/f n s zero n R f. 4
Now consder surjectvty. Pck s O X (U f ). By assumpton, we may cover U f by open sets V such that s s represented by a /g on V. By defnton g / p, for every p V, so that V U g. Now snce sets of the form U h form a base for the topology, we may assume that V = U h. As U h U g t follows that V (g ) V (h ) so that h g. But then h n g, so that h n a = c g. In partcular = c a g h n. Replacng h by h n and a by c a, we may assume that U f s covered by U h, and that s s represented by a /h on U h. Now observe that (5.6), f n = b h, where b 1, b 2,..., b k R and U f can be covered by fntely many of the sets U h. Thus we may assume that we have only fntely many h. Now on U h h j = U h U hj, there are two ways to represent s, one way by a /h and the other by a j /h j. By njectvty, we have a /h = a j /h j n R h h j so that for some n, (h h j ) n (h j a h a j ) = 0. Snce there are only fntely many and j, we may assume that n s ndependent of and j. We may rewrte ths equaton as If we replace h by h n+1 a /h and moreover h n+1 j (h n a ) h n+1 (h n j a j ) = 0. and a by h n a, then s s stll represented by h j a = h a j. Let a = b a, where f n = b h. Then for each j, h j a = b a h j = b h a j = f n a j. But then a/f n = a j /h j on U hj. But then a/f n represents s on the whole of U f. Note that by (2) of (5.8), we have acheved our am of constructng a topologcal space from an arbtrary rng R, whch realses R as a natural subspace of the contnuous functons. 5
Defnton 5.9. A morphsm of schemes s smply a morphsm between two locally rnged spaces whch are schemes. The gves us a category, the category of schemes. Note that the category of schemes contans the category of affne schemes as a full subcategory and that the category of schemes s a full subcategory of the category of locally rnged spaces. Theorem 5.10. There s an equvalence of categores between the category of affne schemes and the category of commutatve rngs wth unty. Proof. Let F be the functor that assocates to an affne scheme, the global sectons of the structure sheaf. Gven a morphsm (f, f # ): (X, O X ) = Spec B (Y, O Y ) = Spec A, of locally rnged spaces then let φ: A B be the nduced map on global sectons. It s clear that F s then a contravarant functor and F s essentally surjectve by (5.8). Now suppose that φ: A B s a rng homomorphsm. We are gong to construct a morphsm (f, f # ): (X, O X ) (Y, O Y ), of locally rnged spaces. Suppose that we are gven p X. Then p s a prme deal of B. But then q = φ 1 (p) s a prme deal of A. Thus we get a functon f : X Y. Now f a s an deal of A, then f 1 (V (a)) = V ( φ(a) ), so that f s certanly contnuous. For each prme deal p of B, there s an nduced morphsm φ p : A φ 1 (p) B p, of local rngs. Now suppose that V Y s an open set. We want to defne a rng homomorphsm f # (V ): O Y (V ) O X (f 1 (V )). Suppose frst that V = U g, where g A. Then O Y (V ) = A g and f 1 (V ) U φ(g). But then there s a restrcton map O X (U φ(g) ) B φ(g) O X (f 1 (V )). On the other hand, composng there s a rng homomorphsm A B φ(g). 6
Snce the mage of g s nvertble, by the unversal property of the localsaton, there s an nduced rng homomorphsm A g B φ(g). Puttng all of ths together, we have defned f # (V ) when V = U g. Snce the sets U g form a base for the topology, and these maps are compatble n the obvous sense, ths defnes a morphsm f # : O Y f O X, of sheaves. Clearly the nduced map on local rngs s gven by φ p, and so (f, f # ) s a morphsm of local rngs. Fnally t suffces to prove that these two assgnments are nverse. The composton one way s clear. If we start wth φ and construct (f, f # ) then we get back φ on global sectons. Conversely suppose that we start wth (f, f # ), and let φ be the map on global sectons. Gven p X, we get a morphsm of local rngs on stalks, whch s compatble wth φ and localsaton, so that we get a commutatve dagram A φ B f # p A f(p) B p. But snce f p # s a morphsm of local rngs, t follows that φ 1 (p) = f(p), so that f concdes wth the map nduced by φ. But then f # s also the map nduced by φ. Defnton 5.11. Let X be a scheme and let x X be a pont of X. The resdue feld of X at x s the quotent of O X,x by ts maxmal deal. Example 5.12. Let R be the local rng of a curve over an algebracally closed feld (or more generally a dscrete valuaton rng). Then Spec R conssts of two ponts; the maxmal deal, and the zero deal. The frst t 0 s closed and has resdue feld the groundfeld k of C, the second t 0 has resdue feld the quotent rng K of R, and ts closure s the whole of X. The ncluson map R K corresponds to a morphsm whch sends the unque pont of Spec K to t 1. There s another morphsm of rnged spaces whch sends the unque pont of Spec K to t 0 and uses the ncluson above to defne the map on structure sheaves. Snce there s only one way to map R to K, ths does not come from a map on rngs. In fact the second map s not 7
a morphsm of locally rnged spaces, and so t s not a morphsm of schemes. Example 5.13. It s nterestng to see an example, n a seemngly esoterc case. Consder the case of a number feld k (that s a fnte extenson of Q, wth ts rng of ntegers A k (that s the ntegral closure of Z nsde k). As a partcular example, take k = Q( 3). Then A = Z Z 3. The pcture s very smlar, as for the case of Z. There are nfntely many maxmal deals, and only one pont whch s not closed, the zero deal. Moreover, as there s a natural rng homomorphsm Z A, by our equvalence of categores, there s an nduced morphsm of schemes Spec A Spec Z. We nvestgate ths map. Consder the fbre over a pont p Spec Z. Ths s just the set of prmes n A contanng the deal pa. It s well known by number theorsts, that three thngs can happen: (1) If p dvdes the dscrmnant of k/q (whch n ths case s 12), that s p = 2 or 3, then the deal p s a square n A. 2 A = 1 + 3 2, and 3 A = 3 2. (2) If 3 s a square modulo p, the prme p factors nto a product of dstnct prmes, 11 A = 4 + 3 3 4 3 3, or 13 A = 4 + 3 4 3, (3) If p > 3 and 3 s not a square mod p (e.g p = 5 and 7), the deal p s prme n A. Defnton 5.14. Let C be a category and let X be an object of C. Let D = C X be the category whose objects consst of pars f : Y X, where f s a morphsm of C, and whose morphsms, consst of commutatve dagrams φ Y Z f g X. D s known as the category over X. If X s a scheme, then a scheme over X s exactly an object of the category of schemes over X. Let R 8
be a rng. Affne n-space over R, denoted A n R, s the spectrum of the polynomal rng R[x 1, x 2,..., x n ]. One of the key deas of schemes, s to work over arbtrary bases. Note that snce there s an ncluson R R[x 1, x 2,..., x n ] of rngs, affne space over R s a scheme over Spec R. Thus we may defne affne space over any affne scheme. Now we turn to the defnton of projectve schemes. The defnton mrrors that for affne schemes. Frst we start wth a graded rng S, S = d N S d. We set S + = d>0 S d, and we let Proj S denote the set of all homogeneous prme deals of S, whch do not contan S +. We put a topology on Proj S analogously to the way we put a topology on Spec S; f a s a homogeneous deal of S, then we set V (a) = { p Proj S a p }. The Zarsk topology s the topology where these are the closed sets. If p s a homogeneous prme deal, then S (p) denotes the elements of degree zero n the localsaton of S at the set of homogenous elements whch do not belong to p. We defne a sheaf of rngs O X on X = Proj S by consderng, for an open set U X, all functons s: U p U S (p), such that s(p) S (p), whch are locally represented by quotents. That s gven any pont q U, there s an open neghbourhood V of p n U and homogeneous elements a and f n S of the same degree, such that for every p V, f / p and s(p) s represented by the class of a/f S (p). Proposton 5.15. Let S be a graded rng and set X = Proj S. (1) For every p X, the stalk O X,p s somorphc to S (p). (2) For any homogeneous element f S +, set U f = { p Proj S f / p }. Then U f s open n Proj S, these sets cover X and we have an somorphsm of locally rnged spaces (U f, O X Uf ) Spec S (f). 9
where S (f) conssts of all elements of degree zero n the localsaton S f. In partcular Proj S s a scheme. Proof. The proof of (1) follows smlar lnes to the affne case and s left as an exercse for the reader. U f = X V ( f ) and so U f s certanly open and these sets certanly cover X. We are gong to defne an somorphsm (g, g # ): (U f, O X Uf ) Spec S (f). If a s any homogeneous deal of S, consder the deal as f S (f). In partcular f p s a prme deal of S, then φ(p) = ps f S (f) s a prme deal of S (f). It s easy to see that φ s a bjecton. Now a p ff as f S (f) ps f S (f) = φ(p), so that φ s a homeomorphsm. If p U f then S (p) and (S (f) ) φ(p) are naturally somorphc. As n the proof of (5.10), ths nduces a morphsm g # of sheaves whch s easly seen to be an somorphsm. Defnton 5.16. Let R be a rng. Projectve n-space over R, denoted P n R, s the proj of the polynomal rng R[x 1, x 2,..., x n ]. Note that P n R s a scheme over Spec R. Defnton-Lemma 5.17. If X s a topologcal space, then let t(x) be the set of rreducble closed subsets of X. Then t(x) s naturally a topologcal space and f we defne a map α: X t(x) by sendng a pont to ts closure then α nduces a bjecton between the closed sets of X and t(x). Proof. Observe that If Y X s a closed subset, then t(y ) t(x), f Y 1 and Y 2 are two closed subsets, then t(y 1 Y 2 ) = t(y 1 ) t(y 2 ), and f Y α s any collecton of closed subsets, then t( Y α ) = t(y α ). The defnes a topology on t(x) and the rest s clear. Theorem 5.18. Let k be an algebracally closed feld. Then there s a fully fathful functor t from the category of varetes over k to the category of schemes. For any varety V, the set of ponts of V may be recovered from the closed ponts of t(v ) and the sheaf of regular functons s the restrcton of the structure sheaf to the set of closed ponts. 10
Proof. We wll show that (t(v ), α O V ) s a scheme, where O V s the sheaf of regular functons on V. As any varety has an open affne cover, t suffces to prove ths for an affne varety, wth coordnate rng A. Let X be the spectrum of A. We are gong to a defne a morphsm of locally rnged spaces, β = (f, f # ): (V, O V ) (X, O X ). If p V, then let f(p) = m p X be the maxmal deal of elements of A vanshng at p. By the Nullstellensatz, f nduces a bjecton between the closed ponts of X and the ponts of V. It s easy to see that f s a homeomorphsm onto ts mage. Now let U X be an open set. We need to defne a rng homomorphsm f # (U): O X (U) f O V (f 1 (U)). Let s O X (U). We want to defne r = f # (U)(s). Pck p U. Then we defne r(p) to be the mage of s(m p ) A mp nsde the quotent A mp /m p k. It s easy to see that r s a regular functon and that f # (U) s a rng somorphsm. As the rreducble subsets of V are n bjecton wth the prme deals of A, t follows that (X, O X ) s somorphc to (t(v ), α O V ), and so the latter s an affne scheme. Note that there s a natural ncluson k A, whch assocates to a scalar the constant functon on V. But then X s a scheme over Spec k. It s easy to check that t s fully fathful. 11