DIFFERENTIAL FORMS BRIAN OSSERMAN

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1 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 the genus of a curve, and to analyze the ramfcaton of morphsms between curves. Although dfferentals reman mportant for arbtrary varetes, we wll restrct our treatment to the case of nonsngular varetes. 1. Dfferental forms A dfferental form s not a functon, but can be defned n an analogous manner. Defnton 1.1. Let X be a nonsngular varety, and U X an open subset. A dfferental form on U assocates to each pont P U an element of the Zarsk cotangent space T P (X). Dfferental forms as defned above play a role analogous to that of arbtrary functons: we need to restrct to a much smaller collecton of them n order to obtan a useful concept. We do ths by observng that for every regular functon, we have an assocated dfferental form. Defnton 1.2. Gven U X an open subset of a nonsngular varety, and f O(U), the dfferental form df assocated to f s defned as follows: for P U, let df(p ) m P /m 2 P be the equvalence class of U, f f(p ). A dfferental form ω on U s regular f for every P U, there exst an open neghborhood V U of P and regular functons f 1,..., f m, g 1,..., g m O(V ) such that ω V = f dg. Notaton 1.3. We denote by Ω(U) the set of regular dfferental forms on U. It s clear that Ω(U) s a module over O(U). Moreover, d defnes a map O(U) Ω(U) whch s vsbly k-lnear, but not O(U)-lnear. Instead, we have the Lebnz rule: Exercse 1.4. Show that for any f, g O(U), we have d(fg) = fdg + gdf. Ths n turn gves us a chan rule for dfferental forms. Exercse 1.5. Suppose that g k(t 1,..., t n ), and f 1,..., f n are regular on U X. Then away from the zero set of the denomnator of g, we have n g d(g(f 1,..., f n )) = (f 1,..., f n )df. t =1 Because of the nonsngularty hypothess, locally on X the modules of dfferental forms are free of rank equal to the dmenson of X. Lemma 1.6. Gven P X, f f 1,..., f n O P gve a bass of m P /m 2 P, there exsts an open set U P on whch all the f are regular, and such that for every open subset V U, every ω Ω(V ) can be wrtten unquely as g df for some g O(V ). For every Q U, we have that the f f (Q) gve a bass of m Q /m 2 Q. 1

2 Proof. Frst let U be an affne open neghborhood of P n X on whch all the f are regular. Then extendng f 1,..., f n to a set of generators of A(U ), we obtan an mbeddng U A m wth coordnates t 1,..., t m such that t U = f for = 1,..., n. Let g 1,..., g d be a set of generators of I(U ) A m. Then for each, f we restrct to U we have 0 = dg = j g t j dt j. Because X s nonsngular at P, we have that the rank of the Jacoban matrx ( g / t j (P )) s equal to m n, and by our hypothess that the f j = t j U generate m P /m 2 P, we fnd that every dt j U can be expressed n terms of df 1,..., df n, wth coeffcents that are ratonal functons on X, regular at P. If we let U U be an open neghborhood of P on whch all the coeffcent functons are regular, we clam that for any V U open, and ω Ω(V ), there exst unque g O(V ) wth ω = g df. We observe that at any pont Q U, the dt j U for j = 1,..., m span m Q /m 2 Q, and t follows that f Q U, n fact the df j for j = 1,..., n span m Q /m 2 Q, so they must be a bass. We conclude that the desred g are unque, f they exst. On the other hand, snce every regular functon on any open subset of U s a ratonal functon n the t, usng Exercse 1.5 we know that ω can be wrtten locally near any pont Q V as a sum of the form h dt, where the h are ratonal functons n the t j, regular at Q. But we can smlarly express each dt for > n as a combnaton of the dt 1,..., dt n wth coeffcents beng ratonal functons n the t j, regular at Q, so we obtan the desred express. It s then clear that we have: Corollary 1.7. Gven f and U as n Lemma 1.6, we have that ω = g df vanshes at P U f and only all the g vansh at P. In partcular, for any U open n X, the locus on whch any regular dfferental form ω Ω(U) vanshes s closed n U. We conclude mmedately from the second statement that regular dfferental forms satsfy the same rgdty property as regular functons. Lemma 1.8. Suppose U V X are open subsets. If two regular dfferental forms on V are equal after restrcton to U, then they are equal on V. We can thus defne a ratonal dfferental form just as we dd a ratonal functon. Defnton 1.9. A ratonal dfferental form on X s an equvalence class of pars U, ω where U X s open, and ω s a regular dfferental form on U. The equvalence relaton s that U, ω V, ω f ω U V = ω U V. The ratonal dfferental forms are clearly a vector space over K(X). We conclude our general dscusson of dfferentals wth a descrpton of the ratonal dfferental forms on X. Proposton The ratonal dfferental forms on X have dmenson over K(X) equal to dm X, and ndeed f P X s any pont, and t 1,..., t n O P,X a bass of m P /m 2 P, where n = dm X, then dt 1,..., dt n form a bass of the ratonal dfferental forms on X over K(X). 2

3 Proof. We know from Lemma 1.6 that there exsts an open neghborhood U of P on whch every ω Ω(U) can be wrtten unquely as f dt for f O(U), and that n fact the same holds for every V U. But then the desred statement s clear, snce every ratonal dfferental form has a representatve on some V U, as does every ratonal functon. 2. Dfferental forms on curves Just as wth ratonal functons, f X s a nonsngular curve, and ω a ratonal dfferental form on X, we have a noton of order of zeroes or poles of ω at ponts on X, and we can assocate a dvsor D(ω) to ω. We assume throughout ths secton that X s a nonsngular. Defnton 2.1. If ω s a nonzero ratonal dfferental form on X, we defne the assocated dvsor D(ω) on X as follows: for any P X, let t be a local coordnate, and wrte ω = fdt for some f K(X). Then the coeffcent of [P ] n D(ω) s ν P (f). Proposton 2.2. The dvsor D(ω) s a well-defned dvsor on X. It has nonnegatve coeffcent at P f and only f ω s regular n a neghborhood of P, and strctly postve coeffcent at P f and only f ω vanshes at P. Proof. By Proposton 1.10 we have that ω = fdt for a unquely determned f, and by Lemma 1.8 f t, t are two local coordnates, then dt and dt can each be wrtten as regular multples of one another, so we must have dt = gdt for some g regular and nonvanshng at P, and we fnd that D(ω) s well-defned at each pont. If D(ω) has nonnegatve coeffcent at P, then f s regular at (and therefore n a neghborhood of) P, so ω s as well. Conversely, f ω s regular at P, we know from Lemma 1.6 that ω can be wrtten as fdt for f regular on a neghborhood of P, so D(ω) has nonnegatve coeffcent at P. Smlarly, snce dt spans m P /m 2 P, t s clear that ω vanshes at P f and only f f does, f and only f the coeffcent of D(ω) at P s postve. Fnally, we see that D(ω) s ndeed a dvsor, because t can have nonzero coeffcent at only fntely many ponts: the ponts at whch ω s not regular, and the ponts at whch ω vanshes. We can thus defne partcular spaces of ratonal dfferental forms subject to vanshng condtons: Defnton 2.3. Denote by Ω(D) the k-vector space of ratonal dfferental forms ω on X such that ω = 0 or D(ω) + D 0. We have the followng easy consequence of the defnton of D(ω): Proposton 2.4. Gven any f K(X) and nonzero ratonal dfferental form ω, we have D(fω) = D(f) + D(ω). Two key facts, both followng from Proposton 1.10, are the followng: Corollary 2.5. Gven any two nonzero ratonal dfferental forms ω, ω on X, we have that D(ω) and D(ω ) are lnearly equvalent. Proof. Gven Proposton 2.4, ths follows mmedately from Proposton 1.10 whch mples that ω = fω for some f K(X). Corollary 2.6. For any dvsor D, the space Ω(D) s fnte-dmensonal over k. Proof. Let K = D(ω) for some nonzero ratonal dfferental form ω on X. By Proposton 1.10, every other such form ω can be wrtten unquely as fω for some f K(X), and by Proposton 2.4 we have D(ω ) = D(f) + K. It follows mmedately that Ω(D) s somorphc to L(K + D) va the map ω f. 3

4 In partcular, we are now able to make the followng fundamental defnton. Defnton 2.7. If X s a nonsngular projectve curve, we defne the genus of X to be the dmenson over k of Ω(X). Example 2.8. If X = P 1, let t be a coordnate on A 1 P 1. Then a dfferental form regular on A 1 s of the form fdt for some f k[t], but one checks easly that dt has a pole of order 2 at, and therefore no matter what f s, the form fdt cannot be regular at. Thus P 1 has genus 0. In fact, we wll see later that up to somorphsm, P 1 s the only nonsngular projectve curve of genus Dfferental forms and ramfcaton We next want to study the structure of ramfcaton of morphsms. Defnton 3.1. A nonconstant morphsm ϕ : X Y of curves s separable f the nduced feld extenson K(X)/K(Y ) s separable. Otherwse, we say ϕ s nseparable. In partcular, f k has characterstc 0, every (nonconstant) morphsm s separable. We am to prove the followng theorem relatng ramfcaton to separablty. Theorem 3.2. Let ϕ : X Y be a nonconstant morphsm of nonsngular curves. Then ϕ s nseparable f and only f every P X s a ramfcaton pont of f, f and only f nfntely many ponts of X are ramfcatons ponts of f. In order to prove the theorem, we have to nvestgate the behavor of dfferental forms under morphsms. Snce a morphsm ϕ : X Y nduces lnear maps Tϕ(P ) T P for every P X, t s clear that gven a dfferental form ω on V, we obtan from ϕ a dfferental form ϕ (ω) on ϕ 1 (V ). Moreover, we see that ϕ g df = ϕ g dϕ f, so f ω s regular, then ϕ ω s lkewse regular. We wll be nterested n the behavor of pullback of dfferental forms for morphsms of nonsngular curves. An mportant prelmnary defnton s: Defnton 3.3. A nonconstant morphsm ϕ : X Y of curves s wldly ramfed at P f char k = p > 0 and p e P. If P s a ramfcaton pont at whch ϕ s not wldly ramfed, we say ϕ s tamely ramfed at P. We say ϕ s tamely ramfed f every P s ether unramfed or tamely ramfed. We can then state the relatonshp between ramfcaton and pullback of dfferental forms as follows. Proposton 3.4. Gven a nonconstant morphsm ϕ : X Y of nonsngular curves, and P X, and t a local coordnate at ϕ(p ), then P s a ramfcaton pont of ϕ f and only f ϕ dt vanshes at P. More precsely, f ϕ dt 0, then t vanshes to order at least e P 1 at P, and ϕ s wldly ramfed at P f and only f we ether have strct nequalty, or ϕ dt = 0. Proof. By defnton, we have ϕ dt = dϕ t. On the other hand, ϕ t = gs e P, where s s a local coordnate at P, and g s a nonvanshng regular functon on a neghborhood of P. Thus, ϕ dt = d(gs e P ) = s e P dg + e P s e P 1 gds. We see that f ths s nonzero, t vanshes to order at least e P 1, as clamed. Morever, s e P dg vanshes to order at least e P, and s e P 1 gds vanshes to order exactly e P 1, so we conclude that ϕ dt s ether dentcally zero or vanshes to order strctly greater than e P 1 f and only f e P = 0 n k, whch s exactly the case of wld ramfcaton. 4

5 The basc behavor of nseparable extensons n the case of transcendence degree 1 s the followng: Exercse 3.5. Suppose L/K s an algebrac extenson of felds of characterstc p > 0, and f L has a mnmal polynomal h(t) K[t] such that each coeffcent of h s a pth power n K, then f = g p for some g L. Proposton 3.6. Gven f K(X) k, we have df = 0 f and only f char k = p > 0 and f = g p for some g K(X). Proof. If f = g p, we have df = pg p 1 dg = 0 n characterstc p, by Exercse 1.5. For the converse, let t be a local coordnate at any pont of X, so that we know from Proposton 1.10 that dt s a bass over K(X) for the ratonal dfferental forms on X; n partcular, gdt = 0 on any open subset f and only f g = 0. Now, k(t) has transcendence degree 1, so K(X) s algebrac over k(t); n partcular, f satsfes a polynomal relaton h(f) = 0 for some h k(t)[z]. We may assume that h s the mnmal polynomal of f, and n partcular rreducble. Snce f k, and k s algebracally closed, we have that at least one coeffcent of h s not n k. Clearng denomnators f necessary, we may assume that the coeffcents of h are n k[t], wth no common factors. Wrtng h(z) = h z and applyng Exercse 1.5 (consderng h as a polynomal n t and z) and the hypothess that df = 0, we have 0 = d(h(f)) = dh dh (f)df + dz dt (f)dt = dh dt f dt, so dh dt f = 0. We conclude that dh dt z s a polynomal havng f as a root, but t has at most the same degree as the mnmal polynomal for f, and the degree n t of the coeffcents s strctly smaller, whch by unqueness of the mnmal polynomal s not possble unless dh dt = 0 for all. We conclude that each h has nonzero coeffcents only for powers of t whch are multples of p; snce k s algebracally closed, each h s a pth power. Thus, we conclude f = g p for some g K(X) by Exercse 3.5. Exercse 3.7. Suppose k has characterstc p > 0, and K s fntely generated of transcendence degree 1 over k. (a) Prove that K p s a subfeld of K, and K has degree p over K p. (b) Prove that f L s a subfeld of K, also of transcendence degree 1 over k, then K/L s nseparable f and only f L K p. Remark 3.8. The geometrc content of Exercse 3.7 s that a nonconstant morphsm X Y of nonsngular curves s nseparable f and only f t factors through a certan Frobenus map X X (p). We can now prove the theorem. Proof of Theorem 3.2. It follows from and Exercse 3.7 that f ϕ s nseparable, then for any f K(Y ), there exsts g K(Y ) such that ϕ f = g p. Gven P X, applyng ths to the case that f s a local coordnate at ϕ(p ), we see that ϕ f must vansh to order a multple of p at P, and therefore f s a ramfcaton pont. We thus wsh to show that f ϕ s separable, t s ramfed at only fntely many ponts of X. Let t be a local coordnate at some pont Q Y ; then snce t has valuaton 1 at Q, we have that t s not a pth power n K(Y ). It follows from separablty of K(X) over K(Y ) that ϕ t s not a pth power n K(X), and thus ϕ dt = dϕ t 0 by Proposton 3.6. Now, we know that t t(q ) s a local coordnate at Q for Q n some open neghborhood V of Q, so by Proposton 3.4, on ϕ 1 (V ) the ramfcaton of ϕ s determned by the vanshng of ϕ d(t t(q )) = ϕ dt. But because ϕ dt 0, t vanshes at only fntely many ponts of ϕ 1 (V ), and we conclude that ϕ can only be ramfed at those ponts or n X ϕ 1 (V ), whch s also a fnte set. Snce we cannot have nseparable extensons n characterstc 0, we conclude the followng mmedately from Theorem

6 Corollary 3.9. If char k = 0, and ϕ : X Y s a nonconstant morphsm of nonsngular curves, then ϕ has only fntely many ramfcaton ponts. 6