Chapter 16 The tangent space and the notion of smoothness We will always assume K algebraically closed. In this chapter we follow the approach of ˇ n Safareviˇc[S]. We define the tangent space TX,P at a point P of an affine variety X A as ⇢ the union of the lines passing through P and “ touching” X at P .Itresultstobeanaffine n subspace of A .Thenwewillfinda“local”characterizationofTX,P ,thistimeinterpreted as a vector space, the direction of T ,onlydependingonthelocalring :thiswill X,P OX,P allow to define the tangent space at a point of any quasi–projective variety. 16.1 Tangent space to an affine variety Assume first that X An is closed and P = O =(0,...,0). Let L be a line through P :if ⇢ A(a1,...,an)isanotherpointofL,thenageneralpointofL has coordinates (ta1,...,tan), t K.IfI(X)=(F ,...,F ), then the intersection X L is determined by the following 2 1 m \ system of equations in the indeterminate t: F (ta ,...,ta )= = F (ta ,...,ta )=0. 1 1 n ··· m 1 n The solutions of this system of equations are the roots of the greatest common divisor G(t)of the polynomials F1(ta1,...,tan),...,Fm(ta1,...,tan)inK[t], i.e. the generator of the ideal they generate. We may factorize G(t)asG(t)=cte(t ↵ )e1 ...(t ↵ )es ,wherec K, − 1 − s 2 ↵ ,...,↵ =0,e, e ,...,e are non-negative, and e>0ifandonlyifP X L.The 1 s 6 1 s 2 \ number e is by definition the intersection multiplicity at P of X and L. If G(t)is identically zero, then L X and the intersection multiplicity is, by definition, + . ⇢ 1 115 Note that the polynomial G(t)doesn’tdependonthechoiceofthegeneratorsF1,...,Fm of I(X), but only on the ideal I(X)andonL. Definition 16.1.1. The line L is tangent to the variety X at P if the intersection multiplicity of L and X at P is at least 2 (in particular, if L X). The tangent space to ⇢ X at P is the union of the lines that are tangent to X at P ;itisdenotedTP,X. n We will see now that TP,X is an affine subspace of A . Assume that P X: then the 2 polynomials Fi may be written in the form Fi = Li + Gi,whereLi is a homogeneous linear polynomial (possibly zero) and G contains only terms of degree 2. Then i ≥ Fi(ta1,...,tan)=tLi(a1,...,an)+Gi(ta1,...,tan), 2 where the last term is divisible by t .LetL be the line OA,withA =(a1,...,an). We note that the intersection multiplicity of X and L at P is the maximal power of t dividing the greatest common divisor, so L is tangent to X at P if and only if Li(a1,...,an)=0forall i =1,...,m. Therefore the point A belongs to TP,X if and only if L (a ,...,a )= = L (a ,...,a )=0. 1 1 n ··· m 1 n n This shows that TP,X is a linear subspace of A , whose equations are the linear components of the equations defining X. n n Example 16.1.2. (i) TO,An = A ,becauseI(A )=(0). (ii) If X is a hypersurface, with I(X)=(F ), we write as above F = L + G;then n TO,X = V (L): so TO,X is either a hyperplane if L =0,orthewholespaceA if L =0.For 6 3 2 2 2 instance, if X is the affine plane cuspidal cubic V (x y ) A , TO,X = A . − ⇢ Assume now that P X has coordinates (y ,...,y ). With an affine transformation 2 1 n we may translate P to the origin (0,...,0), taking as new coordinates functions on An x y ,...,x y .ThiscorrespondstoconsideringtheK-isomorphism K[x ,...,x ] 1 − 1 n − n 1 n ! K[x y ,...,x y ], which takes a polynomial F (x ,...,x )toitsTaylorexpansion 1 − 1 n − n 1 n G(x y ,...,x y )=F (y ,...,y )+d F + d(2)F + ..., 1 − 1 n − n 1 n P P (i) th where dP F denotes the i di↵erential of F at P :itisahomogeneouspolynomialofdegree i in the variables x y ,...,x y .Inparticularthelineartermis 1 − 1 n − n @F @F dP F = (P )(x1 y1)+ + (P )(xn yn). @x1 − ··· @xn − 116 n We get that, if I(X)=(F1,...,Fm), then TP,X is the affine subspace of A defined by the equations d F = = d F =0. P 1 ··· P m The affine space An,whichmayidentifiedwithKn,canbegivenanaturalstructureof K-vector space with origin P ,soinanaturalwayTP,X is a vector subspace (with origin n P ). The functions x y ,...,x y form a basis of the dual space (K )⇤ and their 1 − 1 n − n restrictions generate TP,X⇤ . Note moreover that dim TP,X =dimTP,X⇤ = k if and only if n k is the maximal number of polynomials linearly independent among d F ,...,d F . − P 1 P m If dP F1,...,dP Fn k are these polynomials, then they form a basis of the orthogonal TP,X? of − n the vector space TP,X in (K )⇤,becausetheyvanishonTP,X. 16.2 Zariski tangent space Let us define now the di↵erential of a regular function. Let f (X)bearegularfunction 2O on X.Wewanttodefinethedi↵erentialoff at P .SinceX is closed in An, f is induced by a polynomial F K[x ,...,x ]aswellasbyallpolynomialsoftheformF + G with 2 1 n G I(X). Fix P X:thend (F + G)=d F + d G so the di↵erentials of two polynomials 2 2 P P P inducing the same function f on X di↵er by the term d G with G I(X). By definition, P 2 dP G is zero along TP,X,sowemaydefinedpf as a regular function on TP,X,thedi↵erential of f at P :itisthefunctiononTP,X induced by dP F .SincedP F is a linear combination of x y ,...,x y , d f can also be seen as an element of T ⇤ . 1 − 1 n − n p P,X There is a natural map d : (X) T ⇤ ,whichsendsf to d f.Becauseoftherules p O ! P,X p of derivation, it is clear that dP (f + g)=dP f + dP g and dP (fg)=f(P )dP g + g(P )dP f.In particular, if c K, d (cf)=cd f.Sod is a linear map of K-vector spaces. We denote 2 p P p again by dP the restriction of dP to IX (P ), the maximal ideal of the regular functions on X which are zero at P .Sinceclearlyf = f(P )+(f f(P )) then d f = d (f f(P )), so this − P P − restriction doesn’t modify the image of the map. 2 Proposition 16.2.1. The map dP : IX (P ) TP,X⇤ is surjective and its kernel is IX (P ) . 2 ! Therefore T ⇤ I (P )/I (P ) as K-vector spaces. P,X ' X X n Proof. Let ' T ⇤ be a linear form on T . ' is the restriction of a linear form on K : 2 P,X P,X λ (x y )+...+ λ (x y ), with λ ,...,λ K.LetG be the polynomial of degree 1 1 1 − 1 n n − n 1 n 2 λ (x y )+...+ λ (x y ): the function g induced by G on X is zero at P and coincides 1 1 − 1 n n − n with its own di↵erential, so ' = dP g and dP is surjective. 117 Let now g IX (P )suchthatdpg =0,g induced by a polynomial G. Note that dP G may 2 n be interpreted as a linear form on K which vanishes on TP,X,henceasanelementofTP,X? . So dP G = c1dpF1 + ...+ cmdpFm (c1,...,cm suitable elements of K). Let us consider the polynomial G c F ... c F :sinceitsdi↵erentialatP is zero, it doesn’t have any term − 1 1 − − m m of degree 0 or 1 in x y ,...,x y ,soitbelongstoI(P )2.SinceG c F ... c F 1 − 1 n − n − 1 1 − − m m defines the function g on X,weconcludethatg I (P )2. 2 X ⇤ 2 Corollary 16.2.2. The tangent space TP,X is isomorphic to (IX (P )/IX (P ) )⇤ as an abstract K-vector space. Corollary 16.2.3. Let ' : X Y be an isomorphism of affine varieties and P X, ! 2 Q = '(P ). Then the tangent spaces TP,X and TQ,Y are isomorphic. Proof. ' induces the comorphism '⇤ : (Y ) (X), which results to be an isomorphism O2 !O2 such that '⇤IY (Q)=IX (P )and'⇤IY (Q) = IX (P ) .Sothereisaninducedhomomorphism I (Q)/I (Q)2 I (P )/I (P )2. Y Y ! X X which is an isomorphism of K-vector spaces. By dualizing we get the claim. ⇤ The above map from TP,X to TQ,Y is called the di↵erential of ' at P and is denoted by dP '. Now we would like to find a “more local” characterization of TP,X.Tothisendweconsider the local ring of P in X: .Werecallthenaturalmap (X) = (X) ,the OP,X O !OP,X O IX (P ) last one being the localization. It is natural to extend the map d : (X) T ⇤ to P O ! P,X OP,X setting f g(P )d f f(P )d g d = P − P . P g g(P )2 ⇣ ⌘ As in the proof of Proposition 16.2.1 one proves that the map dP : P,X TP,X⇤ induces 2 O ! an isomorphism P,X/ P,X TP,X⇤ ,where P,X is the maximal ideal of P,X.Soby M M ! 2 M O duality we have: T ( / )⇤.ThisprovesthatthetangentspaceT is a local P,X ' MP,X MP,X P,X invariant of P in X.