Commutative And Homologi al Algebra by ALG Kernel of le ture ‚1 Commutative algebra draught Noetherian rings. We write (f1;f2;:::;fm) for the ideal fg1f1 +g2f2 +(cid:1)(cid:1)(cid:1)+gmfmj g(cid:23) 2 Ag spanned (as A-module) by ff1;f2;:::;fmg (cid:26)A. A ommutative ring A is alled Noetherian, if it satis(cid:12)es the next lemma: Lemma 1.1. The following properties of a ommutative ring A are mutually equivalent 1) any set of elements f(cid:23) ontains some (cid:12)nite subset that generates the same ideal as the initial set; 2) any ideal admits a (cid:12)nite set of generators; 3) for any in(cid:12)nite hain of embedded ideals I1(cid:26)I2(cid:26)I3 (cid:26) (cid:1)(cid:1)(cid:1) there exists n2N su h that I(cid:23) =In 8(cid:23) >n. S Proof. Clearly, (1) ) (2). To dedu e (3) from (2), take a (cid:12)nite set of generatorsfor the ideal I = I(cid:23); sin e they belong to some In, we have I(cid:23) = In = I 8(cid:23) > n. Finally, (1) follows from (3) applied to the hain In = (f1;f2;:::;fn), where fi are hosen from ff(cid:23)g in order to have f(cid:23) 62(f1;f2;:::;f(cid:23)(cid:0)1). (cid:3) Theorem 1.2. (Hilbert's base theorem). If A is Noetherian, then A[x℄ is Noetherian too. Proof.LetI(cid:26)A[x℄beanideal.WewriteLd (cid:26)Aforasetofleading oeÆ ientsofalldegreedpolynomialsinI.Clearly,ea h def S (1) (1) (1) Ld and L1 = dLd areidealsin A. LetL1 begeneratedby a1;a2;:::;as 2A omingfrom f1 ;f2 ;:::;fs1 2I (k) (k) (k) and let max(cid:23)(degf(cid:23)) = m. Similarly, write f1 ;f2 ;:::;fsk for the polynomials whose leading oeÆ ients span the ideal Lk for 06k 6m(cid:0)1. To (cid:12)nish the proof, make: (cid:3) ((cid:22)) Exer i e 1.3. Show that I is spanned by s0+(cid:1)(cid:1)(cid:1)+sm(cid:0)1+s1 polynomials f(cid:23) just onstru ted. Corollary 1.4. A polynomial ring A[x1;x2;:::;xn℄ overa Noetherian ring A is Noetherian as well. (cid:3) Corollary 1.5. Any (cid:12)nitely generated k-algebra is Noetherian for any (cid:12)eld k. Proof. A polynomial algebra k[x1;x2;:::;xn℄ is Noetherian by the previous orollary.Any its fa tor algebraA is Noethe- rian as well: full preimage of any ideal I (cid:26) A under the fa torizing morphism k[x1;x2;:::;xn℄ -- A is an ideal in k[x1;x2;:::;xn℄, i.e.admits a (cid:12)nite set of generators,whose lasses span I over A, ertainly. (cid:3) Integrality. Let A (cid:26) B be two ommutative rings. An element b2B is alled integer over A, if it satis(cid:12)es the onditions from Lemma 1.6 below. If all b 2B are integer over A, then B is alled an integer extension of A or an integer A-algebra. Lemma 1.6. The following properties of an element b2B are pairwise equivalent: m m(cid:0)1 1) b =a1b + (cid:1)(cid:1)(cid:1) +am(cid:0)1b+a0 for some m2N and some a1;a2;:::;am 2A; i 2) A-module spanned by all nonnegative powers fb gi>0 admits a (cid:12)nite set of generators; 1 3) there exist a (cid:12)nitely generated faithful A-submodule M (cid:26)B su h that bM (cid:26)M. Proof. The impli ations (1) =) (2) =) (3) are trivial. To dedu e (1) from (3), let fe1;e2;:::;emg generate M over A m7!bm and let the multipli ation M - M be presented by a matrix Y, i.e.write (be1;be2; ::: ;bem)=(e1;e2;:::;em)(cid:1)Y. { Note that if A-linear map M - M takes(e1;e2;:::;em)7(cid:0)!(e1;e2;:::;em)(cid:1)X, where X is a squarematrix with entries in A, then there is an in lusion (detX)(cid:1)M (cid:26) im{, whi h follows from the Lapla e identity detX (cid:1)Id = Xb (cid:1)X applied to (e1;e2;:::;em). { In our ase the zero operator M - 0 an be presented by the matrix X = b(cid:1)Id (cid:0)Y. So, the multipli ation by det(b(cid:1)Id (cid:0)Y) annihilates M. Sin e M is faithful, det(b(cid:1)Id (cid:0)Y) = 0. This is a polynomial equation on b with the n oeÆ ients in A and the leading term b as required in (1). (cid:3) 2 Example 1.7:Integeralgebrai numbers. LetK (cid:27)Q bea(cid:12)nite dimensional (cid:12)eldextension;thenelementsz2K are alled algebrai numbers. Su h a number z is integer overZi(cid:11) there are some #1;#2;:::;#m 2K su h that the multipli ation by z sends their Q-linear span to itself and is presented there by a matrix whose entries belong to Z. Example 1.8: Invariants of a (cid:12)nite group a tion. Let a (cid:12)nite group G a t on a k-algebra B via k-algebra automorphisms B g- B, g2G, and let A = BG = fa2Bj ga = a 8g2G g be the subalgebra of G-invariants. Then B is an integer Q extensionofA. Indeed, if b1;b2;:::;bs 2B formaG-orbitofanygivenb=b1 2B,thenthepolynomial(cid:12)(t)= (t(cid:0)bi) 3 is moni , lies in A[t℄, and annihilates b. 1 A-moduleM is alled faithful,if aM =0 implies a=0 for a2A 2 as a ve tor spa e over Q 3 a polynomialis alled moni or unitary, if its leading oeÆ ient equals 1 Integer losures. A set of all b2B that are integer over a subring A (cid:26) B is alled an integer losure of A in B. If this losure oin ides with A, then A is alled integrally losed in B. Lemma 1.9. The integer losure of A is a subring in B (in parti ular, ab is integer for any a2A as soon as b is integer). If C (cid:27)B is an other ommutative ring and 2C is integer over an integer losure of A in B, then is integer over A as well (in parti ular, any integer B-algebra is an integer A-algebra as soon as B is an integer A-algebra). m m(cid:0)1 n n(cid:0)1 Proof. If p = xm(cid:0)1p + (cid:1)(cid:1)(cid:1) +x1p+x0, q = yn(cid:0)1q + (cid:1)(cid:1)(cid:1) +y1q+y0 for p;q 2 B, x(cid:23);y(cid:22) 2 A, then A-module i j spannedbyp q with06i6(m(cid:0)1),06j 6(n(cid:0)1)isfaithful(it ontains1)andgoestoitself underthemultipli ation r r(cid:0)1 by both p+q and pq. Similarly, if =zr(cid:0)1 + (cid:1)(cid:1)(cid:1) +z1 +z0 and all z(cid:23) are integer overA, then a multipli ation by preservesa faithful A-module spanned by a suÆ ient number of produ ts iz1j1z2j2 (cid:1)(cid:1)(cid:1) zrjr. (cid:3) 4 Corollary 1.10. For any two ommutative rings A (cid:26) B let f(x);g(x) 2 B[x℄ be two moni polynomials. Then all oeÆ ients of h(x)=f(x)g(x) are integer over A i(cid:11) all oeÆ ients of both f(x), g(x) are integer overA. Q Q 5 Proof. There exists a ring C (cid:27) B su h that f(x) = (t(cid:0)(cid:11)(cid:23)) and g(x) = (t(cid:0)(cid:12)(cid:22)) in C[x℄ for some (cid:11)(cid:23);(cid:12)(cid:22) 2 C. By Q Q Lemma 1.9, all oeÆ ients of h(x) = (t(cid:0)(cid:11)(cid:23)) (t(cid:0)(cid:12)(cid:22)) are integer over A () all (cid:11)(cid:23);(cid:12)(cid:22) are integer over A () all oeÆ ients of f(x) and g(x) are integer over A. (cid:3) Lemma 1.11. Let B (cid:27) A be integer over A. If B is a (cid:12)eld, then A is a (cid:12)eld. Vise versa, if A is a (cid:12)eld and there are no zero divisors in B, then B is a (cid:12)eld. (cid:0)1 (cid:0)m Proof. If B is a (cid:12)eld integer over A, then any non zero a2A has an inverse a 2B, whi h satisfy an equation a = 1(cid:0)m (cid:0)1 m(cid:0)1 (cid:0)1 m(cid:0)2 (cid:11)1a + (cid:1)(cid:1)(cid:1) +(cid:11)m(cid:0)1a +(cid:11)0 with(cid:11)(cid:23)2A. Wemultiplythe both sidesbya and geta =(cid:11)1+ (cid:1)(cid:1)(cid:1) +(cid:11)m(cid:0)1a + m(cid:0)1 i (cid:11)0a 2 A. Conversely, if A is a (cid:12)eld and B is an integer A-algebra, then all non negative integer powers b of any b2B form a (cid:12)nite dimensional ve tor spa e V over A. If b 6=0 and there are no zero divisors in B, then x 7(cid:0)! bx is an (cid:0)1 inje tive linear operator on V, that is, an isomorphism. A preimage of 12V is b . (cid:3) Example 1.12: algebrai elementsand minimalpolynomials.If A=k is a(cid:12)eld and B(cid:27)k is ak-algebra,then b2B isinteger overk i(cid:11) b satisfy a polynomial equationf(b)=0 for some f2k[x℄. Traditionally, su h b is alled algebrai overk rather n than integer. We write k[b℄ for a k-linear span of nonnegative integer powers fb gn>0 and k(b) for a k-linear span of all n integer powers fb gn2Z(if su h exist in B). Clearly, k[b℄ (cid:26) B is the minimal k-subalgebra ontaining 1 and b; in other f(x)7!f(b) terms,k[b℄=im(evb)=k[x℄=ker(evb),whereevb :k[x℄ - B isan evaluation homomorphism. Ifbisalgebrai , def then ker(evb)=(f) = f(cid:1)k[x℄ for some06=f 2k[x℄, be ause k[x℄ is a prin ipal idealdomain. This f is (cid:12)xed uniquely as a moni polynomial of lowest degree su h that f(b)=0; it is alled the minimal polynomial of b overk. Note that k[b℄ is 2 deg(f)(cid:0)1 a deg(f)-dimensional ve tor spa e over k with a basis 1; b; b ; ::: ; b . If B does not have zero divisors, then k[b℄ is a (cid:12)eld by Lemma 1.11; in parti ular, the minimal polynomial of b is irredu ible in this ase. If b is not algebrai , then ker(evb)=0 and k[b℄'k[x℄ is a polynomial ring. In parti ular, it is an in(cid:12)nite dimensional ve tor spa e over k and it is not a (cid:12)eld. This phenomenon will be generalized in Lemma 1.17 below. Lemma 1.13. Letk =Q(A)bethe fra tion(cid:12)eldfora ommutativeringAwithoutzerodivisors,B beanyk-algebra,and b 2 B be algebrai over k with minimal polynomial f 2 k[x℄. If b is integer over A, then all oeÆ ients of f are integer over A. Proof. Sin e b is integer, g(q) = 0 for some moni g 2 A[x℄. Then g = fh in k[x℄ for some moni h 2 k[x℄ and all the oeÆ ients of g, h are integer over A by Corollary1.10. (cid:3) Normal rings. A ommutative ring A without zero divisors is alled normal, if there are no b2Q(A) nA integer over A. Certainly, any (cid:12)eld is normal. Exer i e 1.14. Show that the ring Zis normal. m m(cid:0)1 Hint. Apolynomiala0t +a1t +(cid:1)(cid:1)(cid:1)+am(cid:0)1t+am 2Z[t℄annihilatesa fra tionp=q2Q with oprimep;q2Zonlyif qja0 and pjam Corollary 1.15. Let A be a normal ring and f 2A[x℄ be fa torized in Q(A)[x℄ as f = gh. If both g, h are moni , then ne essarily g;h2A[x℄. Proof. Indeed, all the oeÆ ients of g;h are integer over A by Corollary1.10. (cid:3) Corollary 1.16. LetAbenormalringwiththefra tion(cid:12)eldk =Q(A)andB beanyk-algebra.Then b2B isintegerover A i(cid:11) it is algebrai over k and its minimal polynomial (over k) lies in A[x℄. Proof. This follows immediately from Lemma 1.13. (cid:3) 4 itis known as Gauss -Krone ker -Dedekind lemma 5 For any ommutative ring A and any moni non onstant f(x) 2 A[x℄ there exists a ommutative ring C (cid:27) A su h that Q f(x) = (x(cid:0) (cid:23)) in C[x℄ for some (cid:23) 2 C. It is onstru ted indu tively as follows. Consider a fa tor ring B = A[x℄=(f) (whi h def ontains A as the ongruen e lasses of onstants) and put b = x (mod f) 2 B. Then f(b) = 0 in B[x℄. Hen e the residue after dividingf(x)by(x(cid:0)b)inB[x℄vanishesandwegetthefa torizationf(x)=(x(cid:0)b)h(x)withh(x)2B[x℄.Nowrepeatthepro edure for h, B instead of f, A e.t. . Kernel of le ture ‚1, page 2 Finitely generated ommutative k-algebras. Letkbeany(cid:12)eld.A ommutativek-algebraB is alled(cid:12)nitely (cid:25) generated, if there is a k-algebra epimorphism k[x1;x2;:::;xm℄ -- B. Images bi = (cid:25)(xi) 2 B are alled algebra generators for B over k. Lemma 1.17. A (cid:12)nitely generated k-algebra B is a (cid:12)eld only if ea h b2B is algebrai over k. Proof. Let B be generated by fb1;b2;:::;bmg. We use indu tion over m. The base ase: m=1, B =k[b℄, was onsidered in Example 1.12.Let m>1. If bm is algebrai overk, then k[bm℄ is a (cid:12)eld and B is algebrai overk[bm℄ by the indu tive assumption. Hen e, by Lemma 1.9, B is algebrai over k as well. So, we only have to show that bm must be algebrai over k. Suppose the ontrary: let bm be not algebrai , that is k(bm) be isomorphi to the (cid:12)eld k(x), of rational fun tions in 1 variable, via sending bm 7(cid:0)! x. Then, by the indu tive assumption, B is algebrai over k(bm) and ea h of b1;b2;:::;bm(cid:0)1 satis(cid:12)es a polynomial equation with oeÆ ients in k(bm). Multiplying these equations by appropriate polynomials in bm, we an put their oeÆ ients into k[bm℄ and make all their leading oeÆ ients to be equal to the same polynomial, whi h we denote by p(bm)2k[bm℄. Now, B is integer over the subalgebra F (cid:26)B generated over k by bm and q =1=p(bm). By Lemma 1.11,F is a (cid:12)eld. So, there exists a polynomial g 2 k[x1;x2℄ su h that g(bm;q) is inverse to 1+q in F. Let us write the rational fun tion k g(x; 1=p(x))2k(x) as h(x)=p (x), where h2k[x℄ is oprime to p2k[x℄. Multiplying the both sides of (cid:18) (cid:19) 1 h(bm) 1+ =1 p(bm) pk(bm) k+1 k+1 byp (bm),wegetforbmapolynomialequationh(bm)(p(bm)+1)=p (bm),whi hisnontrivial,be auseh(x)(1+p(x)) is not divisible by p(x). Hen e, bm should be algebrai over k. (cid:3) n Hilbert's Nullstellensatz. Let us write V(I) = fa 2 An j f(a) = 0 8f 2I g (cid:26) A for aÆne algebrai 6 n variety de(cid:12)ned by a system of polynomial equations I (cid:26) k[x1;x2;:::;xn℄. Vise versa, for any subset V (cid:26) A we write I(V) = ff 2 k[x1;x2;:::;xn℄j fjV (cid:17) 0g for a set of all polynomials vanishing along V. Clearly, I(V) is always an ideal and I(V(I)) (cid:27) I for any ideal I. In general, then last in lusion is proper. For example, if 2 1 I =(x ) 2C[x℄, then V(I) =f0g (cid:26)A (C) and I(V(I)) =(x). Theorem 1.18. (week Nullstellensatz). If k is algebrai ally losed, then V(I)=?()12I for an ideal I. Proof. We only have to (cid:12)nd a point p 2 An su h that f(p) = 0 for any polynomial f from a given proper ideal I (cid:26) k[x1;x2;:::;xn℄. Moreover, it is enough to suppose that I is maximal, that is, any f 62I is invertible modulo I. Indeed, otherwiseanidealJ generatedbyf andI wouldbeproperandstri tlylargerthanI,i.e.we ouldrepla eI byJ;a(cid:12)nite hainofsu hrepla ementsleadstosomemaximalideal.AssoonasI ismaximal,thefa toralgebraK =k[x1;x2;:::;xn℄=I isa(cid:12)eld.Hen e,anyelementofK isalgebrai overk (cid:26)K byLemma1.17.Sin ek isalgebrai ally losed,thismeansthat any polynomial is (mod I)- ongruent to some onstant. Let #1;#2;:::;#n be the onstants presenting the basi linear forms x1;x2;:::;xn (mod I). Then any polynomial f 2 k[x1;x2;:::;xn℄ is (mod I) ongruent to f(#1;#2;:::;#n)2k. In parti ular, f(#1;#2;:::;#n)=0 for any f2I as required. (cid:3) k Corollary 1.19. (strong Nullstellensatz). Under onditions of Theorem 1.18, f 2I(V(I))()9k2N :f 2I. n n Proof. Only the impli ation `)' for nonempty V(I) (cid:26) A is nontrivial. To prove it, we identify A with the hyperplane n+1 t = 0 in A oordinated by (t; x1;x2;:::;xn ). If f 2 k[x1;x2;:::;xn℄ (cid:26) k[t; x1;x2;:::;xn ℄ vanishes along V(I), then an ideal J (cid:26) k[t; x1;x2;:::;xn℄ spanned by I and a polynomial g(t;x) = 1(cid:0)tf(x) has empty zero set V(J) (cid:26) n+1 A , be ause of g(x;t) (cid:17) 1 on V(I). So, 1 2 J, i.e.9 q0;q1;:::;qs (cid:26) k[t; x1;x2;:::;xn ℄, f1;f2;:::;fs (cid:26) I : q0(x;t)(1(cid:0)tf(x))+q1(t;x)f1(x)+(cid:1)(cid:1)(cid:1)+qs(x;t)fs(x)=1.Thenahomomorphismk[t;x1;x2;:::;xn℄ - k(x1;x2;:::;xn) sending t 7(cid:0)! 1=f(x), x(cid:23) 7(cid:0)! x(cid:23) takes this equality to q1(1=f(x); x)f1(x)+(cid:1)(cid:1)(cid:1)+qs(1=f(x); x)fs(x) = 1. Sin e I is k proper, some of q(cid:23)(1=f(x);x) should have nontrivial denominators. Hen e, multipli ation by appropriate power f leads k to an expression we are looking for: qe1(x)f1(x)+(cid:1)(cid:1)(cid:1)+qes(x)fs(x)=f (x), where qe(cid:23) 2k[x1;x2;:::;xn℄. (cid:3) 6 ertainly, we analways extendI to anideal spanned byI |this does not e(cid:11)e t on V(I) Kernel of le ture ‚1, page 3 Commutative And Homologi al Algebra by ALG Kernel of le ture ‚2 Tensor Guide Multilinear maps. Let V1;V2;:::;Vn and W be ve tor spa es of dimensions d1;d2;:::;dn and m over an ' arbitrary (cid:12)eld k with hark 6=2. A map V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn (cid:0)(cid:0)!W is alled multilinear, if in ea h argument 0 00 0 00 '(::: ; (cid:21)v +(cid:22)v ; :::)=(cid:21)'(::: ; v ; :::) + (cid:22)'(::: ; v ; :::) when all the other remain to be (cid:12)xed. The multilinearmaps V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn - W form a ve tor spa e of Q (i) (i) (i) dimension m(cid:1) d(cid:23). Namely, if we (cid:12)x a base fe1 ;e2 ;:::;edi g for ea h Vi and a base fe1;e2;:::;emg for W, then any multilinear map ' is uniquely de(cid:12)ned by its values at all ombinations of the base ve tors: (cid:0) (cid:1) X (1) (2) (n) ((cid:11)1;(cid:11)2;:::;(cid:11)n) (cid:23) ' e(cid:11)1 ; e(cid:11)2 ; ::: ; e(cid:11)n = a(cid:23) (cid:1)e 2 W (cid:23) Q ((cid:11)1;(cid:11)2;:::;(cid:11)n) As soon as m(cid:1) d(cid:23) numbers a(cid:23) 2 k are given, the map ' is well de(cid:12)ned by the multilinearity. It Pdi (i) (i) sends a olle tion of ve tors (v1;v2;:::;vn), where vi = x(cid:11)i e(cid:11)i 2Vi for 1 6i 6n, to (cid:11)i=1 Xm (cid:16) X (cid:17) ((cid:11)1;(cid:11)2;:::;(cid:11)n) (1) (2) (n) (cid:23) '(v1;v2;:::;vn) = a(cid:23) (cid:1)x(cid:11)1 (cid:1)x(cid:11)2 (cid:1) (cid:1)(cid:1)(cid:1) (cid:1)x(cid:11)n (cid:1)e 2 W j=1 (cid:11)1;(cid:11)2;:::;(cid:11)n ((cid:11)1;(cid:11)2;:::;(cid:11)n) the numbers a(cid:23) an be onsidered as elements of some €(n+1)-dimensional format matrix of the size 1 m(cid:2)d1(cid:2)d2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)dn(cid:129), if you an imagine su h a thing . Exer i e 2.1. Che k that a olle tion (v1;v2;:::;vn)2V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn doesn't ontain zero ve tor i(cid:11) there exists a multilinear map ' (to somewhere) su h that '(v1;v2;:::;vn)6=0. ' F Exer i e 2.2. Che k that a multilinear map V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn - U followed by a linear operator U - W is the FÆ' multilinear map V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn - W as well. (cid:28) Tensor produ t of ve tor spa es. Let V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn - U be a (cid:12)xed multilinear map. Then for any ve tor spa e W we have a omposition operator ! ! the spa e Hom(U;W)of all F7(cid:0)!FÆ(cid:28) thespa e of all multilinearmaps - (1) F ' linear operators U - W V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn - W (cid:28) AmultilinearmapV1(cid:2)V2(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)Vn (cid:0)(cid:0)!U is alleduniversal ifthe ompositionoperator(1)isanisomorphism for any ve tor spa e W. In other words, the multilinear map (cid:28) is universal, if for any W and any multilinear ' F map V1 (cid:2)V2 (cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn - W there exist a unique linear operator U - W su h that ' = FÆ(cid:28), i. e. the ommutative diagram - U (cid:28) V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn F ' - ? W an be always losed by a unique linear dotted row. Claim 2.3. Let V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn (cid:28)-1 U1 É V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn (cid:28)-2 U2 be two universal multilinear maps. Then there (cid:19) exists a unique linear isomorphism U1 (cid:0)(cid:0)!U2 su h that (cid:28)2 =(cid:19)(cid:28)1. 1usual d(cid:2)m - matri es, whi h present thelinear mapsV - W,have just 2-dimensional format Proof. Sin e both U1;U2 are universal, there are unique linear operators U1 F2-1 U2 and U2 F1-2 U1 mounted in the diagrams IdU1 U61 (cid:27) (cid:28)1 (cid:28)1 - U1 (cid:28)2 (cid:27)...........F-...1.2....U2 F12 (cid:27) V1(cid:28)(cid:2)2 V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)(cid:28)2Vn - F2?1 =) V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn (cid:28)-(cid:28)12 U1...........F..2-..1..-.. IdU2 U2 U2 U2 IdU2 So,the ompositionF21F12 =IdU2,be auseoftheuniquenesspropertyintheuniversalityofU2.Similarly,F12F21 =IdU1. (cid:3) (i) (i) (i) Q Claim 2.4. Let fe1 ;e2 ;:::;edig 2 Vi be a base (for 1 6 i 6 n). Denote by V1(cid:10)V2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)Vn a ( di) - dimensional ve tor spa e whose base ve tors are the symbols (1) (2) (n) e(cid:11)1 (cid:10)e(cid:11)2 (cid:10) ::: (cid:10)e(cid:11)n ; 16(cid:11)i 6di (2) ((cid:22)) (all possible formal €tensor produ ts(cid:129) of base ve tors e(cid:23) ). Then the multilinear map (cid:28) V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn (cid:0)(cid:0)!V1(cid:10)V2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)Vn whi h sends a base ve tor olle tion (e(cid:11)1;e(cid:11)2;:::;e(cid:11)n) 2 V1 (cid:2)V2 (cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn to the orresponding base ve tor (2) is universal. Proof. LetV1(cid:2)V2(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)Vn (cid:0)(cid:0)'!W beamultilinearmapandV1(cid:10)V2(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)Vn F- W bealinearoperator.Comparing the values at the base ve tors, we see that (1) (2) (n) (1) (2) (n) '=FÆ(cid:28) () F(e(cid:11)1 (cid:10)e(cid:11)2 (cid:10) ::: (cid:10)e(cid:11)n )='(e(cid:11)1;e(cid:11)2; ::: ;e(cid:11)n): (cid:3) The Segre embedding. The ve tor spa e V1(cid:10)V2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)Vn is alled a tensor produ t of V1;V2;:::;Vn . The (cid:28) universal multilinear map V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn - V1(cid:10)V2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)Vn is alled a tensor multipli ation. For a olle tionofve tors(v1;v2;:::;vn) 2V1(cid:2)V2(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)Vn theimage(cid:28)(v1;v2;:::;vn)isdenotedbyv1(cid:10)v2(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)vn and alled a tensor produ t of these ve tors. All su h produ ts are alled de omposable tensors. Of ourse, not all the ve tors of V1(cid:10)V2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)Vn are de omposable and im(cid:28) isn't a ve tor subspa e in V1(cid:10)V2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)Vn, be ause (cid:28) is multilinear but not linear. However, the linear span of de omposable tensors exhausts the whole of V1(cid:10)V2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)Vn. Geometri ally, the tensor multipli ation gives a map s P(V1)(cid:2)P(V2)(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)P(Vn) (cid:26) - P(V1(cid:10)V2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)Vn) alled a Segre embedding. Ifdi =dimVi =mi+1, thenthe Segreembeddingisa bije tionbetween Pm1(cid:2)Pm2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Pmn and a Segre subvariety whiP h onsist of all de omposable tensors onsidered up to proportionality. The Segre subvariety has dimension ( mi), doesn't belong to any hyperplane, is ruled by n families of linear Q subspa es, and lives in PN with N =(cid:0)1+ (1+mi). Example 2.5: the Segre embedding Pm1 (cid:2) Pm2 (cid:26) - Pm1+m2+m1m2 sends x = (x0: x1: ::: : xm1) 2 Pm1 and y = (y0: y1: ::: : ym2) 2 Pm2 to the point s(x;y) 2 Pm1+m2+m1m2 whose (1+m1)(1+m2) homogeneous oordinates are (cid:3) all possible produ ts xiyj with 0 6 i 6 m1 and 0 6 j 6 m2. To visualize this thing, take Pm1 = P(V ), Pm2 = P(W), and Pm1+m2+m1m2 =P(Hom(V;W)), where Hom(V;W) is the spa eof all linear maps. Then the Segremap sends a pair (cid:3) ((cid:24);w)2V (cid:2)W to the linear map (cid:24)(cid:10)w, whi h a ts by the rule v 7(cid:0)!(cid:24)(v)(cid:1)w. Exer i e 2.6. Che k that a map V(cid:3) (cid:2)W - Hom(V;W) whi h sends ((cid:24);w) to the operator v 7(cid:0)! (cid:24)(v)(cid:1)w is the (cid:3) universal bilinear map (so, there is a anoni al isomorphism V (cid:10)W 'Hom(V;W)) (cid:3) Exer i e 2.7. Che k that for(cid:24) =(x0;x1; :::; xm1)2V and w =(y0;y1; :::; ym2)2W operator(cid:24)(cid:10)w hasthe matrix aij =xjyi. Sin e any operator (cid:24) (cid:10)w has 1-dimensional image, the orresponding matrix has rank 1. On the other side, any rank 1 matrix has proportional olumns. Hen e, the orresponding operator has 1-dimensional image, say spaned by w2W, and takes v 7(cid:0)! (cid:24)(v)w, where the oeÆ ient (cid:24)(v) 2 k depends on v linearly. So, the image of the Segre embedding onsists of all rank 1 operators up to proportionality. In parti ular, it may be presented as the interse tion of quadri s (cid:18) (cid:19) aij aik det =aija`k(cid:0)aika`j =0 (i. e. by vanishing all 2(cid:2)2 - minors for the matrix (a(cid:22)(cid:23))). a`j a`k Kernel of le ture ‚2, page 2 (cid:10)n def Tensor algebra of a ve tor spa e. If V1 = V2 = (cid:1)(cid:1)(cid:1) = Vn = V, then V = V (cid:10)V (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)V is alled an | {z } n n-th tensor power of V. All tensor powers are ombined in the in(cid:12)nite dimensional non ommutative graded (cid:15) (cid:10)n (cid:10)0 def algebra T V = (cid:8) V , where V = k. n>0 Exer i e 2.8. Using the universality, show that there are anoni al isomorphisms (cid:0) (cid:1) (cid:0) (cid:1) (cid:10)n1 (cid:10)n2 (cid:10)n3 (cid:10)n1 (cid:10)n2 (cid:10)n3 (cid:10)(n1+n2+n3) V (cid:10)V (cid:10)V 'V (cid:10) V (cid:10)V 'V (cid:15) whi h make the ve tor's tensoring to be well de(cid:12)ned asso iative multipli ation on T V. (cid:15) 2 Algebrai ally, T V is what is alled €a free asso iative k-algebra generated by V(cid:129) ( ). Pra ti ally, this means (cid:15) that as soon as a basis fe1;e2;:::;edg (cid:26) V is (cid:12)xed, then T V turns into the spa e of the formal (cid:12)nite linear ombinations of words whi h onsist of the letters ei separated by (cid:10). The words are multipliedby writing after one other onsequently. This multipli ation is extended on the linear ombinations of the words by the usual distributivity rules. (cid:10)n (cid:3)(cid:10)n (cid:3) (cid:3) (cid:3) Duality. The spa es V =V (cid:10)V (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)V and V =V (cid:10)V (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)V are anoni ally dual to ea h | {z } | {z } n n (cid:10)n (cid:3)(cid:10)n other. The pairing between v =v1(cid:10)v2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)vn 2V and (cid:24) =(cid:24)1(cid:10)(cid:24)2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)(cid:24)n 2V is given by a full ontra tion n (cid:10) (cid:11) def Y v; (cid:24) = (cid:24)i(vi) : (3) i=1 (cid:3) Let fe1;e2;:::;eng (cid:26) V and f(cid:24)1;(cid:24)2;:::;(cid:24)ng (cid:26) V be dual bases, i. e. (cid:24)i(ej) = 0 for i 6= j and (cid:24)i(ei) = 1. (cid:15) Then the base words fei1 (cid:10)ei2 (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)eirg and f(cid:24)j1 (cid:10)(cid:24)j2 (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)(cid:24)jsg form the dual bases for T V and (cid:15) (cid:3) (cid:10)n(cid:3) (cid:3)(cid:10)n (cid:10)n (cid:3) T V with respe t to the full ontra tion. So, V 'V . On the other side, the spa e (V ) is naturally identi(cid:12)ed with the spa e of all multilinearforms V (cid:2)V (cid:2) (cid:1)(cid:1)(cid:1) (cid:2)V - k, be ause V(cid:10)n is universal. So, there | {z } n (cid:3)(cid:10)n (cid:3) (cid:3) (cid:3) exists a anoni al isomorphism between V = V (cid:10)V (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)V and the spa e of multilinear forms in n | {z } n Qn (cid:3)(cid:10)n arguments from V. It sends a tensor (cid:24) =(cid:24)1(cid:10)(cid:24)2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)(cid:24)n 2V to the form (v1;v2;:::;vn) 7(cid:0)! (cid:24)i(vi). i=1 I J Partial ontra tions. Let f1;2; ::: ; pg (cid:27) (cid:27)f1;2; ::: ; mg (cid:26) - f1;2; ::: ; qg be two inje tive (not ne es- sary monotonous) maps. We write i(cid:23) and j(cid:23) for I((cid:23)) and J((cid:23)) respe tively and onsiderI =(i1;i2;:::;im) and J = (j1;j2;:::;jm) as two ordered (but not ne essary monotonous) index olle tions of the same ardinality. A linear operator V(cid:3)(cid:10)V(cid:3)(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)V(cid:3)(cid:10)V (cid:10)V (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)V IJ- V(cid:3)(cid:10)V(cid:3)(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)V(cid:3)(cid:10)V (cid:10)V (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)V | {z } | {z } | {z } | {z } p q p(cid:0)m q(cid:0)m Qm N N whi h sends (cid:24)1 (cid:10)(cid:24)2 (cid:10) (cid:1)(cid:1)(cid:1) (cid:10) (cid:24)p (cid:10)v1 (cid:10)v2 (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)vq to (cid:24)i(cid:23)(vj(cid:23)) (cid:1) (cid:24)i (cid:10) vj is alled a partial (cid:23)=1 i62im(I) j62im(J) ontra tion in the indexes I and J. Example 2.9: the ontra tion between a ve tor and a multilinear form. Consider a multilinear form '(v1;v2;:::;vn) as the (cid:3)(cid:10)n (cid:3)(cid:10)(n(cid:0)1) tensor from V and onta t it in the (cid:12)rst index with a ve tor v 2V. The result belongs to V and gives a multilinear form in (n(cid:0)1) arguments. This form is denoted by iv' and alled an inner produ t of v and '. Exer i e 2.10. Che k that iv'(w1;w2;:::;wn(cid:0)1) = '(v; w1;w2;:::;wn(cid:0)1), i. e. the inner multipli ation by v is just the (cid:12)xation of v in the (cid:12)rst argument. Linear span of a tensor. Let U;W (cid:26) V be any two subspa es. Writing down the standard monomial bases, (cid:10)n (cid:10)n (cid:10)n (cid:10)n (cid:10)n we see immediately that (U \W) = U \W in V . So, for any t 2 V there is a minimal subspa e span(t)(cid:26)V whosen-thtensorpower ontainst.Itis alleda linear span oftand oin ideswiththeinterse tion (cid:10)n of all W (cid:26) V su h that t2W . To des ribe span(t) more onstru tively, for any inje tive (not ne essary monotonous) map J =(j1;j2;:::;jn(cid:0)1) : f1; 2; ::: ; (n(cid:0)1)g (cid:26) - f1; 2; ::: ; ng 2 Ifyoulike it, makethefollowing formal exer ise: dedu efromtheuniversality thatfor anyasso iative k-algebra A and ve tor spa e mapV f- A there exists a uniquealgebra homomorphismT(cid:15)V (cid:11)- A su h that(cid:11)jV =f Kernel of le ture ‚2, page 3 onsider a linear map V(cid:3)(cid:10)(n(cid:0)1) Jt- V de(cid:12)ned by omplete ontra tion with t: it sends a de omposable tensor ' = (cid:24)1 (cid:10)(cid:24)2 (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)(cid:24)n(cid:0)1 to a ve tor obtained by oupling (cid:23)-th fa tor (cid:24)(cid:23), of ', with j(cid:23)-th fa tor of t for all 1 6(cid:23) 6(n(cid:0)1), i.e. J (1;2;:::;(n(cid:0)1)) t(') = (j1;j2;:::;jn(cid:0)1) ('(cid:10)t): (cid:16) (cid:17) J (cid:3)(cid:10)(n(cid:0)1) Claim 2.11. span(t)(cid:26)V is spanned by the images t V (cid:26)V taken for all possible J. (cid:0) (cid:1) (cid:10)n J Proof. Letspan(t)=W (cid:26)V.Thent2W andim(cid:0) t(cid:1) (cid:26)W 8J.ItremainstoprovethatW isannihilatedbyanylinear (cid:3) J (cid:3) form (cid:24)2V whi h an(cid:16)nihilate all(cid:17)the subspa es im t . Suppose the ontrary:let (cid:24) 2V have non zero restri tion on W J (cid:3)(cid:10)(n(cid:0)1) (cid:3) but annihilate all t V . Then there exist a base fw1;w2;:::;wkg for W and a base f(cid:24)1;(cid:24)2;:::;(cid:24)dg for V su h (cid:3) that: (cid:24)1 =(cid:24), therestri tionsof (cid:24)1;(cid:24)2;:::;(cid:24)k ontoW formthe baseofW dualto fw(cid:23)g, and(cid:24)k+1; ::: ; (cid:24)d annihilateW. Now, for any J and (cid:24)i1;(cid:24)i2;:::;(cid:24)in(cid:0)1 we have (cid:10) (cid:0) (cid:1)(cid:11) (cid:10) (cid:11) J 0= (cid:24); t (cid:24)i1(cid:10)(cid:24)i2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)(cid:24)in(cid:0)1 = (cid:24)ij1(cid:0)1 (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)(cid:24)ijs(cid:0)(cid:0)11 (cid:10)(cid:24)(cid:10)(cid:24)ijs(cid:0)+11 (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)(cid:24)ijn(cid:0)1 ; t (4) (cid:0)1 (cid:0)1 (cid:0)1 (cid:0)1 where s = f1; 2; ::: ; ngnim(J) and J = (j1 ;j2 ;:::;jn ) is the inverse to J map from im(J) (cid:26) f1; 2; ::: ; ng (cid:3)(n(cid:0)1) to f1; 2; ::: ; (n(cid:0)1)g. Note that ea h basi monomial of V ontaining as a fa tor (cid:24)1 = (cid:24) an appear as the (cid:12)rst operandintherightsideof (4).Butifweexpandttroughthebasi monomialswi1(cid:10)wi2(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)win,thenthe oeÆ ients of this expansion an be omputed as full ontra tions of t with the orresponding elements (cid:24)i1(cid:10)(cid:24)i2 (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)(cid:24)in from (cid:3)(cid:10)n the dual base for W . By (4), su h a ontra tionequals zero as soon one of (cid:24)i(cid:11) equals (cid:24)1 =(cid:24), whi h is dual to w1. So, span(t) is ontained in the linear span of w2; ::: ; wk but this ontradi ts our assumption. (cid:3) ' Symmetry properties. A multilinear map V (cid:2)V (cid:2) (cid:1)(cid:1)(cid:1) (cid:2)V - W is alled symmetri if it doesn't hange | {z } n its value under any permutations of the arguments. If the value of ' is stable under the even permutations and hanges the sign under the odd ones, then ' is alled skew symmetri . Sin e the omposition operator (1) preserves the symmetry properties, for the (skew)symmetri ' the omposition operator (1) turns into 0 1 ! thespa e of all (skew)symmetri the spa e Hom(U;W)of all F7(cid:0)!FÆ' F -  multilinear mapsV (cid:2)V (cid:2) (cid:1)(cid:1)(cid:1) (cid:2)V - W A (5) linear operators U - W | {z } n ' A (skew)symmetri multilinear map V (cid:2)V (cid:2) (cid:1)(cid:1)(cid:1) (cid:2)V - U is alled universal if (5) is an isomorphism for | {z } n n any W. In the symmetri ase the universal target spa e is denoted by S V and alled n-th symmetri power n of V. In the skew symmetri ase it is alled n-th exterior power of V and denoted by (cid:3) V. n n Exer i e 2.12. Show that, if exist, S V and (cid:3) V are unique up to unique isomorphism ommuting with the universal maps. (cid:15) (cid:15) Symmetri algebra S V of V is a fa tor algebra of the free asso iative algebra T V by a ommutation (cid:15) relations vw =wv. More pre isely, denote by Isym (cid:26)T V a linear span of all tensors (cid:1)(cid:1)(cid:1) (cid:10)v(cid:10)w(cid:10) (cid:1)(cid:1)(cid:1) (cid:0) (cid:1)(cid:1)(cid:1) (cid:10)w(cid:10)v(cid:10) (cid:1)(cid:1)(cid:1) ; where the both terms are de omposable, have the same degree, and di(cid:11)er only in order of v;w. Clearly, Isym (cid:15) is a double-sided ideal in T V generated by a linear span of all the di(cid:11)eren es v (cid:10)w(cid:0)w(cid:10)v 2 V (cid:10)V. The (cid:15) def (cid:15) fa tor algebra S V = T V=Isym is alled a symmetri algebra of the ve tor spa e V. By the onstru tion, it is 3 (cid:10)n ommutative . Sin e Isym = (cid:8) (Isym\V ) isthe dire t sum of its homogeneous omponents, the symmetri n>0 (cid:15) L n n def (cid:10)n (cid:10)n algebra is graded: S V = S V, where S V = V =(Isym\V ). n>0 Claim 2.13. The tensor multipli ation followed by the fa torization map: V (cid:2)V (cid:2) (cid:1)(cid:1)(cid:1) (cid:2)V (cid:28) - V(cid:10)n (cid:25)-- Sn(V) (6) | {z } n gives the universal symmetri multilinear map. 3Again, if you like it, prove that for any ommutative k-algebra A and a ve tor spa e map V f- A there exists a unique homomorphismof ommutativealgebras S(cid:15)V (cid:11)- A su h that(cid:11)jV =f Kernel of le ture ‚2, page 4 Proof. Any multilinear map V (cid:2)V (cid:2) (cid:1)(cid:1)(cid:1) (cid:2)V '- W is uniquely de omposed as '=FÆ(cid:28), where V(cid:10)n F- W is linear. F is fa tored through (cid:25) i(cid:11) F((cid:1)(cid:1)(cid:1) (cid:10)v(cid:10)w(cid:10) (cid:1)(cid:1)(cid:1))=F((cid:1)(cid:1)(cid:1) (cid:10)w(cid:10)v(cid:10) (cid:1)(cid:1)(cid:1)), i. e. i(cid:11) '(::: ;v;w; :::)='(::: ;w;v; :::) (cid:3) n The graded omponents S V are alled symmetri powers of V and the map (6) is alled a symmetri multipli- n ation. Ifabasefe1;e2;:::;edg(cid:26)V is(cid:12)xed,thenS V isnaturallyidenti(cid:12)edwiththespa eofallhomogeneous polynomials of degree n in ei. Namely, onsider the polynomial ring k[e1;e2;:::;ed℄ (whose €variables(cid:129) are the base ve tors ei) and identify V with the spa e of all linear homogeneous polynomials in ei. Exer i e 2.14. Che k that the multipli ation map Qn ! (`1;`2;:::;`n)7(cid:0)! `(cid:23) (cid:23)=1 thehomogeneous polynomials V (cid:2)V (cid:2) (cid:1)(cid:1)(cid:1) (cid:2)V - | {z } of degree n in ei n (cid:0) (cid:1) n d+n(cid:0)1 is universal and show that dimS V = . n (cid:15) (cid:15) Exterior algebra (cid:3) V of V is a fa tor algebra of the free asso iative algebra T V by a skew ommutation (cid:15) relations vw =(cid:0)wv. More pre isely, onsider a double-sided ideal Iskew (cid:26)T V generated by all sums v(cid:10)w+ (cid:15) def (cid:15) w(cid:10)v 2V (cid:10)V and put (cid:3) V = T V=Iskew. Exa tly as in the symmetri ase, the ideal Iskew is homogeneous: (cid:10)n (cid:10)n Iskew = (cid:8) (Iskew\V ), where (Iskew\V ) is the linear span of all sums n>0 (cid:1)(cid:1)(cid:1) (cid:10)v(cid:10)w(cid:10) (cid:1)(cid:1)(cid:1) + (cid:1)(cid:1)(cid:1) (cid:10)w(cid:10)v(cid:10) (cid:1)(cid:1)(cid:1) (cid:15) (the both items have degree n and di(cid:11)er only in the order of v;w). So, the fa tor algebra (cid:3) V is graded by the n def (cid:10)n (cid:10)n subspa es (cid:3) V = V =(Iskew\V ). Exer i e 2.15. Provethat the tensor multipli ation followed by the fa torization V (cid:2)V (cid:2) (cid:1)(cid:1)(cid:1) (cid:2)V ..(cid:28)...-... V(cid:10)n (cid:25)-- (cid:3)n(V) (7) | {z } n gives the universal skew symmetri multilinear map. Themap(7) is alledan exterior orskew multipli ation.Theskewprodu tof ve tors (v1;v2;:::;vn)isdenoted by v1 ^v2^ (cid:1)(cid:1)(cid:1) ^vn. By the onstru tion, it hanges the sign under the transposition of any two onsequent terms. So, under any permutation of terms the skew produ t is multiplied by the sign of the permutation. n n n n n n n n Exer i e 2.16. For any U;W (cid:26)V he k that S U\S W =S (U\U) in S V and (cid:3) U\(cid:3) W =(cid:3) (U \U) in (cid:3) V. (cid:15) Grassmannian polynomials. Let fe1;e2;:::;edg (cid:26) V be a basis. Then the exterior algebra (cid:3) V is identi(cid:12)ed with a grassmannian polynomial ring khe1;e2;:::;edi whose €variables(cid:129) are the base ve tors ei whi h skew ommute, that is, ei ^ej = (cid:0)ej ^ei for all i;j. More pre isely, it is linearly spanned by the grassmannian monomials ei1 ^ ei2 ^ (cid:1)(cid:1)(cid:1) ^ ein. It follows from skew ommutativity that ei ^ ei = 0 for all i, that is, a grassmannian monomialvanishesas soonas itbe omes of degree more then1 insome ei. So, any grassmannian monomial has a unique representation ei1 ^ei2 ^ (cid:1)(cid:1)(cid:1) ^ein with 1 6i1 <i2 < (cid:1)(cid:1)(cid:1) <in 6d. def Claim 2.17. The monomials eI = ei1 ^ ei2 ^ (cid:1)(cid:1)(cid:1) ^ ein, where I = (i1;i2;:::;in) runs through the in reasing(cid:0) n(cid:1)- n n n d element subsets in f1;2; ::: ;dg, form a base for (cid:3) V. In parti ular, (cid:3) V = 0 for n > dimV , dim(cid:3) V = , n d and dim khe1;e2;:::;edi=2 . (cid:0) (cid:1) d Proof. Consider n -dimensional ve tor spa e U whose base onsists of the symbols (cid:24)I, where I = (i1;i2;:::;in) runs through the in reasing n-element subsets in f1;2; ::: ;dg. De(cid:12)ne a skew symmetri multilinear map (cid:11) V1(cid:2)V2(cid:2) (cid:1)(cid:1)(cid:1) (cid:2)Vn (cid:0)(cid:0)!U : (ej1;ej2;:::;ejn) 7(cid:0)! sgn((cid:27))(cid:1)(cid:24)I ; where I =(j(cid:27)(1);j(cid:27)(2); ::: ;j(cid:27)(n);) is an in reasing olle tion obtained from (j1;j2;:::;jn) by a (unique) permutation (cid:27). ' This map is universal. Indeed, for any skew symmetri multilinear map V (cid:2)V (cid:2) (cid:1)(cid:1)(cid:1) (cid:2)V - W there exists at most | {z } n F one linear operator U - W su h that ' = FÆ(cid:11), be ause it has to a t on the base as F((cid:24)I) = '(ei1;ei2;:::;ein)) for all in reasing I = (i1;i2;:::;in). On the other side, su h F really de omposes ', be ause F((cid:11)(ej1;ej2;:::;ejn)) = '(ej1;ej2;:::;ejn))forallnotin reasing base olle tions(ej1;ej2;:::;ejn)(cid:26)V| (cid:2)V (cid:2){z(cid:1)(cid:1)(cid:1) (cid:2)V}aswell.Bytheuniversality, n n there exists a anoni al isomorphism between U and (cid:3) V whi h sends (cid:24)I to ei1 ^ei2 ^ (cid:1)(cid:1)(cid:1) ^ein =eI. (cid:3) deg(f)(cid:1)deg(g) Exer i e 2.18. Che k that f(e)^g(e)=((cid:0)1) g(e)^f(e) for all homogeneous f(e);g(e)2 khe1;e2;:::;edi. In parti ular, ea h even degree homogeneous polynomial ommutes with any grassmannian polynomial. Kernel of le ture ‚2, page 5 Exer i e 2.19. Des ribe the enter of khe1;e2;:::;edi, i. e. all grassmannian polynomials whi h ommute with every- thing. Pd Example 2.20: linear base hange in grassmannian polynomial. Under the linear substitution ei = aij(cid:24)j the base mono- j=1 mials eI are hanged by the new base monomials (cid:24)I as follows: (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) X X X eI =ei1 ^ei2 ^ (cid:1)(cid:1)(cid:1) ^ein = ai1j1(cid:24)j ^ ai2j2(cid:24)j ^ (cid:1)(cid:1)(cid:1) ^ ainjn(cid:24)j = j1 j2 jn X X X = sgn((cid:27))ai1j(cid:27)(1)ai2j(cid:27)(2) (cid:1)(cid:1)(cid:1) ainj(cid:27)(n)(cid:24)j1 ^(cid:24)j2 ^ (cid:1)(cid:1)(cid:1) ^(cid:24)jn = aIJ(cid:24)J ; 16j1<j2<(cid:1)(cid:1)(cid:1)<jn6n(cid:27)2Sn J where aIJ is (n(cid:2)n)-minor of (aij) pla ed at (i1;i2;:::;in) rows and (j1;j2;:::;jn) olumns, and J runs through all in reasing index olle tions of the length #J =n. P Exer i e 2.21. Let jIj = i(cid:23) denote a weight of the in reasing index olle tion I = (i1;i2;:::;in) of length #I = n. (cid:23) Che k that eI ^eIb=((cid:0)1)jIj+12#I(1+#I)(cid:1)e1^e2^ (cid:1)(cid:1)(cid:1) ^ed (8) def for any two omplementary index olle tions I and Ib= f1;2; ::: ;ng n I. Example 2.22: Lapla e relations via grassmannian polynomials. Let us take two omplementary index olle tions I and Xd def Ib = f1;2; ::: ;ng n I and do a base hange ei = aij(cid:24)j in the identity (8). Its left side eI ^eIbturns to j=1 (cid:18) (cid:19) (cid:18) (cid:19) X aIK(cid:24)K ^ X aLIb(cid:24)L =((cid:0)1)21#I(1+#I) X ((cid:0)1)jKjaIKaIbKb(cid:24)1^(cid:24)2^ (cid:1)(cid:1)(cid:1) ^(cid:24)d ; K: L: K: #K=#I #L=(d(cid:0)#I) #K=#I where Kb =f1;2; ::: ;dgnK. The right side of (8) gives ((cid:0)1)12#I(1+#I)((cid:0)1)jIjdet(aij)(cid:24)1^(cid:24)2^ (cid:1)(cid:1)(cid:1) ^(cid:24)d. So, for any any olle tion I of rows in any square matrix (aij) the following relation holds: X jKj+jIj ((cid:0)1) aIKbaIK =det(aij); (9) K: #K=#I def 4 where baIK = aIbKb denotes the (d(cid:0)n)(cid:2)(d(cid:0)n) - minor whi h is omplementary to aIK and the summation runs over all (n(cid:2)n) - minors aIK ontained in the rows (i1;i2;:::;in). If we take I\J 6=?, then starting from eI ^eJ =0 instead of (8) we get by the same al ulation the relation X jKj+jIj ((cid:0)1) aIKbaJK =0; (10) K: #K=#I The identities (9) and (10) areknown asLapla e relations. Let us (cid:12)x, say lexi ogra(cid:12) al,orderon the set of indi es I and (cid:0)arrangeall(n(cid:2)(cid:1)n)-minorsaIJ as(cid:0)nd(cid:1)(cid:2)(cid:0)nd(cid:1) -matrixA(n) d=ef (aIJ). IfwedenotebyAb(n) amatrixwhose(IJ)-entryequals ((cid:0)1)jIj+jJjbaJI , then all the Lapla e relations are expressed by the single matrix equality A(n)(cid:1)Ab(n) =det(aij)(cid:1)E. Example 2.23:Redu tionofgrassmannianquadrati forms. Any homogeneousgrassmannianpolynomial ofdegree2 anbe written as (cid:24)1^(cid:24)2+(cid:24)3^(cid:24)4+ (cid:1)(cid:1)(cid:1) +(cid:24)r(cid:0)1^(cid:24)r (11) 5 in some base (over any (cid:12)eld k). Namely, we an suppose that our grassmannianquadrati form is q(e)=e1^((cid:11)2e2+ (cid:1)(cid:1)(cid:1) +(cid:11)nen)+(terms without e1) def where (cid:11)2 6=0and (cid:24)2 = (cid:11)2e2+ (cid:1)(cid:1)(cid:1) +(cid:11)nen does not ontain e1, that is it an be in luded in the new base f(cid:16)1;(cid:16)2;:::;(cid:16)ng (cid:0)1 with (cid:16)i = ei for i 6= 2. After the substitution e2 = (cid:11)2 ((cid:24)2(cid:0)(cid:11)3(cid:16)3(cid:0) (cid:1)(cid:1)(cid:1) (cid:0)(cid:11)n(cid:16)n), ei = (cid:16)i for i 6= 2, we an write q as q((cid:16)) = (cid:16)1 ^(cid:16)2 +(cid:16)2 ^((cid:12)3(cid:16)3+ (cid:1)(cid:1)(cid:1) +(cid:12)n(cid:16)n)+(terms without (cid:16)1 and (cid:16)2). So, in the next new base: f(cid:24)1;(cid:24)2;:::;(cid:24)ng with (cid:24)1 =(cid:16)1(cid:0)(cid:12)3(cid:16)3(cid:0) (cid:1)(cid:1)(cid:1) (cid:0)(cid:12)n(cid:16)n, (cid:24)i =(cid:16)i for i6=1 our q turns to q((cid:16))=(cid:24)1^(cid:24)2+(terms without (cid:24)1 and (cid:24)2) and this pro edure an be repeated indu tively for the remaining terms. P Exer i e 2.24. Let A=(aij) be a skewsymmetri matrix (i. e. aij =(cid:0)aji) and q(e)= aijei^ej be a grassmannian ij quadrati form. Show that in the representation (11) the number r doesn't depend on the base hoi e and equals rkA. (In parti ular, rkA is always even.) 4 i. e. sitting in the omplementaryrows and olumns 5 maybe after appropriate renumberingof thebase ve tors Kernel of le ture ‚2, page 6 Commutative And Homologi al Algebra by ALG Kernel of le ture ‚3 1 Polarizations and ontra ions for (skew)polynomials (cid:10)n (Skew)symmetri tensors. A symmetri group Sn a ts on V permuting fa tors in the de omposable def tensors: (cid:27)(v1(cid:10)v2(cid:10) (cid:1)(cid:1)(cid:1) (cid:10)vn) = v(cid:27)(1) (cid:10)v(cid:27)(2) (cid:10)(cid:1)(cid:1)(cid:1)(cid:10)v(cid:27)(n) 8(cid:27)2Sn . Subspa es n (cid:10)n Skew V =ft 2V j (cid:27)(t) =sgn((cid:27))(cid:1)t 8(cid:27)2Sn g n (cid:10)n Sym V =ft 2V j(cid:27)(t) =t 8(cid:27)2Sn g are alled the spa es of skew symmetri and symmetri tensors. Claim 3.1. If har(k)=0, then, restri ting the anoni al fa torization maps: V(cid:10)n (cid:25)sk-ew (cid:3)nV , V(cid:10)n (cid:25)sy-m SnV , onto the spa es of (skew)symmetri tensors, we get anoni al isomorphisms: SkewnV (cid:25)sk-ew (cid:3)nV and SymnV (cid:25)sy-m SnV : (12) n Proof. In the skew symmetri ase, a base of Skew V is formed by the tensors X def ehi1;i2;:::;ini = sgn((cid:27))(cid:1)ei(cid:27)(1) (cid:10)ei(cid:27)(2) (cid:10) (cid:1)(cid:1)(cid:1) (cid:10)ei(cid:27)(n) (cid:27)2Sn (sum(cid:0)of all the te(cid:1)nsor monomials sent to the basi Grassmannian monomial eI = ei1 ^ei2 ^ (cid:1)(cid:1)(cid:1) ^ein by (cid:25)skew). So, (cid:25)skew ehi1;i2;:::;ini =n!eI.Inthesymmetri ase,letuswPritee[m1;m2;:::;md℄forthesumofalltensormonomials ontaining m1 fa tors e1, m2 fa tors e2, ::: , md fa tors ed, where m(cid:23) =n. These monomials form one Sn-orbit, whi h onsists (cid:23) n! m1 m2 md of m1!m2!(cid:1)(cid:1)(cid:1)md! elementsand olle tsall the de (cid:0)omposable(cid:1)tensorssentto e1 e2 (cid:1)(cid:1)(cid:1) ed by (cid:25)sym. Asabove,the tensors n n! m1 m2 md e[m1;m2;:::;md℄ form a base for Sym V and (cid:25)sym e[i1;i2;:::;in℄ = m1!m2!(cid:1)(cid:1)(cid:1)md!e1 e2 (cid:1)(cid:1)(cid:1) ed . (cid:3) n Exer i e 3.2. Verify that the above sums ehi1;i2;:::;ini and e[m1;m2;:::;md℄ really give the bases for Sym V (over any (cid:12)eld ofan arbitrary hara teristi ).Alsonotethatif har(k)>0dividesn, thenall thesebasi (skew)symmetri tensors are annihilated by fa torization through (skew)symmetri relations. (cid:10)n (n) n n (n) Exer i e 3.3. Verify that if har(k) = 0, then V = Iskew (cid:8) Skew V = Sym V (cid:8) Isym, where the proje tion V(cid:10)n -- SymnV along Is(ynm) is given by the symmetrization map 1 X symn : (cid:28) 7(cid:0)! (cid:27)((cid:28)) n! (cid:27)2Sn and the proje tion V(cid:10)n -- SkewnV along I(n) given by the alternation map skew 1 X altn : (cid:28) 7(cid:0)! sgn((cid:27))(cid:1)(cid:27)((cid:28)): n! (cid:27)2Sn Polarization of (skew)polynomials. The inverse maps to the isomorphisms (12) take 1 ei1 ^ei2 ^ (cid:1)(cid:1)(cid:1) ^ein 7(cid:0)! (cid:1)ehi1;i2;:::;ini n! (13) m1 m2 md m1!m2! (cid:1)(cid:1)(cid:1) md! e1 e2 (cid:1)(cid:1)(cid:1) ed 7(cid:0)! (cid:1)e[m1;m2;:::;md℄ : n! The both maps are alled omplete polarizations of (skew)polynomials and are denoted by f 7(cid:0)!pl(f). Example 3.4: (skew)polynomials and (skew)symmetri multilinear forms. Full polarization pl(f) of a (skew) homogeneous degree n polynomial f of one argument on V an be onsidered as a multilinear form of n arguments on V. It sends (v1;v2;:::;vn)tothefull ontra tionfe(v1;v2;:::;vn)d=ef h(cid:10)v1(cid:10)v2(cid:10) (cid:1)(cid:11)(cid:1)(cid:1) (cid:10)(cid:10)vn; p(cid:11)l(f)iandhasth(cid:10)esames(cid:11)ymm(cid:10)etryprope(cid:11)r- (cid:10)n (cid:3)(cid:10)n (cid:0)1 tiesasf,be auseforall(cid:27)2Sn,t2V ,(cid:24)2V wehave (cid:27)(t); (cid:27)((cid:24)) = t; (cid:24) ,whi himplies (cid:27)(t); (cid:24) = t; (cid:27) ((cid:24)) . 2 (cid:3) Exer i e 3.5. Che k that for a symmetri quadrati form q(x)2S V we have q(x+y)(cid:0)q(x(cid:0)y) q(x+y)(cid:0)q(x)(cid:0)q(y) 1 dXimV q qe(x;y)= = = y(cid:23) : 4 2 2 x(cid:23) (cid:23)=1 1 warning!!!In this le ture we suppose that theground(cid:12)eld k is algebrai ally losed and has zero hara teristi .