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Determinants of Complexes APPENDIX A Determinants of Complexes Complexes Let k be a field. A complex of k-vector spaces is a graded vector space W· = EBieZ Wi together with a family of linear operators di+l Odi = o. The spaces Wi are called the terms of W·. Throughout this Appendix we shall assume that all but finitely many tenns of a complex are zero. Cohomology spaces of a complex w· are defined as Hi (W·) = Ker(di)/Im(di_l). A complex w· is called exact if all Hi (W·) are equal to zero. Example 1. Any linear operator between two vector spaces, say, w- 1 and Wo, can be regarded as a complex with only two non-zero tenns: Such a complex is exact if and only if d_1 is invertible. For any graded vector space w· and any m E Z we shall denote by W·[m] the same vector space but with the grading shifted by m: The same notation will be used for complexes. Let v· and w· be two complexes of vector spaces. A morphism of complexes J : V· -+ w· is a collection of linear operators /; : Vi -+ Wi which commute with the differentials in v· and w·. The cone of a morphism J is a new complex Cone(f) with tenns Cone(fi = Wi 6) Vi+l and the differential given by d(w, v) = (dw(w) + (-li+1J(v), dv(v»), WE Wi, V E Vi+l. (I) Any morphism of complexes J : V· -+ w· induces a morphism of cohomology spaces Hi (f) : Hi(V.) -+ Hi(W·). Amorphism J is called a quasi-isomorphism if all Hi (f) are isomorphisms. The derived category of vector spaces D(Vect) is obtained from the category of complexes of vector spaces by fonnally inverting the quasi-isomorphisms. Determinants of complexes 481 The proofs of several results in this Appendix and in the main body of the book require the use of more general derived categories involving, for example, complexes of modules over a ring R, complexes of sheaves etc. We refrain from giving more detailed background material on this subject referring the reader to first chapters in [Bo], [KS]. Determinantal vector spaces Let W be a vector space of dimension n over a field k. Denote the one-dimensional vector space 1\"(W) by Det(W). For W = 0, we set Det(O) = k. Let now W· = ffi Wi be a finite-dimensional graded vector space. We set (2) where V-I for a one-dimensional vector space V stands for the dual vector space V*. The correspondence W· t---+ Det(W·) defines a functor Det from the cat­ egory of finite-dimensional graded vector spaces and their isomorphisms to the category of one-dimensional vector spaces and isomorphisms. Let us list some basic properties of this functor. Proposition 2. (a) Det takes multiplication by )., E k* to the multiplication by ).,x(w·), where X(W·) = Li(-I)idim(Wi). (b) Let V· and W· be two graded vector spaces. Then there is a natural isomor­ phism (c) There is a natural isomorphism The proofs are obvious. The Euler isomorphism Let (W·, d) be a finite-dimensional complex of vector spaces. Forgetting the differential, we can regard W· simply as a graded vector space. The cohomology spaces of our complex form another graded vector space H·(W·) = EB Hi (W·). Proposition 3. There is a natural isomorphism 482 Appendix A We call E1ld the Euler isomorphism because of the following corollary. Corollary 4. We have an equality To deduce Corollary 4 from Proposition 3 it suffices to consider the action of A. e k* on Det(We ) and Det(He(we». Proof of Proposition 3. As in the standard proof of the above corollary, we split our complex into several short exact sequences: and 0-+ Im(dl-I) -+ Ker(dj ) -+ H'(we) -+ 0 for all i. The proof then reduces to the following particular case. Lemma S. For an exact sequence of vector spaces there is a natural isomorphism Det(B) ~ Det(A) ® Det(e). Proof. Let m, n, p be the dimensions of A, B, e respectively. We identify A with its image in B, and e with the quotient space B / A. Let m P al A ••• A am e Det(A) = /\(A), CI A··· ACp e Det(e) = /\(e). We define a map Det(A) ® Det(e) -+ Det(B) by sending where Cj e B is some representative of Cj e C = B / A. We leave it to the reader to check that this map is well-defined and establishes the desired isomorphism. Corollary 6. Let (we, d) be an exact complex ofk-vector spaces. Then there is a natural identification Determinants of complexes 483 Indeed, Det(H·(W·» is canonically identified with k. (We can easily avoid appealing to the detenninant of the zero vector space by splitting an exact complex into only one group of short exact sequences and then applying Lemma 5.) Note that even if we fix the tenns Wi, the isomorphism in Corollary 6 depends on d in a very essential way. The determinant of a based exact complex Let e = {ea, a = 1, ... , dim(W)} be a basis of a vector space W. Denote by Det(e) E Det(W) the wedge product el /\ ... /\ edim(W). Clearly, Det(e) is a basis (i.e., a non-zero vector) in Det(W). Now let W· = EB Wi be a graded vector space and suppose that e = {e(i)} is a system of bases in all Wi, so that each e(i) isa basis (e(ih, e(ih, ... , e(i)dim(WI)} in Wi. Let us define a non-zero vector Det(e) E Det(W·) as Det(e) = ®Det(e(i))Hi, i where for any non-zero element I of a I-dimensional vector space L the element I-I E L -I = L* is defined by I-I (I) = 1. We call a based complex a system (W·, d, e), where (W·, d) is a complex and e = (e(i)} is a system of bases in all tenns of w·. Definition 7. Let (W·, d, e) be a based exact complex. We call its determinant the number det(W·, d, e) = EUd(Det(e» E k*. Proposition 8. (a) Let W-I ~ WO be a two-term exact complex (i.e., just an isomorphism of vector spaces). Then det(W·, d, e) is the usual determinant ofthe matrix ofd with respect to the chosen bases. (b) For the shifted complex we have det(W·[I], d, e) = det(W·, d, e)-I. Proposition 9. Lete = {e(i)} and f = {f(i)} be two systems of bases in an exact complex (W·, d). Let {A (i)} be the transition matrices, i.e., we have dimWi f(i)p = L A(i)pqe(i)q. q=1 484 Appendix A Then we have det(W·, d, f) = det(W·, d, e) . ndet(A(i))(-I)i. i The proofs of Propositions 8 and 9 are obvious. Example 10. Suppose we have a three-term exact complex O~ A~ B~ C~O. Let {at. ... , am}, {bt. ... , bn }, {CI' ... , cp } be bases in A, B, C respectively, so n = m + p. Let us fix the grading of the complex so that B lies in degree O. Let us calculate the determinant of this exact complex with respect to the chosen bases. Denote by D-I and Do the matrices of d-l and do in the chosen bases; thus, D-I is an n x m matrix of rank m, and Do is a p x n matrix of rank p. Let D-I be the submatrix of D-I given by all the m columns and first m rows. Permuting if necessary the basis vectors bi (i.e., the rows of D_I) we can assume thatdet(D_I) =1= O. Denote by Do thesubmatrixof Do formed by the last p = n-m columns and all the p rows. It follows from our assumption that det( Do) =1= 0 either. Proposition 11. In the above situation, the determinant ofthe complex 0 ~ A ~ B ~ C ~ 0 with respect to the chosen bases is equal to w~:ri/. Proof. Let CI,.'" cp E B have the same meaning as in the proof of Lemma 5, i.e., do(cj) = Cj. Using the construction in Lemma 5, we see that the determinant in question is the coefficient of proportionality d_l(al) A··· A d_l(am ) A CI A··· A Cp bi A··· Abn Since Do is invertible, we can choose all the Cj to be linear combinations of bm+l , ••• , bn • Hence the transition matrix between the bases {d-I (aI), ... , d_ I (am), CI,"" cp } and {bI, ... , bn } in B has the block form (D;I D~-I). The determinant of our complex is the determinant of this matrix, i.e., is equal to det(D:j) as claimed. det(Do) , Thus we have several different ways to calculate the determinant of a three­ term exact complex: they correspond to different choices of a non-zero minor of order m of the matrix D-I. Determinants of complexes 485 The Cayley formula for the determinant of a complex Proposition 11 can be generalized to the case of an exact complex of arbitrary length. For simplicity we assume that the non-zero terms of the complex are situated in degrees between 0 and r > 0 (the general case can be reduced to this one by using Proposition 8 (b». We shall write our complex in the explicit coordinate form (3) so that a totally ordered set Bi is the indexing set for the chosen basis in the i-th term of the complex, and Di is a matrix with columns Bi and rows Bi+I.
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