4 WAYNE AITKEN 2. The Determinant of Cohomology Functor (2.1) Determinants. Let V be a vector space of dimension n over a field K. We define the determinant of V, written Det V, to be /\n V, the nth order exterior product of V. The determinant of V is a one-dimensional K-vector space. When V is zero-dimensional, Det V is canonically isomorphic to the field K. For V and W two i^-vector spaces and a an isomorphism between them, we define Det a, the associated determinant map, to be the isomorphism from Det V and Det W induced by a. (2.2) The Determinant Ma p of a Sequence. Given an exact sequence (0 o —• vi - ^ v 2 - ^ ... ^ v n ^ o, we define the determinant map of this sequence Det £ to be the isomorphism ^ DetV1 i even jy- 8i0ddDetvi defined as follows: For i = 2,... , n , let 6^i,..., 6^m. be a basis of the kernel of /?*. For each bij, choose an element ci-ij of Vi-\ which maps to bij. Clearly 6^i,..., 6^m., c^i,..., Ci^mi+1 is a basis of V*, and so def = h^ A ... Ablirni ACi^i A... Aciyrni+1 is a non-zero element of Det Vi. We define Det £ by the rule d = X*even— ^ 1- ® i o d d * By the basic properties of exterior products, di is independent of the choice of ele- ments c^i,..., Ci?mi+1. mapping to 62+1,1,..., &i+i,mi+1 In other words, df depends only on the choice of basis (bi+ij), if 1 z n, and the basis (&i,j), if 1 i n. Although both di-i and di (for 1 i n) depend on the choice of (bij), it follows, from basic properties of exterior products, that the "ratio" di-i DetV1"1 dx DetV1 ' and hence d itself, is independent of this choice. Thus Det £ is well-defined. (2.3) The Determinant of Cohomology. Let Y be a scheme proper over a field if, and let J b e a coherent sheaf of Y. Define the determinant of cohomology of T, denoted D(^r), to be the one-dimensional K-vector space def gievenDet/T(^) DCF) ® l o d d D e t ^ ( ^ ) Given an isomorphism a between coherent sheaves T and ?, define T(a) to be the isomorphism between D(^") and D((?) given by the formula def ® - DetfP(a) riff*,} _ ^-^i even v ' 8 , o d d D e t ^ ( a ) ' where Hl(a) is the isomorphism between Hl(T) and Hl(Q) induced by a, and DetHl(a) is the determinant of this isomorphism.
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