# Cohomology of Vector Bundles & Syzgies by Jerzy Weyman By Jerzy Weyman

The valuable subject of this booklet is an in depth exposition of the geometric means of calculating syzygies. whereas this can be an incredible device in algebraic geometry, Jerzy Weyman has elected to put in writing from the viewpoint of commutative algebra with the intention to steer clear of being tied to important instances from geometry. No earlier wisdom of illustration idea is believed. Chapters on numerous functions are integrated, and diverse workouts will supply the reader perception into tips on how to follow this crucial approach.

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Example text

7), because that statement implies that the map ψλ /µ is the dual of the map φλ/µ for E ∗ . 2. Schur Functors and Highest Weight Theory The modules L λ E play a crucial role in the representation theory of the general linear group. In this section we describe this connection. We assume that the commutative ring K is an inﬁnite ﬁeld of arbitrary characteristic. 50 Schur Functors and Schur Complexes Let us denote by T ( by U ) the subgroup of GL(E) of all diagonal matrices (all upper triangular matrices with 1’s on the diagonal) with respect to a ﬁxed basis e1 , .

Exu , V2 = e y1 ∧ . . ∧ e yλa −µa +λa+1 −µa+1 −u−v , V3 = ez1 ∧ . . ∧ ezv is a sum of tableaux, where we put in each tableau x 1 , . . , xu in the u empty boxes in the ﬁrst row, put z 1 , . . , z v in the v empty boxes in the second row, and shufﬂe the elements y1 , . . , yλa +λa+1 −u−v between the ﬁlled boxes in the ﬁrst and second rows, with the appropriate signs coming from exterior diagonal. If the number of parts of λ is bigger than 2, the relations R(λ/µ, E) can be interpreted in terms of tableaux as follows.

We denote by i the weight i (t1 , . . , tn ) = ti . 2. Schur Functors and Highest Weight Theory 51 The next step is to investigate how the elements of U change weights of vectors from V . We denote by Ai, j (x) the elementary endomorphism Ai, j (x)(es ) = es + xδs j ei . 3) Proposition. (a) Let V be a polynomial representation of GL(E), and W a subrepresentation. Let v ∈ Vχ . Then there exists a natural number r and elements v0 , . .