Algebraic Geometry

Get Equivariant Sheaves and Functors PDF

By Joseph Bernstein

ISBN-10: 3540580719

ISBN-13: 9783540580713

The equivariant derived type of sheaves is brought. All ordinary functors on sheaves are prolonged to the equivariant scenario. a few functions to the equivariant intersection cohomology are given. the idea should be precious to experts in illustration concept, algebraic geometry or topology.

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The equivariant derived class of sheaves is brought. All traditional functors on sheaves are prolonged to the equivariant state of affairs. a few purposes to the equivariant intersection cohomology are given. the speculation might be necessary to experts in illustration conception, algebraic geometry or topology.

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9. If Q(v1) = Q(v2))4 0 then Q(vl + V2) = 2Q(vi) + 2B(vi, V2) = 2B(vi, V1 + v2), implying Tv1+v2 (vl) = vl - (vl + v2) = -V2; likewise Tv1 _7J2 (vl) = V2. These facts help prove that the reflections generate the orthogonal group; cf. Exercises A12-A17. We are ready for the two major basic structure theorems of Witt. 10 (Witt Decomposition Theorem). Any quadratic space (V, Q) is an orthogonal sum of a totally isotropic space, a hyperbolic space, and an anisotropic space (any of which can be 0).

3'. 3(1), given submodules {Mi : i E I} of M, we define E Mi to be the set of all finite sums of elements from the Mi; this is the smallest submodule of M containing each Mi. 4. 1) f (ra) = r f (a) for all a, b in M and all r in R. Module homomorphisms are also called maps, and we favor this terminology, in order to avoid confusion with ring homomorphisms. 5. To check that f : M -* N is a map, one needs show that f (a + b) =f(a)+f(b) and f(ra) = r f (a) for all a, b in M and all r in R. Note. Recall that in linear algebra, the vector space V = F(') is often studied in terms of the ring of linear transformations from V to itself, which is identified with the matrix ring M,,(F).

3. The image of a homomorphism should be a structure. 4. The "kernel" of a homomorphism is of special interest to us, since it indicates what information is lost by applying the homomorphism. 5. We want to obtain Noether-type isomorphism theorems which relate the above concepts, via the notion of a factor structure. Rather than proceed formally with this program, we take several illustrative examples. Vector spaces and linear algebra 13 1. A monoid (M, 1, ) has the distinguished element 1 and the binary operation of multiplication, satisfying the sentences VaEM:1 dai E M : (ala2)a3 = al(a2a3) 2.

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Equivariant Sheaves and Functors by Joseph Bernstein

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