By Piotr Pragacz

The articles during this quantity are dedicated to:

- moduli of coherent sheaves;

- primary bundles and sheaves and their moduli;

- new insights into Geometric Invariant Theory;

- stacks of shtukas and their compactifications;

- algebraic cycles vs. commutative algebra;

- Thom polynomials of singularities;

- 0 schemes of sections of vector bundles.

The major goal is to offer "friendly" introductions to the above issues via a sequence of accomplished texts ranging from a truly common point and finishing with a dialogue of present examine. In those texts, the reader will locate classical effects and strategies in addition to new ones. The booklet is addressed to researchers and graduate scholars in algebraic geometry, algebraic topology and singularity thought. many of the fabric provided within the quantity has no longer seemed in books before.

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**Extra info for Algebraic cycles, sheaves, shtukas, and moduli**

**Example text**

Then for every coherent sheaf E on Cn we have R(E ⊗ D) = R(E), Deg(E ⊗ D) = Deg(E) + R(E) deg(D). m . It follows that a coherent sheaf E of positive rank is semi-stable (resp. , it has no subsheaf with ﬁnite support) and if for every proper subsheaf F ⊂ E we have Deg(E) Deg(F ) ≤ R(F ) R(E) (resp. < ). Moduli Spaces of Coherent Sheaves on Multiples Curves 37 4. Quasi locally free sheaves Let P ∈ C and z ∈ On,P be a local equation of C. Let M be a On,P -module of ﬁnite type. Then M is called quasi free if there exist nonnegative integers m1 , .

Ext2 (T, O) = T , which pure of codimension 2, with the Exotic Fine Moduli Spaces of Coherent Sheaves 27 We suppose now that Exti (F ∗∗ , G) = {0} if i ≥ 1 (which is true if G = 0). Then from (∗) we deduce that Ext1 (F, G) Ext2 (T, G) = G0 (p) with p 1 ∗ Hom(G , T ). It is easy to see that the element of Ext (F, G) associated to (∗) is λ. Let Γ = ker(λ). We have an exact sequence 0 −→ F ∗ −→ E ∗ −→ Γ −→ 0, (∗∗) Ext1 (Γ, F ∗ ) Hom(F ∗∗ , T ), and the element of Ext1 (Γ, F ∗ ) associated to (∗∗) is the quotient map F ∗∗ → T .

If OR (1) is very ample, then a linearization is the same thing as a representation of G on the vector space H 0 (OR (1)) such that the natural embedding R → P(H 0 (OR (1))∨ ) is equivariant. 1 (Categorical quotient). Let R be a scheme endowed with a G-action. A categorical quotient is a scheme M with a G-invariant morphism p : R → M such that for every other scheme M , and G-invariant morphism p , there is a unique morphism ϕ with p = ϕ ◦ p RC CC CCp p CC C! 2 (Good quotient). Let R be a scheme endowed with a G-action.