Sheaves on non-reduced curves can appear in moduli spaces of 1-dimensional semistable sheaves over a surface and Higgs bundles as well. We estimate the dimension stack Mx (nC, χ) pure supported at curve nC (n ≽ 2) with C an integral X. prove that Hilbert-Chow morphism $${h_{L,\chi}}:{\cal M}_X^H\left({L,\chi} \right) \to \left| L \right|$$ sending each sheaf to its support has all fibers same f...

08MX In this chapter we write about derived categories associated to algebraic stacks. This means in particular derived categories of quasi-coherent sheaves, i.e., we prove analogues of the results on schemes (see Derived Categories of Schemes, Section 1) and algebraic spaces (see Derived Categories of Spaces, Section 1). The results in this chapter are different from those in [LMB00] mainly be...

In this chapter we write about derived categories associated to algebraic stacks. This means in particular derived categories of quasi-coherent sheaves, i.e., we prove analogues of the results on schemes (see Derived Categories of Schemes, Section 1) and algebraic spaces (see Derived Categories of Spaces, Section 1). The results in this chapter are different from those in [LMB00] mainly because...

0. Algebraic background 1. Projective sets and their ideals; Weak Nullstellensatz 2. Irreducible components 3. Hilbert polynomial. Nullstellensatz 4. Graded modules; resolutions and primary decomposition 5. Dimension, degree and arithmetic genus 6. Product of varieties 7. Regular maps 8. Properties of morphisms 9. Resolutions and dimension 10. Ruled varieties 11. Tangent spaces and cones; smoot...

1. The principal object of this note is to give an extremely simple axiomatic characterization of the Euler characteristic for finite polyhedra. The idea of the construction used was suggested by the definition of the Grothendieck group of sheaves over a space [l]. Unless otherwise noted, "space" always means "triangulable space with base point," and a "pair" is a "triangulable pair of spaces w...

The category of combinatorial systems is a model category for ”scaling up” algorithms. This involves networks of algorithms, herein called schedules, acting on very wide range of inputs types that are outputs from other algorithms. This patching algorithms over networks gives rise to the subcategory of combinatorial modules which are an algebraic analogy for the creation of manifolds from Eucli...

It has been long expected that there exists a deep connection between singularities of certain arc spaces and harmonic analysis over nonarchimedean fields. For instance, certain functions appearing naturally in harmonic analysis can be interpreted as the function attached to the trace of the Frobenius operator on what should be the stalks of the intersection complex of certain arc spaces, see [...

Microlocal perverse sheaves form a stack on the cotangent bundle T ∗X of a complex manifold that is the analogue of the stack of perverse sheaves on the manifold X itself. We give an embedding of the stack of microlocal perverse sheaves into a simpler stack on T ∗X which we call melded Morse systems. The category of melded Morse systems is elementary, in the sense that an object is a collection...