Non-negative least squares

In algebraic geometry, given a smooth projective curve over a finite field $\mathbf {F} _{q}$ and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by $\operatorname {Bun} _{G}$, is an algebraic stack over the category of G-bundles on X that is given by: for any $\mathbf {F} _{q}$-algebra R,

$R\mapsto$ the category of (principal) G-bundles over the relative curve $X\times _{\mathbf {F} _{q}}\operatorname {Spec} R$.

In particular, the category of $\mathbf {F} _{q}$-points of $\operatorname {Bun} _{G}$, that is, $\operatorname {Bun} _{G}(\mathbf {F} _{q})$, is the category of G-bundles over X. Similarly, the moduli stack of principal bundles is defined when the curve X is over the field of complex numbers.

It is known that $\operatorname {Bun} _{G}$ is a smooth stack. It is not of finite type but of locally of finite type; one thus uses a stratification by open subsets of finite type (cf. the Harder–Narasimhan stratification.) If G is a split reductive group, then the set of connected components $\pi _{0}(\operatorname {Bun} _{G})$ is in a natural bijection with the fundamental group $\pi _{1}(G)$.

Behrend's formula

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. This is a version of Lefschetz trace formula for $\operatorname {Bun} _{G}$ when X is over a finite field, introduced by Behrend in 1993.

Notes

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References

• J. Heinloth, Lectures on the moduli stack of vector bundles on a curve, 2009 preliminary version
• J. Heinloth, A.H.W. Schmitt, The Cohomology Ring of Moduli Stacks of Principal Bundles over Curves, 2010 preprint, available at http://www.uni-essen.de/~hm0002/.

Further reading

• http://mathoverflow.net/questions/87171/tamagawa-number-for-functional-fields