Homological percolation and the Euler characteristic

The Euler Characteristic

I will describe a few basic properties of the Euler characteristic and then I use them to prove special case of a cute formula due to Bernstein-Khovanskii-Koushnirenko. 1. Basic properties of the Euler characteristic The Euler characteristic is a function χ which associates to each reasonable topological space X an integer χ(X). For us a reasonable space would be a space which admits a finite s...

Exponentiation and Euler Characteristic

Mixed Characteristic Homological Theorems in Low Degrees

Let R be a locally finitely generated algebra over a discrete valuation ring V of mixed characteristic. For any of the homological properties, the Direct Summand Theorem, the Monomial Theorem, the Improved New Intersection Theorem, the Vanishing of Maps of Tors and the Hochster-Roberts Theorem, we show that it holds for R and possibly some other data defined over R, provided the residual charac...

Secondary Characteristic Classes and the Euler Class

We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We then show that if X is a smooth affine scheme of dimension d over a field k of finite 2cohomological dimension (with char(k) 6= 2) and E is a rank d vector bundle over X , vanishing of the Chow-Witt theoretic Euler class of E is equivalent to vanishing of its top Chern class an...

Asymptotic Homological Conjectures in Mixed Characteristic

In this paper, various Homological Conjectures are studied for local ringswhich are locally finitely generated over a discrete valuation ring V of mixed characteristic.Typically, we can only conclude that a particular Conjecture holds for such a ring providedthe residual characteristic of V is sufficiently large in terms of the complexity of the data, where the complexity is primari...