Computational techniques for sheaf cohomology of locally profinite sets

TL;DR

It is shown that questions of intermediate cohomology degrees can be reduced to questions about top cohomology degrees by exhibiting nontrivial top cocycles as pointwise limits of coboundaries.

Abstract

We compute the sheaf cohomology with constant $\mathbb{Z}_2$ coefficients of a concrete class of locally profinite sets of independent interest. We introduce $k$-Fubini partitions to aid in constructions, which witness a failure of a Fubini theorem analog for these spaces. It is also shown that questions of intermediate cohomology degrees can be reduced to questions about top cohomology degrees by exhibiting nontrivial top cocycles as pointwise limits of coboundaries.

Figures (4)

• Figure 1: The facets $F_A$ and basic open set $\mathcal{N}_A(\vec{x}, \vec{\mathcal{E}})$. Here $\vec{\kappa} = \langle \aleph_0, \aleph_1, \aleph_2 \rangle$ and $A = \{1\}$.
• Figure 2: Given a 2-cocycle $\vec{f} = \langle f_{012}, f_{013}, f_{023}, f_{123} \rangle$ and a $k$-Fubini partition $\langle Y_0,Y_1,Y_2,Y_3 \rangle$, the above piecewise definition of $\langle g_B : B \in [4]^2\rangle$ defines a trivializer of $\vec{f}$. (For brevity the columns are labeled $Y_i$ rather than $\mathcal{D}_A \cap Y_i$.)
• Figure 3: The naive extension of a 1-cochain $\langle f_{01},f_{02},f_{12} \rangle$ defined on $X(\aleph_0,\aleph_0,\aleph_0)^-$ (left) to $\langle f_{01}^\infty,f_{02}^\infty,f_{12}^\infty \rangle$ defined on $X(\aleph_0,\aleph_0,\aleph_1)^-$ (right). Notice that the latter is not necessarily a cocycle.
• Figure 4: Given a 1-cochain $\vec{f} = \langle f_{01}, f_{02}, f_{03}, f_{12},f_{13},f_{23} \rangle$ and a $k$-Fubini partition $\langle Y_0,Y_1,Y_2,Y_3 \rangle$, the above piecewise definition of $\langle f^{\mathop{\mathrm{mod}}\nolimits}_A : A \in [4]^2\rangle$ modifies $\vec{f}$ to a cocycle without affecting its values at any limit points. (Again, for brevity the columns are labeled $Y_i$ rather than $\mathcal{D}_A \cap Y_i$.)

Theorems & Definitions (33)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4: aoki
• Theorem A
• Theorem B
• Corollary 2.5
• Theorem C
• Proposition 3.1
• Lemma 3.2