Homogeneous Groups and Covers
This paper is my 2014 Cornell Senior Thesis, written under the supervision of R. Keith Dennis. It contains results on homogeneous covers and generating sequences of finite groups.
math.GR↗
arXiv subjects
Eric Primozic
Publications and source records attributed to Eric Primozic.
On universal stably free modules in positive characteristic
We study the mod $p$ motivic cohomology of homogeneous varieties such as $GL_{n}/GL_{r}$ or $Sp_{2n}/Sp_{2n-2}$ along with the action of the Steenrod operations, without restrictions on the characteristic of the base field. In particular, we prove that certain quotient maps do not admit sections.
math.AG↗
Motivic cohomology and infinitesimal group schemes
For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ \'etale motivic cohomology of the classifying stack $\mathcal{B}G.$ This also gives a cycle class map into the Hodge cohomology of $\mathcal{B}G.$ We study the cycle class map for some examples, including Frobenius kernels.
math.AG↗
Computations of de Rham cohomology rings of classifying stacks at torsion primes
For the split group $G_{2}$ defined over $\mathbb{Z},$ we show that the de Rham cohomology ring of $B(G_{2})_{\mathbb{F}_{2}}$ is isomorphic to the singular cohomology ring with $\mathbb{F}_{2}$-coefficients of $B(G_{2})_{\mathbb{C}}.$ For the spin groups $\textrm{Spin}(n)$ defined over $\mathbb{Z},$ we show that the de Rham cohomology ring of $B\textrm{Spin}(n)_{\mathbb{F}_{2}}$ is isomorphic to the singular cohomology ring with $\mathbb{F}_{2}$-coefficients of $B\textrm{Spin}(n)_{\mathbb{C}}$ for $n \leq 10.$ For $n=11,$ we make a full computation of the de Rham cohomology ring of $B\textrm{Spin}(11)_{\mathbb{F}_{2}},$ which is not isomorphic to the singular cohomology ring with $\mathbb{F}_{2}$-coefficients of $B\textrm{Spin}(11)_{\mathbb{C}}.$ We also show that the Hodge spectral sequence for $BG_{\mathbb{F}_{2}}$ degenerates for all of the groups $G$ mentioned above.
math.AG↗
Motivic Steenrod operations in characteristic $p$
Using the recent work of Frankland and Spitzweck, we define Steenrod operations $P^{n}$ on the mod $p$ motivic cohomology of smooth varieties defined over a base field of characteristic $p$. We show that $P^{n}$ is the $p$th power on $H^{2n,n}(-,\mathbb{F}_{p})\cong CH^{n}(-)/p$ and prove an instability result for the operations. Restricted to mod $p$ Chow groups, we show that the operations satisfy the expected Adem relations and Cartan formula. For $p=2$, we use the new Steenrod squares to obtain new results on quadratic forms over a base field of characteristic $2$.
math.AG↗