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Marcin Chałupnik

Publications and source records attributed to Marcin Chałupnik.

7 recordsLinked to original sources
arXiv2022

On strict polynomial functors with bounded domain

We introduce a new functor category: the category $\mathcal{P}_{d,n}$ of strict polynomial functors with bounded by $n$ domain of degree $d$ over a field of characteristic $p>0$. It is equivalent to the category of finite dimensional modules over the Schur algebra $S(n,d)$, hence it allows one to apply the tools available in functor categories to representations of the algebraic group $\operatorname{GL}_n$. We investigate in detail the homological algebra in ${\cal P}_{d,n}$ for $d=p$ and establish equivalences between certain subcategories of ${\cal P}_{d,n}$'s which resemble the Spanier-Whitehead duality in stable homotopy theory.

math.RT↗
arXiv2022

On spectra and affine strict polynomial functors

We compare derived categories of the category of strict polynomial functors over a finite field and the category of ordinary endofunctors on the category of vector spaces. We introduce two intermediate categories: the category of $\infty$--affine strict polynomial functors and the category of spectra of strict polynomial functors. They provide a conceptual framework for compuational theorems of Franjou--Friedlander--Scorichenko--Suslin and clarify the role of inverting Frobenius morphism in comparing rational and discrete cohomology.

math.KT↗
arXiv2020

Difference sheaves and torsors

We develop sheaf theory in the context of difference algebraic geometry. We introduce categories of difference sheaves and develop the appropriate cohomology theories. As specializations, we get difference Galois cohomology, difference Picard group and a good theory of difference torsors.

math.AG↗
arXiv2016

On Serre functor in the category of strict polynomial functors

We introduce and study a Serre functor in the category ${\cal P}_d$ of strict polynomial functors over a field of positive characteristic. By using it we obtain the Poincaré duality formula for Ext--groups from [C3] in elementary way. We also show that the derived category of the category of affine strct polynomial functors in some cases carries the structure of Calabi-Yau category.

math.KT↗
arXiv2014

Affine strict polynomial functors and formality

We introduce the notion of affine strict polynomial functor. We show how this concept helps to understand homological behavior of the operation of Frobenius twist in the category of strict polynomial functors over a field of positive characteristic. We also point out for an analogy between our category and the category of representations of the group of algebraic loops on $GL_n$.

math.KT↗