Related papers: The generating hypothesis in the derived category …

We study the category of Sp-equivariant modules over the infinite variable polynomial ring, where Sp denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely…

Given a numerical semigroup $S$ and a positive integer $d$, the fraction $\frac{S}{d}=\{ x \in \mathbb{N} \ | \ dx \in S\}$ is again a numerical semigroup. In this paper we determine a generating set for $\frac{S}{d}$ in terms of the…

We investigate compact projective generators in the category of equivariant $D$-modules on a smooth affine variety. For a reductive group $G$ acting on a smooth affine variety $X$, there is a natural countable set of compact projective…

In this work we construct a compactly generated tensor-triangulated stable category for a large class of infinite groups, including those in Kropholler's hierarchy $\mathrm{LH}\mathfrak{F}$. This can be constructed as the homotopy category…

We determine Grothendieck groups of periodic derived categories. In particular, we prove that the Grothendieck group of the $m$-periodic derived category of finitely generated modules over an Artin algebra is a free $\mathbb{Z}$-module if…

It has been conjectured that finite tensor categories have finitely generated cohomology. We show that this is equivalent to finitely generated Hochschild cohomology for the endomorphism algebras of the projective generators.

In this paper we study the derived categories of coherent sheaves on Grassmannians $\operatorname{Gr}(k,n),$ defined over the ring of integers. We prove that the category $D^b(\operatorname{Gr}(k,n))$ has a semi-orthogonal decomposition,…

We revisit a construction of wide subcategories going back to work of Ingalls and Thomas. To a torsion pair in the category $ R\operatorname{-}\operatorname{mod}$ of finitely presented modules over a left artinian ring $R$, we assign two…

For a non-Archimedean local field $F$ of residue cardinality $q=p^r$, we give an explicit classical generator $V$ for the bounded derived category $D_{fg}^b(\mathsf{H}_1(G))$ of finitely generated unipotent representations of…

We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors ?^{**} and D tensor ^{**}? tensor D^{-1}. This…

We show that the homotopy category of complexes K(B) over any finitely accessible additive category B is locally well generated. That is, any localizing subcategory L in K(B) which is generated by a set is well generated in the sense of…

Let $R$ be a commutative Noetherian ring and let $\mathcal D(R)$ be its (unbounded) derived category. We show that all compactly generated t-structures in $\mathcal D(R)$ associated to a left bounded filtration by supports of Spec$(R)$ have…

We define discrete generating series for arbitrary functions \( f \colon \mathbb{Z}^n \rightarrow \mathbb{C} \) and derive functional relations that these series satisfy. For linear difference equations with constant coefficients, we…

We show that in an essentially small rigid tensor triangulated category with connected Balmer spectrum there are no proper non-zero thick tensor ideals admitting strong generators. This proves, for instance, that the category of perfect…

Let $K[X_d]=K[x_1,\ldots,x_d]$ be the polynomial algebra in $d$ variables over a field $K$ of characteristic 0. The classical theorem of Weitzenb\"ock from 1932 states that for linear locally nilpotent derivations $\delta$ (known as…

Let $\mathcal{C}$ be a small category. We investigate ringed sites $(\mathbf{C},\mathfrak{R})$ on $\mathcal{C}$ and the resulting module categories $\mathfrak{M}{\rm od}\text{-}\mathfrak{R}$. When $\mathcal{C}$ is finite, based on…

We furnish any category of a universal (co)homology theory. Universal (co)homologies and universal relative (co)homologies are obtained by showing representability of certain functors and take values in $R$-linear abelian categories of…

We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolving subcategories of finitely generated…

We consider recollements of derived categories of dg-algebras induced by self orthogonal compact objects obtaining a generalization of Rickard's Theorem. Specializing to the case of partial tilting modules over a ring, we extend the results…

We establish a foundational homotopical framework for ternary $\Gamma$-modules by establishing that $\mathcal{T}\text{-Mod}$ is a Barr-exact, monoidal closed category. We resolve the long-standing "additivity obstruction" in non-binary…