Related papers: The telescope conjecture for semihereditary commut…
The classical honeycomb conjecture asserts that any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. Pappus discusses this problem in his preface to Book V. This paper…
The paper is withdrawn by the author. Parts of the contents are expanded into separate papers; hep-th/0308015 LOCALIZED TACHYON MASS AND A G-THEOREM ANALOGUE, hep-th/0308028 COMMENTS ON THE FATE OF UNSTABLE ORBIFOLDS, hep-th/0308029 CHIRAL…
We develop some foundations of commutative algebra, with a view towards algebraic geometry, in symmetric tensor categories. Most results establish analogues of classical theorems, in tensor categories which admit a tensor functor to some…
The purpose of this article is to prove that Gersten's conjecture for a commutative regular local ring is true. As its applications, we will prove the vanishing conjecture for certain Chow groups, generator conjecture for certain $K$-groups…
In this short paper we outline (mostly without proofs) our new approach to the derived category of sheaves of commutative DG rings. The proofs will appear in a subsequent paper. Among other things, we explain how to form the derived…
We prove descent theorems for semiorthogonal decompositions using techniques from derived algebraic geometry. Our methods allow us to capture more general filtrations of derived categories and even marked filtrations, where one descends not…
We consider Takahashi's categorical interpretation of the Berglund-Hubsch mirror symmetry conjecture for invertible polynomials in the case of chain polynomials. Our strategy is based on a stronger claim that the relevant categories satisfy…
This paper investigates a Tate algebra version of the Jacobian conjecture, referred to as the Tate-Jacobian conjecture, for commutative rings $R$ equipped with an $I$-adic topology. We show that if the $I$-adic topology on $R$ is Hausdorff…
This paper has been withdrawn by the author due to a crucial error in the proof of Lemma 2.2.
We give three proofs that valuation rings are derived splinters: a geometric proof using the absolute integral closure, a homological proof which reduces the problem to checking that valuation rings are splinters (which is done in the…
A new proof of the classification for tensor ideal thick subcategories of the bounded derived category, and the stable category, of modular representations of a finite group is obtained. The arguments apply more generally to yield a…
In this paper, we introduce and investigate \emph{semicorings} over associative semirings and their categories of \emph{semicomodules.} Our results generalize old and recent results on corings over rings and their categories of comodules.…
A commutative ring is reduced when it can be embedded into a direct product of fields. While the category of reduced commutative rings plays a fundamental role in affine geometry, it exhibits several structural deficiencies: it admits…
This paper has been withdrawn due to non-clearness of some technical points, as well as lack of a reasonable statement of quantization conjecture.
The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's…
We determine the Balmer spectrum of dualisable objects in the stable module category for $\mathrm{H}_1\mathfrak{F}$ groups of type $\mathrm{FP}_{\infty}$ and show that the telescope conjecture holds for these categories. We also determine…
Let A be a commutative ring, and let \a = \frak{a} be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived \a-torsion and \a-adic completion functors to be nicely behaved is the weak…
This paper has been withdrawn by the author due to the version of [A complete proof of Hamilton's conjecture] at arXiv:1008.1576
The second author studied arithmetic properties of a class of sequences that generalize the sequence of derangements. The aim of the following paper is to disprove two conjectures stated in \cite{miska}. The first conjecture regards the set…
In this paper we study the probability that the commutator of a randomly chosen pair of elements, one from a subring of a finite ring and other from the ring itself equals to a given element of the ring.