Kai Behrend
We introduce perfect resolving algebras and study their fundamental properties. These algebras are basic for our theory of differential graded schemes, as they give rise to affine differential graded schemes. We also introduce etale morphisms. The purpose for studying these, is that they will be used to glue differential graded schemes from affine ones with respect to an etale topology.
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Graded perfectoid rings
本文提出分级完美环的概念，建立了其与分级完美棱镜的范畴等价性，并证明了分级版本的André平坦性引理，为分级完美环理论奠定了基础。
Reflecting Perfection for Finite Dimensional Differential Graded Algebras
本文将有限维代数的中山引理与单模检测有限投射维数两个事实推广到有限维微分分次代数，证明两个分别关联Gorenstein微分分次代数与Koszul对偶的对偶版本，并给出余可表示性应用。
We construct a 2-category of differential graded schemes. The local affine models in this theory are differential graded algebras, which are graded commutative with unit over a field of characteristic zero, are concentrated in non-positive degrees and have perfect cotangent complex. Quasi-isomorphic differential graded algebras give rise to 2-isomorphic differential graded schemes and a differential graded algebra can be recovered up to quasi-isomorphism from the differential graded scheme it defines. Differential graded schemes can be glued with respect to an etale topology and fibered produc…
Recently, Okounkov, Lazarsfeld and Mustata, and Kaveh and Khovanskii have shown that the growth of a graded linear series on a projective variety over an algebraically closed field is asymptotic to a polynomial. We give a complete description of the possible asymptotic growth of graded linear series on projective schemes over a perfect field. If the scheme is reduced, then the growth is polynomial like, but the growth can be very complex on nonreduced schemes. We also give an example of a graded family of m-primary ideals {I_n} in a nonreduced d-dimensional local ring R, such that the length o…
We prove that for any finite-dimensional differential graded algebra with separable semisimple part the category of perfect modules is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme with a full separable semi-exceptional collection. Moreover, we also show that it gives a characterization of such categories assuming that a subcategory is idempotent complete and has a classical generator.
As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the free affine $p$-adic group scheme functor, as well as the free formal group functor, defined on $k$-algebras for a perfect field $k$ of characteristic $p$, factors through $p$-polar $k$-algebras. It follows that the same is true for any affine $p$-adic or formal group functor, in particular for the functor of $p$-typical Witt vectors. As an application, we sho…
The Jacobian algebra arising from a consistent dimer model is a bimodule $3$-Calabi-Yau algebra, and its center is a $3$-dimensional Gorenstein toric singularity. A perfect matching of a dimer model gives the degree making the Jacobian algebra $\mathbb{Z}$-graded. It is known that if the degree zero part of such an algebra is finite dimensional, then it is a $2$-representation infinite algebra which is a generalization of a representation infinite hereditary algebra. Internal perfect matchings, which correspond to toric exceptional divisors on a crepant resolution of a $3$-dimensional Gorenste…
In this paper we discuss different properties of noncommutative schemes over a field. We define a noncommutative scheme as a differential graded category of a special type. We study regularity, smoothness and properness for noncommutative schemes. Admissible subcategories of categories of perfect complexes on smooth projective schemes provide natural examples of smooth and proper noncommutative schemes that are called geometric noncommutative schemes. In this paper we show that the world of all geometric noncommutative schemes is closed under an operation of a gluing of differential graded cat…
Graded $p$-polar rings and the homology of $Ω^nΣ^nX$
本文定义了分次 $p$-极化环，证明了仿射 $p$-进群概型函子与形式群函子可经由 $p$-极化代数分解，并揭示了自由 $E_n$-代数的同调仅函子化地依赖于底空间的 $p$-极化结构。
Isocrystals and de Rham-Witt connections
本文在拓扑空间的微分分次代数层上引入可积联络概念，并在形式概形及正特征光滑概形的 de Rham-Witt 复形等有限局部投射情形下加以刻画，从而为带 Frobenius 结构的收敛与过收敛等晶体提供了新描述。
Let $k$ be a perfect field of characteristic $p>0$, and $S$ an scheme over $k$. An $F$-zip is basically a locally free $O_S$-module of finite rank endowed with two filtration and an Frobenius-linear isomorphism between their graded pieces. The natural generalization of this notion for a reductive algebraic group $G/k$ is an "$F$-zip with $G$-structure", a so-called $G$-zip introduced by R. Pink, T. Wedhorn, P. Ziegler. A $G$-zip $I$ over $S$ yields the stratification of the base scheme in loci, where $I$ has locally a constant isomorphism class for the fppf topology. We show that these strata …
Following the approach of Haiden-Katzarkov-Kontsevich arXiv:1409.8611, to any homologically smooth graded gentle algebra $A$ we associate a triple $(Σ_A, Λ_A; η_A)$, where $Σ_A$ is an oriented smooth surface with non-empty boundary, $Λ_A$ is a set of stops on $\partial Σ_A$ and $η_A$ is a line field on $Σ_A$, such that the derived category of perfect dg-modules of $A$ is equivalent to the partially wrapped Fukaya category of $(Σ_A, Λ_A ;η_A)$. Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of $Σ_A$ on the h…
We show that the multivariate additive higher Chow groups of a smooth affine $k$-scheme $\Spec (R)$ essentially of finite type over a perfect field $k$ of characteristic $\not = 2$ form a differential graded module over the big de Rham-Witt complex $\W_mΩ^{\bullet}_{R}$. In the univariate case, we show that additive higher Chow groups of $\Spec (R)$ form a Witt-complex over $R$. We use these structures to prove an étale descent for multivariate additive higher Chow groups.
The Derived Auslander-Iyama Correspondence
本文研究了导出的Auslander-Iyama对应关系，通过高维版本将扭曲的(d+2)-周期代数与具有dZ-簇化对象的三角范畴联系起来，并证明了这些范畴具有唯一的微分格增强。
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