Derived Algebraic Geometry III: Commutative Algebra

Discrete probabilistic and algebraic dynamics: a stochastic commutative Gelfand-Naimark Theorem

We introduce a category of stochastic maps (certain Markov kernels) on compact Hausdorff spaces, construct a stochastic analogue of the Gelfand spectrum functor, and prove a stochastic version of the commutative Gelfand-Naimark Theorem. This relates concepts from algebra and operator theory to concepts from topology and probability theory. For completeness, we review stochastic matrices, their relationship to positive maps on commutative $C^*$-algebras, and the Gelfand-Naimark Theorem. No knowledge of probability theory nor $C^*$-algebras is assumed and several examples are drawn from physics.

A characterisation of Lie algebras amongst anti-commutative algebras

Let $\mathbb{K}$ be an infinite field. We prove that if a variety of anti-commutative $\mathbb{K}$-algebras - not necessarily associative, where $xx=0$ is an identity - is locally algebraically cartesian closed, then it must be a variety of Lie algebras over $\mathbb{K}$. In particular, $\mathsf{Lie}_{\mathbb{K}}$ is the largest such. Thus, for a given variety of anti-commutative $\mathbb{K}$-algebras, the Jacobi identity becomes equivalent to a categorical condition: it is an identity in~$\mathcal{V}$ if and only if $\mathcal{V}$ is a subvariety of a locally algebraically cartesian closed var…

Model structure on differential graded commutative algebras over the ring of differential operators

We construct a cofibrantly generated model structure on the category of differential non-negatively graded quasi-coherent commutative $D_X$-algebras, where $D_X$ is the sheaf of differential operators of a smooth afine algebraic variety X. The paper contains an extensive appendix on D-modules, sheaves versus global sections, some more technical model categorical issues, as well as on relative Sullivan algebras. This article is the first of a series of works -located at the interface of homotopical algebra, algebraic geometry, and mathematical physics - on a derived D-geometric approach to the …

Tensors, !-graphs, and non-commutative quantum structures (extended version)

!-graphs provide a means of reasoning about infinite families of string diagrams and have proven useful in manipulation of (co)algebraic structures like Hopf algebras, Frobenius algebras, and compositions thereof. However, they have previously been limited by an inability to express families of diagrams involving non-commutative structures which play a central role in algebraic quantum information and the theory of quantum groups. In this paper, we fix this shortcoming by offering a new semantics for non-commutative !-graphs using an enriched version of Penrose's abstract tensor notation.

Holomorphic Functions of Exponential Type and Duality for Stein Groups with Algebraic Connected Component of Identity

We suggest a generalization of Pontryagin duality from the category of commutative Stein groups to the category of (not necessarily commutative) Stein groups with algebraic connected component of identity. In contrast to the other similar generalizations, in our approach the enveloping category consists of Hopf algebras (in a proper symmetrical monoidal category).

On the moduli stack of commutative, 1-parameter formal Lie groups

We attempt to develop a general algebro-geometric study of the moduli stack of commutative, 1-parameter formal Lie groups. We emphasize the pro-algebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of n-buds, and these latter stacks are algebraic. Our main results pertain to aspects of the height stratification relative to fixed prime p on the stacks of buds and formal Lie groups. We conclude with a largely expository account of some foundational material on limits in bicategories.

非交换Gelfand对偶：代数情形

Noncommutative Gelfand Duality: the algebraic case

该研究借助导出代数几何技术定义了非交换Gelfand对偶，证明环范畴与预环化位点子范畴反等价，提出的新谱的相对精细化版本在复有限生成交换代数情形与Grothendieck谱一致，为量子时空几何研究奠基。

Reflexive functors of modules in Commutative Algebra

Reflexive functors of modules naturally appear in Algebraic Geometry, mainly in the theory of linear representations of group schemes, and in "duality theories". In this paper we study and determine reflexive functors and we give many properties of reflexive functors.