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Let $\mathsf{k}$ be an algebraically closed field and $X$ an abstract variety (an integral separated scheme of finite type over $\mathsf{k}$). Is there any way to define the usual dimension of $X$ usi …

Expanding the last entry on noncommutative transcendence degrees: When your algebra $A$ is prime Goldie (such as Noetherian domains) there are two recent analogues of noncommutative transcendence degr …

Noether's problem is a famous problem in invariant theory, introduced in the 1910's by Emmy Noether in relation to the inverse Galois problem. It is as follows: Noether's Problem: Let $F=k(x_1,\dotsc, …

Let $\mathbb{k}$ be an algebraically closed field of positive characteristic, $X$ an affine smooth variety over it. Then the ring of crystalline differential operators on $X$ is generated by $\mathcal …

In this question, I am going to consider noncommutative projective algebraic geometry, as introduced by Artin and Zhang in the seminal paper Noncommutative projective schemes. The $\operatorname{Proj} …

To each $3d \, N=4$ supersymmetric quantum field theory $\mathcal{T}$, there is a related space called the Coulomb branch of this theory, $\mathcal{M}_C(\mathcal{T})$ (it is a piece of the moduli spac …

Let $k$ be any algebraically closed field of zero characteristic. Let $A_n$ be the n rank Weyl algebra, and $M$ a finitely generated module. We have that $GK(M)$ is always a positive integer and (Bers …

For the sake of simplicity, let $\mathsf{k}$ be algebraically closed and of zero characteristic. Varieties are irreducible. The (linear) Noether's Problem (which goes back to the early 1910's in Burns …

As mentioned in the answer by user6976, there is the idea of development of algebraic geometry to (essentialy) any general algebraic system. This is carried out(following Plotkin's work) by E. Daniyar …

To give you some personal background: I am a ring theorist, and most of my research focus on invariant theory of noncommutative rings. Recently I became interested in a certain problem that requires a …

Suppose the base field algebraically closed and of zero characteristic. There are two fascinating questions in the intersection of ring theory and algebraic geometry (for which an excellent discussion …

This question is a follow up to my previous question on rings of crystaline differential operators, to which I refer for the adequate definitions. First, let's consider the case of an algebraically cl …

In his seminal paper, Some open problems on three-dimensional graded domains, M. Artin proposed a very small list of possible division rings of fractions that can appear as 'noncommutative function fi …

Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector fiel …

I will follow the definition of Coulomb branches of $3d$ $\mathcal{N}=4$ gauge theories from the paper by Braverman, Finkelberg and Nakajima, Towards a mathematical definition of Coulomb branches of 3 …