When a ring is a polynomial ring?

In the paper (2.11) the authors show that if k∗ is a separable algebraic extension of k and x1,x2,…,xn are indeterminates over k∗ and a normal one dimensional ring A with k⊂A⊂k∗[x1,x2,…,xn] then A has the form k′[t] where k′ is the algebraic closer of k in A. The above is a very strict sufficient … Read more

Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite k-algebras (k is a (preferably, but not necessarily finite) field) with one or more of the following properties: 1) the Auslander-Reiten quiver contains two cylinders (so, there are periodic modules), but also many non-periodic modules 2) the Auslander-Reiten quiver contains … Read more

Equivariant sheaves over affine schemes

Let k be a field, let G be a linear algebraic group over k and let A be a commutative k-algebra which is acted on by G. We say that an A-module M is a (G,A)-module if it satisfies the following two properties: 1) M is a rational G-module (over k) 2) The multiplication map … Read more

A right adjoint to the truncated Witt functor?

For any ring A, let wEtA be the category of weakly etale A-algebras ; it is a cocomplete category. By a theorem of Van der Kallen, the truncated Witt vector functor Wr:wEtA⟶wEtWr(A) is well-defined and commutes with all colimits. Can one explicit a right adjoint to Wr ? Answer AttributionSource : Link , Question Author … Read more

Does a polynomial system with precisely e solutions have a Groebner basis of degree bounded by e?

Let k be a field and let R=k[X1,…,Xn] be a polynomial ring. Let F⊂R be a finite subset generating a radical ideal I with precisely e solutions over an algebraic closure of k. Is there a monomial order on R with a Groebner basis of degree ≤e for I? Answer AttributionSource : Link , Question … Read more

When is a given polynomial a square of another polynomial?

I meet a problem in which I hope to show a special polynomial is not a square of another polynomial. More precisely, let’s consider the polynomial f(x):=1−x+2bxn−2bxn+1−b2x2n−1+2b2x2n−b2x2n+1−2bx3n−1+2bx3n−x4n−1+x4n∈k[x], where k is a field of characteristic p>0, n>2 is an integer, and b∈k with b≠0,1,−1. Indeed, in the context I meet, the field k is just the … Read more

polynomials satisfying the Plücker relation

Let $S_{12}$, $S_{13}$, $S_{14}$, $S_{23}$, $S_{24}$, $S_{34}$ be complex homogeneous polynomials in 4 variables satisfying the Plücker relation : $$S_{12}S_{34}-S_{13}S_{24}+S_{14}S_{23}=0 .$$ Suppose also that they don’t have any common non trivial zero. Let $d_{ij}=\deg(S_{ij})$. Let $d$ be the integer $$d=d_{12}+d_{34}=d_{13}+d_{24}=d_{14}+d_{23}.$$ By an indirect way (I used this to construct vector bundles on $\mathbb{P}_3$), I find … Read more

Formalism behind local characterizations of formal smoothness/unramifiedness/étaleness over algebraically closed fields

In synthetic differential geometry, one way to define formally étale morphisms is as follows. Say f:M→N is formally étale if TM≅TN×NM, in other words if the unique map from TM to the pullback is an isomorphism. (A generalization of) this approach is taken e.g in definition 8.10 of Kostecki’s notes. One could also say f … Read more

Methods to check if an ideal of a polynomial ring is prime

Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell – 1$ and $z_1, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, non-homogeneous) polynomials $q_i = z_{\ell}x_i^r – z_ix_{\ell}^r – (z_{\ell} – z_i)$ for $1 \leq … Read more