# Tagged Questions

Questions about commutative rings, their ideals, and their modules.

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### Any characterization for commutative rings over which “projective modules” equal “free modules”?

As far as I know, over any PID, an polynomial rings over a field, or an local ring, projective modules are always free. This kind of results make me curious about if there are any overall ...
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### Showing a Variety is Rational?

I'm trying to show that the following varieties are rational: $V_1=V(y^2z-x^3)$ and $V_2=V(xyz-x^3-y^3)$. But I can't think of how to show they are birationally equivalent to $\mathbb{A}^n$ or ...
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### Definition algebra over commutative ring (injectivity needed?)

I'm aware that many differently looking definitions exist for an algebra over a commutative ring $A$. For me, the most natural definition of an algebra over a commutative ring $A$ consists of a tuple ...
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### Why are Newton polygons only ever treated over complete DVRs?

Let $R$ be a discrete valuation ring, and $f(x) := c_nx^n + \cdots + c_1x + c_0\in R[x]$ a polynomial with $c_nc_0\ne 0$. Then the newton polygon of $f(x)$ is the lower convex hull of the points ...
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### Completion of a local ring, Vakil 29.3A

If $p$ is a point of $X$, which is a $\bar{k}$ variety of dimension $1$, $p$ is a node if the completion of $\mathcal{O}_{X,p}$ at $m_{X,p}$ is isomorphic to $\bar{k}[[x,y]]/(xy)$. If now ...
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### Are rings of fractions of integral domains closed under finite intersection?

Let $D$ be an integral domain with fraction field $K$. Let $V$, $W$ be multiplicatively closed subsets of $D$. Consider the rings of fractions $V^{-1}D$ and $W^{-1}D$ as subrings of $K$. Is ...
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### If $f$ is a polynomial and $g(n+1)-g(n)=f(n)$, then $g$ is a polynomial. [closed]

Assume that $f$ is a polynomial of degree $s$ which is not constant, and that for sufficiently large positive integers $n$, $g(n+1)-g(n)=f(n)$. Here $g$ is defined on the positive integers. Must ...
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### Conditions for a ring to be a direct product of local rings

I recently came across a property of commutative rings which I could prove only for rings that are (isomorphic to) a direct product of (possibly infinitely many) local rings. It might be that my ...
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### Minimal graded free resolution of the ideal $I = (x^r, y^s) \subset k[x,y]$

What is the minimal free-graded resolution of the ideal $I = (x^r, y^s) \subset k[x,y]=R$ for $r,s \in \mathbb{N}$? I tried reducing this down to $r = s = 1$ and I think it is 0 \to R(-2) \to ...
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### dimension of an affine variety over $\mathbb C.$ [closed]

Consider the polynomials $f,g,h \in \mathbb C [x_1, \dots,x_n]$ defined by $f(x)=x_1^2 + \cdots + x_{n-2}^2, \quad g(x)=x_{n-1},$ and $h(x)=x_n.$ How can I compute the dimension of the variety ...
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### Variant of Nakayama's lemma

I am trying to prove that if $M$ is an $R$-module, with $R$ complete w.r.t. an ideal $\mathfrak{m}$, and $M$ is separated ($\cap_k \mathfrak{m}^k M=0$) and the images of $m_1,\dots,m_n$ generate ...
I have encountered a lot the concept of zero-dimensional ideal: Let $k$ be a field. An ideal $I\subseteq k[x_1,...,x_m]$ is said to be zero-dimensional if its zero set $Z(I)$ has a finite number ...
### Induced maps between coordinate rings of $\mathcal Z(xy-z)$ and $\mathbb A^2$
I'm trying to understand the answer here to the question of finding the induced maps of coordinate rings corresponding to explicit isomorphisms between $\mathcal Z(xy-z)$ and $\mathbb A^2$. A ...