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Let $R$ a commutative ring with one and $I, J \triangleleft R$ two ideals. The saturated ideal $I^{sat}_J$ with respect $J$ is the ideal $$(I : J^\infty )= \cup_{n \geq 1} (I:J^n)$$ where $(I:J^n)= \ …

Let $f: X \to S$ a finite morphism between affine schemes $X=Spec(A), S= Spec(R)$. Denote by $\phi:R \to A$ the corresponding ring map. I'm looking for pure ring theoretical/algebraic tools/criterion …

Sorry for a not research level question asking for a definition but unfortunately I nowhere found a source which explains the construction presented below in a satisfactory way. This question refers …

Let $g: A \to B$ be a local ring morphism between local Noetherian (commutative) rings $A,B$ (so $g(m_A) \subset m_B$ for the unique maximal ideals of the corresponding rings). Assume that the induced …

Let $R$ be integral domain and $p \neq 0$ a prime ideal. It's well known that in category of $R/p$ modules the injective hull of $R/p$ is $K=\operatorname{Frac}(R/p)$. Is there a successful theory kno …

That's the second part of my coarse becoming acquainted with Henselizations of fields and local rings. (in this question we focus on local rings as it is more algebro geometric motivated). So let $(R …

Consider two schemes $X,Y$ over a locally noetherian scheme $S$. Let $p \in X$ and assume that $X$ is irreducible and not affine spectrum of a semilocal ring. We assume moreover we have a morphism $ …

I have following question about so-called "principle of degeneration" in algebraic geometry (which in modern terms is an immediate consequence of Zariski's main theorem and goes in it's original form …

Let $\mathbb{Z}_p$ the ring of $p$-adic numbers. It's known that the multiplicative unit group $\mathbb{Z}_p ^\times$ can be set theoretically described as $\bigcup _{1 \le a \le p-1} a+ p\mathbb{Z}_p …

Let $X$ be a proper $n$-dimensional $k$-scheme and $x \in X$ a closed point. Consider the stalk $\mathcal{O}_{X,x}$. We consider now it's completion $O_{X,x}^{\wedge}$ wrt it's maximal ideal $m_x$. M …

Consider the complete intersection ideal ${\displaystyle (f,g_{1},g_{2},g_{3})\subset \mathbb {C} [x_{0},\ldots ,x_{n}]}$ and let $X$ be a projective variety defined by the (product) ideal sheaf ${\di …

I have question about a statement from Etale Cohomology and the Weil Conjecture by Freitag, Kiehl at the top of page 16. It seemingly uses the same notations as introduced at the bottom of page 15 and …

Let $\phi:A \to B $ an injective finite ring map between two noetherian integral domains $A,B$. Let $ C \subset B$ a subring of $B$ and assume that there exist a prime ideal $\mathfrak{p} \subset A$ , …

Let $A= \bigoplus_{n \ge 0} A_n$ be a commutative Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined map …

I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces": Here the relevant excerpt: Remark …