# Questions tagged [local-rings]

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Consider the power series ring $\mathbf{Z}_p[[T]]$, where $\mathbf{Z}_p$ denotes the $p$-adic integers. I'll call a function $f(T) \in \mathbf{Z}_p[[T]]$ a rational function if I can write it as: f(... 1 vote 0 answers 116 views ### Artin-Winters proof of semi-stable reduction theorem: details I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail— Let \... 5 votes 1 answer 298 views ### About the structure of unit groups appearing in number theory I think the following statement is not true in the general situations, but consider it: R is a ring, \mathfrak{p} is a prime ideal, then the unit group of \dfrac{R}{\mathfrak{p}^nR} is ... 0 votes 0 answers 51 views ### Freeness of injective hull of finite module over Gorenstein Artinian ring This is Lemma 4.1 in Brochard, Khare, Iyengar: Wiles defect and criteria for freeness. I have managed to understand the proof, except for the following part: Why is F, the injective hull of a finite ... 2 votes 1 answer 75 views ### The quotient of an algebra with an ideal whose generators are decomposed as the product of irreducible elements I would like to find reference for the following statement. I'm also interested in a more general form of this statement. Let A be a commutative algebra (over \mathbb{R} or \mathbb{C}), I is ... 3 votes 0 answers 79 views ### On the descent of noetherianess along completion Let A be a commutative local ring with maximal ideal m and \hat{A} be its m-adic completion. Are there any non-trivial conditions on A, under which \hat{A} noetherian implies A ... 2 votes 1 answer 158 views ### Strict henselianization and branches of explicit curve at singularity Let A be a local ring, which we can assume is reduced. Let k be the residue field of A. In the Stacks project (https://stacks.math.columbia.edu/tag/06DT), I have learned some notion of the ... 3 votes 1 answer 166 views ### Vanishing of \operatorname{Ext}_R(\operatorname{Tr} M,N) and freeness criteria \DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\coker{coker}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Tor{Tor}I am investigating the interplay between freeness ... 8 votes 0 answers 184 views ### Image of multiplication map in tensor powers of finite-dimensional ring Let R be a (commutative, unital) ring of dimension n over a field k. Assume the characteristic of k is greater than n. Then R^{\otimes n} has a natural ring structure, together with an ... 2 votes 0 answers 75 views ### Order of the symplectic group over \mathbb{Z}/4\mathbb{Z} [duplicate] Let p be a prime number and q some power of it. It is well-known that the order of the symplectic group \text{Sp}_{2g}(\mathbb{F}_q) over the finite field \mathbb{F}_q equals q^{g^2}\prod_{i=... 3 votes 3 answers 316 views ### On the map \Phi_M: M\otimes_RM^*\xrightarrow{x\otimes y\mapsto \left\{f\mapsto f(x)y\right\}}\text{Hom}_R(M^*,M^*)  \DeclareMathOperator\Hom{Hom}Let M be a finitely generated module over a Noetherian local ring (R,\mathfrak m). Denote (\_)^*:=\Hom_R(\_,R). There is a natural map \begin{align} \Phi_M: M \... 2 votes 0 answers 50 views ### Division algorithm for multivariable power series Let \mathbb{Z}_p be the ring of p-adic integers. Consider the ring R=\mathbb{Z}_p[[T]]. Let f,g \in R and assume that f=a_0+a_1T+... with a_i \in p\mathbb{Z}_p for 0\le i \le n-1, but ... 2 votes 2 answers 246 views ### Why is M torsion-free? I am studying the following article https://www.math.nagoya-u.ac.jp/~takahashi/tc9.pdf The main theorem is the Theorem 3.3. Howewer, I have the following questions about the proof: How does it help ... 4 votes 0 answers 129 views ### Structure of \bigwedge^{2}_{\mathbb{Z}}(A) with A a local integral domain I am trying to see the structure of \bigwedge^{2}_{\mathbb{Z}}(A) where A is a local integral domain with small residue field. Let A be a local integral domain with maximal ideal M, residue ... 0 votes 0 answers 201 views ### When is u \circ v=v \circ u for p-adic power series u and v in two power series rings A and B respectively? Let K \supset \mathbb{Q}_p be the p-adic field with ring of integers O_K and maximal ideal m_K. Let \bar K be the algebraic closure and \bar{m}_K be the integral closure of m_K with ... 1 vote 1 answer 86 views ### Symbolic power of an ideal associated to non-singular algebraic set Let Z\subset \mathbb P^n be a reduced non-singular algebraic set and I denote the saturated homogeneous ideal of Z. I have seen the following result without proof: For all  n\geq 1, I^{(n)}=(... 0 votes 0 answers 82 views ### Complete local Noetherian ring whose normalized dualizing complex has depth equal to dimension of the ring Let (R, \mathfrak m,k) be a complete Noetherian local ring. Let D be a normalized dualizing complex of R. If \text{depth}_R D=\dim R, then must R be Cohen-Macaulay (i.e. must D be actually ... 2 votes 1 answer 162 views ### Good prime ideals in tensor products of local rings Let L/K be a field extension. Let R,S be two local commutative K-algebras and let \varphi : R \to S be a homomorphism of K-algebras, not assumed to be local. Let's call a prime ideal \... 4 votes 2 answers 284 views ### Ideals in Artinian Gorenstein local ring (R,\mathfrak m) with \mu(\mathfrak m)=2, \mathfrak m^2\ne 0 and \mathfrak m^3=0 Let (R,\mathfrak m,k) be an Artinian Gorenstein local ring such that\mu(\mathfrak m)=2, \quad\mathfrak m^2\ne 0,\quad\text{and}\quad \mathfrak m^3=0.$$Then, is it true that every non-maximal ... 5 votes 0 answers 206 views ### Unbounded derived Nakayama lemma Let R be a (commutative) local ring, which I don't assume to be noetherian. Let m be its maximal ideal, and k its residue field. Let X be a complex of R-modules with finitely generated ... 1 vote 0 answers 164 views ### Is the following local map unramified? Let (R,m) and (S,n) be two local rings, R \subseteq S, R regular, S a finitely generated and flat R-algebra, and mS=n. In comments to this question it was claimed that in such situation ... 1 vote 0 answers 103 views ### A \to B with A regular imply that B is CM The answer to this question says the following: "The general statement is if A \to B is finite and injective, and A is noetherian and regular, then B is CM if and only if A \to B is flat. ... 1 vote 1 answer 180 views ### Local rings R \subsetneq S with R regular and S Cohen-Macaulay, non-regular Let R \subseteq S be local rings with maximal ideals m_R and m_S. Assume that: (1) R and S are (Noetherian) integral domains. (2) \dim(R)=\dim(S) < \infty, where \dim is the Krull ... 1 vote 1 answer 180 views ### Flat and algebraic (non-integral) local rings extension R \subseteq S with m_RS=m_S Let R \subseteq S be two Noetherian local rings, not necessarily regular, which are integral domains, with m_RS=m_S, namely, the ideal in S generated by m_R (= the maximal ideal of R) is ... 5 votes 1 answer 223 views ### Hakim's definition of a locally ringed topos In Hakim's book "Topos annelés et schémas relatifs", Chap. III, Def. 2.3 states that a ringed topos (X,A) is a locally ringed topos when two equivalent conditions are satisfied: (i) For ... 0 votes 1 answer 231 views ### Given a unitary commutative ring R, what are the rings R\langle x,y\rangle/(x^2-A,y^2-B,yx-a-bx-cy-dxy) called We are studying the rings$$ R \langle x, \, y \rangle\,\big/\left(x^2-A, \, y^2-B, \, yx-a-bx-cy-dxy \right) $$Do you know if they have a name? 1 vote 1 answer 135 views ### Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property? Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property? The examples of rings not isomorphic to their opposite that I know of are not ... 6 votes 1 answer 173 views ### Cohomology of finite p-groups over integers in local fields Let p be a prime, G be a finite group of order p^a. Let M be a \mathbb{Z}[G]-module. Then H^n(G, M) is annihilated by p^a for all n \geq 1 (see e.g. Brown, Corollary III.10.2). In ... 5 votes 1 answer 267 views ### Kähler differentials on an Artinian local ring Suppose R is a commutative Artinian local ring over an algebraically closed characteristic 0 field k. Suppose f\in R is such that df=0 (in the sense that the element df vanishes in the ... 4 votes 0 answers 157 views ### Sections of smooth morphisms over henselian rings Let (A,\mathfrak m) be a henselian local ring. Let R and S be A-algebras of finite type and f\colon R\to S be a smooth morphism. Assume that the induced morphism R/\mathfrak m R\to S/\... 5 votes 1 answer 463 views ### Do you know which is the minimal local ring that is not isomorphic to its opposite? The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite. 1 vote 0 answers 75 views ### What would be the quotient groups U_{\mathrm{gen}}/U_{\mathrm{gen}}^{(n)} and U_{\mathrm{gen}}^{(n)}/U_{\mathrm{gen}}^{(n+1)}? Let K \supseteq \mathbb{Q}_p be a p-adic field with ring of integer O and maximal ideal m. Let O^* be the group of units in O. Consider the group of units U^{(0)}=U=O^* and U^{(n)}=1+m^... 2 votes 1 answer 118 views ### When can we decompose a multivariable p-adic power series into product of single variable power series? Is there any known result of decomposing multivariable power series over p-adic field into product of single variable power series ? For example, consider the following power series in n variables:... 9 votes 2 answers 572 views ### Ideal norm in orders Let \overline{T} be a Dedekind ring such that \overline{T}/\overline{I} is finite for every nonzero ideal \overline{I} of \overline{T}. Let T be a subring of \overline{T} with the same ... 8 votes 0 answers 204 views ### Finitely generated commutative rings with the same profinite completion Let R_1 and R_2 be two finitely generated commutative rings. Assume that their profinite completions are isomorphic: \widehat{R_1}\cong \widehat{R_2}. Suppose that R_1 is a domain. Does ... 6 votes 2 answers 323 views ### Does the category of local rings with residue field F have an initial object? Let F be a field. Does the category C_F of local rings R equipped with a surjective morphism R\longrightarrow F have an initial object? This is, for instance, true if F=\mathbb{F}_{p} for ... 3 votes 2 answers 310 views ### Isomorphism between finite algebras over {\Bbb Z}_p Let \pi \colon R \twoheadrightarrow {\Bbb T} be a surjective ring homomorphism between finite algebras over {\Bbb Z}_p. Further, we suppose the following three conditions\colon R is a ... 4 votes 1 answer 143 views ### injective hulls in mixed characteristic Let R=\underleftarrow\lim (R/\mathfrak m^i) be a complete local ring, with residue field k=R/\mathfrak m, and let's assume that R is Noetherian. If R is a k-algebra, then I believe that ... 10 votes 2 answers 707 views ### Krull dimension of a local ring and completion Let A be a local ring (not noetherian) of finite Krull dimension such that its maximal ideal \mathfrak{m} is of finite type. Let \hat{A} be its \mathfrak{m}-adic completion. Do we have that \... 0 votes 0 answers 288 views ### A question about strict henselian local rings Let f: X\to Y be a finite morphism of schemes, let y\in Y, Y(\bar{y}):={\rm Spec}(\mathscr{O}_{Y, \bar{y}}), X_{\bar{y}}:=X\times_Y{\rm Spec}(\kappa_y^s)=X_y\otimes_{\kappa_y}\kappa_y^s, X({\bar{y}}... 1 vote 0 answers 60 views ### Generators for Ideals in ring of multivariate Laurent Polynomials Consider the following problem: Find an ideal I \subset \mathbb{Q}[x^{\pm}_1,x^{\pm}_2,x^{\pm}_3] such that I_{aff} \subset \mathbb{Q}[x_1, x_2, x_3] = I \cap k[x_1, x_2, x_3] requires more ... 1 vote 1 answer 176 views ### Power series ring \Theta[[X_1,\ldots,X_d]] and prime ideals Let \Theta be a domain. We shall choose d elements \theta_1,\ldots,\theta_d \in \Theta such that any chosen j elements \theta_{i_1},\ldots,\theta_{i_j} form a prime ideal (\theta_{i_1},\... 0 votes 0 answers 269 views ### On the product in the power series ring Let A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]] be a power series ring over a field K in 2n variables and {\frak m}_{A_n} be the unique maximal ideal of A_n. Suppose we have two ... 1 vote 0 answers 147 views ### Structure of Complete Local Rings 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. ... 7 votes 1 answer 252 views ### Additive group of local rings Is there a theory or characterization for those finite p-groups that can be considered as the additive group of a finite local commutative ring with identity? 0 votes 1 answer 166 views ### Finite extension of K[[X]] and the norm Let R \colon= K[[X]] be a formal power series ring over a field K. We consider a monic polynomial f(T) \in R[T] as follows\colon$$ f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0.  ...
Consider two local morphisms $f,g: B\rightarrow A$ of noetherian complete local rings and $f$ surjective. Does there exist an integer $n\in\mathbb{N}$, such that if $f=g \mod \mathfrak{m}_{A}^{n}$ ...