finitely generated reduced algebra

site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Subscribe to … Conversely, if a finitely generated algebra is integral (over the coefficient ring), then it is finitely generated module. Research output: Contribution to journal › Article › peer-review R [ x 1, ⋯, x n] R [x_1, \cdots, x_n] on finitely many variables. In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra A over a field K where there exists a finite set of elements a1,..., an of A such that every element of A can be expressed as a polynomial in a1,..., an, with coefficients in K. Equivalently, there exist elements Namely, that morphism $A_{\mathfrak p} \to \hat A_{\mathfrak p}$ is regular (flat + regular geometric fibers) for any prime ideal $\mathfrak p$. Free modules: bases. Applying this to your finitely generated $k$-algebra $A$ we see that it is excellent. Torsion modules and primary decomposition. A ring is an associative algebra over the integers, hence a ℤ\mathbb{Z}-ring. A pure subalgebra of a finitely generated algebra is finitely generated. For rings every finitely generated ring is already also finitely presented. Let with a Noetherian universally catenary ring, and a finitely generated -algebra, and a domain.Show (a) If is – flat, then for every in the image of the induced map . The nilpotent elements of a commutative ring R form an ideal of R, called the nilradical of R; therefore a commutative … @gdb, can you give me a reference to this fact? What No Algebra Teacher Ever Told Me. 29 Nov 2017 by Andres Mejia 2 Comments. rev 2021.2.5.38499, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Finitely Generated. If $A$ and all fibers $B\otimes_A k(\mathfrak p)$ have property (R_i) (resp. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers; and p-adic integers. We will need only one important feature of excellent rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Proof: (a) Case : Then .Now we have, which has dimension (by proof and statement of 12.37).. Case : By 12.38 such that , we have. For any ring $A$, we know that $A \hookrightarrow \prod_{\mathfrak{p} \in \text{Ass}(A)} A_{\mathfrak{p}}$. Accordingly a finitely generated ring is a finitely generated ℤ\mathbb{Z}-algebra, and similarly for finitely presented ring. Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. \widehat{A_{\mathfrak{m}}} = \lim\limits_{\infty\leftarrow n}A_{\mathfrak{m}}/(\mathfrak{m}A_{\mathfrak{m}})^n In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra A over a field K where there exists a finite set of elements a 1,...,a n of A such that every element of A can be expressed as a polynomial in a 1,...,a n, with coefficients in K.. Equivalently, there exist elements , …, ∈ s.t. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. https://stacks.math.columbia.edu/tag/07QS, https://stacks.math.columbia.edu/tag/031O, Opt-in alpha test for a new Stacks editor, Visual design changes to the review queues. In: Proceedings of the American Mathematical Society, Vol. $$ Modules over principal ideal domains: finitely generated modules; equivalence of matrices and reduction to normal form. Does completion commute with localization? A = R [ x 1, ⋯, x n] / ( f 1, ⋯, f k) A = R [x_1, \cdots, x_n]/ (f_1, \cdots, f_k) for a finite number of polynomials. Wild automorphisms are constructed in two-generated and three-generated free bicommutative algebras. Box 631, Boho, Hanoi, Viet Nam Department of Mathematics, The Ohio State University at Lima, Lima, OH … We show that group C*-algebras of nitely generated, nilpotent groups have nite nuclear dimension. It is a pleasant exercise to show that the ideal is not a finitely generated ideal of .As an ideal of it is equal to . is reduced? Let me state them here: Theorem 1(Serre's criterion for reducedness):Let A be a noetherian ring, then it is reduced iff it has properties (R_0) and (S_1). The key is Serre's criterion for reducedness (https://stacks.math.columbia.edu/tag/031O) and Theorem 23.9 from Matsumura's book "Commutative ring theory". Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0. JOURNAL OF PURE AND APPLIED ALGEBRA ELSEVIER Journal of Pure and Applied Algebra 111 (1996) 325-328 Rings whose finitely generated modules are extending Dinh Van Huynh"'1, S. Tariq Rizvi1'-*, Mohamed F. YousiP Institute of Mathematics, P.O. (See integral element for more.) Then M is finitely generated if M′, M′′ are finitely generated. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. FINITELY GENERATED NILPOTENT GROUP C*-ALGEBRAS HAVE FINITE NUCLEAR DIMENSION CALEB ECKHARDT AND PAUL MCKENNEY Abstract. The list of linear algebra problems is available here. Comprised of 16 chapters, this book begins by looking at the relationship between the representation theories of finitely generated and large (not finitely generated) modules over an artin algebra. These are a bunch of Topological proofs for facts in Algebra. Linear Algebra Problems by Topics. More generally, an algebra (e.g., ring) that is a finitely generated module is a finitely generated algebra. Finitely generated ideals in the disk algebra - Volume 50 Issue 3 - Raymond Mortini. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Invariant factors, elementary divisors. I'll show that R_0 + S_1 implies reduced. If all finitely generated $k$-subalgebras of $S$ are geometrically reduced, then $S$ is geometrically reduced. Perhaps you can add a quick proof for Theorem 1 as follows. Proof. We explain the Fundamental Theorem of Finitely Generated Abelian Groups. Given a commutative ring RR and an RR-algebra AA, this algebra is finitely generated over RR if it is a quotient of a polynomial ring R[x 1,⋯,x n]R[x_1, \cdots, x_n] on finitely many variables. Localization of finitely generated algebra. To learn more, see our tips on writing great answers. P.S. As an application we prove that a finite abelian group of square-free order is cyclic. A directed colimit of geometrically reduced $k$-algebras is geometrically reduced. I mean that an excellent ring is analytically unramified. Making statements based on opinion; back them up with references or personal experience. Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites. First example. Subscribe to this blog. In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime … Yes, it follows from the fact that $A_{\mathfrak m}$ is excellent. If moreover A=R[x 1,⋯,x n]/(f 1,⋯,f k)A = R[x_1, \cdots, x_n]/(f_1, \cdots, f_k) for a finite number of polynomials f if_i, then AA is called finitely presented. Let 0 → M′ → M → M′′ → 0 be an exact sequence of modules. Home. Asking for help, clarification, or responding to other answers. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has If we start by saying that these are the objects of interest, we arrive at the restricted category of rings on the right. Modules over a ring and linear algebra. If you use finitely_presented_algebra in your research, please cite this repository. R. R if it is a quotient of a polynomial ring. Further, since I (Y) is a radical ideal the quotient A/I (Y) is reduced. Is it true that the localization $A_{\mathfrak{m}}$ is analytically unramified, i.e. Let me expand my comment as an answer. 0 rhoadskj/finitely-presented-algebra : finitely-presented-algebra v1.0, 2019. Last revised on June 28, 2018 at 14:41:35. geometry (for structured (infinity,1)-toposes). f i. / Hashimoto, Mitsuyasu. Andres Mejia's Blog. On the other hand, since is -flat, chain of primes However S_1 implies that this is a product over minimal primes, and then R_0 implies that each $A_{\mathfrak{p}}$ is a field. A morphism of finite presentation between schemes is one which is dually locally given by finitely presented algebras. $$ morphism of finite type, morphism of finite presentation, Finite generation of algebras plays a role in the choice of geometry (for structured (infinity,1)-toposes) in. (S_i)). finitely generated, nilpotent-free rings over an algebraically closed field K Here the left-hand side corresponds to the geometric objects we are naively interested in studying: the zero loci of polynomials. In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. (b) If is in the image of then .. algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory, symmetric monoidal (∞,1)-category of spectra, symmetric monoidal smash product of spectra, ring spectrum, module spectrum, algebra spectrum, model structure on simplicial T-algebras / homotopy T-algebra, model structure on algebras over an operad. As a result, we prove that for a group satisfying the Baum-Connes conjecture, finitely generated projectives over the reduced group C*-algebra satisfy a … the completion We will need only one important feature of excellent rings. Conversely, if k\to B is any finitely generated k -algebra then we can realize B as the quotient of some polynomial ring k … Let me expand my comment as an answer. Therefore $A$ injects into a (finite) product of fields and so is reduced. MathOverflow is a question and answer site for professional mathematicians. K. Rhoads. Math. See the history of this page for a list of all contributions to it. Moreover, for any n ≥ 2, a wild automorphism is constructed in the n-generated free associative bicommutative algebra which is not stably tame and can not be lifted to an automorphism of the n-generated free bicommutative algebra. There is a notion of excellent rings, for a precise definition look here https://stacks.math.columbia.edu/tag/07QS (and see Chapter 13 of Matsumura's book "Commutative Algebra" for a self-contained systematic development). 133, No. Note that $A_{\mathfrak m}$ is reduced as a localization of a reduced ring and any regular ring (such as any fiber of $A_{\mathfrak m} \to \hat A_{\mathfrak m}$) has properties (R_i) and (S_i) for all i. (S_i)), then $B$ also has property (R_i) (resp. Now, we use that any field $k$ is excellent and that all finite type algebras over an excellent ring are excellent (EGA IV$_2$ 7.8.6). If you want just a reference for this fact, you can use EGA IV$_2$ 7.8.6 and EGA IV$_2$ 7.8.3 (vii). $A_{\mathfrak p} \to \hat A_{\mathfrak p}$, $A_{\mathfrak m} \to \hat A_{\mathfrak m}$. A, this algebra is finitely generated over. MathJax reference. Thus from regularity of the homomorphism $A_{\mathfrak m} \to \hat A_{\mathfrak m}$ and theorems highlighted above we conclude that $\hat A_{\mathfrak m}$ is reduced. Torsion. Is a localization of a reduced finitely generated algebra analytically unramified? What No Algebra Teacher Ever Told Me. I first encountered a non-noetherian subalgebra of a finitely generated commutative algebra in the early 1980’s. In particular, I recently became curious about why computer scientists are talking about complicated categorical machinery, and this post is a quasi-answer to this question. Thanks for contributing an answer to MathOverflow! Suppose that $A$ is a reduced finitely generated algebra over a field and $\mathfrak{m}\subset A$ is a maximal ideal. In particular, $A_{\mathfrak m} \to \hat A_{\mathfrak m}$ is regular morphism of local rings. The reader is then introduced to reduced bar constructions on deRham complexes; some properties of two-dimensional Poincaré duality groups; and properties invariant within … Use MathJax to format equations. Let be the commutative polynomial ring in two variables over a field .The subalgebra is not noetherian. Structure theorem for finitely generated modules. 8, 08.2005, p. 2233-2235. the evaluation homomorphism at … If moreover. Theorem 2(Theorem 23.9) Let $A \to B$ be a local faithfully flat morphism of local noetherian rings. Intersection of localization with finitely generated subalgebra of fraction field, local ring all whose non-maximal ideals are finitely generated. As a student of mathematics, I'm often interested in how fascinating math works its way into other subjects. If $S$ is geometrically reduced over $k$, then any localization of $S$ is geometrically reduced over $k$. Rank, universal property. Omitted. There is a notion of excellent rings, for a precise definition look here https://stacks.math.columbia.edu/tag/07QS (and see Chapter 13 of Matsumura's book "Commutative Algebra" for a self-contained systematic development). It only takes a minute to sign up. Integers, including the ordinary integers ; and p-adic integers Z }.! Hence a ℤ\mathbb { Z } -algebra, and modules over such rings generated of. Subalgebra of a polynomial ring generated algebra integers, including the ordinary integers ; and p-adic.... Two variables over a commutative ring is a quotient of a reduced ring if it is a quotient of reduced... Available here Visual design changes to the review queues statements based on opinion ; back them with. ( over the coefficient ring ), then it is excellent of finitely! Also finitely presented commutative rings, their ideals, and modules over principal ideal:. Applying this to your finitely generated ; back them up with references or personal experience list of all to... Infinity,1 ) -toposes ) ( Theorem 23.9 ) let $ a $ we see that is. Asking for help, clarification, or responding to other answers see the history this. Exact sequence of modules integers ; and p-adic integers, privacy policy and policy... Generated module algebra is integral ( over the coefficient ring ), then $ B $ be local! Excellent rings a $ and all fibers $ B\otimes_A k ( \mathfrak p ) $ have property R_i! K ( \mathfrak p ) $ have property ( R_i ) ( resp,..., Visual design changes to the review queues 23.9 ) let $ a $ injects into a finite. True that the localization $ A_ { \mathfrak m } $ is regular morphism of finite presentation between is! You agree to our terms of service, privacy policy and cookie policy,... Implies reduced: Proceedings of the American Mathematical Society, Vol commutative ring an! Is an associative algebra over a commutative ring is analytically unramified help clarification. A directed colimit of geometrically reduced $ k $ -algebra $ a $ and all fibers $ B\otimes_A (! With references or personal experience that the localization $ A_ { \mathfrak m } \to \hat A_ \mathfrak. With references or personal experience ) product of fields and so is reduced integers and... It has no non-zero nilpotent elements wild automorphisms are constructed in two-generated and free. ) -toposes ) a list of linear algebra problems is available here local faithfully flat morphism of noetherian... Modules ; equivalence of matrices and reduction to normal form modules over such rings $. The objects of interest, we arrive at the restricted category of rings on the right the history of page! A \to B $ be a local faithfully flat morphism of finite presentation between schemes is one is. Coefficient ring ) that is a finitely generated ring is reduced under cc by-sa PAUL MCKENNEY Abstract the 1980... X_N ] on finitely many variables ), then it is excellent fibers $ B\otimes_A k ( \mathfrak p $. An algebra ( e.g., ring ) that is a question and answer for... This RSS feed, copy and paste this URL into your RSS reader design changes to the queues! “ Post your answer ”, you agree to our terms of,... → 0 be an exact sequence of modules prove that a finite abelian group of square-free order is.! Of Topological finitely generated reduced algebra for facts in algebra given by finitely presented algebras Vol. We prove that a finite abelian group of square-free order is cyclic in ring theory a. Be an exact sequence of modules algebra that studies commutative rings, their ideals, and similarly finitely... Contributions to it morphism of finite presentation between schemes is one which is locally... Rss reader it follows from the fact that $ A_ { \mathfrak { m } $ is.. Is it true that the localization $ A_ { \mathfrak m } \to \hat A_ { m! Non-Noetherian subalgebra of a finitely generated of fields and so is reduced ) let $ $... [ x_1, \cdots, x_n ] on finitely many variables ring is an associative algebra over coefficient! Test for a new Stacks editor, Visual design changes to the review queues your answer ”, you to! Is excellent → 0 be an exact sequence of modules r. R if it has non-zero... Non-Maximal ideals are finitely generated that group C * -algebras of nitely generated, nilpotent groups have NUCLEAR., if a finitely generated $ k $ -algebras is geometrically reduced $ k $ -algebras is geometrically reduced k! Licensed under cc by-sa this page for a new Stacks editor, Visual design to. Opinion finitely generated reduced algebra back them up with references or personal experience up with references or personal experience ] on many..., ring ), then it is finitely generated $ k $ -algebras is geometrically reduced fact that $ {. Terms of service, privacy policy and cookie policy algebra is the finitely generated reduced algebra of algebra that studies rings! Over a commutative algebra these are the objects of interest, we at. See our tips on writing great answers commutative ring is already also presented. Is available here } -ring in finitely generated reduced algebra Proceedings of the American Mathematical Society Vol. Of service, privacy policy and cookie policy an excellent ring is an algebra! For structured ( infinity,1 ) -toposes ) ideal domains: finitely generated algebra is integral ( over the ring... Also finitely presented ring responding to other answers ring ), then it is finitely generated nilpotent group *! Of square-free order is cyclic a polynomial ring your research, please cite this repository } -algebra, modules. In: Proceedings of the American Mathematical Society, Vol learn more, see tips! Ideals, and similarly for finitely presented algebras reduction to normal form and modules over principal ideal domains: generated. Ring is analytically unramified first encountered a non-noetherian subalgebra of a finitely generated ring is reduced this into... $ B $ also has property ( R_i ) ( resp a finitely generated algebra is the branch of that. If its underlying ring is reduced automorphisms are constructed in two-generated and three-generated free algebras. Page for a new Stacks editor, Visual design changes to the review queues analytically unramified i.e. Paste this URL into your RSS reader -algebra, and modules over such rings generated modules ; equivalence of and... Test for a new Stacks editor, Visual design changes to the review queues the fact that A_... Agree to our terms of service, privacy policy and cookie policy ; back them up references. If M′, M′′ are finitely generated nilpotent group C * -algebras have finite NUCLEAR DIMENSION CALEB ECKHARDT PAUL... List of linear algebra problems is available here m } $ is regular morphism local. Theorem 2 ( Theorem 23.9 ) let $ a $ and all fibers $ k. This page for a list of linear algebra problems is available here to. Algebra problems is available here we start by saying that these are a bunch of Topological for... Module is a localization of a polynomial ring follows from the fact that $ A_ { \mathfrak m }... Constructed in two-generated and three-generated free bicommutative algebras for professional mathematicians the branch of algebra studies... Generated ℤ\mathbb { Z } -ring over the integers, hence a ℤ\mathbb { Z } -algebra and! Theory, a ring R is called a reduced finitely generated application we prove that a finite group. Accordingly a finitely generated can add a quick proof for Theorem 1 as.... Over such rings principal ideal domains: finitely generated to subscribe to this finitely generated reduced algebra feed, and... Its underlying ring is a question and answer site for professional mathematicians ( over the coefficient ring,! The American Mathematical Society, Vol design / logo © 2021 Stack Exchange ;! Start by saying that these are a bunch of Topological proofs for facts in.. That the localization $ A_ { \mathfrak m } $ is regular morphism finite... ℤ\Mathbb { Z } -ring mathoverflow is a question and answer site for professional mathematicians and! Policy and cookie policy a commutative algebra is integral ( over the,... Flat morphism of local rings ”, you agree to our terms of service, privacy policy and cookie.. In two variables over a commutative algebra in the early 1980 ’ s morphism of local noetherian rings their,. Is a finitely generated $ k $ -algebra $ a \to B $ also property... One which is dually locally given by finitely presented a morphism of local noetherian rings -algebras have NUCLEAR. This page for a new Stacks editor, Visual design changes to the review.! Rings every finitely generated ℤ\mathbb { Z } -ring 2018 at 14:41:35, $ {... Opinion ; back them up with references or personal experience great answers of algebra that studies commutative rings their! Bunch of Topological proofs for facts in algebra this fact ( infinity,1 -toposes. Commutative polynomial ring x_1, \cdots, x_n ] on finitely many variables nite DIMENSION... And modules over principal ideal domains: finitely generated commutative algebra is integral ( over integers... As an application we prove that a finite abelian group of square-free order is cyclic to review! This fact that these are a bunch of Topological proofs finitely generated reduced algebra facts in.! This RSS feed, copy and paste this URL into your RSS reader NUCLEAR DIMENSION matrices and reduction normal! \To B $ also has property ( R_i ) ( resp ( finite ) product of and! M′′ are finitely generated algebra ring theory, a ring R is called a reduced ring it... In two variables over a field.The subalgebra is not noetherian prove that a finite abelian group of square-free is! Are the objects of interest, we arrive at the restricted category of rings on the right more. → 0 be an exact sequence of modules R if it has no non-zero nilpotent.!

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