{"id":461,"date":"2023-02-10T10:26:44","date_gmt":"2023-02-10T10:26:44","guid":{"rendered":"https:\/\/milkyeggs.com\/?p=461"},"modified":"2023-10-30T22:53:48","modified_gmt":"2023-10-30T22:53:48","slug":"localizations-of-a-ring-at-specific-elements-correspond-to-sections-of-a-sheaf","status":"publish","type":"post","link":"https:\/\/milkyeggs.com\/math\/localizations-of-a-ring-at-specific-elements-correspond-to-sections-of-a-sheaf\/","title":{"rendered":"Localizations of a ring at specific elements correspond to sections of a sheaf"},"content":{"rendered":"\n

<\/p>\n\n\n\n

One fundamental observation in algebraic geometry is that there are a number of powerful correspondences between ring localizations and algebraic sheaves. Although elementary, this post will outline one such correspondence, which is that the localizations of a ring to a specific element \"f\" can be exactly thought of as the sections of the sheaf of regular functions over the distinguished open set of \"f\". (Consider this post to be a self-directed refresher! It has been a long time since I learned this material.)<\/p>\n\n\n\n

Suppose that \"R\" is a finitely generated algebra over an algebraically closed field \"k\". One might naturally wonder if there is any geometric interpretation of the localization of this ring at a single one of its elements. Say we take \"f \in R\" (where \"f \neq 0\") and consider the local ring \"R_f\", i.e.<\/em> quotients of the form \"g/f^n\" for \"g \in R\" and nonnegative integers \"n\". If we consider \"f\" and \"g\" as polynomials to be evaluated, then these quotients only make sense if \"f\" is always nonzero. We can in fact show that the local ring \"R_f\" corresponds exactly to the sections of the sheaf \"\mathcal{O}\" over the distinguished open set \"D(f)\".<\/p>\n\n\n\n

Consider the natural homomorphism \"\phi : R_f \to \mathcal{O}(D(f))\" which maps a formal fraction \"g/f^n \in R_f\" to the regular function which is given by the actual quotient of polynomials \"g/f^n \in \mathcal{O}(D(f))\". This mapping is straightforwardly injective: suppose that \"g/f^n = 0\" on \"D(f)\"; then \"g = 0\" on \"D(f)\" and \"f = 0\" on the complement of \"D(f)\" by assumption, so \"fg = 0\" everywhere. Hence the preimage of such a function \"g/f^n \in \mathcal{O}(D(f))\" is exactly 0.<\/p>\n\n\n\n

Surjectivity is less trivial. Obviously, every function of the form \"g/f^n \in \mathcal{O}(D(f))\" is regular, but is every function in \"\mathcal{O}(D(f))\" globally representable in the form \"g/f^n\"? Yes. Suppose that \"\psi : D(f) \to k\" is a regular function in \"\mathcal{O}(D(f))\". I will sketch out the argument in broad strokes.<\/p>\n\n\n\n

First, by the definition of a regular function, \"\psi\" is representable as a quotient of polynomials, \"g_a/f_a\", in the neighborhood of every point \"a \in D(f)\". Distinguished open sets are in a sense the “smallest open sets” of the Zariski topology (i.e.<\/em> they form a basis of the topology), so if we shrink these neighborhoods enough, they become distinguished open sets in their own right, say \"D(f_a)\" for every such point \"a \in D(f)\". These open neighborhoods together cover \"D(f)\" and, conversely, we may say that the variety \"V(f)\" is the intersection of every \"V(f_a)\" or, equivalently, the single variety \"V({f_a})\".<\/p>\n\n\n\n

We have essentially managed to glue together all our little pieces of information about the regular function \"\psi\" into a single piece of information about \"f\", but how can we go further? Translating into the algebraic setting, we know of course that \"f \in I(V(f))\", but here we can use the Nullstellensatz to yield a more tractable result; in particular, it tells us that \"f \in \sqrt{\langle f_a\rangle}\", or that \"f\" (as an element of \"R\") is a member of the radical of the ideal which is generated by finitely many elements \"f_a\".<\/p>\n\n\n\n

Now we know that \"f_n = \sum_a k_a f_a\" for finitely many \"a \in D(f)\". Let us define \"g := \sum_a k_a g_a\", recalling from before that each \"g_a\" comes from the local representation of \"f\" as \"g_a/f_a\" near \"a \in D(f)\". Now taking any<\/em> arbitrary point \"b \in D(f)\" with a local representation \"g_b/f_b\" of \"f\", we can check algebraically that \"gf_b = g_bf^n\" and hence \"g_b/f_b = g/f^n\". Therefore \"g/f^n\" is a valid global representation of \"\psi\" over \"D(f)\".<\/p>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

One fundamental observation in algebraic geometry is that there are a number of powerful correspondences between ring localizations and algebraic sheaves. Although elementary, this post will outline one such correspondence, which is that the localizations of a ring to a specific element \"f\" can be exactly thought of as the sections of the sheaf of regular functions over the distinguished open set of \"f\". (Consider this post to be a self-directed refresher! It has been a long time since I learned this material.)<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"_links":{"self":[{"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/posts\/461"}],"collection":[{"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/comments?post=461"}],"version-history":[{"count":13,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/posts\/461\/revisions"}],"predecessor-version":[{"id":482,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/posts\/461\/revisions\/482"}],"wp:attachment":[{"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/media?parent=461"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/categories?post=461"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/tags?post=461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}