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Spectrum of a ring

In commutative algebra, the spectrum of a commutative ring R, denoted \operatorname{Spec}(R), is the set of all prime ideals of R equipped with the Zariski topology, where the closed sets are those of the form V(I) = \{\mathfrak{p} \in \operatorname{Spec}(R) \mid I \subseteq \mathfrak{p}\} for ideals I \subseteq R.^[1] This construction provides a geometric interpretation of the algebraic structure of R, transforming abstract ring-theoretic data into a topological space that captures information about the ring's ideals and their relationships.^[1] The Zariski topology is defined such that the basic open sets are the D(f) = \{\mathfrak{p} \in \operatorname{Spec}(R) \mid f \notin \mathfrak{p}\} for f \in R, forming a basis for the topology, and the space is quasi-compact with a basis of quasi-compact open sets.^[1] Notably, \operatorname{Spec}(R) is empty if and only if R is the zero ring, and the map V(\cdot) satisfies properties like V(I) = V(\sqrt{I}) and D(f) \sqcup V(f) = \operatorname{Spec}(R).^[1] The spectrum functor \operatorname{Spec}: \mathbf{Ring}^{\mathrm{op}} \to \mathbf{Top} is contravariant and continuous, sending ring homomorphisms to continuous maps between spectra, with the induced map on \operatorname{Spec}(R \to S) being a homeomorphism onto its image under certain localizations, such as \operatorname{Spec}(R_f) \cong D(f).^[1] Introduced by Alexander Grothendieck in his foundational work on algebraic geometry, the spectrum serves as the underlying space for the affine scheme associated to R, enabling the uniform treatment of classical varieties and more general objects through sheaf theory and morphisms of schemes. This framework unifies commutative algebra with geometry, allowing prime ideals to correspond to points in a "space" whose structure sheaves recover the original ring via global sections.^[1]

Fundamentals

Definition

In commutative algebra, the prime spectrum of a commutative ring R with identity, denoted \operatorname{Spec}(R), is defined as the set of all prime ideals of R.^[1] This construction provides a foundational space-theoretic object associated to the ring, capturing its prime ideal structure in a purely set-theoretic manner.^[2] Elements of \operatorname{Spec}(R) are typically denoted by \mathfrak{p} (fraktur p) or \mathbf{p} (bold p).^[1] The concept of the spectrum was introduced by Alexander Grothendieck in the 1960s as a key component of the foundations of scheme theory, appearing in his seminal work Éléments de géométrie algébrique.^[3] The maximal spectrum, denoted \operatorname{MaxSpec}(R), is the subset of \operatorname{Spec}(R) consisting of all maximal ideals of R.^[4] By definition, every maximal ideal is prime, so \operatorname{MaxSpec}(R) \subseteq \operatorname{Spec}(R).^[1]

Prime and Maximal Ideals

In commutative algebra, the prime ideals of a ring R are precisely those proper ideals \mathfrak{p} \subset R for which the quotient ring R/\mathfrak{p} is an integral domain. This characterization highlights the role of prime ideals in preserving the absence of zero-divisors in the quotient. Similarly, the maximal ideals \mathfrak{m} \subset R are those proper ideals for which R/\mathfrak{m} is a field, emphasizing their position as the "largest" proper ideals. The collection of all prime ideals in R determines key radicals of the ring. Specifically, the nilradical \mathrm{Nil}(R), which is the set of all nilpotent elements of R (or equivalently, the radical of the zero ideal \sqrt{(0)}), equals the intersection of all prime ideals of R. In contrast, the Jacobson radical J(R) is defined as the intersection of all maximal ideals of R, capturing the elements that are "quasi-regular" in a certain sense across all residue fields.^[5] For any ideal I \subset R, the spectrum of the quotient ring \mathrm{Spec}(R/I) can be identified with the subset of \mathrm{Spec}(R) consisting of those prime ideals containing I; the map sending a prime \mathfrak{q} \subset R/I to \mathfrak{q} + I \subset R is a bijection onto this subset.^[6] This embedding reflects how ideals in the quotient correspond to primes "above" I in the original ring. Finally, the spectrum \mathrm{Spec}(R) is empty if and only if R is the zero ring, as every nonzero ring admits at least one prime ideal (in fact, a maximal one by Zorn's lemma).^[7]

Topology

Zariski Topology

The Zariski topology on the spectrum \operatorname{Spec}(R) of a commutative ring R is defined by taking as closed sets the subsets of the form V(I) = \{ \mathfrak{p} \in \operatorname{Spec}(R) \mid I \subseteq \mathfrak{p} \}, where I is any ideal of R. These sets satisfy the axioms of a topology: the empty set is V(R) and the whole space is V(0); arbitrary intersections of closed sets are closed, since V(\bigcap I_\alpha) = \bigcap V(I_\alpha); and finite unions are closed, since V(I_1 \cup I_2) = V(I_1) \cap V(I_2).^[1]^[8] The open sets in this topology are complements of the closed sets, and they admit a convenient basis consisting of the principal open subsets D(f) = \{ \mathfrak{p} \in \operatorname{Spec}(R) \mid f \notin \mathfrak{p} \} for f \in [R](/page/R). These form a basis because any open set is a union of such D(f), and the intersection of two basis elements satisfies D(f) \cap D(g) = D(fg). Moreover, the closed sets satisfy the distinguishability property V(I) = V(\sqrt{I}), where \sqrt{I} = \{ r \in [R](/page/R) \mid r^n \in I \text{ for some } n \geq 1 \} is the radical of I, ensuring that the topology depends only on the radical ideals.^[1]^[9]^[8] Each principal open set D(f) is naturally identified with the spectrum of the localization R_f, via the homeomorphism \operatorname{Spec}(R_f) \to D(f) given by \mathfrak{p} R_f \mapsto \mathfrak{p}, where \mathfrak{p} is a prime ideal of R_f. This bijection is continuous and open with respect to the respective Zariski topologies. The Zariski topology is thus the coarsest topology on \operatorname{Spec}(R) that renders all the sets D(f) open.^[1]^[8]

Topological Properties

The spectrum \operatorname{Spec}(R) of a commutative ring R, equipped with the Zariski topology, exhibits several notable topological properties that distinguish it from more familiar spaces like those in classical topology. One fundamental property is quasi-compactness: for any open cover of \operatorname{Spec}(R) by standard open sets D(f_i) where f_i \in R, there exists a finite subcover.^[10] This follows from the fact that such a cover implies the ideal generated by the f_i is the unit ideal, which can be witnessed by finitely many generators, yielding a finite subcollection whose D(f_i) cover \operatorname{Spec}(R).^[10] Moreover, \operatorname{Spec}(R) is a spectral space, meaning it is quasi-compact and admits a basis of quasi-compact open sets, with the additional sobriety condition that every irreducible closed subset has a unique generic point. This structure, characterized by Hochster, underscores the interplay between the algebraic structure of R and the topological features of its spectrum. In general, \operatorname{Spec}(R) fails to be Hausdorff. For instance, consider R = k where k is an infinite field; the spectrum includes the generic point (0) and closed points (t - a) for a \in k. Any nonempty open set containing the generic point must intersect every nonempty open set containing a closed point, as opens are cofinite in the closed points, preventing disjoint neighborhoods for these points. This non-Hausdorff behavior arises because the Zariski topology is coarse, with closed sets being algebraic varieties that cannot separate generic and special points in infinite-dimensional settings.^[11] Regarding separation axioms, \operatorname{Spec}(R) satisfies the T0 (Kolmogorov) property: for distinct prime ideals \mathfrak{p} \neq \mathfrak{q}, without loss of generality assume \mathfrak{p} \not\subseteq \mathfrak{q}; then there exists f \in \mathfrak{p} \setminus \mathfrak{q}, so the open set D(f) contains \mathfrak{q} but not \mathfrak{p}, separating them.^[12] However, it is generally not T1, as singletons of non-maximal primes are not closed unless the prime is maximal.^[13] The space \operatorname{Spec}(R) is irreducible if and only if R has a unique minimal prime ideal, in which case the nilradical \mathrm{Nil}(R) is prime. In this situation, the entire space is an irreducible closed subset with the zero ideal (or the unique minimal prime) as its generic point.^[14] The irreducible components of \operatorname{Spec}(R) correspond precisely to the closed subsets V(\mathfrak{p}) where \mathfrak{p} runs over the minimal primes of R.^[14] Finally, \operatorname{Spec}(R) is connected if and only if R has no nontrivial idempotents, i.e., the only idempotents in R are 0 and 1. This equivalence arises because clopen subsets of \operatorname{Spec}(R) correspond bijectively to idempotents via the sets D(e) for idempotents e \in R, and connectedness precludes nontrivial such decompositions.^[15]

Geometric Structure

Structure Sheaf

The structure sheaf \mathcal{O}_{\operatorname{Spec}(R)} on the spectrum \operatorname{Spec}(R) of a commutative ring R with identity equips the topological space with a sheaf of rings, allowing the assignment of algebraic sections to open sets that reflect the local ring structure of R. This sheaf is defined such that its sections over the distinguished open sets D(f) = \{ \mathfrak{p} \in \operatorname{Spec}(R) \mid f \notin \mathfrak{p} \} for f \in R are given by \mathcal{O}_{\operatorname{Spec}(R)}(D(f)) = R_f, the localization of R at the multiplicative set \{1, f, f^2, \dots \}.^[16]^[17] These sections consist of rational functions a/f^n with a \in R and n \geq 0, providing a ring of "regular functions" on D(f).^[18] For a general open set U \subseteq \operatorname{Spec}(R), which can be covered by distinguished opens U = \bigcup_i D(f_i) for some f_i \in R, the sections \mathcal{O}_{\operatorname{Spec}(R)}(U) are the elements of the equalizer

\mathcal{O}_{\operatorname{Spec}(R)}(U) = \left\{ (s_i)_i \in \prod_i R_{f_i} \;\middle|\; \operatorname{res}_{D(f_i), D(f_i f_j)}(s_i) = \operatorname{res}_{D(f_j), D(f_i f_j)}(s_j) \;\forall\, i,j \right\},

where the restriction maps ensure compatibility on the intersections D(f_i f_j).^[18]^[19] This construction satisfies the sheaf axioms, gluing local sections consistently while preserving the ring structure.^[16] The stalks of the structure sheaf at a point \mathfrak{p} \in \operatorname{Spec}(R) are given by \mathcal{O}_{\operatorname{Spec}(R),\mathfrak{p}} = R_{\mathfrak{p}}, the localization of R at the prime ideal \mathfrak{p}, obtained as the direct limit of sections over opens containing \mathfrak{p}.^[16]^[17] The global sections over the entire space are \Gamma(\operatorname{Spec}(R), \mathcal{O}_{\operatorname{Spec}(R)}) = R, corresponding to the case where D(1) = \operatorname{Spec}(R) and R_1 \cong R.^[18]^[19] Restriction maps between sections are induced by localization: for D(f) \subseteq D(g) (which holds if some power of g lies in the ideal generated by f), the map \mathcal{O}_{\operatorname{Spec}(R)}(D(g)) = R_g \to \mathcal{O}_{\operatorname{Spec}(R)}(D(f)) = R_f is the natural homomorphism sending a/g^m to a/g^m viewed in R_f, or more generally from R_g to R_{fg} when refining to intersections.^[16]^[17] This sheaf is unique up to isomorphism as the sheaf of rings on \operatorname{Spec}(R) whose stalks are precisely R_{\mathfrak{p}} at each prime \mathfrak{p}, ensuring it captures the local algebraic structure without redundancy.^[18]^[19]

Affine Schemes

In algebraic geometry, the spectrum of a ring R, equipped with its structure sheaf \mathcal{O}_{\Spec(R)}, forms a locally ringed space denoted (\Spec(R), \mathcal{O}_{\Spec(R)}), which serves as the prototypical example of an affine scheme associated to R.^[20] An affine scheme is defined as any locally ringed space isomorphic to (\Spec(R), \mathcal{O}_{\Spec(R)}) for some commutative ring R.^[21] This construction bridges commutative algebra to scheme theory, where the points of \Spec(R) correspond to prime ideals of R, and the sheaf \mathcal{O}_{\Spec(R)} encodes the ring's localizations.^[22] The locally ringed space (\Spec(R), \mathcal{O}_{\Spec(R)}) has stalks that are local rings: at a point p \in \Spec(R) corresponding to a prime ideal \mathfrak{p} \subset R, the stalk \mathcal{O}_{\Spec(R), p} = R_{\mathfrak{p}}, with maximal ideal \mathfrak{p} R_{\mathfrak{p}}.^[22] The residue field at p is then \kappa(p) = R_{\mathfrak{p}} / \mathfrak{p} R_{\mathfrak{p}}, which captures the "field of fractions" of the quotient ring R / \mathfrak{p}.^[22] These properties ensure that affine schemes are locally ringed in a manner reflecting the local structure of the ring R.^[21] Morphisms between affine schemes arise naturally from ring homomorphisms. Given a ring homomorphism \phi: R \to S, it induces a morphism of locally ringed spaces \Spec(\phi): \Spec(S) \to \Spec(R), defined on points by \Spec(\phi)(\mathfrak{q}) = \phi^{-1}(\mathfrak{q}) for primes \mathfrak{q} \subset S.^[23] This map is continuous with respect to the Zariski topology, as the preimage of a basic open D(g) \subset \Spec(R) (for g \in R) is D(\phi(g)) \subset \Spec(S), which is open.^[24] Moreover, \Spec(\phi) is a morphism of ringed spaces via the sheaf map \mathcal{O}_{\Spec(R)} \to \phi_* \mathcal{O}_{\Spec(S)}, restricting to local homomorphisms on stalks, thus preserving the locally ringed structure.^[23] Two affine schemes \Spec(R) and \Spec(S) are isomorphic as locally ringed spaces if and only if the rings R and S are isomorphic as commutative rings.^[25] This equivalence underscores the functorial nature of the spectrum construction, establishing the category of affine schemes as anti-equivalent to the category of commutative rings.^[26] A fundamental example is the affine line over a field k, denoted \mathbb{A}^1_k = \Spec(k), where x is an indeterminate.^[27] Here, the prime ideals of k are (0) (the generic point) and principal ideals (x - a) for a \in k (closed points), realizing \mathbb{A}^1_k as the scheme-theoretic analogue of the classical line, with morphisms to \Spec(k) corresponding to k-algebra structures on k.^[27]

Categorical Aspects

Functoriality

The spectrum functor \operatorname{Spec} defines a contravariant functor from the opposite category of commutative rings to the category of topological spaces: it sends a commutative ring R to the space \operatorname{Spec}(R) equipped with the Zariski topology.^[1] For a ring homomorphism \phi: R \to S, the induced map \operatorname{Spec}(\phi): \operatorname{Spec}(S) \to \operatorname{Spec}(R) is defined by sending a prime ideal q \subset S to its preimage \phi^{-1}(q) \subset R.^[28] This map is continuous with respect to the Zariski topology, as the preimage of a basic open set D(f) \subset \operatorname{Spec}(R) (primes not containing f \in R) is D(\phi(f)) \subset \operatorname{Spec}(S).^[28] The functor \operatorname{Spec} preserves fiber products: given ring homomorphisms R \to T and S \to T, the fiber product ring R \times_T S (defined via the universal property of the tensor product R \otimes_T S) maps under \operatorname{Spec} to the fiber product of topological spaces \operatorname{Spec}(R) \times_{\operatorname{Spec}(T)} \operatorname{Spec}(S).^[29] In particular, for the direct product of rings R \times S, the spectrum \operatorname{Spec}(R \times S) is homeomorphic to the topological disjoint union \operatorname{Spec}(R) \sqcup \operatorname{Spec}(S), via the maps induced by the projections R \times S \to R and R \times S \to S.^[30] More generally, \operatorname{Spec} induces a contravariant equivalence between the category of commutative rings and the opposite category of affine schemes (locally ringed spaces locally isomorphic to \operatorname{Spec}(R) for some ring R), making it fully faithful on the full subcategory of finitely presented rings.^[25] This equivalence restricts to finitely presented rings and their opposite category of affine schemes of finite presentation, preserving the homomorphisms bijectively.^[25]

Universal Properties

The spectrum functor \Spec: \CommRing^\op \to \Sch is left adjoint to the global sections functor \Gamma: \Sch \to \CommRing, where \Sch denotes the category of schemes. This adjunction establishes a natural bijection

\Hom_\Sch(\Spec(R), X) \cong \Hom_\CommRing(R, \Gamma(X, \mathcal{O}_X))

for any commutative ring R and scheme X, with \mathcal{O}_X the structure sheaf of X.^[31] The unit of the adjunction corresponds to the natural map R \to \Gamma(\Spec(R), \mathcal{O}_{\Spec(R)}) identifying R with the global sections of the affine scheme \Spec(R), while the counit is the canonical morphism \Spec(\Gamma(X, \mathcal{O}_X)) \to X. This universal property characterizes morphisms into affine schemes and extends the representability of the functor of points for affine schemes.^[31] As a consequence of this adjunction and the Yoneda lemma, the functor \Spec embeds the opposite category of commutative rings fully faithfully into the category of schemes, providing a dense subcategory of affine schemes within all schemes. Specifically, \Spec is fully faithful, meaning that for commutative rings R and S,

\Hom_\Sch(\Spec(R), \Spec(S)) \cong \Hom_\CommRing(S, R),

with the isomorphism given by precomposition with the structure map. This embedding realizes the category of affine schemes as representable functors on \CommRing, preserving the categorical structure of ring homomorphisms as scheme morphisms in the opposite direction. The initial object \mathbb{Z} in \CommRing maps under \Spec to the terminal object \Spec(\mathbb{Z}) in \Sch, as there exists a unique morphism \Spec(R) \to \Spec(\mathbb{Z}) for any commutative ring R, corresponding to the unique ring homomorphism \mathbb{Z} \to R. This makes \Spec(\mathbb{Z}) the "big point" or base scheme over which all schemes are defined relative to the integers.^[32] Regarding colimits, the functor \Spec preserves them in the sense that filtered colimits in \CommRing correspond to filtered limits in \Sch; that is, for a filtered system of commutative rings \{R_i\}_{i \in I},

\Spec\left( \varinjlim_{i \in I} R_i \right) \cong \varprojlim_{i \in I} \Spec(R_i),

where the colimit is taken in \CommRing and the limit in \Sch. This property ensures that affine schemes arising from direct limits of rings, such as localizations or completions in filtered systems, form inverse systems in the category of schemes, facilitating constructions like formal schemes or ind-schemes.^[33]

Motivations

From Commutative Algebra

Hilbert's Nullstellensatz establishes a bijection between maximal ideals of the polynomial ring k[x_1, \dots, x_n] over an algebraically closed field k and points in affine n-space \mathbb{A}^n_k, where each maximal ideal corresponds to the kernel of the evaluation map at a point (a_1, \dots, a_n). This correspondence highlights the role of ideals in encoding geometric data, prefiguring the spectrum \operatorname{Spec}(R) by associating prime ideals—rather than just maximals—to points in a topological space, allowing a broader algebraic-geometric duality even for non-polynomial rings. The theorem's proof techniques, such as normalization and Noetherian induction, underscore the need for a framework like \operatorname{Spec}(R) to generalize such ideal-point links beyond maximal ideals.^[2] Points in \operatorname{Spec}(R) correspond to prime ideals \mathfrak{p}, and the localization R_\mathfrak{p} at such a point provides a local ring that captures the behavior of R "near" \mathfrak{p}, enabling the study of properties like invertibility or zero-divisors in a restricted setting. For instance, elements outside \mathfrak{p} become units in R_\mathfrak{p}, allowing analysis of generic behavior at minimal primes, where the localization often yields the quotient field of the integral domain R/\mathfrak{p}, reflecting the "most general" or dense aspects of the ring. This localization technique reduces global questions about modules or ideals to local ones at specific primes, a cornerstone of commutative algebra that motivates viewing \operatorname{Spec}(R) as a space where points encode these localized rings.^[34]^[35] The Krull dimension of a ring R, defined as the supremum of lengths of strictly ascending chains of prime ideals in R, coincides with the dimension of the topological space \operatorname{Spec}(R), measured by the longest such chain. This equivalence provides an intrinsic geometric interpretation of dimension in purely algebraic terms, where the height of a prime \mathfrak{p} is the length of the longest chain descending to \mathfrak{p}, and the dimension is the maximum height over maximal ideals. Such chains visualize the "depth" of the ring's structure, aiding in computations like those for polynomial rings, where \dim k[x_1, \dots, x_n] = n.^[36] In the context of integral dependence, the going-up and going-down theorems describe how prime ideals in an integral extension R \subseteq S can be lifted or extended while preserving chain lengths, offering a geometric visualization on \operatorname{Spec}(S) over \operatorname{Spec}(R) via the continuous map induced by the inclusion. Specifically, for any chain of primes in R, there exists a comparable chain in S of the same length (going-up), and under additional conditions like normality of R, chains can be extended downward (going-down), ensuring the fiber over each point is well-behaved. These results, proved using integral dependence relations, illustrate how \operatorname{Spec}(R) facilitates understanding ramification and decomposition in extensions, such as in number theory.^[37]^[38] For Artinian rings, the spectrum \operatorname{Spec}(R) is finite and discrete, consisting solely of maximal ideals with no inclusions among them, as Artinian rings satisfy the descending chain condition on ideals and are Noetherian with only finitely many primes. This structure implies R decomposes as a finite product of local Artinian rings, each with a single point in its spectrum, simplifying the study of zero-dimensional phenomena like length computations in module theory.^[39]

From Algebraic Geometry

In classical algebraic geometry, affine varieties are associated with radical ideals in polynomial rings over algebraically closed fields, corresponding to reduced, irreducible schemes of finite type. However, this framework excludes non-reduced structures, such as those involving nilpotent elements, which are essential for capturing infinitesimal information and degenerations. The spectrum Spec(R) overcomes this limitation by allowing arbitrary commutative rings R, including those with nilpotents in their structure sheaf, thereby enabling the study of non-reduced schemes that model multiple components or higher-order tangencies without reducing to classical points.^[40] General schemes extend this further by gluing together affine schemes Spec(R_i) along open covers, where the structure sheaf is defined compatibly on overlaps via ring homomorphisms. This construction allows for projective and non-affine geometries, such as the projective line obtained by gluing two copies of Spec(k) along the complement of the origin, facilitating the uniform treatment of global objects beyond affine patches. Affine schemes thus serve as building blocks for arbitrary schemes, preserving the functorial nature of morphisms.^[41] A concrete illustration arises in relative curves, where Spec(k[x,y]/(y^2 - x^3 - x^2)) represents the nodal cubic curve with a singularity at the origin, capturing the node as an ordinary double point in its local ring. This scheme-theoretic view highlights the singularity's resolution via normalization to Spec(k), parametrized by (x,y) \mapsto (t^2 - 1, t(t^2 - 1)), which separates the branches while retaining the geometric embedding. Such examples demonstrate how Spec encodes singularities intrinsically, aiding the study of families of curves over bases.^[42] The étale site on Spec(R), formed by étale morphisms over R with surjective families as coverings, originates étale cohomology theories by generalizing sheaf cohomology algebraically. For Spec(k) with k a field, sheaves on this site correspond to discrete Gal(k^{sep}/k)-modules, yielding cohomology groups isomorphic to Galois cohomology, which probes arithmetic and geometric invariants like the Picard group. This site-theoretic perspective on Spec enables comparisons with topological cohomology for varieties over \mathbb{C}, foundational for l-adic cohomology and the Weil conjectures.^[43] Grothendieck's revolution in Éléments de Géométrie Algébrique (EGA) and Fondements de la Géométrie Algébrique (FGA) reframes schemes via the functor of points, viewing Spec(R) as the representable functor h_R = Hom_{Rings}(-, R) on the category of rings, emphasizing morphisms over points. This functorial approach unifies affine and general schemes, allowing gluing and descent in a categorical framework that extends classical geometry to arbitrary base rings and relative settings.^[44]

Examples

Affine Examples

The spectrum of the ring of integers \mathbb{Z}, denoted \operatorname{Spec}(\mathbb{Z}), consists of the prime ideals (0) and (p) for each prime number p. The point corresponding to (0) is the generic point, dense in the space, while the points (p) are closed and maximal. In the Zariski topology, the closed sets include those of the form V(n\mathbb{Z}) for n \geq 0: V(0) is the entire \operatorname{Spec}(\mathbb{Z}); for n > 0, V(n\mathbb{Z}) is the finite set of maximal ideals (p) where p divides n.^[45] For a field k, the spectrum \operatorname{Spec}(k) of the polynomial ring in one variable comprises the zero ideal (0) and the maximal ideals (x - \alpha) for \alpha \in k, assuming k is algebraically closed for simplicity. The generic point (0) is dense, and the closed points (x - \alpha) correspond to evaluation at \alpha. The basic open sets D(f) for f \in k are the complements of the finite sets of zeros of f, forming a structure akin to the affine line over k.^[46] Consider the quotient ring k[x,y]/(xy) over an algebraically closed field k. The spectrum \operatorname{Spec}(k[x,y]/(xy)) realizes the union of the x-axis and y-axis in the affine plane, with prime ideals including the minimal primes (x) and (y), corresponding to these axes as irreducible components. The maximal ideals are of the form (x - a, y) for a \neq 0 on the x-axis and (x, y - b) for b \neq 0 on the y-axis, while the origin corresponds to (x, y). This space has two irreducible components, each homeomorphic to \operatorname{Spec}(k), intersecting at the closed point of the origin.^[47] An Artin local ring A, which is a local ring of finite length as a module over itself, has a unique prime ideal, namely its maximal ideal \mathfrak{m}. Thus, \operatorname{Spec}(A) consists of a single closed point \{\mathfrak{m}\}, with the Zariski topology being the trivial topology on this singleton set. Examples include finite field extensions or quotient rings like k[\epsilon]/(\epsilon^2) for a field k. By the Chinese Remainder Theorem, if R and S are commutative rings such that the ideals (0) \times S and R \times (0) are comaximal (which holds generally), then \operatorname{Spec}(R \times S) is the topological disjoint union \operatorname{Spec}(R) \sqcup \operatorname{Spec}(S). The prime ideals of R \times S are precisely those of the form \mathfrak{p} \times S for \mathfrak{p} \in \operatorname{Spec}(R) or R \times \mathfrak{q} for \mathfrak{q} \in \operatorname{Spec}(S), making the space disconnected if both R and S are nonzero.^[48]

Non-Affine Examples

The Proj construction provides a fundamental example of a non-affine scheme derived from the spectrum of a graded ring. For the polynomial ring R = k[x, y, z] graded by total degree over an algebraically closed field k, the projective plane \mathbb{P}^2_k = \Proj R consists of homogeneous prime ideals of R that do not contain the irrelevant ideal (x, y, z). These points correspond to equivalence classes [x : y : z] under scalar multiplication by k^\times, where each closed point arises from a maximal homogeneous ideal generated by linear forms, such as (ay - bx, cz - az) for distinct points.^[49]^[50] The affine cone over this projective variety illustrates the relationship between Spec and Proj. The spectrum \Spec R represents the affine cone, including the vertex at the origin corresponding to the irrelevant ideal, which deforms the projective plane into a three-dimensional affine variety. In contrast, quotienting by the irrelevant ideal yields \Spec(k[x,y,z]/(x,y,z)) \cong \Spec k, a single point embodying the cone's apex, highlighting how Proj removes this vertex to form the non-affine projective space.^[50]^[51] A nodal cubic curve exemplifies non-affineness in singular embeddings. The affine scheme \Spec k[t^2 - 1, t(t^2 - 1)] parametrizes the nodal cubic y^2 = x^3 + x^2 in \mathbb{A}^2_k, which is affine but non-normal at the node (0,0). Its normalization \Spec k \to \Spec k[x,y]/(y^2 - x^3 - x^2) resolves the singularity, but the projective closure in \mathbb{P}^2_k requires gluing multiple affine charts, rendering the full curve non-affine despite its affine pieces.^[52]^[42] The Grassmannian \Gr(r, n), parametrizing r-dimensional subspaces of k^n, is constructed as a gluing of affine schemes. It admits an open cover by affines U_I = \Spec k[x_{ij} \mid i \in I, j \notin I] for subsets I \subset of size r, where transition maps identify coordinates on overlaps. The Plücker embedding maps \Gr(r, n) into \mathbb{P}^{\binom{n}{r} - 1}_k via determinants of r \times r minors, realizing it as a closed subscheme defined by quadratic Plücker relations, which is projective and thus non-affine.^[53] These examples underscore a key limitation: the spectrum \Spec R of any commutative ring R is always an affine scheme, serving as a building block for non-affine geometry. Non-affine schemes like Proj arise by quotienting or gluing such spectra, as in projective varieties, to capture phenomena absent in affine settings.^[49]^[50]

Advanced Constructions

Relative Spec

In the context of commutative rings, given a ring homomorphism f: A \to B, the relative spectrum \operatorname{Spec}_A(B) is defined as the spectrum \operatorname{Spec}(B) equipped with the structure morphism \pi: \operatorname{Spec}(B) \to \operatorname{Spec}(A) that sends each prime ideal \mathfrak{p} \subset B to its preimage f^{-1}(\mathfrak{p}) under the induced map on spectra.^[54] This construction endows \operatorname{Spec}_A(B) with the structure of an A-scheme, where the morphism \pi corresponds to the ring map f.^[55] The fibers of this morphism are obtained by base change to residue fields: for a prime ideal \mathfrak{q} \subset A, the fiber over the point corresponding to \mathfrak{q} is \operatorname{Spec}(\kappa(\mathfrak{q}) \otimes_A B), where \kappa(\mathfrak{q}) = \operatorname{Frac}(A/\mathfrak{q}) is the residue field at \mathfrak{q}.^[56] These fibers capture the local structure of the relative spectrum over points of the base \operatorname{Spec}(A).^[57] As a relative affine scheme, \operatorname{Spec}_A(B) consists of \operatorname{Spec}(B) equipped with the structure morphism \pi: \operatorname{Spec}(B) \to \operatorname{Spec}(A) and the usual structure sheaf \mathcal{O}_{\operatorname{Spec}(B)}.^[55] This sheaf structure ensures compatibility with the base change functor in the category of schemes over \operatorname{Spec}(A).^[57] Base change along a ring homomorphism A \to C yields \operatorname{Spec}_C(B \otimes_A C) as the pullback, preserving the relative spectrum construction; for instance, scalar extension from \mathbb{Z} to a ring S gives \operatorname{Spec}_{\operatorname{Spec}(S)}(R \otimes_\mathbb{Z} S) for a ring R.^[54] This property facilitates studying the relative spectrum under extensions of the base ring.^[57] The universal property of \operatorname{Spec}_A(B) states that it represents the functor from the category of A-schemes to sets, sending an A-scheme T = \operatorname{Spec}(C) (with C an A-algebra) to the set of A-algebra homomorphisms \operatorname{Hom}_A(B, C), with the representing morphism induced by the canonical map A \to \Gamma(\operatorname{Spec}_A(B), \mathcal{O}_{\operatorname{Spec}_A(B)}). This makes \operatorname{Spec}_A(B) the initial object in the category of schemes over \operatorname{Spec}(A) equipped with an A-algebra structure compatible with B.^[55]

Other Topologies on Spec

In addition to the Zariski topology, the prime spectrum of a ring admits several Grothendieck topologies that refine it, enabling the construction of sites for descent theory, cohomology, and moduli problems in algebraic geometry. These topologies are defined on the category of schemes over \operatorname{Spec}(R) (or affines over R), where covers are families of morphisms satisfying certain flatness or smoothness conditions, and they induce finer topologies on the underlying set of prime ideals than the Zariski topology, with sieves generated by more general covers.^[58] The flat topology, also known as the fpqc (fidèlement plate et quasi-compacte) topology in its quasi-compact variant, is generated by families of faithfully flat and quasi-compact morphisms as covers. This topology supports faithfully flat descent, allowing the reconstruction of objects over the base from data over the cover, as developed in the context of algebraic spaces and stacks. It is finer than the Zariski topology, with every Zariski open immersion being a flat cover, but includes more general surjective families for gluing modules or schemes.^[58] The étale topology uses étale morphisms—flat, unramified, and locally of finite presentation—as basic covers, providing a framework analogous to the classical topology for manifolds but adapted to algebraic varieties. Introduced by Grothendieck, it underpins étale cohomology, which computes Galois cohomology for number fields and l-adic cohomology for varieties over finite fields, capturing topological invariants like Betti numbers in characteristic zero.^[43] Étale covers refine Zariski opens, as every étale morphism is open, and the topology is subcanonical, preserving representable presheaves as sheaves. The fppf (fidèlement plate de présentation finie) topology refines the flat topology by requiring covers to consist of faithfully flat morphisms that are locally of finite presentation, ensuring compactness suitable for moduli spaces. It is employed in the study of algebraic groups and abelian schemes, where fppf cohomology classifies torsors and extensions, as in the cohomology of the multiplicative group \mathbb{G}_m.^[59] This topology sits between the étale and fpqc topologies in the refinement order, being finer than Zariski but coarser than fpqc, and supports descent for finite presentation properties.^[58] The h-topology, along with its variants like the completely decomposed h-topology (cdh) and Nisnevich topology, extends these by including proper morphisms and blow-ups in blow-up squares as covers, designed for handling singularities in motivic settings. The Nisnevich topology, generated by étale morphisms equating residue fields locally, refines the Zariski topology for local cohomology computations, while the cdh topology adds abstract blow-ups to resolve singularities completely, facilitating Voevodsky's proof of the Milnor conjecture via motivic cohomology groups isomorphic to higher Chow groups.^[60] These are used in motivic homotopy theory to define sheaves on singular schemes, with the h-topology being coarser than étale but essential for descent in stable homotopy categories. Collectively, these topologies are strictly finer than the Zariski topology on \operatorname{Spec}(R), meaning every Zariski cover is a cover in each, but they admit more covers for effective epimorphisms in the corresponding sites. They form part of the hierarchy of Grothendieck topologies on the category of schemes, ordered by refinement (Zariski \subset Nisnevich \subset étale \subset fppf \subset fpqc), enabling the big and small site constructions for coherent sheaves and higher derived categories.^[58]

Applications

Representation Theory

In the context of representation theory for commutative rings, modules over a ring R correspond to quasi-coherent sheaves on the spectrum \operatorname{Spec}(R) through the tilde functor \widetilde{\cdot}, which associates to an R-module M the sheaf \tilde{M} defined by \tilde{M}(D(f)) = M_f for basic open sets D(f) \subseteq \operatorname{Spec}(R), where f \in R. This equivalence identifies the category of R-modules with the category of quasi-coherent \mathcal{O}_{\operatorname{Spec}(R)}-modules, preserving exactness and enabling geometric interpretations of algebraic constructions. The support of an R-module M, denoted \operatorname{Supp}(M), is the closed subset \{ \mathfrak{p} \in \operatorname{Spec}(R) \mid M_{\mathfrak{p}} \neq 0 \} of \operatorname{Spec}(R), which coincides with the variety V(\operatorname{Ann}(M)) consisting of primes containing the annihilator ideal \operatorname{Ann}(M). This geometric realization allows the support to capture the "vanishing locus" of the module in the spectral topology, with \operatorname{Supp}(M) being closed when M is finitely generated.^[61] Finitely generated projective R-modules correspond to vector bundles on \operatorname{Spec}(R), as the tilde functor sends such a module P to a locally free sheaf \tilde{P} of finite rank, reflecting the bundle's trivialization over basic opens. This identification underpins the study of K_0(R), the Grothendieck group of projective modules, which aligns with the group of isomorphism classes of vector bundles on the affine scheme. Tensor products of R-modules M \otimes_R N map under the tilde functor to the tensor product of sheaves \tilde{M} \otimes_{\mathcal{O}_{\operatorname{Spec}(R)}} \tilde{N}, preserving the bilinear structure and facilitating computations of sheaf cohomology via algebraic tensor products.^[62] The Artin-Rees lemma implies that local cohomology modules H^i_{\mathfrak{a}}(M) for an ideal \mathfrak{a} \subseteq R are supported on the closed subset V(\mathfrak{a}) of \operatorname{Spec}(R), meaning their stalks vanish outside V(\mathfrak{a}) and providing bounds on the growth of \mathfrak{a}^n-invariant submodules in filtrations of M. This connection links algebraic depth and dimension to geometric properties of supports in the spectrum.

Functional Analysis

In functional analysis, the spectrum of a ring finds notable analogies through the study of Banach and C*-algebras, where ideals and their topologies mirror algebraic constructions. For commutative Banach algebras, the Gelfand transform provides a representation as functions on the maximal ideal space, establishing a duality between the algebra and a compact Hausdorff space. Specifically, for a unital commutative complex Banach algebra A, the maximal ideal space M(A) consists of all nonzero algebra homomorphisms \phi: A \to \mathbb{C}, equipped with the weak* topology, making M(A) compact and Hausdorff. The Gelfand transform \hat{a}(\phi) = \phi(a) for a \in A embeds A into C(M(A)), the algebra of continuous complex-valued functions on M(A), and this map is a unital algebra homomorphism that is continuous and isometric when A is a C*-algebra.^[63] Extending to noncommutative settings, the primitive spectrum of a C*-algebra A, denoted \Prim(A), comprises the kernels of irreducible *-representations of A on Hilbert spaces, each of which is a primitive ideal. This set is endowed with the hull-kernel topology, where for a subset S \subseteq \Prim(A), the kernel \ker(S) = \bigcap_{P \in S} P and the hull h(I) = \{P \in \Prim(A) \mid I \subseteq P\} for an ideal I, generating closed sets of the form h(\ker(S)). The space \Prim(A) is compact, T_0, and Hausdorff if A is commutative, analogous to the maximal spectrum \MaxSpec(R) of a commutative ring R with the Zariski topology, but adapted to capture irreducible representations rather than characters. In the commutative case, \Prim(A) coincides with the maximal ideal space.^[64] A noncommutative analogue of Stone duality arises in the context of inverse semigroups and étale groupoids, linking algebraic structures to C*-algebras. For the commutative C*-algebra C(X) of continuous functions on a compact Hausdorff space X, the classical Stone duality recovers X as the Stone space, homeomorphic to \Prim(C(X)) under the hull-kernel topology. In the noncommutative generalization, Boolean inverse \wedge-semigroups dualize to Hausdorff Boolean groupoids, inducing a correspondence where the associated C*-algebra of the groupoid recovers the original structure; for finite X, this yields the groupoid X \times X with discrete topology, mirroring the space X. This framework connects to graph C*-algebras, where the duality provides a topological model for the spectrum.^[65] Further parallels emerge in spectral theory for operators, where the approximate point spectrum \sigma_{ap}(T) of a bounded linear operator T on a Banach space—defined as the set of \lambda \in \mathbb{C} such that \lambda - T is not bounded below (i.e., there exists a sequence of unit vectors x_n with \|(T - \lambda)x_n\| \to 0)—analogizes the prime spectrum \Spec(R) of a ring R. \sigma_{ap}(T) refines the usual spectrum by identifying approximate eigenvalues, extending beyond isolated points like maximal ideals. This notion underscores the continuous, analytic flavor of functional analytic spectra compared to the discrete algebraic ones.^[66] These connections trace back to Israel Gelfand's foundational work in the 1940s on normed rings and Banach algebras, where he introduced the spectrum via maximal ideals as "points" of the algebra, prefiguring the prime ideal spectrum in algebraic geometry. Gelfand's approach, developed amid post-war exposure to functional analysis, influenced later geometric interpretations, as Alexander Grothendieck encountered and built upon similar ideas during his early studies around 1945.^[67]

Generalizations

Non-Commutative Settings

In non-commutative rings, the notion of prime ideals extends beyond the commutative case, where a two-sided ideal P of a ring R is prime if, for any two-sided ideals A and B of R, the condition AB \subseteq P implies A \subseteq P or B \subseteq P. However, due to non-commutativity, one also considers left-prime ideals, where P is left-prime if A B \subseteq P for left ideals A and B implies A \subseteq P or B \subseteq P, and similarly for right-prime ideals.^[68] These distinctions arise because left and right modules behave differently, complicating the direct analog of the commutative prime spectrum. In this setting, the spectrum of R, denoted \Prim(R), is typically taken to be the set of primitive ideals, where a two-sided ideal P is primitive if it is the annihilator of a simple left R-module, or equivalently, the kernel of an irreducible representation of R.^[69] The primitive spectrum \Prim(R) equips a natural topology known as the Jacobson topology, defined such that a set U \subseteq \Prim(R) is open if it is the complement of the hull of some ideal, where the hull of an ideal I is the set of primitive ideals containing I.^[70] This topology replaces the Zariski topology of the commutative case, as the full prime spectrum in non-commutative rings often lacks desirable geometric properties like being sober or having a good notion of closed points corresponding to maximal ideals.^[69] Challenges in developing a full Zariski-like topology stem from the asymmetry between left and right structures, leading researchers to focus on the primitive spectrum for its ties to representation theory, where points correspond to irreducible modules.^[71] Non-commutative affine schemes generalize the spectrum to non-commutative rings by associating to R a geometric object whose "points" are the simple left R-modules, with a structure sheaf of quasi-coherent sheaves modeled on modules over R.^[72] One approach reconstructs such schemes via the enveloping algebra R^e = R \otimes_{Z(R)} R^{\mathrm{op}}, where the spectrum of R^e captures bimodule information, allowing a functorial correspondence between non-commutative rings and certain schemes.^[73] In the context of Banach algebras, a non-commutative extension appears in the Gelfand spectrum, which for a unital C*-algebra A is the space of non-zero *-homomorphisms from A to \mathbb{C}, topologized by uniform convergence on compact sets, serving as a non-commutative analog of the maximal ideal space.^[74] A concrete example is the matrix ring M_n(R) over a commutative ring R, where the only primitive ideal is the zero ideal (0), making \Prim(M_n(R)) a single point in the Jacobson topology. This reflects the simplicity of representations of matrix rings, as every simple left module is isomorphic to R^n, annihilated only by zero.^[75]

Topological Analogues

The spectrum of a Boolean algebra B, denoted \Spec(B), consists of the ultrafilters on B, equipped with the topology generated by sets of the form \{ U \in \Spec(B) \mid a \in U \} for a \in B. Stone duality establishes a contravariant equivalence between the category of Boolean algebras and the category of Stone spaces, which are compact, totally disconnected Hausdorff spaces.^[76] Under this duality, \Spec(B) is the Stone space associated to B, and the clopen sets of \Spec(B) correspond bijectively to elements of B.^[77] This duality highlights how the prime spectrum of a Boolean ring (isomorphic to a Boolean algebra) captures its topological structure as a totally disconnected compact space.^[78] Spectral spaces provide a broader topological framework generalizing the Zariski topology on \Spec(R). A spectral space is a sober T_0 topological space possessing a basis of quasi-compact open sets that is closed under finite intersections.^[79] The Zariski topology on \Spec(R) for any commutative ring R is spectral, as it satisfies sobriety (each irreducible closed set has a unique generic point) and has the required basis of basic opens D(f) = \{ \mathfrak{p} \in \Spec(R) \mid f \notin \mathfrak{p} \}, which are quasi-compact.^[80] Hochster's theorem characterizes spectral spaces topologically as precisely those homeomorphic to \Spec(R) for some commutative ring R, equipped with its Zariski topology.^[81] Priestley duality extends Stone duality to bounded distributive lattices, pairing them with ordered Stone spaces, or Priestley spaces, which are compact ordered topological spaces that are totally order-disconnected (for any x \not\leq y, there exists a clopen upset separating them).^[82] For a distributive lattice L, the Priestley space P(L) consists of the prime filters of L, ordered by inclusion and topologized such that the clopen upsets correspond to elements of L.^[83] This duality enriches the spectrum with an order structure, reflecting the lattice operations, and applies to rings whose spectra carry natural orderings, such as ordered rings.^[84] Generalizations to locales shift the perspective to pointless topology, where spaces are replaced by frames (complete distributive lattices satisfying the infinite distributive law) and locales are their duals.^[85] The frame of open sets of a spectral space forms a coherent frame, and the duality between spatial frames and sober spaces extends Stone and spectral dualities to this setting, allowing spectra to be studied without reference to points via ideals in the frame.^[86] In particular, the spectrum of a ring corresponds to a spatial locale whose frame is the lattice of quasi-coherent sheaves or open sets in the structure sheaf.^[87]

References

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The spectrum of a ring R is the set of prime ideals of R, denoted as Spec(R). It is empty only if R is the zero ring.
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[PDF] Contents
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10.19 The Jacobson radical of a ring - Stacks Project
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[PDF] 1 The Zariski prime spectrum 2 Distinguished open subsets
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[PDF] an introduction to the zariski topology - UChicago Math
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Definition 26.5.3 (01HU)—The Stacks project
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Section 26.5 (01HR): Affine schemes—The Stacks project
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Lemma 26.5.4 (01HV)—The Stacks project
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Lemma 26.6.4 (01I1)—The Stacks project
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Lemma 26.6.5 (01I2)—The Stacks project
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26.6 The category of affine schemes - Stacks Project
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[PDF] COMMUTATIVE ALGEBRA 00AO Contents 1. Introduction 4 2 ...
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Section 10.60 (00KD): Dimension—The Stacks project
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[PDF] The Going Up and Going Down Theorems
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[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 14
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[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41
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https://stacks.math.columbia.edu/tag/01LX
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None
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https://stacks.math.columbia.edu/download/topologies.pdf
[59]
[PDF] Topologies on Schemes - Stacks Project
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Section 10.40 (080S): Supports and annihilators—The Stacks project
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Section 17.16 (01CA): Tensor product—The Stacks project
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None
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[65]
[PDF] A (Very) Short Course on C -Algebras - Dartmouth Mathematics
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None
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[PDF] Spectrum (functional analysis)
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[PDF] A country of which nothing is known but the name Grothendieck and ...
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[PDF] a one-sided prime ideal principle for noncommutative rings
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[PDF] arXiv:2008.06605v1 [math.RA] 14 Aug 2020
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[PDF] Some Remarks on the Prime Spectrum of a Noncommutative Ring
Throughout this paper, we assume that R is an associative ring (not neces- sarily commutative) with unity. Let R be a ring. A proper left ideal P of R is ...
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[PDF] Prime spectrum and primitive Leavitt path algebras - UCA
Mar 24, 2009 · In noncommutative Ring Theory, one-sided conditions tend not to be left-right symmetric (perhaps with the remarkable exception of semisim-.
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Noncommutative algebraic geometry - Project Euclid
The points of such an affine scheme are the simple modules of the algebra, and the local structure of the scheme at a finite family of points, is expressed in ...<|separator|>
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Noncommutative Affine Schemes | SpringerLink
This chapter might be regarded as an introduction to noncommutative affine algebraic geometry. In other words, we consider here facts which are naturally ...
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Gelfand spectrum in nLab
Aug 19, 2025 · The Gelfand spectrum (originally Гельфанд) of a commutative C*-algebra A A is a topological space X X such that A A is the algebra of complex- ...Missing: 1940s work prefiguring<|control11|><|separator|>
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[PDF] Matrix ring
Nov 19, 2012 · If R is commutative, the matrix ring has a structure of a *-algebra over R, where the involution * on. Mn(R) is the matrix transposition.
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[PDF] Stone Duality for Boolean Algebras - The University of Manchester
Stone Duality is nowadays a term describing a tight relation between classes of algebraic structures and classes of topological spaces.
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[PDF] Stone duality for Boolean algebras - Sam van Gool
Mar 10, 2024 · The aim of this note is to give a detailed proof of Stone duality for Boolean algebras [1] to facilitate its formalization in Mathlib.
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[PDF] An Introduction to Stone Duality - Alexander Kurz
Apr 9, 2004 · We prove two classical theorems, Stone's theorem representing Boolean algebras as topological spaces and Goldblatt's theorem representing modal ...
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[PDF] arXiv:2001.00808v3 [math.RA] 10 Dec 2021
Dec 10, 2021 · Abstract. Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings.
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[PDF] prime ideal structure in commutative rings(¹)
In this section we define and construct a space-preserving functor from A to B such that the image objects are simple. In the next section we obtain our goal by ...
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[PDF] Ring constructions on spectral spaces - The University of Manchester
spectral spaces, defined as those topological spaces which are T0, quasi-compact and sober, whose quasi-compact and open subsets form a basis for the topology.
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[PDF] Remarks on spectra, supports, and Hochster duality
5.1 Hochster's Theorem (1969). Definition. A space is spectral if it is sober, and if the quasi-compact open sets form a sub-lattice that is a basis for the ...Missing: T0 | Show results with:T0
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Duality for Distributive Lattices: The Priestley Topology
The Priestley duality between bounded distributive lattices and ordered Boolean spaces can easily be restricted to a duality between the category of. BoolAlg.
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[PDF] arXiv:2009.00168v1 [math.LO] 1 Sep 2020
Sep 1, 2020 · By Priestley duality, each bounded distributive lattice is represented as the lattice of clopen upsets of a Priestley space, and by Esakia ...
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[PDF] Priestley duality for distributive semilattices - PoPuPS
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[PDF] the point of pointless topology1 - by peter t. johnstone
It is here that the real point of pointless topology begins to emerge; the difference between locales and spaces is one that we can (usually) afford to ignore ...
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[PDF] Borel and analytic sets in locales - arXiv
Nov 1, 2020 · Locale theory, also known as pointless or point-free topology, is the “dual” algebraic study of topological spaces and generalizations ...