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Künneth theorem

The is a fundamental result in that relates the or groups of the of two topological spaces to those of the factors, providing an algebraic formula typically involving tensor products and (or Ext) groups. Named after the German mathematician Hermann Künneth, the theorem originated in the study of the of product manifolds in the early (specifically, Künneth's 1922 on Betti numbers of product manifolds) but was formalized in its modern form through developments in by and others in the 1940s. In its standard version for with coefficients in a R (such as the integers \mathbb{Z} or a field), and for spaces X and Y that are CW complexes, the theorem asserts the existence of a natural short exact sequence $0 \to \bigoplus_{p+q=n} H_p(X; R) \otimes_R H_q(Y; R) \to H_n(X \times Y; R) \to \bigoplus_{p+q=n-1} \Tor^R_1(H_p(X; R), H_q(Y; R)) \to 0, which splits as R-modules (though not naturally). The proof relies on the Eilenberg–Zilber theorem, which establishes a chain homotopy equivalence between the singular chain complex of the product and the tensor product of the individual chain complexes, combined with the algebraic Künneth theorem for chain complexes over a PID. When R is a (so all Tor terms vanish, as homology groups are vector spaces), the sequence yields a direct isomorphism H_n(X \times Y; R) \cong \bigoplus_{p+q=n} H_p(X; R) \otimes_R H_q(Y; R), greatly simplifying computations for product spaces like the torus T^n = (S^1)^n or S^m \times S^k. A cohomology analogue exists, where the external cup product (or cross product) induces a ring isomorphism H^*(X \times Y; R) \cong H^*(X; R) \otimes_R H^*(Y; R) if X and Y are CW complexes and the homology groups H_*(Y; R) are finitely generated free R-modules in each degree, preserving the ring structure via the diagonal approximation. Relative versions apply to pairs of spaces, and the theorem extends to filtered complexes via spectral sequences, as well as to generalized cohomology theories under additional flatness or projectivity assumptions. The Künneth theorem is indispensable for determining topological invariants of products, enabling inductive calculations on complex spaces, and underpinning applications in manifold topology, fiber bundles, and equivariant theories; for instance, it facilitates the computation of Betti numbers and torsion in the homology of Lie groups or configuration spaces.

Introduction and history

General concept

The Künneth theorem provides a fundamental tool in for computing the or groups of a product X \times Y by relating them to the corresponding groups of the individual spaces X and Y. Direct computation of these invariants for product spaces often proves difficult due to the intricate combinatorial structure arising from the , which complicates simplicial or cellular decompositions. Instead, the theorem leverages algebraic constructions on the or of the factors to simplify the process, transforming a topological problem into a more manageable algebraic one. Intuitively, under appropriate conditions on the spaces and rings, the (or ) of the product space is assembled from the tensor products of the (or ) groups of X and Y, with additional terms incorporating corrections for torsional phenomena in the groups. This structure captures how the topological features—such as holes and connectivity—of the individual spaces interact and combine in the product. The approach reflects the multiplicative nature of products in , allowing the overall invariant to emerge from pairwise combinations across dimensions. Applications of the Künneth theorem are widespread in simplifying calculations for familiar product spaces like tori or spheres, enabling the determination of key topological properties without exhaustive direct analysis. For instance, when coefficients lie in a , the theorem implies that Betti numbers—the ranks of the groups—multiply in a polynomial-like manner, providing a quick way to quantify the number of holes in each dimension for the product. This utility extends to broader classifications of manifolds and bundles, where products frequently arise.

Development and key contributors

The Künneth theorem traces its origins to the work of German mathematician Hermann Künneth, who in his 1923 dissertation published as a paper addressed the of product manifolds, expressing the Betti number b_n(X × Y) as the sum over p + q = n of the products b_p(X) b_q(Y) of the of the individual factors. Künneth's result provided an early algebraic tool for understanding the of products and laid the groundwork for later generalizations in combinatorial topology. In the 1940s, extended Künneth's formula to encompass groups with torsion coefficients for products of simplicial complexes, building on techniques he developed during that period. This generalization addressed limitations in the original version by incorporating the full structure of integer , including non-free abelian groups, and reflected the growing emphasis on algebraic invariants in amid wartime collaborations among mathematicians. By the early , Eilenberg and Norman Steenrod integrated the Künneth theorem into their axiomatic framework for homology theories in their seminal 1952 monograph, ensuring its applicability to any theory satisfying the Eilenberg-Steenrod axioms. Mid-century advancements further refined the theorem for general coefficients through the inclusion of Tor terms, as detailed in and Samuel Eilenberg's 1956 treatment of , which provided the algebraic foundation for exact sequences involving tensor products and derived functors. Concurrently, William S. Massey contributed to the formulation in the late 1950s via his development of exact couples, enabling iterative computations of product homologies in more complex settings. Key contributors to the theorem's evolution include Hermann Künneth for the initial Betti number version, Samuel Eilenberg for torsion-inclusive extensions and axiomatic integration, Norman Steenrod for the foundational axioms, Henri Cartan for algebraic refinements with Tor, and William Massey for the spectral sequence approach.

Künneth theorems in singular homology

Coefficients in a field

The Künneth theorem achieves its simplest form when singular homology is computed with coefficients in a field. For topological spaces X and Y and a field F, there exists a natural isomorphism of graded vector spaces over F, H_*(X \times Y; F) \cong H_*(X; F) \otimes_F H_*(Y; F), or more explicitly in each degree, H_k(X \times Y; F) \cong \bigoplus_{i+j=k} H_i(X; F) \otimes_F H_j(Y; F). This isomorphism is induced by the cross product map in homology, which is defined using the Eilenberg–Zilber chain map and the tensor product of chain complexes. The naturality of the isomorphism means that for any continuous maps f \colon X' \to X and g \colon Y' \to Y, the following diagram commutes: \begin{CD} H_k(X' \times Y'; F) @>{\cong}>> \bigoplus_{i+j=k} H_i(X'; F) \otimes_F H_j(Y'; F) \\ @V{f \times g}_*VV @VV{f_* \otimes g_*}V \\ H_k(X \times Y; F) @>{\cong}>> \bigoplus_{i+j=k} H_i(X; F) \otimes_F H_j(Y; F). \end{CD} This property ensures the theorem respects the category structure of topological spaces and continuous maps. A key application arises when F = \mathbb{Q} or \mathbb{R}, where the homology groups are vector spaces whose dimensions are the Betti numbers b_k(Z) = \dim H_k(Z; F). The isomorphism then implies that the Poincaré polynomial of the product is the product of the individual polynomials: p_{X \times Y}(t) = p_X(t) \cdot p_Y(t), with p_Z(t) = \sum_k b_k(Z) t^k. This multiplicative property simplifies computations for products of manifolds or CW complexes, as the Betti numbers satisfy b_k(X \times Y) = \sum_{i+j=k} b_i(X) b_j(Y). The theorem holds for arbitrary topological spaces using singular homology, without requiring compact supports or finite type, because the singular chain complexes are free abelian groups (hence flat over F) and tensor products over fields preserve exactness, yielding no higher derived functor terms in the homology computation.

Coefficients in a principal ideal domain

The Künneth theorem for singular homology with coefficients in a R provides a short exact sequence relating the homology of the product space X \times Y to the homologies of X and Y. Specifically, for topological spaces X and Y, and integers k \geq 0, there is a natural short $0 \to \bigoplus_{i+j=k} H_i(X; R) \otimes_R H_j(Y; R) \to H_k(X \times Y; R) \to \bigoplus_{i+j=k-1} \operatorname{Tor}_1^R \bigl( H_i(X; R), H_j(Y; R) \bigr) \to 0. This sequence is natural in the pair (X, Y), meaning that continuous maps X \to X' and Y \to Y' induce commutative diagrams of the corresponding sequences. The sequence splits as a short exact sequence of R-modules, though the splitting is not canonical or natural. Consequently, the homology group of the product decomposes up to isomorphism as H_k(X \times Y; R) \cong \Bigl( \bigoplus_{i+j=k} H_i(X; R) \otimes_R H_j(Y; R) \Bigr) \oplus \Bigl( \bigoplus_{i+j=k-1} \operatorname{Tor}_1^R \bigl( H_i(X; R), H_j(Y; R) \bigr) \Bigr). This splitting implies that the Tor term captures potential torsion or non-free contributions in the homology of the product that do not arise from simple tensor products of the individual homologies. The theorem holds for general topological spaces under mild assumptions, such as one of X or Y having finitely generated homology groups over R, ensuring the relevant chain complexes admit suitable resolutions. The Tor term vanishes if R is a field, in which case \operatorname{Tor}_1^R(-, -) = 0, reducing the sequence to the isomorphism H_k(X \times Y; R) \cong \bigoplus_{i+j=k} H_i(X; R) \otimes_R H_j(Y; R), as established in the case of field coefficients. More generally, the Tor term is zero whenever at least one of the homology groups H_*(X; R) or H_*(Y; R) is a free R-module, since \operatorname{Tor}_1^R(M, N) = 0 if M or N is free over the PID R.

Example

To illustrate the utility of the Künneth theorem for coefficients in the principal ideal domain \mathbb{Z}, consider the computation of the singular homology groups H_*(\mathbb{RP}^2 \times \mathbb{RP}^2; \mathbb{Z}). The homology groups of the real projective plane \mathbb{RP}^2 with integer coefficients are H_0(\mathbb{RP}^2; \mathbb{Z}) \cong \mathbb{Z}, H_1(\mathbb{RP}^2; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}, and H_2(\mathbb{RP}^2; \mathbb{Z}) \cong 0 (with all higher groups zero). The Künneth theorem provides a short exact sequence for each degree n \geq 0: $0 \to \bigoplus_{i+j=n} H_i(\mathbb{RP}^2; \mathbb{Z}) \otimes H_j(\mathbb{RP}^2; \mathbb{Z}) \to H_n(\mathbb{RP}^2 \times \mathbb{RP}^2; \mathbb{Z}) \to \bigoplus_{i+j=n-1} \operatorname{Tor}_1^\mathbb{Z}(H_i(\mathbb{RP}^2; \mathbb{Z}), H_j(\mathbb{RP}^2; \mathbb{Z})) \to 0. This sequence splits (though not naturally), so the homology groups of the product are direct sums of the tensor and terms. For n=0, the tensor term is H_0 \otimes H_0 \cong \mathbb{Z} \otimes \mathbb{Z} \cong \mathbb{Z}, and the Tor term vanishes (as it arises from degree -1). Thus, H_0(\mathbb{RP}^2 \times \mathbb{RP}^2; \mathbb{Z}) \cong \mathbb{Z}. For n=1, the tensor term is H_0 \otimes H_1 \oplus H_1 \otimes H_0 \cong (\mathbb{Z} \otimes \mathbb{Z}/2\mathbb{Z}) \oplus (\mathbb{Z}/2\mathbb{Z} \otimes \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}, and the Tor term from degree 0 is \operatorname{Tor}_1^\mathbb{Z}(H_0, H_0) \cong \operatorname{Tor}_1^\mathbb{Z}(\mathbb{Z}, \mathbb{Z}) \cong 0. Thus, H_1(\mathbb{RP}^2 \times \mathbb{RP}^2; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}. For n=2, the tensor term is H_0 \otimes H_2 \oplus H_1 \otimes H_1 \oplus H_2 \otimes H_0 \cong 0 \oplus ([\mathbb{Z}/2\mathbb{Z} \otimes \mathbb{Z}/2\mathbb{Z}](/page/Z)) \oplus 0 \cong \mathbb{Z}/2\mathbb{Z} (since \mathbb{Z}/m\mathbb{Z} \otimes \mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/\gcd(m,n)\mathbb{Z}), and the Tor term from degree 1 is \operatorname{Tor}_1^\mathbb{Z}(H_0, H_1) \oplus \operatorname{Tor}_1^\mathbb{Z}(H_1, H_0) \cong 0 \oplus 0 \cong 0. Thus, H_2(\mathbb{RP}^2 \times \mathbb{RP}^2; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}. For n=3, the tensor term is H_1 \otimes H_2 \oplus H_2 \otimes H_1 \cong 0 \oplus 0 \cong 0, and the Tor term from degree 2 is \operatorname{[Tor](/page/Tor)}_1^\mathbb{Z}(H_1, H_1) \cong \operatorname{[Tor](/page/Tor)}_1^\mathbb{Z}(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}/2\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}. Thus, H_3(\mathbb{RP}^2 \times \mathbb{RP}^2; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}. For n=4 and higher, both tensor and Tor terms vanish, so H_n(\mathbb{RP}^2 \times \mathbb{RP}^2; \mathbb{Z}) \cong 0.

The Künneth spectral sequence

Formulation for homology

The Künneth spectral sequence in relates the homology groups of a product space to those of its factors using derived functors, providing a computational tool beyond the exact sequences available under restrictive assumptions. For topological spaces X and Y, and coefficients in a R, the C_*(X; R) and C_*(Y; R) give rise to a first-quadrant whose E^2 page is E^2_{p,q} = \bigoplus_{q_1 + q_2 = q} \Tor_p^R \bigl( H_{q_1}(X; R), H_{q_2}(Y; R) \bigr), converging to H_{p+q}(X \times Y; R). This spectral sequence arises from the double complex structure on the tensor product of projective resolutions of the homology groups, filtered appropriately to ensure convergence. It converges strongly when the filtrations are finite or when the homology groups satisfy suitable boundedness and finiteness conditions, such as being finitely generated over R. The E^2 term directly incorporates the algebraic structure of the classical Künneth formula: the row p=0 consists of the graded tensor products \bigoplus_{q_1 + q_2 = q} [H_{q_1}(X; R) \otimes_R H_{q_2}(Y; R)](/page/Homology), while the row p=1 captures the \Tor_1^R terms measuring torsion or non-projectivity; higher rows involve higher \Tor_p^R groups for p \geq 2. Under conditions where \Tor_p^R([H_*(X; R)](/page/Homology), [H_*(Y; R)](/page/Homology)) = 0 for all p \geq 2, such as when one of the modules is flat over R, the collapses at the E^2 page, reproducing the short of the classical Künneth theorem for coefficients in a . The sequence's utility lies in handling scenarios where the classical theorem's hypotheses fail, including products of spaces with infinitely generated groups or non-projective modules, allowing indirect computation via differentials on higher pages.

Formulation for

The Künneth spectral sequence for singular relates the groups of the product space X \times Y to those of X and Y via the \Ext_R, where R is a . This sequence arises from the structure of the cochain complex for the of the product, using the adjunction between tensor and Hom functors applied to the singular complexes, combined with a that yields the . Assuming the singular chain complexes C_*(X; R) and C_*(Y; R) satisfy appropriate boundedness conditions (e.g., one is projective or both have homology of finite type), there exists a first-quadrant spectral sequence E_2^{p,q} = \bigoplus_{i+j=q} \Ext_R^p \bigl( H_i(X; R), H^j(Y; R) \bigr) \quad \Rightarrow \quad H^{p+q}(X \times Y; R), with differentials d_r: E_r^{p,q} \to E_r^{p+r, q-r+1}. The sequence converges to the cohomology of the product space, and the edge homomorphism induces the natural map from the E_2^{0,q} term (isomorphic to \bigoplus_{i+j=q} \Hom_R(H_i(X; R), H^j(Y; R))) to H^q(X \times Y; R). This formulation is the cohomological analog of the homological Künneth , replacing the with Ext to account for the Hom-structure in computations. When R is a , higher Ext terms vanish (\Ext_R^p = 0 for p > 0), so the degenerates at the E_2 page, yielding the H^*(X \times Y; R) \cong H^*(X; R) \otimes_R H^*(Y; R). For general PID coefficients, if the groups are free, the sequence further simplifies to a split short exact sequence $0 \to \bigoplus_{p+q=n} \Tor_1^R \bigl( H^p(X; R), H^q(Y; R) \bigr) \to H^n(X \times Y; R) \to \bigoplus_{p+q=n} H^p(X; R) \otimes_R H^q(Y; R) \to 0, reflecting the contributions from \Ext^0 = \Hom and \Ext^1, with higher terms absent. The conditions mirror those for the homological version, requiring the homology groups to be finitely generated or one space to have projective chains for the sequence to converge strongly.

Relation to homological algebra and proof sketch

Background in homological algebra

In , the provides a fundamental construction for combining over a . For a R with right R- A and left R- B, the A \otimes_R B is the generated by symbols a \otimes b subject to bilinearity relations (a + a') \otimes b = a \otimes b + a' \otimes b, a \otimes (b + b') = a \otimes b + a \otimes b', and (ra) \otimes b = a \otimes (rb) for r \in R. This -\otimes_R B: {}_R\mathrm{Mod} \to \mathrm{Ab} is right exact, meaning that for a short $0 \to A' \to A \to A'' \to 0 of right R-modules, the induced sequence A' \otimes_R B \to A \otimes_R B \to A'' \otimes_R B \to 0 is exact. The failure of left exactness is measured by the derived functors of the . The Tor functors, denoted \mathrm{Tor}^R_p(A, B) for p \geq 0, are the left derived functors of -\otimes_R -. They are computed by taking a projective resolution P_\bullet \to A of A, tensoring with B to form P_\bullet \otimes_R B, and setting \mathrm{Tor}^R_p(A, B) = H_p(P_\bullet \otimes_R B); thus, \mathrm{Tor}^R_0(A, B) \cong A \otimes_R B. In particular, \mathrm{Tor}^R_1(A, B) detects the obstruction to exactness on the left and classifies extensions of modules; for instance, if \mathrm{Tor}^R_1(A, B) = 0, then tensoring preserves the exactness of short exact sequences involving A and B. These functors arise naturally in the study of chain complexes: for chain complexes C_\bullet and D_\bullet of R-modules, the tensor product complex is defined by (C_\bullet \otimes D_\bullet)_n = \bigoplus_{p+q=n} C_p \otimes_R D_q with differential d(c \otimes d) = dc \otimes d + (-1)^p c \otimes dd, and the homology H_n(C_\bullet \otimes D_\bullet) is related to \mathrm{Tor}^R_*(H_*(C_\bullet), H_*(D_\bullet)) via a or under suitable conditions. Dually, the Ext functors provide cohomology-like information. The functor \mathrm{Ext}^R_p(A, B) for p \geq 0 is the right derived functor of \mathrm{Hom}_R(A, -), computed using an injective resolution $0 \to B \to I^\bullet of B and setting \mathrm{Ext}^R_p(A, B) = H^p(\mathrm{Hom}_R(A, I^\bullet)), with \mathrm{Ext}^R_0(A, B) \cong \mathrm{Hom}_R(A, B). Specifically, \mathrm{Ext}^R_1(A, B) classifies equivalence classes of short exact sequences $0 \to B \to E \to A \to 0 up to congruence, serving as the extension group. While Tor is bivariant and measures tensor obstructions, Ext is contravariant in the first argument and covariant in the second, balancing the homological structure. Flat resolutions simplify these computations. A module F over R is flat if -\otimes_R F is exact, i.e., preserves all exact sequences. Free modules and modules over fields are flat, as tensoring with a field vector space preserves exactness due to the absence of torsion. Over a principal ideal domain (PID), every finitely generated module admits a free resolution, allowing Tor and Ext to be computed via free (hence flat) approximations, which vanish in higher degrees under flatness conditions.

Outline of the proof

The proof of the classical Künneth theorem begins with the Eilenberg-Zilber theorem, which establishes a natural chain homotopy equivalence between the singular chain complex of the product space C_*(X \times Y) and the tensor product of the individual singular chain complexes C_*(X) \otimes C_*(Y). This equivalence is constructed using the Alexander-Whitney map as one direction and the shuffle product (Eilenberg-Zilber map) as the inverse, both of which are chain maps inducing isomorphisms on homology. The theorem relies on the method of acyclic models to ensure uniqueness up to chain homotopy. At the algebraic level, the homology of the tensor product chain complex H_*(C \otimes D) is analyzed using the algebraic Künneth short exact sequence over a principal ideal domain R: $0 \to \bigoplus_{p+q=n} H_p(C) \otimes_R H_q(D) \to H_n(C \otimes D) \to \bigoplus_{p+q=n-1} \Tor^R_1(H_p(C), H_q(D)) \to 0. This sequence splits (non-naturally) when the chain complexes consist of free R-modules, providing an exact description of the homology of the tensor product in terms of tensor products and Tor terms of the individual homologies. Combining this with the Eilenberg-Zilber equivalence yields the topological Künneth theorem, relating H_*(X \times Y) to H_*(X) and H_*(Y). For the spectral version, the tensor product C \otimes D is viewed as a double complex, equipped with a (typically by columns or rows) on the complex. This induces a whose E_2-page is given by \Tor(H_*(C), H_*(D)), converging to H_*(C \otimes D). The differentials arise from the double complex structure, and under suitable boundedness assumptions on the , the sequence abuts to the desired . Exactness in both classical and forms requires conditions such as one of the complexes being projective (or flat) over R, ensuring vanishing of higher terms or acyclicity in the tails; for instance, when coefficients are in a , [\Tor](/page/Tor) = 0, yielding a direct .

Generalizations to other theories

Generalized homology theories

, also known as theories, are covariant functors h_* from the homotopy category of pointed connected CW-complexes (or spaces) to graded abelian groups that satisfy the Eilenberg-Steenrod axioms of additivity, exactness, homotopy invariance, excision, and the wedge axiom, but may violate the dimension axiom. These theories arise from the homotopy groups of smash products with an Omega-spectrum E, where h_n(X) = [ \Sigma^\infty X, E_n ]_*, and prominent examples include complex K_* and complex MU_*. In complex K-theory, Atiyah established a Künneth theorem stating that if K_*(X) is a and Y is a cell complex, then there is a natural $0 \to K_*(X) \otimes_\mathbb{Z} K_*(Y) \to K_*(X \times Y) \to \Tor_1^\mathbb{Z} (K_*(X), K_*(Y)) \to 0, which splits (but not naturally). This result, originally for and extended to via Bott periodicity, holds under the finite generation condition to ensure the Tor term vanishes in many cases, such as when one space has torsion-free K-theory. For complex cobordism MU_*, there are foundational results leading to a Künneth-type spectral sequence converging to MU_*(X \times Y), with E_2-term involving Tor groups over MU_*(\pt), accounting for the rich ring structure of MU_*. In general, for a generalized homology theory represented by a ring spectrum E, the Künneth theorem takes the form of a spectral sequence E_2^{p,q} = \Tor_p^{E_*} (E_*(X), E_*(Y))_q \implies E_{p+q *}(X \wedge Y), relating the homology of the smash product to a derived tensor product, since E_*(X \times Y) \cong E_*(X \wedge Y) up to suspension for finite CW-complexes. This holds for any E-module spectra X and Y, with convergence under flatness conditions on E_*(X) or E_*(Y) over E_*. Modern extensions appear in equivariant , where Greenlees and May construct equivariant Künneth spectral sequences for RO_G-graded theories using Mackey functors and geometric fixed points, generalizing the classical case to G-spectra. In motivic , analogous Künneth properties hold for motivic spectra over a , ensuring that motivic of products decomposes via a , as realized in the stable motivic category.

Cohomology in generalized theories

In generalized cohomology theories, which satisfy the Eilenberg-Steenrod axioms except the axiom, versions of the Künneth theorem relate the of a product or to the cohomologies of the factors, often via tensor products or spectral sequences. These theories include complex K^*, represented by the BU \times \mathbb{Z}, and sheaf on algebraic varieties, where the theorem facilitates computations for products. A foundational extension to sheaf cohomology appears in Grothendieck's framework using and hypercohomology, as developed in the . For separated schemes of finite type over a k, the Künneth formula provides a quasi-isomorphism in the : R\Gamma(X \times_k Y, p_1^* \mathcal{F} \otimes p_2^* \mathcal{G}) \simeq R\Gamma(X, \mathcal{F}) \otimes^\mathbb{L} R\Gamma(Y, \mathcal{G}) for quasi-coherent sheaves \mathcal{F} on X and \mathcal{G} on Y, where p_1, p_2 are projections and \otimes^\mathbb{L} denotes the . This relies on the projection formula and proper base change in . In étale cohomology, a specific instance of sheaf cohomology, the Künneth theorem holds for products of schemes over a separably closed field. For proper X over k and quasi-compact quasi-separated Y over k, with bounded above complexes E, K of torsion sheaves, there is an isomorphism R\Gamma(X \times_k Y, \pr_1^{-1} E \otimes_\mathbb{Z} \pr_2^{-1} K) \simeq R\Gamma(X, E) \otimes_\mathbb{Z}^\mathbb{L} R\Gamma(Y, K) in the derived category of \mathbb{Z}-modules. For torsion coefficients \mathbb{Z}/n\mathbb{Z} with n \geq 1 and Y of finite type over k, this simplifies to R\Gamma(X \times_k Y, \pr_1^{-1} E \otimes_{\mathbb{Z}/n\mathbb{Z}} \pr_2^{-1} K) \simeq R\Gamma(X, E) \otimes_{\mathbb{Z}/n\mathbb{Z}} R\Gamma(Y, K). These results, building on Grothendieck's étale site, use Leray spectral sequences and base change properties. The universal coefficient theorem in generalized relates with coefficients in an G to the underlying via Ext functors over the ring R = \pi_0(E) of the representing E. For an ordinary theory h^*, it yields a short $0 \to \Ext_R^1(h_{n-1}(X; R), G) \to h^n(X; G) \to \Hom_R(h_n(X; R), G) \to 0, which often splits unnaturally. In spectrum-based theories, this extends to h^n(X; G) \cong [X \wedge S^0, E \wedge K(G, n)]_*, linking contravariant to covariant . This duality to generalized versions holds under flatness assumptions on . In algebraic K-theory, Quillen's plus-construction and the definition of K_*(X) for schemes X via the homotopy groups of the K-theory space lead to Künneth-type results for products, often in the form of spectral sequences; short exact sequences like $0 \to K_*(X) \otimes K_*(Y) \to K_*(X \times Y) \to \Tor_1^\mathbb{Z}(K_*(X), K_*(Y)) \to 0 hold only under strong conditions such as torsion-freeness, with counterexamples otherwise. Thomason's work on higher K-theory of schemes formalizes related structures using Waldhausen categories and assembly maps. In , Artin and Mazur's foundational work on the supports the cohomology Künneth via comparisons to singular cohomology, ensuring compatibility for varieties over finite fields. Spectral sequences provide a general tool for these Künneth theorems in . The Anderson spectral sequence, arising in complex via the Eilenberg-Moore resolution, converges to K^*(X \times Y) with E_2-term involving Tor over the cohomology ring, generalizing to spectrum-based theories. Similarly, the Adams spectral sequence, when adapted for generalized computations, incorporates Künneth isomorphisms in its E_2-page via Ext groups, aiding calculations in stable categories. In , modern generalizations reformulate the Künneth theorem using tensor products of dg-categories. For smooth and proper dg-categories \mathcal{A}, \mathcal{B} over a , Blumberg and Mandell establish a Künneth for topological periodic cyclic (a theory on dg-categories): \TP(\mathcal{A} \otimes \mathcal{B}) \simeq \TP(\mathcal{A}) \otimes \TP(\mathcal{B}), extending to inertia-invariant versions and categories via methods. Toën and Vezzosi further develop \ell-adic and Künneth formulas for dg-categories of singularities, relating them to motivic measures in . These results underpin applications in noncommutative motives and derived stacks.

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