nLab
Beck module

Contents

Context

Category theory

Higher algebra

Contents

Idea

One usually defines cohomology with respect to some coefficient objects:

Beck modules are a simultaneous generalisation of all three types of module.

Definition

Let 𝒞\mathcal{C} be a category with pullbacks and let AA be an object in 𝒞\mathcal{C}. A Beck module over AA is an abelian group object in the slice category 𝒞 /A\mathcal{C}_{/ A}. In particular, if AA is the terminal object, this reduces to the notion of an abelian group object in 𝒞\mathcal{C}. We write Ab(𝒞 /A)\mathbf{Ab}(\mathcal{C}_{/ A}) for the category of Beck modules over AA.

(Beck 67, def. 5)

Properties

Proposition

Let 𝒞\mathcal{C} be an effective regular category (resp. locally presentable category) and let AA be an object in 𝒞\mathcal{C}. Then Ab(𝒞 /A)\mathbf{Ab}(\mathcal{C}_{/ A}) is an abelian category (resp. locally presentable category).

Proof

If 𝒞\mathcal{C} is a effective regular category (resp. locally presentable category), then so is 𝒞 /A\mathcal{C}_{/ A}. Thus, the claim reduces to the fact that the category of abelian group objects in an effective regular category (resp. locally presentable category) is an abelian category (resp. locally presentable category).

Proposition

Let 𝒞\mathcal{C} be an effective regular category with filtered colimits and let AA be an object in 𝒞\mathcal{C}. If filtered colimits in 𝒞\mathcal{C} preserve finite limits, then Ab(𝒞 /A)\mathbf{Ab}(\mathcal{C}_{/ A}) (is an abelian category and) satisfies axiom AB5.

Proof

The forgetful functor 𝒞 /A𝒞\mathcal{C}_{/ A} \to \mathcal{C} creates pullbacks and filtered colimits, so filtered colimits in 𝒞 /A\mathcal{C}_{/ A} also preserve finite limits. The forgetful functor Ab(𝒞 /A)𝒞 /A\mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A} creates limits and filtered colimits, so filtered colimits in Ab(𝒞 /A)\mathbf{Ab}(\mathcal{C}_{/ A}) preserve kernels. In view of the earlier proposition, it follows that Ab(𝒞 /A)\mathbf{Ab}(\mathcal{C}_{/ A}) satisfies axiom AB5.

Corollary

Let 𝒞\mathcal{C} be a locally finitely presentable effective regular category and let AA be an object in 𝒞\mathcal{C}. Then Ab(𝒞 /A)\mathbf{Ab}(\mathcal{C}_{/ A}) is a Grothendieck category.

Proof

Combine the two propositions above.

Derivations

Let 𝒞\mathcal{C} be a category with pullbacks, let AA be an object in 𝒞\mathcal{C}, and let U:Ab(𝒞 /A)𝒞 /AU : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A} be the forgetful functor. Given a Beck module MM over AA, an MM-valued derivation of AA is a morphism 1 AUM1_A \to U M in 𝒞 /A\mathcal{C}_{/ A}, where 1 A1_A is the terminal object in 𝒞 /A\mathcal{C}_{/ A}, and we write

Der(A,M)=𝒞 /A(1 A,UM)Der (A, M) = \mathcal{C}_{/ A} (1_A, U M)

for the set of MM-valued derivations of AA. The Beck module of differentials over AA is an object Ω A\Omega_A in Ab(𝒞 /A)\mathbf{Ab}(\mathcal{C}_{/ A}) representing the functor Der(A,):Ab(𝒞 /A)SetDer (A, -) : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathbf{Set}.

The Beck module Ω A\Omega_A is not guaranteed to exist in general. When the functor U:Ab(𝒞 /A)𝒞 /AU : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A} has a left adjoint, Ω A\Omega_A is simply the value of the left adjoint at 1 A1_A.

Proposition

Let 𝒞\mathcal{C} be a locally presentable category and let AA be an object in 𝒞\mathcal{C}. Then the forgetful functor U:Ab(𝒞 /A)𝒞 /AU : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A} has a left adjoint.

Proof

The forgetful functor Ab(𝒞 /A)𝒞 /A\mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A} creates limits and κ\kappa-filtered colimits (for some κ\kappa large enough), so we may apply the accessible adjoint functor theorem.

Examples

Beck modules over associative algebras

Proposition

Let 𝒞\mathcal{C} be the category of (not necessarily commutative) rings and let AA be a ring. Then Ab(𝒞 /A)\mathbf{Ab}(\mathcal{C}_{/ A}) is equivalent to the category of AA-bimodules.

Proof

Let ϵ:BA\epsilon : B \to A be ring homomorphism. To give it the structure of a Beck module over AA, we must give ring homomorphisms η:AB\eta : A \to B and μ:B× ABB\mu : B \times_A B \to B such that ϵη=id A\epsilon \circ \eta = id_A, ϵ(μ(b 0,b 1))=ϵ(b 0)=ϵ(b 1)\epsilon (\mu (b_0, b_1)) = \epsilon (b_0) = \epsilon (b_1), as well as various other equations. Given elements b 0,b 1,b 2,b 3b_0, b_1, b_2, b_3 of BB such that ϵ(b 0)=ϵ(b 2)\epsilon (b_0) = \epsilon (b_2) and ϵ(b 1)=ϵ(b 3)\epsilon (b_1) = \epsilon (b_3), we have the following interchange law:

μ(b 0+b 1,b 2+b 3)=μ(b 0,b 2)+μ(b 1,b 3)\mu (b_0 + b_1, b_2 + b_3) = \mu (b_0, b_2) + \mu (b_1, b_3)

Hence, if a=ϵ(b 1)=ϵ(b 2)a = \epsilon (b_1) = \epsilon (b_2),

μ(b 1,b 2)=μ(η(a)+b 1η(a),η(a)+b 2η(a))=μ(η(a),η(a))+μ(b 1η(a),b 2η(a))=η(a)+μ(b 1η(a),b 2η(a))\mu (b_1, b_2) = \mu (\eta (a) + b_1 - \eta (a), \eta (a) + b_2 - \eta (a)) = \mu (\eta (a), \eta (a)) + \mu (b_1 - \eta(a), b_2 - \eta (a)) = \eta (a) + \mu (b_1 - \eta(a), b_2 - \eta (a))

but ϵ(b 1η(a))=ϵ(b 2η(a))=0\epsilon (b_1 - \eta (a)) = \epsilon (b_2 - \eta (a)) = 0, so

μ(b 1η(a),b 2η(a))=μ(0,b 2η(a))+μ(b 1η(a),0)=b 2η(a)+b 1η(a)\mu (b_1 - \eta(a), b_2 - \eta(a)) = \mu (0, b_2 - \eta (a)) + \mu (b_1 - \eta (a), 0) = b_2 - \eta (a) + b_1 - \eta (a)

and we conclude that

μ(b 1,b 2)=b 1η(ϵ(b 1))+b 2=b 1+b 2η(ϵ(b 2))\mu (b_1, b_2) = b_1 - \eta (\epsilon (b_1)) + b_2 = b_1 + b_2 - \eta (\epsilon (b_2))

and in particular, μ\mu is entirely determined by η\eta and ϵ\epsilon. We also have the following interchange law,

μ(b 0b 1,b 2b 3)=μ(b 0,b 2)μ(b 1,b 3)\mu (b_0 b_1, b_2 b_3) = \mu (b_0, b_2) \mu (b_1, b_3)

and in particular,

μ(η(ϵ(b 2))b 1,b 2η(ϵ(b 1)))=μ(η(ϵ(b 2)),b 2)μ(b 1,η(ϵ(b 1)))=b 2b 1\mu (\eta (\epsilon (b_2)) b_1, b_2 \eta (\epsilon (b_1))) = \mu (\eta (\epsilon (b_2)), b_2) \mu (b_1, \eta (\epsilon (b_1))) = b_2 b_1

hence,

b 2b 1=η(ϵ(b 2))b 1η(ϵ(b 2b 1))+b 2η(ϵ(b 1))b_2 b_1 = \eta (\epsilon (b_2)) b_1 - \eta (\epsilon (b_2 b_1)) + b_2 \eta (\epsilon (b_1))

so if ϵ(b 1)=ϵ(b 2)=0\epsilon (b_1) = \epsilon (b_2) = 0, then b 2b 1=0b_2 b_1 = 0.

Let M=kerϵM = \ker \epsilon. The above shows that the internal abelian group structure on BB restricts to the pre-existing abelian group structure on kerϵ\ker \epsilon. In addition, the homomorphism η:AB\eta : A \to B gives MM the structure of an AA-bimodule, and we see that BB is naturally isomorphic to the square-0 extension ring AMA \oplus M, with componentwise addition and the multiplication given below,

(a 0,m 0)(a 1,m 1)=(a 0a 1,a 0m 1+m 0a 1)(a_0, m_0) \cdot (a_1, m_1) = (a_0 a_1, a_0 m_1 + m_0 a_1)

regarded as a Beck module over AA by defining ϵ:AMA\epsilon : A \oplus M \to A, η:AAM\eta : A \to A \oplus M, and μ:(AM)× A(AM)AM\mu : (A \oplus M) \times_A (A \oplus M) \to A \oplus M as follows:

ϵ(a,m)=a\epsilon (a, m) = a
η(a)=(a,0)\eta (a) = (a, 0)
μ((a,m 0),(a,m 1))=(a,m 0+m 1)\mu ((a, m_0), (a, m_1)) = (a, m_0 + m_1)

Thus, we have an equivalence between Ab(𝒞 /A)\mathbf{Ab}(\mathcal{C}_{/ A}) and the category of AA-bimodules, as claimed.

Proposition

Let AA be a ring. Then the Beck module Ω A\Omega_A is isomorphic to the AA-bimodule of Kähler differentials (relative to \mathbb{Z}).

Proof

Let MM be an AA-bimodule, regard AMA \oplus M as a ring as above, and let ϵ:AMA\epsilon : A \oplus M \to A be the obvious projection. A ring homomorphism ϕ:AAM\phi : A \to A \oplus M satisfying ϵϕ=id A\epsilon \circ \phi = id_A is the same thing as an additive homomorphism δ:AM\delta : A \to M satisfying the following equations,

δ(a 0a 1)=δ(a 0)a 1+a 0δ(a 1)\delta (a_0 a_1) = \delta (a_0) a_1 + a_0 \delta (a_1)

i.e. a derivation AMA \to M (over \mathbb{Z}). Thus, the Beck module Ω A\Omega_A has the same universal property as the AA-bimodule of Kähler differentials.

Beck modules over groups

Proposition

Let 𝒞\mathcal{C} be the category of (not necessarily abelian) groups and let GG be a group. Then Ab(𝒞 /G)\mathbf{Ab}(\mathcal{C}_{/ G}) is equivalent to the category of left GG-modules.

Proof

Let ϵ:HG\epsilon : H \to G be group homomorphism. To give it the structure of a Beck module over GG, we must give group homomorphisms η:GH\eta : G \to H and μ:H× GHH\mu : H \times_G H \to H such that ϵη=id A\epsilon \circ \eta = id_A, ϵ(μ(h 0,h 1))=ϵ(h 0)=ϵ(h 1)\epsilon (\mu (h_0, h_1)) = \epsilon (h_0) = \epsilon (h_1), as well as various other equations. Given elements h 0,h 1,h 2,h 3h_0, h_1, h_2, h_3 of HH such that ϵ(h 0)=ϵ(h 2)\epsilon (h_0) = \epsilon (h_2) and ϵ(h 1)=ϵ(h 3)\epsilon (h_1) = \epsilon (h_3), we have the following interchange law:

μ(h 0h 1,h 2h 3)=μ(h 0,h 2)μ(h 1,h 3)\mu (h_0 h_1, h_2 h_3) = \mu (h_0, h_2) \mu (h_1, h_3)

and in particular,

μ(η(ϵ(h 2))h 1,h 2η(ϵ(h 1)))=μ(η(ϵ(h 2)),h 2)μ(h 1,η(ϵ(h 1)))=h 2h 1\mu (\eta (\epsilon (h_2)) h_1, h_2 \eta (\epsilon (h_1))) = \mu (\eta (\epsilon (h_2)), h_2) \mu (h_1, \eta (\epsilon (h_1))) = h_2 h_1

but on the other hand, if g=ϵ(h 1)=ϵ(h 2)g = \epsilon (h_1) = \epsilon (h_2), then

μ(h 1,h 2)=μ(η(g)η(g) 1h 1,η(g)η(g) 1h 2)=μ(η(g),η(g))μ(η(g) 1h 1,η(g) 1h 2)=η(g)μ(η(g) 1h 1,η(g) 1h 2)\mu (h_1, h_2) = \mu (\eta (g) \eta (g)^{-1} h_1, \eta (g) \eta (g)^{-1} h_2) = \mu (\eta (g), \eta (g)) \mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2) = \eta (g) \mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2)

and writing ee for the unit of GG and HH, we have ϵ(η(g) 1h 1)=ϵ(η(g) 1h 2)=e\epsilon (\eta (g)^{-1} h_1) = \epsilon (\eta (g)^{-1} h_2) = e, hence

μ(η(g) 1h 1,η(g) 1h 2)=μ(e,η(g) 1h 2)μ(η(g) 1h 1,e)=η(g) 1h 2η(g) 1h 1\mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2) = \mu (e, \eta (g)^{-1} h_2) \mu (\eta (g)^{-1} h_1, e) = \eta (g)^{-1} h_2 \eta (g)^{-1} h_1

so we conclude that

μ(h 1,h 2)=h 2η(g) 1h 1\mu (h_1, h_2) = h_2 \eta (g)^{-1} h_1

and in particular, μ\mu is entirely determined by η\eta.

Let M=kerϵM = \ker \epsilon. The above shows that the internal abelian group structure on BB restricts to the pre-existing group structure on kerϵ\ker \epsilon. (In particular, MM is an abelian group!) We make MM into a left GG-module as follows:

gm=η(g)mη(g) 1g \cdot m = \eta (g) m \eta (g)^{-1}

We can then construct the semi-direct product MGM \rtimes G, which has the following multplication:

(m 0,g 0)(m 1,g 1)=(m 0η(g 0)m 1η(g 0) 1,g 0g 1)(m_0, g_0) \cdot (m_1, g_1) = (m_0 \eta (g_0) m_1 \eta (g_0)^{-1}, g_0 g_1)

There is a group homomorphism MGHM \rtimes G \to H defined by (m,g)mη(g)(m, g) \mapsto m \eta (g), and it is bijective: surjectivity is clear, and injectivity is a consequence of the fact that Mimη={e}M \cap \operatorname{im} \eta = \{ e \}. We may regard MGM \rtimes G as a Beck module over GG by defining ϵ:MGG\epsilon : M \rtimes G \to G, η:GMG\eta : G \to M \rtimes G, and μ:(MG)× G(MG)MG\mu : (M \rtimes G) \times_G (M \rtimes G) \to M \rtimes G as follows:

ϵ(m,g)=g\epsilon (m, g) = g
η(g)=(0,g)\eta (g) = (0, g)
μ((m 0,g),(m 1,g))=(m 1m 0,g)\mu ((m_0, g), (m_1, g)) = (m_1 m_0, g)

Thus, we have an equivalence between Ab(𝒞 /G)\mathbf{Ab}(\mathcal{C}_{/ G}) and the category of left GG-modules, as claimed.

Proposition

Let GG be a group and let MM be a left GG-module. Under the above identification of Beck modules over GG with left GG-modules, MM-valued derivations of GG are precisely crossed homomorphisms GMG \to M, i.e. maps δ:GM\delta : G \to M satisfying the following equation:

δ(g 0g 1)=δ(g 0)+g 0δ(g 1)\delta (g_0 g_1) = \delta (g_0) + g_0 \cdot \delta (g_1)
Proof

Let ϵ:MGG\epsilon : M \rtimes G \to G be the evident projection. A group homomorphism ϕ:GMG\phi : G \to M \rtimes G such that ϵϕ=id G\epsilon \circ \phi = id_G is the same thing as a map δ:GM\delta : G \to M satisfying the equation below,

(δ(g 0),g 0)(δ(g 1),g 1)=(δ(g 0g 1),g 0g 1)(\delta (g_0), g_0) \cdot (\delta (g_1), g_1) = (\delta (g_0 g_1), g_0 g_1)

which is equivalent to the defining equation for crossed homomorphisms.

The tangent category

One may assemble the individual categories of Beck modules over the objects of 𝒞\mathcal{C} into a category fibred over 𝒞\mathcal{C}, called the tangent category.

References

The concept is due to

  • Jon Beck, Triples, algebras and cohomology, Ph.D. thesis, Columbia University, 1967, Reprints in Theory and Applications of Categories, No. 2 (2003) pp 1-59 (TAC)

and was popularized in

  • Daniel G. Quillen, On the (co-)homology of commutative rings, in Proc. Symp. on Categorical Algebra, 65 – 87, American Math. Soc., 1970.

See also

An application to knot theory is given in

  • Markus Szymik, Alexander-Beck modules detect the unknot, Fund. Math. 246 (2019) 89-108.

Last revised on May 6, 2019 at 15:56:55. See the history of this page for a list of all contributions to it.