Shapiro's lemma
Encyclopedia
In mathematics
, especially in the areas of abstract algebra
dealing with group cohomology
or relative homological algebra, Shapiro's lemma, also known as the Eckmann–Shapiro lemma, relates extensions of modules over one ring to extensions over another, especially the group ring
of a group
and of a subgroup
. It thus relates the group cohomology
with respect to a group to the cohomology with respect to a subgroup. Shapiro's lemma is named after Arnold Shapiro, who proved it in 1961; however, Beno Eckmann
had discovered it earlier, in 1953.
, so that S becomes a left and right R-module. Let M be a left S-module and N a left R-module. By restriction of scalars, M is also a left R-module.
See . The projectivity conditions can be weakened into conditions on the vanishing of certain Tor- or Ext-groups: see .
in G, then the group ring R[G] is finitely generated projective as a left and right R[H] module, so the previous applies in a simple way. Let M be a finite dimensional representation of G and N a finite dimensional representation of H. In this case, the module S ⊗R N is called the induced representation
of N from H to G, and RM is called the restricted representation
of M from G to H. One has that:
When n = 0, this is called Frobenius reciprocity for completely reducible modules, and Nakayama reciprocity in general. See , which also contains these higher versions of the Mackey decomposition.
of N from H to G using the tensor product
, and for H* the group homology:
Similarly, for NG the co-induced representation of N from H to G using the Hom functor
, and for H* the group cohomology
:
When H is finite index in G, then the induced and coinduced representations coincide and the lemma is valid for both homology and cohomology.
See .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the areas of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
dealing with group cohomology
Group cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
or relative homological algebra, Shapiro's lemma, also known as the Eckmann–Shapiro lemma, relates extensions of modules over one ring to extensions over another, especially the group ring
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
and of a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
. It thus relates the group cohomology
Group cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
with respect to a group to the cohomology with respect to a subgroup. Shapiro's lemma is named after Arnold Shapiro, who proved it in 1961; however, Beno Eckmann
Beno Eckmann
Beno Eckmann was a Swiss mathematician who was a student of Heinz Hopf.Born in Bern, Eckmann received his master's degree from Eidgenössische Technische Hochschule Zürich in 1931. Later he studied there under Heinz Hopf, obtaining his Ph.D. in 1941...
had discovered it earlier, in 1953.
Statement for rings
Let R → S be a ring homomorphismRing homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
, so that S becomes a left and right R-module. Let M be a left S-module and N a left R-module. By restriction of scalars, M is also a left R-module.
- If S is projective as a right R-module, then:
- If S is projective as a left R-module, then:
See . The projectivity conditions can be weakened into conditions on the vanishing of certain Tor- or Ext-groups: see .
Statement for group rings
When H is a subgroup of finite indexIndex of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
in G, then the group ring R[G] is finitely generated projective as a left and right R[H] module, so the previous applies in a simple way. Let M be a finite dimensional representation of G and N a finite dimensional representation of H. In this case, the module S ⊗R N is called the induced representation
Induced representation
In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...
of N from H to G, and RM is called the restricted representation
Restricted representation
In mathematics, restriction is a fundamental construction in representation theory of groups. Restriction forms a representation of a subgroup from a representation of the whole group. Often the restricted representation is simpler to understand...
of M from G to H. One has that:
When n = 0, this is called Frobenius reciprocity for completely reducible modules, and Nakayama reciprocity in general. See , which also contains these higher versions of the Mackey decomposition.
Statement for group cohomology
Specializing M to be the trivial module produces the familiar Shapiro's lemma. Let H be a subgroup of G and N a representation of H. For NG the induced representationInduced representation
In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...
of N from H to G using the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
, and for H* the group homology:
- H*(G, NG) = H*(H, N)
Similarly, for NG the co-induced representation of N from H to G using the Hom functor
Hom functor
In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.-Formal...
, and for H* the group cohomology
Group cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
:
- H*(G, NG) = H*(H, N)
When H is finite index in G, then the induced and coinduced representations coincide and the lemma is valid for both homology and cohomology.
See .