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# 数学代写|代数拓扑代考Algebraic Topology代考|MATH625 The Category of A -Modules

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## 数学代写|代数拓扑代考Algebraic Topology代考|The Category of A -Modules

Let $\mathscr{M}$ denote the category of (left) $\mathscr{A}$-modules. This is an abelian category with additional structure; namely, the fact that $\mathscr{A}$ is a Hopf algebra implies that the tensor product (as graded vector spaces) of two $\mathscr{A}$-modules has a natural $\mathscr{A}$-module structure. Explicitly, the Steenrod squares act via:
$$S q^n(x \otimes y)=\sum_{i+j=n} S q^i(x) \otimes S q^j(y)$$
this corresponds to the fact that the diagonal $\Delta: \mathscr{A} \rightarrow \mathscr{A} \otimes \mathscr{A}$ is determined by $\Delta S q^n=\sum_{i+j=n} S q^i \otimes S q^j$

Since $\mathscr{A}$ is a connected algebra (concentrated in non-negative degrees, with $\mathscr{A}^0=\mathbb{F}$ ) the Hopf algebra conjugation (or antipode) $\chi: \mathscr{A}^{\circ} \rightarrow \mathscr{A}$ is determined by the diagonal [MM65] and is an isomorphism of algebras, where $\mathscr{A}^{\circ}$ is $\mathscr{A}$ equipped with the opposite algebra structure ( $\chi$ is an anti-automorphism of $\mathscr{A}$ ).

## 数学代写|代数拓扑代考Algebraic Topology代考|Unstable Modules and Destabilization

Whereas the cohomology of a spectrum (object from stable homotopy theory which represents a cohomology theory) is simply an $\mathscr{A}$-module, the cohomology of a space has further structure; it is an algebra (via the cup product) and the underlying $\mathscr{A}$-module is unstable.

Definition 2.2.3.1 An $\mathscr{A}$-module $M$ is unstable if $S q^i x=0, \forall i>|x|$. The full subcategory of unstable modules is denoted $\mathscr{U} \subset \mathscr{M}$.

Proposition 2.2.3.2 The category $\mathscr{U}$ is an abelian subcategory of $\mathscr{M}$ and is closed under the tensor product $\otimes$ of $\mathscr{M}$.

Proof From the definition of instability, it is clear that a submodule (respectively quotient) of an unstable module is unstable. This implies that $\mathscr{U}$ is an abelian subcategory of $\mathscr{M}$.

Closure under $\otimes$ is seen as follows. By definition, $S q^n(x \otimes y)=$ $\sum_{i+j=n} S q^i(x) \otimes S q^j(y)$; if $n>|x \otimes y|$ and $i+j=n$, then either $i>|x|$ or $j>|y|$, so that the right hand expression is zero, as required.

Remark 2.2.3.3 The duality functor $(-)^{\vee}: \mathscr{M}^{\mathrm{op}} \rightarrow \mathscr{M}$ does not preserve $\mathscr{U}$, since the relation $S q^0=1$ implies that an unstable module is concentrated in degrees $\geq 0$. The dual $M^{\vee}$ of a module $M$ concentrated in degrees $\geq 0$ is concentrated in degrees $\geq 0$ if and only if $M=M^0$; for example, the dual of $\Sigma \mathbb{F}$ is not unstable.
Example 2.2.3.4 For $n \in \mathbb{N}$, the suspension functor $\Sigma^n: \mathscr{M} \rightarrow \mathscr{M}$ restricts to an exact functor $\Sigma^n: \mathscr{U} \rightarrow \mathscr{U}$ (given by $\Sigma^n \mathbb{F} \otimes-$ ). This is not an equivalence of categories if $n>0$.

For later use, the following definition is recalled, which uses the tensor product of $\mathscr{U}$.

## 数学代写|代数拓扑代考Algebraic Topology代考|The Category of A -Modules

$$S q^n(x \otimes y)=\sum_{i+j=n} S q^i(x) \otimes S q^j(y)$$

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