derived tensor product

In algebra, given a differential graded algebra ''A'' over a commutative ring ''R'', the derived tensor product functor is
:$- \otimes_A^ - : D(\mathsf_A) \times D(_A \mathsf) \to D(_R \mathsf)$
where $\mathsf_A$ and $_A \mathsf$ are the categories of right ''A''-modules and left ''A''-modules and ''D'' refers to the homotopy category (i.e., derived category). By definition, it is the left derived functor of the tensor product functor $- \otimes_A - : \mathsf_A \times _A \mathsf \to _R \mathsf$.

Derived tensor product in derived ring theory

If ''R'' is an ordinary ring and ''M'', ''N'' right and left modules over it, then, regarding them as discrete spectra, one can form the smash product of them:
:$M \otimes_R^L N$
whose ''i''-th homotopy is the ''i''-th Tor:
:$\pi_i (M \otimes_R^L N) = \operatorname^R_i(M, N)$.
It is called the derived tensor product of ''M'' and ''N''. In particular, $\pi_0 (M \otimes_R^L N)$ is the usual tensor product of modules ''M'' and ''N'' over ''R''.

Geometrically, the derived tensor product corresponds to the intersection product (of derived schemes).

Example: Let ''R'' be a simplicial commutative ring, ''Q''(''R'') → ''R'' be a cofibrant replacement, and $\Omega_^1$ be the module of Kähler differentials. Then
:$\mathbb_R = \Omega_^1 \otimes^L_ R$
is an ''R''-module called the cotangent complex of ''R''. It is functorial in ''R'': each ''R'' → ''S'' gives rise to $\mathbb_R \to \mathbb_S$. Then, for each ''R'' → ''S'', there is the cofiber sequence of ''S''-modules
:$\mathbb_ \to \mathbb_R \otimes_R^L S \to \mathbb_S.$
The cofiber $\mathbb_$ is called the relative cotangent complex.

See also

* derived scheme (derived tensor product gives a derived version of a scheme-theoretic intersection.)

Notes

References

*Lurie, J.,
Spectral Algebraic Geometry (under construction)
'
*Lecture 4 of Part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry
*Ch. 2.2. o
Toen-Vezzosi's HAG II

Algebraic geometry

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