Derived tensor product

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:

:$M\; \backslash otimes\_R^L\; N$

whose i-th homotopy is the i-th Tor:

:$\backslash pi\_i\; (M\; \backslash otimes\_R^L\; N)\; =\; \backslash operatorname\{Tor\}^R\_i(M,\; N)$.

It is called the derived tensor product of M and N. In particular, $\backslash pi\_0\; (M\; \backslash otimes\_R^L\; N)$ 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 $\backslash Omega\_\{Q(R)\}^1$ be the module of Kähler differentials. Then

:$\backslash mathbb\{L\}\_R\; =\; \backslash Omega\_\{Q(R)\}^1\; \backslash otimes^L\_\{Q(R)\}\; R$

is an R-module called the cotangent complex of R. It is functorial in R: each R → S gives rise to $\backslash mathbb\{L\}\_R\; \backslash to\; \backslash mathbb\{L\}\_S$. Then, for each R → S, there is the cofiber sequence of S-modules

:$\backslash mathbb\{L\}\_\{S/R\}\; \backslash to\; \backslash mathbb\{L\}\_R\; \backslash otimes\_R^L\; S\; \backslash to\; \backslash mathbb\{L\}\_S.$

The cofiber $\backslash mathbb\{L\}\_\{S/R\}$ is called the relative cotangent complex.

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

{{reflist}}

• Lurie, J., [http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf Spectral Algebraic Geometry (under construction)]
• Lecture 4 of Part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry
• Ch. 2.2. of [https://arxiv.org/abs/math/0404373 Toen-Vezzosi's HAG II]

Category:Algebraic geometry

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