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YCor
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(A soft question) Is there some relations between derived category and intersection theory?

After learning the trditionaltraditional intersection theory (W. Fulton's Intersection Theory and D. Eisenbud & J. Harris's 3264 and all that), I have some biased thinking about what I have learned in this period of time:

The first is the excess intersection formula:

Theorem 1. (Excess intersection formula) For regularly embedded subschemes $X_i\subset Y$ , a closed subscheme $V\subset Y$ and a connected component $Z\subset V\cap\bigcap_iX_i$, we have $$(X_1\cdot\ldots\cdot X_r\cdot V)^Z=\left\{s(Z,V)\cap\prod_{i=1}^rc(N_{X_i}Y|_Z)\right\}_{\dim V-\sum\mathrm{codim}(X_i,Y)}$$

Using this formula we can compute the contribution of degree support on the wrong dimension connected component. But is there some explanations using derived-philosophy? I guess may be the intersection information of bad dimensions (like self-intersection and excess intersection formula as above) can be reflected in some higher ranks of the derived intersection product.

Second, we all know the Serre's formula about intersection multiplicity:

Theorem 2. (Serre) If $A,B\subset X$ are two closed subschemes of smooth variety $X$ with expected dimensions and $Z$ be an irreducible component of $A\cap B$, then $$i(Z;A,B;X)=\sum_{i=0}^{\dim X}(-1)^i\mathrm{length}_{\mathscr{O}_{A\cap B,Z}}\mathrm{Tor}_i^{\mathscr{O}_{X,Z}}(\mathscr{O}_{A,Z},\mathscr{O}_{B,Z}).$$

Note that the zero-degree term is just $\mathrm{length}_{\mathscr{O}_{A\cap B,Z}}\mathscr{O}_{X,Z}/(\mathscr{I}_{A},\mathscr{I}_{B})$. Now is there relationship between this formula and some derived tensor product (maybe $\mathscr{O}_{A,Z}\otimes^{\mathbf{L}}_{\mathscr{O}_{X,Z}}\mathscr{O}_{B,Z}$)? What is the meaning of this tensor product? (or some derived intersection product?)

Maybe these things are related to the derived algebraic geometry (or the derived intersection theory I guess), is there some references (papers/notes/books) about derived-aspect intersection theory (char$=0$ is OK)?

Any help will be grateful!

(A soft question) Is there some relations between derived category and intersection theory?

After learning the trditional intersection theory (W. Fulton's Intersection Theory and D. Eisenbud & J. Harris's 3264 and all that), I have some biased thinking about what I have learned in this period of time:

The first is the excess intersection formula:

Theorem 1. (Excess intersection formula) For regularly embedded subschemes $X_i\subset Y$ , a closed subscheme $V\subset Y$ and a connected component $Z\subset V\cap\bigcap_iX_i$, we have $$(X_1\cdot\ldots\cdot X_r\cdot V)^Z=\left\{s(Z,V)\cap\prod_{i=1}^rc(N_{X_i}Y|_Z)\right\}_{\dim V-\sum\mathrm{codim}(X_i,Y)}$$

Using this formula we can compute the contribution of degree support on the wrong dimension connected component. But is there some explanations using derived-philosophy? I guess may be the intersection information of bad dimensions (like self-intersection and excess intersection formula as above) can be reflected in some higher ranks of the derived intersection product.

Second, we all know the Serre's formula about intersection multiplicity:

Theorem 2. (Serre) If $A,B\subset X$ are two closed subschemes of smooth variety $X$ with expected dimensions and $Z$ be an irreducible component of $A\cap B$, then $$i(Z;A,B;X)=\sum_{i=0}^{\dim X}(-1)^i\mathrm{length}_{\mathscr{O}_{A\cap B,Z}}\mathrm{Tor}_i^{\mathscr{O}_{X,Z}}(\mathscr{O}_{A,Z},\mathscr{O}_{B,Z}).$$

Note that the zero-degree term is just $\mathrm{length}_{\mathscr{O}_{A\cap B,Z}}\mathscr{O}_{X,Z}/(\mathscr{I}_{A},\mathscr{I}_{B})$. Now is there relationship between this formula and some derived tensor product (maybe $\mathscr{O}_{A,Z}\otimes^{\mathbf{L}}_{\mathscr{O}_{X,Z}}\mathscr{O}_{B,Z}$)? What is the meaning of this tensor product? (or some derived intersection product?)

Maybe these things are related to the derived algebraic geometry (or the derived intersection theory I guess), is there some references (papers/notes/books) about derived-aspect intersection theory (char$=0$ is OK)?

Any help will be grateful!

Is there some relations between derived category and intersection theory?

After learning the traditional intersection theory (W. Fulton's Intersection Theory and D. Eisenbud & J. Harris's 3264 and all that), I have some biased thinking about what I have learned in this period of time:

The first is the excess intersection formula:

Theorem 1. (Excess intersection formula) For regularly embedded subschemes $X_i\subset Y$ , a closed subscheme $V\subset Y$ and a connected component $Z\subset V\cap\bigcap_iX_i$, we have $$(X_1\cdot\ldots\cdot X_r\cdot V)^Z=\left\{s(Z,V)\cap\prod_{i=1}^rc(N_{X_i}Y|_Z)\right\}_{\dim V-\sum\mathrm{codim}(X_i,Y)}$$

Using this formula we can compute the contribution of degree support on the wrong dimension connected component. But is there some explanations using derived-philosophy? I guess may be the intersection information of bad dimensions (like self-intersection and excess intersection formula as above) can be reflected in some higher ranks of the derived intersection product.

Second, we all know the Serre's formula about intersection multiplicity:

Theorem 2. (Serre) If $A,B\subset X$ are two closed subschemes of smooth variety $X$ with expected dimensions and $Z$ be an irreducible component of $A\cap B$, then $$i(Z;A,B;X)=\sum_{i=0}^{\dim X}(-1)^i\mathrm{length}_{\mathscr{O}_{A\cap B,Z}}\mathrm{Tor}_i^{\mathscr{O}_{X,Z}}(\mathscr{O}_{A,Z},\mathscr{O}_{B,Z}).$$

Note that the zero-degree term is just $\mathrm{length}_{\mathscr{O}_{A\cap B,Z}}\mathscr{O}_{X,Z}/(\mathscr{I}_{A},\mathscr{I}_{B})$. Now is there relationship between this formula and some derived tensor product (maybe $\mathscr{O}_{A,Z}\otimes^{\mathbf{L}}_{\mathscr{O}_{X,Z}}\mathscr{O}_{B,Z}$)? What is the meaning of this tensor product? (or some derived intersection product?)

Maybe these things are related to the derived algebraic geometry (or the derived intersection theory I guess), is there some references (papers/notes/books) about derived-aspect intersection theory (char$=0$ is OK)?

Any help will be grateful!

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DVL-WakeUp
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(A soft question) Is there some relations between derived category and intersection theory?

After learning the trditional intersection theory (W. Fulton's Intersection Theory and D. Eisenbud & J. Harris's 3264 and all that), I have some biased thinking about what I have learned in this period of time:

The first is the excess intersection formula:

Theorem 1. (Excess intersection formula) For regularly embedded subschemes $X_i\subset Y$ , a closed subscheme $V\subset Y$ and a connected component $Z\subset V\cap\bigcap_iX_i$, we have $$(X_1\cdot\ldots\cdot X_r\cdot V)^Z=\left\{s(Z,V)\cap\prod_{i=1}^rc(N_{X_i}Y|_Z)\right\}_{\dim V-\sum\mathrm{codim}(X_i,Y)}$$

Using this formula we can compute the contribution of degree support on the wrong dimension connected component. But is there some explanations using derived-philosophy? I guess may be the intersection information of bad dimensions (like self-intersection and excess intersection formula as above) can be reflected in some higher ranks of the derived intersection product.

Second, we all know the Serre's formula about intersection multiplicity:

Theorem 2. (Serre) If $A,B\subset X$ are two closed subschemes of smooth variety $X$ with expected dimensions and $Z$ be an irreducible component of $A\cap B$, then $$i(Z;A,B;X)=\sum_{i=0}^{\dim X}(-1)^i\mathrm{length}_{\mathscr{O}_{A\cap B,Z}}\mathrm{Tor}_i^{\mathscr{O}_{X,Z}}(\mathscr{O}_{A,Z},\mathscr{O}_{B,Z}).$$

Note that the zero-degree term is just $\mathrm{length}_{\mathscr{O}_{A\cap B,Z}}\mathscr{O}_{X,Z}/(\mathscr{I}_{A},\mathscr{I}_{B})$. Now is there relationship between this formula and some derived tensor product (maybe $\mathscr{O}_{A,Z}\otimes^{\mathbf{L}}_{\mathscr{O}_{X,Z}}\mathscr{O}_{B,Z}$)? What is the meaning of this tensor product? (or some derived intersection product?)

Maybe these things are related to the derived algebraic geometry (or the derived intersection theory I guess), is there some references (papers/notes/books) about derived-aspect intersection theory (char$=0$ is OK)?

Any help will be grateful!